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fd75f919cbb6-0	Park et al. (2023) Title: Future sea-level projections with a coupled atmosphere-ocean-ice-sheet model Key Points: Presents sea-level projections using an Earth system model of intermediate complexity where the model interacts with both the Greenland Ice Sheet and Antarctic Ice Sheet. Exploration of potential sea-level contribution from these regions under SSP 1-1.9, 2-4.5, 5-8.5 future emission scenarios. The model predicts ice-sheet contributions to global sea-level rise by 2150 of 0.2 ± 0.01, 0.5 ± 0.01 and 1.4 ± 0.1 m under SSP 1-1.9, 2-4.5, 5-8.5, respectively. Only the most substantial climate mitigation scenario SPP1-1.9 avoids a long-term sustained sea-level contribution from Greenland and Antarctica. Keywords: Sea-level rise, Climate modeling, Coupled atmosphere-ocean-ice-sheet model, Ice-sheet dynamics, Greenhouse warming simulations, Shared Socioeconomic Pathways, Antarctic Ice Sheet, Greenland Ice Sheet Corresponding author: Jun-Young Park Citation: Park, J.-Y., Schloesser, F., Timmermann, A., Choudhury, D., Lee, J.-Y., & Nellikkattil, A. B. (2023). Future sea-level projections with a coupled atmosphere-ocean-ice-sheet model. Nature Communications, 14(1), 636. doi:10.1038/s41467-023-36051-9 URL: https://www.nature.com/articles/s41467-023-36051-9 Abstract Climate-forced, offline ice-sheet model simulations have been used extensively in assessing how much ice-sheets can contribute to future global sea-level rise. Typically, these model projections do not account for the two-way interactions between ice-sheets and climate. To quantify the impact of ice-ocean-atmosphere feedbacks, here we conduct greenhouse warming simulations with a coupled global climate-ice-sheet model of intermediate complexity. Following the Shared Socioeconomic Pathway (SSP) 1-1.9, 2-4.5, 5-8.5 emission scenarios, the model simulations ice-sheet contributions to global sea-level rise by 2150 of 0.2 ± 0.01, 0.5 ± 0.01 and 1.4 ± 0.1 m, respectively. Antarctic ocean-ice-sheet-ice-shelf interactions enhance future subsurface basal melting, while freshwater-induced atmospheric cooling reduces surface melting and iceberg calving. The combined effect is likely to decelerate global sea-level rise contributions from Antarctica relative to the uncoupled climate-forced ice-sheet model configuration. Our results demonstrate that estimates of future sea-level rise fundamentally depend on the complex interactions between ice-sheets, icebergs, ocean and the atmosphere. Introduction Global mean sea-level (SL) has risen over the past century by about 20 cm, in part due to the thermal expansion of seawater, glacier and ice-sheet melt and changes in groundwater storage1,2,3. This trend is likely to accelerate in response to increasing atmospheric greenhouse gas concentrations and anthropogenic warming. With a considerable fraction of the world’s population living near coastlines, it is crucial to provide accurate projections of global and regional future SL trends and constrain remaining uncertainties. The largest uncertainty originates from the response of the Antarctic ice-sheet (AIS) to greenhouse warming. Recent model-based estimates of the 21st century AIS contribution to future SL rise for a Representative Concentration Pathway (RCP) 8.5 scenario range from 0 to 1.4 m3,4,5,6,7,8. These estimates were obtained with offline ice-sheet models that use atmospheric and oceanic forcings from future climate model projections. Accordingly, several issues need to be considered: (i) the transient and equilibrium climate sensitivities of the current generation of earth system models still have remaining uncertainties of 1.5–2.2 K and 2.6–4.1 K, respectively9, (ii) several processes such as iceberg calving or basal melting are not well constrained and represented; also different ice-sheet models show varying sensitivities to warming scenarios5,10,11, and (iii) the impact of climate-ice-sheet coupling is not included in offline ice-sheet model simulations, even though it may play an important role in Southern Hemisphere climate change12,13. Here, we present a new suite of coupled future earth system model simulations, which captures important interactions between atmosphere, ocean, ice-sheets, ice-shelves, and icebergs in both hemispheres. Observational data shows that AIS meltwater discharge has increased over the past decades14,15. This in turn can increase Southern Ocean (SO) stratification and subsurface warming due to reduced vertical heat exchange. Subsurface Southern Ocean (SSO) warming enhances sub-shelf melting16,17,18,19 which can lead to a reduction of the buttressing effect of ice-shelf on grounded ice20,21. As a consequence the flow of ice streams can accelerate toward the ocean22, which would translate to SL rise. These processes can be further amplified by other feedbacks, including the Marine Ice-Sheet Instability (MISI) along retrograde slopes23, hydrofracturing and the Marine Ice-Cliff Instability (MICI)5—all of which remain poorly constrained8,10. To quantify the effect of these interactions on future SL projections, one needs to employ coupled global climate-ice-sheet models24,25, which capture the complex interactions between climate components and ice-sheet, ice-shelf and iceberg dynamics and thermodynamics. Our study focuses on the AIS and Greenland ice-sheet (GrIS) contributions to future SL projections using an ensemble of simulations conducted with the coupled three-dimensional climate-ice-sheet-iceberg modeling system26 LOVECLIP (Supplementary Fig. S1). LOVECLIP is based on the climate model of intermediate complexity LOVECLIM27 and the Penn State University Ice-Sheet model (PSUIM)5,28,29. Biases of surface air temperature, precipitation, and SSO are corrected from LOVECLIM to PSUIM26. PSUIM is forced by surface air temperature, precipitation, evaporation, solar radiation, and annual mean subsurface ocean temperature. Surface air temperature and precipitation are downscaled vertically to PSUIM grid with applied lapse rate corrections28. The simulations include an 8000-years-long coupled pre-industrial spin-up run for initialization and 10 member ensemble of simulations forced by increasing CO2 concentrations following the Shared Socioeconomic Pathway (SSP) 1–1.9, 2–4.5 and 5–8.5 scenarios30 until 2150 CE. To further elucidate the effect of AIS meltwater flux on polar climate, the stability of the ice-sheets and SL we performed an additional idealized sensitivity experiment (Supplementary Table S1) for which we scaled the amplitudes of AIS liquid runoff and iceberg calving to balance net precipitation over Antarctica (experiment SSP5-8.5_MWOFF). In addition, to quantify the impact of AIS hydrofracturing and ice-cliff failure on the ice-sheet evolution, these parameterizations are turned off (CREVLIQ = 0 m per (m/year)−2 and VCLIF = 0 km/year) with and without meltwater flux coupling (experiments SSP5-8.5_HFCMOFF and SSP5-8.5_MWHFCMOFF). Furthermore, to estimate the sensitivity of Antarctic ice-shelf mass loss to SSO warming, we doubled the sub-shelf SSO temperature anomaly (relative to 1850 CE) in the SSP5-8.5 scenario with (Re_SSP5-8.5_2xSOTA) and without meltwater fluxes (Re_SSP5-8.5_2xSOTA_MWOFF). Results Recent trend and interannual variability of GrIS and AIS	https://sealeveldocs.readthedocs.io/en/latest/park23.html
fd75f919cbb6-1	The simulations include an 8000-years-long coupled pre-industrial spin-up run for initialization and 10 member ensemble of simulations forced by increasing CO2 concentrations following the Shared Socioeconomic Pathway (SSP) 1–1.9, 2–4.5 and 5–8.5 scenarios30 until 2150 CE. To further elucidate the effect of AIS meltwater flux on polar climate, the stability of the ice-sheets and SL we performed an additional idealized sensitivity experiment (Supplementary Table S1) for which we scaled the amplitudes of AIS liquid runoff and iceberg calving to balance net precipitation over Antarctica (experiment SSP5-8.5_MWOFF). In addition, to quantify the impact of AIS hydrofracturing and ice-cliff failure on the ice-sheet evolution, these parameterizations are turned off (CREVLIQ = 0 m per (m/year)−2 and VCLIF = 0 km/year) with and without meltwater flux coupling (experiments SSP5-8.5_HFCMOFF and SSP5-8.5_MWHFCMOFF). Furthermore, to estimate the sensitivity of Antarctic ice-shelf mass loss to SSO warming, we doubled the sub-shelf SSO temperature anomaly (relative to 1850 CE) in the SSP5-8.5 scenario with (Re_SSP5-8.5_2xSOTA) and without meltwater fluxes (Re_SSP5-8.5_2xSOTA_MWOFF). Results Recent trend and interannual variability of GrIS and AIS According to satellite observations, the GrIS and AIS have been losing mass at a rate of ~286 Gt/year in 2010–2018 CE and ~252 Gt/year in 2009–2017 CE, respectively31,32. Quantifying the natural and anthropogenic contributions to this trend remains difficult because of our limited understanding of naturally occurring low-frequency ice-sheet dynamics and the relatively short observational period. Here we compare the observed 19-year trend of ice mass balance from the Gravity Recovery and Climate Experiment (GRACE)33 for the period 2002–2020 CE to the corresponding values in forced experiments as well as a 5000-year-long pre-industrial control run (CTR) (Supplementary Fig. S2) conducted with LOVECLIP (Fig. 1). In Fig. 1, each 19-year chunk of mass balance in CTR is cut after high-pass filtering over than 80 years and then, 19-year trends of nature variability are extracted. Those trends are expressed here in terms of sea-level-equivalent (SLE, 1 m SLE = 3.62 × 1014 m3). Consistent with the GRACE measurements, changes in the mass balance are calculated only from the grounded parts of the ice-sheets for LOVECLIP. Interannual variability of the mass balance recorded by GRACE and simulated by the forced LOVECLIP experiments during 2002–2020 CE fall within the range of natural variability exhibited by the CTR (Supplementary Fig. S3). This indicates that the range of interannual mass balance changes of ice-sheets are represented realistically in the model simulation on the global scale. However, the observed GrIS 19-year trend (−0.075 cm/year SLE) lies outside the respective 95% confidence interval range of CTR (Fig. 1a), which suggests that the current observed mass loss in Greenland is inconsistent with natural variability, as estimated from LOVECLIP. Although the simulated GrIS trend (−0.13 to −0.08 cm/year SLE) is slightly overestimated than the observed GRACE estimate for the same period (Fig. 1a, red range), we can still conclude that greenhouse warming contributed to GrIS melting over the past decades. On the other hand, the AIS mass balance trend recorded by GRACE (−0.04 cm/year SLE) and the forced AIS trend (−0.1 to −0.02 cm/year SLE) lie within the 95% confidence range of the LOVECLIP CTR simulation due to the fact that AIS natural variability amplitude exceeds that of the GrIS by a factor of 7. Figure 1: 19-year trends of observed and simulated mass balance of Greenland ice-sheet (GrIS) and Antarctic ice-sheet (AIS). a Histogram of each extracted 19-year trend of Greenland mass balance after 80-year-high-pass filtering in the 5000-year-long pre-industrial control run (CTR, gray histogram) with 95% confidence interval range of CTR (black dashed line), and observed estimates of 19-year trend for 2002–2020 CE from the Gravity Recovery and Climate Experiment (GRACE)33 (blue dashed line) and simulated by the forced LOVECLIP ensemble (red line) in sea-level-equivalent (SLE); b same as a, but for Antarctica. Consistent with the GRACE measurements, mass balance terms for LOVECLIM are calculated in this figure using only the grounded ice-sheet portion. Future change of global surface temperature and SL The projected ensemble average of global surface temperature rise in 2100 CE (2150 CE) relative to the pre-industrial levels (1850–1900 CE) amounts to 1.4 ± 0.17 °C (1.2 ± 0.14 °C), 2.4 ± 0.15 °C (2.7 ± 0.16 °C) and 4.0 ± 0.15 °C (5.3 ± 0.09 °C) for the SSP1-1.9, SSP2-4.5, and SSP5-8.5 scenarios, respectively (Fig. 2a). The uncertainty values are calculated at 95% confidence interval in this paper. Relative to the recent past (1995–2014 CE) the simulated end-of-century warming (2081–2100 CE) attains values of 0.3 °C for SSP1-1.9 and 2.6 °C for SSP5-8.5, which is at the lower end of the multi-model range in projected changes obtained from the respective Coupled Model Intercomparison 6 (CMIP6) models34,35. Figure 2: Global surface temperature and sea-level (SL) projections, and their tendencies. a–d Annual anomalies (relative to the 1850–1900 CE mean) of (a) the global surface temperature, (b) SL, and (c) SL contributions from the Greenland ice-sheet (GrIS) and (d) Antarctic ice-sheet (AIS). e–h are the respective time derivatives of a–d (change per year). Solid lines of a–d indicate the ensemble mean and shading the ensemble range. The solid line in e represents the 9-year moving average of the time derivative of global surface temperature, with the dashed line indicating 0 °C/year. Different colors represent the historical (black line; period 1850–2014 CE), and SSP1-1.9 (blue line), SSP2-4.5 (pink line), SSP5-8.5 (red line) and SSP5-8.5_MWOFF (orange line) simulations during the period 2014–2150 CE.	https://sealeveldocs.readthedocs.io/en/latest/park23.html
fd75f919cbb6-2	The projected ensemble average of global surface temperature rise in 2100 CE (2150 CE) relative to the pre-industrial levels (1850–1900 CE) amounts to 1.4 ± 0.17 °C (1.2 ± 0.14 °C), 2.4 ± 0.15 °C (2.7 ± 0.16 °C) and 4.0 ± 0.15 °C (5.3 ± 0.09 °C) for the SSP1-1.9, SSP2-4.5, and SSP5-8.5 scenarios, respectively (Fig. 2a). The uncertainty values are calculated at 95% confidence interval in this paper. Relative to the recent past (1995–2014 CE) the simulated end-of-century warming (2081–2100 CE) attains values of 0.3 °C for SSP1-1.9 and 2.6 °C for SSP5-8.5, which is at the lower end of the multi-model range in projected changes obtained from the respective Coupled Model Intercomparison 6 (CMIP6) models34,35. Figure 2: Global surface temperature and sea-level (SL) projections, and their tendencies. a–d Annual anomalies (relative to the 1850–1900 CE mean) of (a) the global surface temperature, (b) SL, and (c) SL contributions from the Greenland ice-sheet (GrIS) and (d) Antarctic ice-sheet (AIS). e–h are the respective time derivatives of a–d (change per year). Solid lines of a–d indicate the ensemble mean and shading the ensemble range. The solid line in e represents the 9-year moving average of the time derivative of global surface temperature, with the dashed line indicating 0 °C/year. Different colors represent the historical (black line; period 1850–2014 CE), and SSP1-1.9 (blue line), SSP2-4.5 (pink line), SSP5-8.5 (red line) and SSP5-8.5_MWOFF (orange line) simulations during the period 2014–2150 CE. Higher surface temperatures increase ice-sheet surface melting and subsequent meltwater discharge, and ice-sheet calving in both hemispheres. For the SSP1-1.9, SSP2-4.5 and SSP5-8.5 scenarios the GrIS contributes about 12 ± 1, 18 ± 0.9 and 23 ± 1.6 cm and the AIS adds 3 ± 0.8, 7 ± 1.4, and 15 ± 1.5 cm to SL by the year 2100 relative to pre-industrial levels (Fig. 2c, d). 2100 CE (2150 CE) LOVECLIP simulates for the respective scenarios a total ice-sheet contribution to SL of 15 ± 0.9, 24 ± 1.3, 39 ± 2 (19 ± 1.4, 48 ± 1.4, 136 ± 6.2) cm (Fig. 2b). The GrIS and AIS contributions lie within the range of estimates obtained from uncoupled scenario-forced models for Greenland36,37,38 and Antarctica6,8,39,40. One factor impacting the LOVECLIP ice-sheet response is the relatively weak temperature sensitivity to greenhouse forcing compared to most CMIP6 models (Supplementary Fig. S8). With lower sensitivity, nonetheless, our LOVECLIP shows both Arctic and Antarctic amplification. On the other hand, CMIP6 models do not show the aspect of Antarctic amplification. From a climate sensitivity point of view, our model results can therefore be regarded as conservative estimates. Our simulated SL is also substantially lower than the projected 1 m end-of-century AIS contribution to SL presented in a series of offline ice-sheet model simulations conducted with the PSUIM5. In our coupled model simulations, which use the same ice-sheet model, but a different climate model, different ice-sheet and coupling parameters and at lower resolution, the rate of global temperature change slows down (negative second derivative with respect to time) around 2100 CE for SSP2-4.5 and SSP5-8.5 (Fig. 2a). This strongly contrasts the continued acceleration (positive second derivative) of SL (Fig. 2b) for these scenarios. This behavior illustrates the combined effect of long response timescales of the ice-sheets, the effect of positive feedbacks and their prolonged contribution to SL, even long after CO2 emissions have started to decline. Reductions in future greenhouse gas emissions can help slowdown global warming trends for the high-end emission scenarios (Fig. 2e). However, they are unlikely to stop the ice-sheet-driven SL rise acceleration (Fig. 2f–h) and the apparent run-away in SL for the next 130 years. Only the much more aggressive SSP1-1.9 scenario can lead to a gradual slow-down of SL rise acceleration (Fig. 2f), which implies that, according to our simulations, the 2°C warming (above the pre-industrial level) target emphasized by the Paris agreement41 is insufficient to prevent accelerated SL rise over the next century [42]. Ice loss from GrIS and AIS In our model simulations, future warming leads to an increase in snow accumulation and ice thickness in the central part of GrIS (Fig. 3c–e, Fig. 4a) and West Antarctica (Fig. 3f–h, Fig. 4e). However, the negative mass balance terms together are considerably larger (Fig. 4b–d, f–h), leading to a net projected 80-year mass loss for the different scenarios of 14 ± 1.5, 20 ± 0.9 and 25 ± 1.5 cm SLE for GrIS and 46 ± 6, 94 ± 8 and 152 ± 8 cm for AIS, respectively (Fig. 3a, b). Although the AIS shows significantly more ice melting until 2100 CE in comparison to the GrIS, its contribution to SL is similar or even lower (Fig. 2c, d) because most of the GrIS melting occurs at the surface as ice ablation (Fig. 3d, e, Fig. 4b), whereas the AIS loses mass primarily below-SL and at the shelves through melting and calving, especially in the Ross ice-shelf and Ronne-Filchner ice-shelf regions (Fig. 3g, h, Fig. 4g, h). Ice-shelf and submarine ice-loss do not directly contribute to SL rise (except for marginal contributions from the difference in ice and seawater density). In our simulations warmer Circumpolar Deep Water (CDW) reaches the continental shelf regions which in turn increases basal melting, sub-shelf melting and potential grounding line retreat (Fig. 3g, h). Due to the larger extent of ice-shelves, basal melting is more important for the AIS than for the GrIS (Fig. 4c, g). Note that global coarse-resolution ocean models, such as the one used here with a 3° × 3° degree horizontal resolution cannot fully resolve the small-scale coastal ocean circulation processes around Antarctica43 and ignore sub-cavity flows, which are important to explicitly resolve basal melting processes. In our modeling framework basal melting is parameterized using open ocean temperatures interpolated on the finer ice-sheet model grid28. Figure 3: Projected changes in mass balance, ice thickness and subsurface ocean temperature. a, b Time series of the annual mean (a) Greenland ice-sheet (GrIS) and (b) Antarctic ice-sheet (AIS) net mass balance in sea-level-equivalent (SLE) (including contributions from ice shelves), respectively; c GrIS 1850–1860 CE mean ice thickness (grayscale colormap) and 400 m Arctic Ocean (AO) temperature (red-yellow colormap); d 2090–2100 CE change in SSP5-8.5 scenario of GrIS ice thickness with respect to 1850–1860 CE and change in mean AO subsurface temperature; e same as d, but for period 2140–2150 CE; f, g same as c–e, but for AIS with 400 m Southern Ocean (SO) temperature. Black contours indicate simulated grounding lines for different periods. Cyan contours indicate the edge lines of ice-shelves for different periods.	https://sealeveldocs.readthedocs.io/en/latest/park23.html
fd75f919cbb6-3	Figure 3: Projected changes in mass balance, ice thickness and subsurface ocean temperature. a, b Time series of the annual mean (a) Greenland ice-sheet (GrIS) and (b) Antarctic ice-sheet (AIS) net mass balance in sea-level-equivalent (SLE) (including contributions from ice shelves), respectively; c GrIS 1850–1860 CE mean ice thickness (grayscale colormap) and 400 m Arctic Ocean (AO) temperature (red-yellow colormap); d 2090–2100 CE change in SSP5-8.5 scenario of GrIS ice thickness with respect to 1850–1860 CE and change in mean AO subsurface temperature; e same as d, but for period 2140–2150 CE; f, g same as c–e, but for AIS with 400 m Southern Ocean (SO) temperature. Black contours indicate simulated grounding lines for different periods. Cyan contours indicate the edge lines of ice-shelves for different periods. Figure 4: Individual mass balance terms for Greenland ice-sheet (GrIS) and Antarctic ice-sheet (AIS). a–d Represent the individual GrIS mass balance terms for (a) the accumulation, (b) surface melting, (c) basal melting and (d) ice calving expressed as sea-level-equivalent (SLE) per year; e–h same as a–d, but for AIS. Solid lines indicate the ensemble mean and shading the ensemble range. Different colors represent the historical (black line; period 1850–2014 CE), and SSP1-1.9 (blue line), SSP2-4.5 (pink line), SSP5-8.5 (red line) and SSP5-8.5_MWOFF (orange line) simulations during the period 2014–2150 CE. According to our numerical experiments, the Ross ice-shelf completely disappears in the SSP5-8.5 scenario after 2100 CE (Fig. 3g, h). At this time basal melting and calving rates peak (Fig. 4g, h). A secondary simulated increase in these fluxes at the beginning of the 22nd century is associated with an accelerated retreat of the Ronne-Filchner ice-shelves (Figs. 3h, 4g, h). Even though the AIS contribution to SL rise is initially smaller than that of the GrIS (before 2100 CE), the rapid loss of stabilizing ice-shelves leads to a gradual increase of ice flow across the grounding lines that will initiate positive ice-sheet feedbacks associated with the MISI23, hydrofracturing and MICI5,29. The AIS calving fluxes, which attain values of ~2 cm/year SLE by 2080 CE (corresponding to a freshwater flux into the ocean of ~0.34 Sv; 1 Sv = 106 m3/s), dominate the negative mass balance and global SL contribution. The accelerated mass loss over the AIS is related to a combination of surface melting, basal melting and grounding line retreat which contributes to the massive ice calving fluxes (Fig. 4f–h) – each component with their individual temporal contributions to the total freshwater and SL effect. In contrast to the AIS, the GrIS shows a gradual decrease in basal melting and ice calving fluxes (Fig. 4c, d), interrupted only by an abrupt GrIS ice calving event around 2090 CE in SSP5-8.5, which is associated with a complete loss of small ice-shelf areas. In Greenland the dominant mass loss and the contribution to SL are due to the positive trend in surface melting, which attains values of up to 2.1 ± 0.3 cm/year SLE by 2150 CE (Fig. 4b)—a ~30-fold increase compared to the recent observed interannual rates of GrIS mass loss (Supplementary Fig. S3a). Ice-sheet/climate feedbacks in Southern Hemisphere To further quantify the effects of climate-ice-sheet coupling in the Southern Hemisphere, and test the previously hypothesized positive CDW/MISI feedback16,17,18,19,44 we performed idealized SSP5-8.5 ensemble sensitivity experiments in which the freshwater coupling from the Antarctic meltwater is decoupled (experiment SSP5-8.5_MWOFF). Increased AIS meltwater fluxes in the fully coupled model experiment (experiment SSP5-8.5) reduce surface ocean salinity in the SO relative to SSP5-8.5_MWOFF (Fig. 5a). In turn this increases ocean stratification and reduces vertical heat exchange between cold surface and warmer subsurface waters. As a result, annual mean subsurface temperatures around Antarctica increase by 1.5 °C over the 21st century in SSP5-8.5 (Fig. 5b). In contrast, in SSP5-8.5_MWOFF the AIS melting does not directly impact the SO stratification, which leads to a temporary 30% reduction in subsurface ocean warming (Fig. 5b) and a 50% reduction in basal melting (Fig. 4g). At the surface, increased stratification in SSP5-8.5 and reduced vertical heat exchange lead to cooling and increased sea-ice production6,12,13,45 (Fig. 5d), relative to the SSP5-8.5_MWOFF experiment. The 21st century annual mean surface air temperatures around Antarctica are about 1.4 °C colder in SSP5-8.5 as compared to SSP5-8.5_MWOFF (Fig. 5c). This cooling effect provides a negative feedback for AIS surface melting6,42. Moreover, without meltwater coupling temperatures, precipitation and snow accumulation increase over Antarctica by about 0.1 cm/year SLE (Fig. 4e) around 2100 CE. At the ice-sheet margins, higher temperatures and increased precipitation in SSP5-8.5_MWOFF contribute to hydrofracturing and the simulated increased calving rates5 (Fig. 4h). Overall, in the fully coupled simulation reduced surface melting (Fig. 4f) and calving rates (Fig. 4h) outweigh reduced accumulation rates, and hence the freshwater-induced surface cooling (Fig. 5c) provides a net-negative feedback to ice-sheet melting. As a consequence of the substantial differences in AIS mass balance between SSP5-8.5 and SSP5-8.5_MWOFF, the rate of total ice volume loss and the corresponding rate of SL contribution are decelerated when accounting for the fully coupled system (Fig. 2d, Fig. 3b) and the surface temperature effects (Fig. 2a). Although it has been suggested that AIS meltwater fluxes could remotely impact the GrIS via changes in Atlantic Meridional Overturning circulation and interhemispheric heat fluxes46, we do not find any noticeable changes in the GrIS response between SSP5-8.5 and SSP5-8.5_MWOFF. A higher-resolution climate simulation may be required to explain the teleconnection at the end of 21st century shown in Supplementary Fig. S6d. Figure 5: Climate-ice-sheet feedbacks in Southern Hemisphere. a–c Annual anomalies (relative to the 1850–1900 CE mean) of (a) the Southern Ocean (SO) surface salinity, (b) 400 m subsurface Southern Ocean (SSO) temperature and (c) surface air temperature averaged between 60°S and 90°S. d is the SO sea-ice area averaged between 60°S and 90°S. Solid lines indicate the ensemble mean and shading the ensemble range. Different colors represent the SSP5-8.5 (red line) and SSP5-8.5_MWOFF (blue line) simulations during the period 2014–2150 CE. When hydrofracturing and ice-cliff failure parameterizations are turned off in the additional model experiments (Supplementary Table S1), the AIS meltwater flux still decelerates global warming (experiments SSP5-8.5_HFCMOFF and SSP5-8.5_MWHFCMOFF, Supplementary Fig. S4a orange and blue lines). However, the negative and positive coupled feedbacks on SL rise related to the meltwater flux are more in balance (Supplementary Figs. S4b, S5). Despite meltwater and calving fluxes being substantially reduced relative to SSP5-8.5, the surface cooling is nearly as strong in SSP5-8.5_HFCMOFF, due to the cooling becoming less efficient with increasing meltwater flux amplitude13. Without hydrofracturing, however, increased surface temperatures and rainfall do not directly impact the calving flux, therefore surface temperature-related feedbacks are weaker (calving is still stronger in SSP5-8.5_MWHFCMOFF than in SSP5-8.5_HFCMOFF to compensate for changes in other fluxes, in particular reduced basal melting). Sensitivity of subsurface Southern Ocean warming	https://sealeveldocs.readthedocs.io/en/latest/park23.html
fd75f919cbb6-4	Figure 5: Climate-ice-sheet feedbacks in Southern Hemisphere. a–c Annual anomalies (relative to the 1850–1900 CE mean) of (a) the Southern Ocean (SO) surface salinity, (b) 400 m subsurface Southern Ocean (SSO) temperature and (c) surface air temperature averaged between 60°S and 90°S. d is the SO sea-ice area averaged between 60°S and 90°S. Solid lines indicate the ensemble mean and shading the ensemble range. Different colors represent the SSP5-8.5 (red line) and SSP5-8.5_MWOFF (blue line) simulations during the period 2014–2150 CE. When hydrofracturing and ice-cliff failure parameterizations are turned off in the additional model experiments (Supplementary Table S1), the AIS meltwater flux still decelerates global warming (experiments SSP5-8.5_HFCMOFF and SSP5-8.5_MWHFCMOFF, Supplementary Fig. S4a orange and blue lines). However, the negative and positive coupled feedbacks on SL rise related to the meltwater flux are more in balance (Supplementary Figs. S4b, S5). Despite meltwater and calving fluxes being substantially reduced relative to SSP5-8.5, the surface cooling is nearly as strong in SSP5-8.5_HFCMOFF, due to the cooling becoming less efficient with increasing meltwater flux amplitude13. Without hydrofracturing, however, increased surface temperatures and rainfall do not directly impact the calving flux, therefore surface temperature-related feedbacks are weaker (calving is still stronger in SSP5-8.5_MWHFCMOFF than in SSP5-8.5_HFCMOFF to compensate for changes in other fluxes, in particular reduced basal melting). Sensitivity of subsurface Southern Ocean warming To analyze the sensitivity of the Antarctic ice-shelves to SSO warming, first, we obtained new equilibrium conditions from the 10 member initial conditions by doubling the SSO temperature anomaly (with respect to 1850 CE) near the Antarctic ice-shelves during 650 years without greenhouse forcing. SSO temperatures in the Antarctic ice model TIM are calculated using T^{IM} = 2 × (T^{LC} - T^{LC}_{1850}) + T^{LC}_{1850} (1) where T^{LC} is the 400 m ocean temperature simulated in LOVECLIM and T^{LC}_{1850} is the corresponding LOVECLIM temperature in the year 1850. Subsequently, we ran a 10 member ensemble covering the historical period and the SSP5-8.5 scenario with/without Antarctic meltwater flux (experiments Re_SSP5-8.5_2xSOTA and Re_SSP5-8.5_2xSOTA_MWOFF). Warming SSO temperature (Fig. 6m) increases basal melting under the Antarctic ice-shelves, thereby accelerating grounding line retreat (Fig. 6e–h) relative to SSP5-8.5 (Fig. 6a–d). This is most evident in Ross ice-shelf which vanishes completely by 2100 CE, leading to an integrated freshwater input of 2 ± 0.35 m SLE total AIS mass by 2100 CE (Fig. 6p) and de facto SL rise of 0.5 ± 0.04 m (Fig. 6o). Ice calving is the largest term in the mass balance over AIS (Fig. 4e–h). However, as the shelf retreat accelerates mainly due to the basal melting, the role of ice calving term diminishes (Supplementary Fig. S6g, h). Figure 6: Transections of the Antarctic Ross ice-shelf, and global temperature and SL. a–l Transects of Antarctic Ross-ice-shelf simulated in (a–d) SSP5-8.5, (e–h) Re_SSP5-8.5_2xSOTA and (i–l) Re_SSP5-8.5_2xSOTA_MWOFF experiments in 1850, 1950, 2015 and 2100 CE. Dashed lines indicate grounding lines. m–p Time series of annual anomalies (relative to the 1850–1900 CE mean) of (m) subsurface Southern Ocean (SSO) temperature, (n) global surface temperature, (o) sea-level (SL) and (p) Antarctic ice-sheet (AIS) net mass balance in sea-level-equivalent (SLE). Different colors represent the SSP5-8.5 with historical (black line), Re_SSP5-8.5_2xSOTA (red line) and Re_SSP5-8.5_2xSOTA_MWOFF (blue line) simulations. Horizontal scales of a–l are shown in Supplementary Fig. S9 as a red line. The result of increased Antarctic meltwater fluxes by enhanced SSO warming concurs with our previous discussion of a global warming slowdown by 0.4 °C, relative to a simulation without such coupling (Fig. 6n red and blue lines). However, this sensitivity experiment increases SL by an additional 3 cm SL rise (Fig. 6o). Not unexpectedly, SSO warming does not show any significant influence on the GrIS (Supplementary Fig. S6a–d). Discussion Here we used the coupled three-dimensional climate-ice-sheet model LOVECLIP to better understand the impact of ice-sheet/ice-shelf/ocean/atmosphere coupling processes on the future evolution of GrIS and AIS, and to estimate their respective contributions to SL rise. In our high-end emission scenario, the GrIS and AIS each contribute about 60–70 cm to global mean SL rise over the next 130 years. Even though for SSP2-4.5 and SSP5-8.5 global surface temperatures are projected to increase at a reduced rate after 2100 CE (Fig. 2e), the ice-sheet contributions to SL continue to accelerate beyond 2100 CE (Fig. 2g, h), mostly driven by accelerated surface melting in case of the GrIS and due to a combination of effects for the AIS. According to our simulations, limiting 21st century global surface temperature rise to 2 °C above the pre-industrial level41,47 would be insufficient to slowdown the rate of global SL rise over the next 130 years42. Only the more aggressive low greenhouse gas emission scenario (SSP1-1.9), with temperatures leveling off below 1.5 °C (Fig. 2a), avoids SL rise acceleration (Fig. 2f). A longer-term warming and SL perspective until 2500 CE (Supplementary Table S1) illustrates that for the SSP2-4.5 scenario, SL rise due to GrIS melting accelerates for over 250 years after maximum global warming rates occur, and peaks at over 0.3 cm/year shortly before 2300 CE (Supplementary Fig. S7g). The Antarctic SL contribution for SSP2-4.5 fluctuates between 0.2 and 0.3 cm/year from 2200–2500 CE (Supplementary Fig. S7h). This indicates an even more prolonged response and larger commitment to SL rise due to 21st century warming, with the total AIS contribution reaching about 1.1 m by 2500 CE. In contrast, the aggressive greenhouse gas reduction scenario SSP1-1.9 with temperatures leveling off at less than 1.5 °C (Supplementary Fig. S7a, e) is sufficient to prevent substantial ice-loss in Antarctica (Supplementary Fig. S7h) over the next centuries. Our results are to some extent consistent with recent uncoupled single-hemisphere ice-sheet model simulations5,8,38 which also show the tendency for unabated SL acceleration over the next two centuries in response to strong greenhouse gas forcing. One of the key advantages of our coupled model setup, even though it uses lower oceanic resolution, is that it allows us to quantify the role of meltwater forcing and calving fluxes on the stability of the ice-sheet. This is particularly timely because recent observations already show a marked increase in AIS meltwater fluxes14,15, which could increase SO stratification, reduce vertical heat exchange, increase subsurface temperatures and lead to enhanced basal melting. Numerous studies have suggested that, in combination with MISI and MICI, this may set in motion a run-away effect for ice loss. We can clearly see the positive feedback between ice-sheet melting, warm CDW intrusions and basal melting in our model (Figs. 4g, 5b). According to our study, however, the impact of MICI shows a weaker influence on the future SL contributions from AIS, as compared to earlier studies5,10 (Supplementary Fig. S4).	https://sealeveldocs.readthedocs.io/en/latest/park23.html
fd75f919cbb6-5	Our results are to some extent consistent with recent uncoupled single-hemisphere ice-sheet model simulations5,8,38 which also show the tendency for unabated SL acceleration over the next two centuries in response to strong greenhouse gas forcing. One of the key advantages of our coupled model setup, even though it uses lower oceanic resolution, is that it allows us to quantify the role of meltwater forcing and calving fluxes on the stability of the ice-sheet. This is particularly timely because recent observations already show a marked increase in AIS meltwater fluxes14,15, which could increase SO stratification, reduce vertical heat exchange, increase subsurface temperatures and lead to enhanced basal melting. Numerous studies have suggested that, in combination with MISI and MICI, this may set in motion a run-away effect for ice loss. We can clearly see the positive feedback between ice-sheet melting, warm CDW intrusions and basal melting in our model (Figs. 4g, 5b). According to our study, however, the impact of MICI shows a weaker influence on the future SL contributions from AIS, as compared to earlier studies5,10 (Supplementary Fig. S4). In our high-emission scenario model simulations that include parameterizations for hydrofracturing, ice-cliff instabilities, and capture sea-ice and atmospheric responses, the net impact of ice-sheet/climate feedbacks on SL rise is negative. These processes strongly contribute to the fast AIS response to warming temperatures, slightly less so in the coupled model than in meltwater-decoupled sensitivity experiments due to reduced surface temperature warming, rainfall, and surface melt. Given this sensitivity, it is plausible that the net effect of including the coupling and even its sign is strongly dependent by models: negative feedbacks related to meltwater fluxes are reduced when hydrofracturing and ice-cliff failure parameterizations are turned off in the model in our sensitivity experiments, where positive and negative feedbacks nearly balance. Furthermore, the net effect of including meltwater coupling is positive in the experiments with increased SSO warming. A more dramatic fall in SL, 22 mm by 2100 in the SSP5-8.5 scenario, was found in the U.K. Earth System Model coupled to BISICLES dynamic ice-sheet model25. Thereby, standalone ice-sheet simulations may create misleading projections about the GrIS and AIS contributions to global SL rise. Coupled simulations document that trajectories of future climate change and SL rise depend on the complex and delicate balance of climate-ice-sheet coupling processes, some of which are not yet well constrained observationally. The results presented here from our earth system model of intermediate complexity multi-parameter simulations may provide further guidance in designing new coupled climate-ice-sheet model simulations using the next generation of coupled general circulation models. Whereas the LOVECLIP model employed here captures climate-ice-sheet coupling in both hemispheres, as well as iceberg routing and thermodynamic effects of melting icebergs, it does not resolve sub-shelf ocean dynamical processes48,49, interactive changes in ocean bathymetry, as well as narrow coastal currents, which can play an important role in basal melting50. Moreover, several important ice-sheet processes are still not well constrained, and both the ice-sheet/ice-shelf model and the atmospheric model use relatively low horizontal resolutions of ~20 km in Antarctica and ~5.6°, respectively. This can lead to additional biases in ice-stream dynamics and poleward moisture transport, respectively. Additionally, glacial-isostatic adjustment51 other than vertical bedrock response by elastic lithosphere is not fully implemented in LOVECLIP. Nevertheless, our simulations also illustrate that ice-sheet ocean/atmosphere coupling, which can account for individual mass balance differences of 0.5-0.8 cm/year SLE over the next ~100 years (Fig. 4f–h), is a first order process that needs to be included in future SL assessments. Methods The coupled climate-ice-sheet-ice-shelf-iceberg model LOVECLIP26 is based on the earth-system climate model LOVECLIM27 and the Penn State University Ice-Sheet/Shelf model (PSUIM)5,28,29. Here we review its main components, the coupling algorithm, experimental setup, and highlight modeling differences to previous studies. LOVECLIM earth system model The earth system model of intermediate complexity [27] used here, LOVECLIM model (version 1.3), includes ocean, atmosphere, sea-ice, iceberg components as well as a vegetation model. The atmospheric model ECBILT [52] uses a T21 spectral truncation (corresponding to ~5.6° × 5.6° horizontal resolution) and the prognostic quasi-geostrophic atmospheric equations are solved on three vertical levels. The model includes parameterizations of ageostrophic terms [53] to better capture tropical dynamics. The free-surface primitive equation ocean model, CLIO [54,55,56], which is also coupled to a thermodynamic-dynamic sea-ice model, adopts a 3° × 3° horizontal resolution and 20 vertical levels. The coupling between atmosphere and ocean is expressed in terms of freshwater, momentum and heat flux exchanges. An iceberg model integrates iceberg trajectories, melting and freshwater release along individual simulated iceberg trajectories [57,58]. VECODE [59], the terrestrial vegetation model of LOVECLIP calculates the temporal evolution of annual mean desert, tree and grass fractions in each land grid cell. LOVECLIM has been used extensively to study the earth system response to a variety of boundary conditions [60,61,62,63]. Here the model is configured with a present-day land mask and an open Bering Strait. PSUIM ice-sheet/shelf model The ice-sheet/shelf model PSUIM [5,28,29] is used here in a bi-hemispheric configuration. By adopting shallow ice and shallow shelf approximations, the model retains the ability to simulate streaming and stretching flow and to capture ice streams and floating ice-shelves. Floating ice-shelves, grounding line migration, and basal ice fluxes are parameterized [21]. PSUIM estimates the surface energy and ice mass balance by accounting for contributions from changes in temperature and radiation [64,65]. Similar to previous versions of the model [5,29], we include parameterizations for enhanced calving caused by rainwater-driven hydrofracturing and surface melting, as well as a representation of marine ice-cliff failure. Through calculating changes of ice calving, floating-ice, grounding line migration and pinning by bathymetric bedrock perturbations, the SL is estimated. A horizontal resolution of 1° latitude and 0.5° longitude is used in the Northern Hemisphere and a stereographic grid is adopted for Antarctica with a resolution of 20 km. The model version employed here differs from the one used in a recent SL study [5] in that our spatial resolution is lower over Antarctica. Moreover, different parameters were used, namely those characterizing sub-ice-shelf ocean melting (OCFAC), the coefficient in the parameterization of hydrofracturing due to surface liquid (CREVLIQ), and the maximum rate of horizontal wastage due to ice-cliff failure (VCLIF). The default parameter values are OCFAC = 1.0, CREVLIQ = 100 m per (m/year)−2 and VCLIF = 3 km/year5. Here we use a value of OCFAC = 1.5, which were chosen such that the AIS has a realistic extent under pre-industrial conditions and for the corresponding LOVECLIM climate forcing. Additionally, LOVECLIP realistically simulates the Antarctic ice velocity and shape of ice-shelves compared to the mean of 1996–2016 Antarctic ice velocity obtained from the satellite data66,67 (Supplementary Fig. S11). Although, the size of the ice-shelves and the associated outflow velocities are overestimated. Climate-ice-sheet model coupling (LOVECLIP) A coupling algorithm exchanges variables and boundary conditions between LOVECLIM and PSUIM in both hemispheres (Supplementary Fig. S1), in a series of alternating climate and ice-sheet model runs (“chunks”) [26,68,69,70]. The chunk length is set to 1 year for LOVECLIM and PSUIM. The ice model is forced by monthly LOVECLIM surface air temperature, precipitation, evaporation, solar radiation and annual mean subsurface ocean temperature. LOVECLIM has polar temperature and precipitation biases, similar to those documented for more complex CMIP5 models [71]. Present-day climatological surface air temperature and precipitation biases, as well as subsurface ocean temperature biases near Antarctica, are removed in the coupler through a bias correction [70]. Surface air temperature and precipitation are downscaled vertically to the PSUIM grid with applied lapse rate corrections26,28. Subsurface ocean temperature is interpolated under ice-shelves on the PSUIM grid.	https://sealeveldocs.readthedocs.io/en/latest/park23.html
fd75f919cbb6-6	The model version employed here differs from the one used in a recent SL study [5] in that our spatial resolution is lower over Antarctica. Moreover, different parameters were used, namely those characterizing sub-ice-shelf ocean melting (OCFAC), the coefficient in the parameterization of hydrofracturing due to surface liquid (CREVLIQ), and the maximum rate of horizontal wastage due to ice-cliff failure (VCLIF). The default parameter values are OCFAC = 1.0, CREVLIQ = 100 m per (m/year)−2 and VCLIF = 3 km/year5. Here we use a value of OCFAC = 1.5, which were chosen such that the AIS has a realistic extent under pre-industrial conditions and for the corresponding LOVECLIM climate forcing. Additionally, LOVECLIP realistically simulates the Antarctic ice velocity and shape of ice-shelves compared to the mean of 1996–2016 Antarctic ice velocity obtained from the satellite data66,67 (Supplementary Fig. S11). Although, the size of the ice-shelves and the associated outflow velocities are overestimated. Climate-ice-sheet model coupling (LOVECLIP) A coupling algorithm exchanges variables and boundary conditions between LOVECLIM and PSUIM in both hemispheres (Supplementary Fig. S1), in a series of alternating climate and ice-sheet model runs (“chunks”) [26,68,69,70]. The chunk length is set to 1 year for LOVECLIM and PSUIM. The ice model is forced by monthly LOVECLIM surface air temperature, precipitation, evaporation, solar radiation and annual mean subsurface ocean temperature. LOVECLIM has polar temperature and precipitation biases, similar to those documented for more complex CMIP5 models [71]. Present-day climatological surface air temperature and precipitation biases, as well as subsurface ocean temperature biases near Antarctica, are removed in the coupler through a bias correction [70]. Surface air temperature and precipitation are downscaled vertically to the PSUIM grid with applied lapse rate corrections26,28. Subsurface ocean temperature is interpolated under ice-shelves on the PSUIM grid. LOVECLIM’s surface land-ice cover and orography are updated using the simulated ice-sheet and vertical bedrock evolution from PSUIM which is based on elastic lithosphere response with fixed bedrock response time [28]. The spatial distribution of liquid runoff into the ocean is calculated based on model topography and the calving flux is released as icebergs into the LOVECLIM iceberg model. Both liquid runoff from the surface and basal melting are released into the surface ocean. Ocean currents and wind-drag subsequently steer the icebergs and along their pathways they melt and cool the ocean. Contributions to SL changes are calculated in PSUIM in both hemispheres, and take into account the bedrock response28. The total SL evolution is calculated in the coupler based on the Northern Hemisphere and Antarctic contributions. Spin-up and initial conditions The model is initialized from constant pre-industrial conditions. Because of the different equilibration timescales of the ice-sheet and climate components, asynchronous coupling is used to obtain equilibrated initial conditions. In particular, the model has been integrated for 120 chunk lengths with chunk lengths of 50 years for PSUIM and 5 years for LOVECLIM, and then 2000 chunk lengths with chunk length of 1 year for each ice model and LOVECLIM, which results in 8000 years of spin-up for the ice component and 2600 years for LOVECLIM (Supplementary Table S1), after climate trends are negligible (Supplementary Fig. S10). Ensembles of 10 members with different initial conditions have been conducted for each of the experiments, with the initial conditions taken from the last 100 chunks of the spin-up run. Statistics of forced and unforced ice-sheet mass balance To compare whether the observed ice-sheet mass balance estimated from the Gravity Recovery and Climate Experiment (GRACE)33 for the period 2002–2020 CE is consistent with the null hypothesis of unforced ice-sheet variability, we conducted a 5000 years LOVECLIP control experiment with constant, pre-industrial CO2 concentrations (CTR) (Supplementary Table S1). There is still a remaining very weak drift in the unforced simulation, which amounts to −1.5 cm for GrIS and −4 cm for AIS of mass balance in SLE over 5000 years (Supplementary Fig. S2). The drift is removed by high-pass filtering over than 80 years, the net mass balance and the resulting high-frequency components are then used as an estimate for the unforced ice-sheet variability. Each 19-year chunk is cut after high-pass filtering to get 19-year trends of natural variability (Fig. 1). Scenario simulations To assess the ice-sheet sensitivity to different greenhouse gas emission pathways, we conducted a suite of coupled scenario simulations (each with 10 individual ensemble members), in which CO2 concentrations in LOVECLIP follow the historical from the year 1850 to 2014, and SSP1-1.9, SSP2-4.5 and SSP5-8.5 from the year 2015 to 2150 (Supplementary Table S1). In terms of the Antarctic contribution to SL by 2100 CE the LOVECLIP model projections (Fig. 2) lie within the range simulated by offline models which were forced by RCP2.6, 4.5 and 8.5 climate scenarios6,39,40. For the Greenland ice-sheet we find a similar agreement with other modeling studies6,36,37 and the “likely range” of provided by the 5th assessment report, WG1 of the Intergovernmental Panel on Climate Change (Chapter 13)4. The SSP1-1.9 and SSP2-4.5 simulations were further extended until 2500 CE (Supplementary Fig. S7). To further quantify the impact of climate-ice-sheet coupling in the Southern Hemisphere in global SL rise we conducted an additional SSP5-8.5 sensitivity experiment, for which the AIS liquid runoff and iceberg calving balance net precipitation over Antarctica (experiment SSP5-8.5_MWOFF) (Fig. 5 and Supplementary Table S1). To explore the impact of AIS hydrofracturing and ice-cliff failure parameterizations, we also obtained ensembles with these parameterizations turned off (CREVLIQ = 0 m per (m/year)−2 and VCLIF = 0 km/year) with and without meltwater flux coupling (experiments SSP5-8.5_HFCMOFF and SSP5-8.5_MWHFCMOFF) (Supplementary Figs. S4, S5). Another sensitivity experiment of the Antarctic ice-shelves to SSO warming is conducted by doubling the SSO temperature anomaly (relative to 1850 CE) to the Antarctic ice shelves in the SSP5-8.5 scenario with/without Antarctic meltwater flux (experiments Re_SSP5-8.5_2xSOTA and Re_SSP5-8.5_2xSOTA_MWOFF). Specifically, SSO temperatures in the Antarctic ice model TIM are calculated using Eq. (1). Applying doubled SSO temperature anomaly would lead to a different climate equilibrium. Therefore, new initial conditions were created. To get the new equilibrium conditions we added the SSO temperature anomalies from the 10 member initial conditions for 650 years, with fixed pre-industrial CO2 concentration. Supplementary information Figure S1: Coupling algorithm. This schematic shows how the LOVECLIP is coupled and exchanges variables and boundary conditions between LOVECLIM and PSUIM. Figure S2: Ice-sheet volume change in control experiment with pre- industrial CO2 concentrations. (a) total ice-sheet volume change (relative to initial condition) over Greenland simulated by control experiment with constant, pre-industrial CO2 concentrations; (b) same as (a), but for Antarctica. Shading indicates ensemble range, solid line the 19-year moving average of time series result and dashed line the linear regression by the Least Squares Method. Figure S3: Observed and simulated interannual mass balance of Greenland ice-sheet (GrIS) and Antarctic ice-sheet (AIS). (a) the histogram of Greenland interannual mass balance in the 5,000-year-long pre-industrial control run (CTR, gray histogram) with the 95% confidence interval range (black dashed line), and observed estimates for interannual change during 2002-2020 CE from the Gravity Recovery and Climate Experiment (GRACE) (blue line) and simulated by the forced LOVECLIP ensemble (red line); (b) same as (a), but for Antarctica. Consistent with the GRACE measurements, mass balance changes for LOVECLIM are calculated in this figure using only the grounded ice-sheet portion.	https://sealeveldocs.readthedocs.io/en/latest/park23.html
fd75f919cbb6-7	Another sensitivity experiment of the Antarctic ice-shelves to SSO warming is conducted by doubling the SSO temperature anomaly (relative to 1850 CE) to the Antarctic ice shelves in the SSP5-8.5 scenario with/without Antarctic meltwater flux (experiments Re_SSP5-8.5_2xSOTA and Re_SSP5-8.5_2xSOTA_MWOFF). Specifically, SSO temperatures in the Antarctic ice model TIM are calculated using Eq. (1). Applying doubled SSO temperature anomaly would lead to a different climate equilibrium. Therefore, new initial conditions were created. To get the new equilibrium conditions we added the SSO temperature anomalies from the 10 member initial conditions for 650 years, with fixed pre-industrial CO2 concentration. Supplementary information Figure S1: Coupling algorithm. This schematic shows how the LOVECLIP is coupled and exchanges variables and boundary conditions between LOVECLIM and PSUIM. Figure S2: Ice-sheet volume change in control experiment with pre- industrial CO2 concentrations. (a) total ice-sheet volume change (relative to initial condition) over Greenland simulated by control experiment with constant, pre-industrial CO2 concentrations; (b) same as (a), but for Antarctica. Shading indicates ensemble range, solid line the 19-year moving average of time series result and dashed line the linear regression by the Least Squares Method. Figure S3: Observed and simulated interannual mass balance of Greenland ice-sheet (GrIS) and Antarctic ice-sheet (AIS). (a) the histogram of Greenland interannual mass balance in the 5,000-year-long pre-industrial control run (CTR, gray histogram) with the 95% confidence interval range (black dashed line), and observed estimates for interannual change during 2002-2020 CE from the Gravity Recovery and Climate Experiment (GRACE) (blue line) and simulated by the forced LOVECLIP ensemble (red line); (b) same as (a), but for Antarctica. Consistent with the GRACE measurements, mass balance changes for LOVECLIM are calculated in this figure using only the grounded ice-sheet portion. Figure S4: Global surface temperature and sea-level (SL) projections, and their tendencies. (a-d) annual anomalies (relative to the 1850–1900 CE mean) of (a) the global surface temperature, (b) SL, and (c) SL contributions from the Greenland ice-sheet (GrIS) and (d) Antarctic ice-sheet (AIS). (e-h) are the respective time derivatives of (a-d) (change per year). Solid lines of (a-d) indicate the ensemble mean and shading the ensemble range. The solid line in (e) represents the 9-year moving average of the time derivative of global surface temperature, with the dashed line indicating 0 °C/year. Different colors represent the SSP5-8.5 with historical (black line; period 1850– 2150 CE), and SSP5-8.5_MWOFF (cyan line), SSP5-8.5_HFOFF (red line), SSP5- 8.5_CMOFF (pink line), SSP5-8.5_HFCMOFF (orange line) and SSP5- 8.5_MWHFCMOFF (blue line) simulations during the period 2014–2150 CE. The sudden drops seen in (f and g) are due to the changes in parameters associated with hydrofracturing and ice-cliff failure. Figure S5: Individual mass balance terms for Antarctic ice-sheet (AIS). (a-d) represent the individual AIS mass balance terms for (a) the accumulation, (b) surface melting, (c) basal melting and (d) ice calving expressed as sea-level-equivalent (SLE) per year. Solid lines indicate the ensemble mean and shading the ensemble range. Different colors represent the SSP5-8.5 with historical (black line; period 1850–2150 CE), and SSP5-8.5_MWOFF (cyan line), SSP5-8.5_HFOFF (red line), SSP5-8.5_CMOFF (pink line), SSP5-8.5_HFCMOFF (orange line) and SSP5-8.5_MWHFCMOFF (blue line) simulations during the period 2014–2150 CE. Figure S6: Individual mass balance terms for Greenland ice-sheet (GrIS) and Antarctic ice-sheet (AIS) by subsurface Southern Ocean warming. (a-d) represent the individual GrIS mass balance terms for (a) the accumulation, (b) surface melting, (c) basal melting and (d) ice calving expressed as sea-level-equivalent (SLE) per year; (e-h), same as (a-d), but for AIS. Solid lines indicate the ensemble mean and shading the ensemble range. Different colors represent the SSP5-8.5 (black line) and Re_SSP5- 8.5_2xSOTA (red line). Figure S7: Long-term Global surface temperature and sea-level (SL) projections, and their tendencies. (a-d) annual anomalies (relative to the 1850–1900 CE mean) of (a) the global surface temperature, (b) SL, and (c) SL contributions from the Greenland ice-sheet (GrIS) and (d) Antarctic ice-sheet (AIS). (e-h) are the respective time derivatives of (a-d) (change per year). Solid lines of (a-d) indicate the ensemble mean and shading the ensemble range. The solid line in (e) represents the 19-year moving average of the time derivative of global surface temperature, with the dashed line indicating 0 °C/year. Different colors represent the historical (black line; period 1850–2014 CE), and SSP1-1.9 (blue line) and SSP2-4.5 (pink line) simulations during the period 2014–2500 CE. Figure S8: Arctic and Antarctic sensitivities compared to global surface temperature increase by the end of 21st century. (a) Arctic amplification of LOVECLIP and CMIP6 under the historical and SSP5-8.5 scenarios. ∆temperature is the anomalous mean surface temperature in 2090-2100 relative to 1850-1900. (b) same as (a), but for Antarctic amplification. The Arctic region is defined as latitudes from 60oN to 90oN, and the Antarctic region from 60oS to 90oS. Figure S9: Transection of Ross ice-shelf. The red line in this map indicates the location of Ross ice-shelf transection shown in figure 6. Figure S10: Global temperature and ice height of Greenland ice- sheet (GrIS) and Antarctic ice-sheet (AIS) by the spin-up simulation. (a-c) global surface air temperature at (a) the starting point, (b) ending point and (c) difference between (b) and (a). (d-f) ice height of the GrIS at (d) the starting point, (e) ending point and (f) difference between (e) and (d); (g-i) same as (d-f), but for AIS. Figure S11: Ice velocity over Antarctica. (a) Annual average of the 1996-2016 ice velocity over Antarctica (a) observed NSIDC-0484 satellite data1,2 and simulated by (b) LOVECLIP. Table S1: List of experiments conducted with LOVECLIP. This table shows the experiments that we conducted in this study. Ensembles of 10 members with different initial conditions were simulated for historical and SSP experiments, with the initial conditions taken from the last 100 chunks of the spin-up run.	https://sealeveldocs.readthedocs.io/en/latest/park23.html
83d9565fb013-0	Palmer et al. (2020) Title: Exploring the Drivers of Global and Local Sea-Level Change Over the 21st Century and Beyond Key Points: We have developed a new set of global and local sea-level projections for the 21st century and extended to 2300 that are rooted in CMIP5 climate model simulations, including more comprehensive treatment of uncertainty than previously reported in IPCC AR5 Analysis of local sea-level projections and tide gauge data suggests that local variability will dominate the total variance in sea-level change for the coming decades at all locations considered The extended sea-level projections highlight the substantial multicentury sea-level rise commitment under all RCP scenarios and the dependence of modeling uncertainty on geographic location, time horizon, and climate scenario Corresponding author: Palmer Citation: Palmer, M. D., Gregory, J. M., Bagge, M., Calvert, D., Hagedoorn, J. M., Howard, T., et al. (2020). Exploring the Drivers of Global and Local Sea‐Level Change Over the 21st Century and Beyond. Earth’s Future, 8(9). doi:10.1029/2019ef001413 URL: https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2019EF001413 Abstract We present a new set of global and local sea-level projections at example tide gauge locations under the RCP2.6, RCP4.5, and RCP8.5 emissions scenarios. Compared to the CMIP5-based sea-level projections presented in IPCC AR5, we introduce a number of methodological innovations, including (i) more comprehensive treatment of uncertainties, (ii) direct traceability between global and local projections, and (iii) exploratory extended projections to 2300 based on emulation of individual CMIP5 models. Combining the projections with observed tide gauge records, we explore the contribution to total variance that arises from sea-level variability, different emissions scenarios, and model uncertainty. For the period out to 2300 we further breakdown the model uncertainty by sea-level component and consider the dependence on geographic location, time horizon, and emissions scenario. Our analysis highlights the importance of local variability for sea-level change in the coming decades and the potential value of annual-to-decadal predictions of local sea-level change. Projections to 2300 show a substantial degree of committed sea-level rise under all emissions scenarios considered and highlight the reduced future risk associated with RCP2.6 and RCP4.5 compared to RCP8.5. Tide gauge locations can show large ( > 50%) departures from the global average, in some cases even reversing the sign of the change. While uncertainty in projections of the future Antarctic ice dynamic response tends to dominate post-2100, we see substantial differences in the breakdown of model variance as a function of location, time scale, and emissions scenario. Introduction The IPCC Special Report on Oceans and Cryosphere in a Changing Climate (SROCC) estimates that global-mean sea level (GMSL) increased by 0.16 ± 0.05 m over the period 1902–2015 (IPCC, 2019). Furthermore, GMSL rise is accelerating: The estimated rate for 2006–2015 (3.6 mm yr − 1) is about 2.5 times the rate for 1901–1990 (1.4 mm yr − 1) with the contribution from melting ice sheets and glaciers exceeding that of thermal expansion for the recent period. Sea-level rise exacerbates extreme sea-level events and coastal hazards and has numerous adverse impacts on marine coastal ecosystems (IPCC, 2019). Information on future sea-level rise is therefore a key component for climate change impacts studies and informing coastal decision makers, particularly for adaptation planning. A number of recent studies have considered potential future changes in both local mean sea-level change and drivers of extreme sea-level events (such as waves and storm surges) to explore changes in future coastal flood risk (e.g., Cannaby et al., 2016; Howard et al., 2019; Vousdoukas et al., 2018). While changes in the drivers of extreme sea levels can make a substantive contribution, the overwhelmingly dominant factor in projections of future coastal flood risk is mean sea-level rise, which results primarily from melting of land-based ice and the expansion of seawater as the oceans warm (Church et al., 2013). Therefore, the work presented here focuses on projections of mean sea-level change at global and local scales. Throughout the manuscript, we adopt the sea-level nomenclature and definitions recently put forward by Gregory et al. (2019). The sea-level projections presented here have their origins in research carried out as part of UKCP18 (Lowe et al., 2018), a government-funded project to deliver state-of-the-art climate projections primarily for the United Kingdom. Detailed methods on the UKCP18 sea-level projections, including consideration of changes in surges, tides, and coastal waves, are described in Palmer, Howard, et al. (2018), with a synthesis of the results for the 21st century presented by Howard et al. (2019). The UKCP18 mean sea-level projections were rooted in the CMIP5 model simulations and Monte Carlo approach used for GMSL projections in the IPCC 5th Assessment Report of Working Group I (Church et al., 2013; hereafter, AR5), with several extensions and innovations, as described below. First, the contribution to future GMSL rise from dynamic ice input from Antarctica was updated based on the scenario-dependent projections from Levermann et al. (2014). Second, a scaling approach was adopted for local projections of sterodynamic sea-level change that better isolates the forced response from internal variability (e.g., Bilbao et al., 2015; Perrette, Landerer, Riva, Frieler, & Meinshausen, 2013). Third, the AR5 Monte Carlo approach was extended to the local sea-level projections to ensure traceability to the GMSL projections and preserve the correlations among the different terms. Fourth, a more comprehensive treatment of uncertainty was devised, by including several different estimates of (i) the sea-level change associated with glacial isostatic adjustment (GIA) and (ii) the barystatic-GRD fingerprints, that is, the spatial patterns of sea-level change associated with projections of land-based ice loss that arise from gravitational, rotational, and deformation effects. Note that the term “GRD” has been introduced by Gregory et al. (2019). Previously these effects have been referred to as the “sea-level equation” or “(gravitational) fingerprints,” for example. In addition, UKCP18 provided an additional set of projections based on an emulated ensemble of CMIP5 models that extend to 2300 (Palmer, Harris, et al., 2018). These exploratory projections have a high degree of consistency with the UKCP18 21st century projections and maintain traceability to the CMIP5 models. The methods presented here are almost identical to those used for UKCP18 and described by Palmer, Howard, et al. (2018). Here, we make use of global GIA estimates, rather than regional solutions developed specifically for the United Kingdom. Only two of the three sets of GRD “fingerprints” presented here were available for UKCP18, but this limitation does not make any substantive difference to the results (see section 4). Whereas UKCP18 considered only local sea-level projections for the United Kingdom, this study is global in scope, and we include new analysis of the drivers of variance for both GMSL and local projections. The local sea-level projections presented here correspond to a limited number of example tide gauge locations around the world. These locations are selected based on the availability of tide gauge records long enough to estimate local sea-level variability and to span a range of future projection regimes that illustrate important geographic differences. While we also make use of satellite altimeter observations, tide gauge records are particularly useful for estimating local interannual variability owing to thelonger records available and more direct monitoring of coastal sea level. In addition, tide gauge records include vertical land motion associated with glacial isostatic adjustment—a process included in our projections but absent from satellite altimeter observations. The focus on a limited set of tide gauges allows a deeper exploration of the drivers and uncertainties in future local sea-level change through computation of the covariance matrix of our large Monte Carlo simulations.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-1	In addition, UKCP18 provided an additional set of projections based on an emulated ensemble of CMIP5 models that extend to 2300 (Palmer, Harris, et al., 2018). These exploratory projections have a high degree of consistency with the UKCP18 21st century projections and maintain traceability to the CMIP5 models. The methods presented here are almost identical to those used for UKCP18 and described by Palmer, Howard, et al. (2018). Here, we make use of global GIA estimates, rather than regional solutions developed specifically for the United Kingdom. Only two of the three sets of GRD “fingerprints” presented here were available for UKCP18, but this limitation does not make any substantive difference to the results (see section 4). Whereas UKCP18 considered only local sea-level projections for the United Kingdom, this study is global in scope, and we include new analysis of the drivers of variance for both GMSL and local projections. The local sea-level projections presented here correspond to a limited number of example tide gauge locations around the world. These locations are selected based on the availability of tide gauge records long enough to estimate local sea-level variability and to span a range of future projection regimes that illustrate important geographic differences. While we also make use of satellite altimeter observations, tide gauge records are particularly useful for estimating local interannual variability owing to thelonger records available and more direct monitoring of coastal sea level. In addition, tide gauge records include vertical land motion associated with glacial isostatic adjustment—a process included in our projections but absent from satellite altimeter observations. The focus on a limited set of tide gauges allows a deeper exploration of the drivers and uncertainties in future local sea-level change through computation of the covariance matrix of our large Monte Carlo simulations. Since the publication of AR5 several studies have highlighted the need for more comprehensive information on potential future sea-level change “tail risk” to complement IPCC-like sea-level projections that characterize the central part of the probability distribution, in order to facilitate effective coastal planning (e.g., Hinkel et al., 2019; Stammer et al., 2019). This requirement has motivated the development of probabilistic projections that aim to provide more comprehensive information on the projected probability density functions (PDFs; e.g., Garner et al., 2018; Jevrejeva et al., 2019). However, probabilistic sea-level projections are sensitive to the assumptions made about the tails of the PDFs. For example, using different methods, both Kopp et al. (2014) and Jackson and Jevrejeva (2016) drew on the expert elicitation study of Bamber and Aspinall (2013) to introduce non-Gaussian uncertainty into the tails of Greenland and Antarctic ice sheet contributions. The substantial differences in their PDFs of projected global and local sea-level at 2100 (Jevrejeva et al., 2019) are indicative of the uncertainty associated with our understanding of key ice sheet processes and low scientific confidence in the extreme percentiles. An alternative approach to exploring tail risk is through consideration of possible high-end scenarios of future sea-level rise (Stammer et al., 2019), such as the “H + + ” scenario developed for UKCP09 (Lowe et al., 2018). Shepherd et al. (2018) suggested an event-orientated storyline approach with no requirement for a priori probability assessment. Ideally, these physicallybased narratives should be testable with future observations (e.g., marine ice cliff instability; DeConto & Pollard, 2016) and can be a useful framework to aid the communication and interpretation of risk. The UKCP18 Marine Report (Palmer, Howard, et al., 2018) recommended that information from high-end scenarios be used alongside climate-model-derived sea-level projections, such as those presented here, to more fully sample future possibility space (Rohmer, Le Cozannet, & Manceau, 2019). The outline of the paper is as follows. In section 2 we describe the observational and model data used in this study and present the tide gauge locations used for our local sea-level projections. In section 3 we present an overview of the methods used in our global and local sea-level projections. GMSL projections are presented in section 4.1, including a breakdown of the component uncertainties and discussion of the correlations among the different components. In section 4.2 we present sea-level projections at several tide gauge locations and explore the relative importance of variability, scenario, and model uncertainty over the 21st century following Hawkins and Sutton (2009). Section 4.3 focuses on model uncertainty and how the breakdown of total variance into the different components varies by geographic location, scenario, and time scale. Finally, in section 5, we discuss our key findings and present a summary. Data Tide Gauge Data The local sea-level projections presented in section 4.2 are premised on several example tide gauge locations around the world (Figure 1). These locations are chosen to span a range of future sea-level change regimes and to provide reasonable tide gauge time series with which to estimate the local interannual variability. Data are sourced from the Permanent Service for Mean Sea Level (Holgate et al., 2013; https://www.psmsl.org/). The latitude and longitude of each tide gauge location are summarized in Table S1. The tide gauge records used have not been corrected for vertical land motions. This is appropriate, since our local sea-level projections include an estimate of local glacial isostatic adjustment (GIA), and therefore, we do not want to remove this signal from the tide gauge record. Figure 1: Locations of tide gauge data used in this study. The same locations are used for extraction of satellite altimeter observations and the local sea-level projections presented in section 4.2. Satellite Altimeter Data The satellite altimeter data used in this study come from v2.0 of the European Space Agency (ESA) Climate Change Initiative for observations of sea level (http://www.esa-sealevel-cci.org), as described by Legeais et al. (2018). This data product is based on reprocessed and homogenized gridded observations from nine altimeter missions over the period 1993–2015 and provides monthly mean values for GMSL and two-dimensional fields on a ¼ × ¼° latitude-longitude grid. Monthly mean timeseries of GMSL anomaly are converted to annual means for comparison with our projections of GMSL. Similarly, we convert monthly mean two-dimensional fields of gridded sea-level anomaly to annual-mean values. We extract the annual-mean time series from the closest available grid box to the tide gauge locations shown in Figure 1. The only exception to this is for Palermo, where we select values from two grid boxes further east in order to avoid apparent data issues that may be associated with land-proximity effects. CMIP5 Data The sea-level projections presented in this study are rooted in climate model simulations carried out as part of the Coupled Model Intercomparison Project Phase 5 project (CMIP5; Taylor et al., 2012). A full list of the CMIP5 models used and their various applications is summarized in Table S2. The 21st century projections presented here are based on the same CMIP5 model ensemble as used for the GMSL projections presented in AR5. These projections make use of simulations of global-mean surface temperature (tas) and global-mean thermosteric sea-level (zostoga) rise from 21 CMIP5 models under the representative concentration pathway climate change scenarios (RCPs, Meinshausen et al., 2011). Time series of zostoga have been drift-corrected using a quadratic fit to the corresponding pre-industrial control simulation for each model. This step is performed to remove any artificial signals associated with ongoing spin-up deep ocean and/or limitations in the representation of energy conservation in the model domain, as discussed by Sen Gupta et al. (2013) and Hobbs et al. (2016). Further information is provided in the supplementary materials of AR5 (Church et al., 2013). Our extended sea-level projections to 2300 are based on an ensemble of two-layer energy balance model (TLM) simulations with parameter settings that have been tuned to emulate the forced response of individual CMIP5 models in idealized CO2 experiments models following Geoffroy et al. (2013). This ensemble also provides time series of tas and zostoga under the extended RCP scenarios (Meinshausen et al., 2011), and it is based on 14 CMIP5 models, with 11 models common to the AR5 CMIP5 ensemble. Full details of the methods and evaluation of the TLM simulations are described by Palmer, Harris, et al. (2018).	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-2	CMIP5 Data The sea-level projections presented in this study are rooted in climate model simulations carried out as part of the Coupled Model Intercomparison Project Phase 5 project (CMIP5; Taylor et al., 2012). A full list of the CMIP5 models used and their various applications is summarized in Table S2. The 21st century projections presented here are based on the same CMIP5 model ensemble as used for the GMSL projections presented in AR5. These projections make use of simulations of global-mean surface temperature (tas) and global-mean thermosteric sea-level (zostoga) rise from 21 CMIP5 models under the representative concentration pathway climate change scenarios (RCPs, Meinshausen et al., 2011). Time series of zostoga have been drift-corrected using a quadratic fit to the corresponding pre-industrial control simulation for each model. This step is performed to remove any artificial signals associated with ongoing spin-up deep ocean and/or limitations in the representation of energy conservation in the model domain, as discussed by Sen Gupta et al. (2013) and Hobbs et al. (2016). Further information is provided in the supplementary materials of AR5 (Church et al., 2013). Our extended sea-level projections to 2300 are based on an ensemble of two-layer energy balance model (TLM) simulations with parameter settings that have been tuned to emulate the forced response of individual CMIP5 models in idealized CO2 experiments models following Geoffroy et al. (2013). This ensemble also provides time series of tas and zostoga under the extended RCP scenarios (Meinshausen et al., 2011), and it is based on 14 CMIP5 models, with 11 models common to the AR5 CMIP5 ensemble. Full details of the methods and evaluation of the TLM simulations are described by Palmer, Harris, et al. (2018). At regional scales, changes in ocean dynamic sea level (arising from changes in ocean circulation and/or density) are important determinants of local sea-level change. To account for this, we make use of CMIP5 model simulations of global-mean thermosteric sea level (zostoga) and ocean dynamic sea level (zos) from 21 CMIP5 models under the RCP climate change scenarios. Following previous studies (Cannaby et al., 2016; Palmer, Howard, et al., 2018), both zostoga and zos are drift-corrected using a linear fit to the corresponding pre-industrial control simulations. These data are then used to establish regression relationships between the local sterodynamic sea-level change (zostoga + zos) and global-mean thermosteric sea-level change (zostoga) across the CMIP5 ensemble at each tide gauge location. The spatial pattern of sterodynamic sea-level change is illustrated for RCP4.5 (Figure 2). The characteristic multimodel mean response includes an increase in sea-level gradient across the Southern Ocean and enhanced sea-level rise in the North Atlantic and Arctic Oceans. The multimodel spread is largest in the North Atlantic and Arctic Oceans. Analysis of AOGCM experiments conducted for the flux-anomaly-forced model intercomparison project (FAFMIP, Gregory et al., 2016) shows that the change in the Southern Ocean is due to a combination of increases in wind stress and heat input, in the North Atlantic due to reduced heat loss and the consequent weakening of the Atlantic meridional overturning circulation (especially along the North American coast; Bouttes et al., 2014; Yin et al., 2009), and in the Arctic due to increased freshwater input from precipitation and river inflow. Figure 2: Projections of sterodynamic sea level change for the period 2081–2100 relative to the 1986–2005 average from an ensemble of 21 CMIP5 models: (a) ensemble mean; (b) ensemble spread (90% confidence interval based on the ensemble standard deviation). The spatial patterns arise from the forced response of ocean dynamic sea level across the CMIP5 ensemble. Adapted from IPCC AR5 (Church et al., 2013; Figure 13.16). GRD Estimates Changes in the amount of ice and water stored on land give rise to spatial patterns of mean sea-level (MSL) change associated with the effects on Earth’s gravity, rotation, and solid earth deformation (e.g., Tamisiea & Mitrovica, 2011). Gregory et al. (2019) refer to these effects collectively as GRD (Gravity, Rotation, Deformation), and we adopt their nomenclature here. We use three different estimates of GRD for the different ice mass terms following Slangen et al. (2014), Spada and Melini (2019), and Klemann and Groh (2013) extended to include rotational deformation following Martinec and Hagedoorn (2014). We use a single GRD estimate for changes in land water storage based on the projections of Wada et al. (2012), following Slangen et al. (2014). The geographic distributions of mass change for each component come from Slangen et al. (2014). Note that, while our results incorporate some uncertainty arising from different GRD model solutions, they do not account for uncertainties in the geographic distribution of mass change. Further details on the GRD calculations are available in the supporting information. The GRD estimates are expressed as the local MSL change per unit GMSL rise from each of the following barystatic (i.e., GMSL mass addition/loss) terms: (i) Antarctic surface mass balance, (ii) Antarctic ice dynamics, (iii) Greenland surface mass balance, (iv) Greenland ice dynamics, (v) worldwide glaciers, and (vi) changes in land water storage (Figure 3). Loss of ice from the Antarctic and Greenland ice sheets are characterized by a near-field MSL fall and a greater-than-unity rise in the far-field (e.g., Figures 3a and 3g), with notable differences in the GRD estimates for surface mass balance and ice dynamics, owing to different geographic distributions of mass change. The GRD estimates associated with worldwide glaciers and land water storage are spatially more complex, owing to the more geographically widespread mass distributions (Figures 3c and 3i). The glacier GRD pattern assumes a fixed distribution of the ratios of glacier mass loss between the glacier regions based on the projected distribution in 2100 under RCP8.5 (Church et al., 2013). Previous analysis showed that this pattern does not vary much over the 21st century and the amount of mass closely related to the initial glacier mass for a given region. We acknowledge that this is a simplistic approach, and recent studies have shown that the mass loss distribution to be model and scenario dependent (Hock et al., 2019). For the local sea-level projections presented here, we expect the uncertainty in the total glacier contribution to dominate. However, future sea-level projections could be improved by more comprehensive representation of the uncertainties associated with the spatial pattern of future glacier mass loss. Figure 3: Estimates of the combined effect of mass changes on Earth’s gravity, rotation, and solid earth deformation (GRD) on local relative sea level. Panels (a), (b), (c), (g) and (h) show the mean of three sets of estimates with corresponding standard deviations across estimates shown in (d), (e), (f), (j) and (k). only a single estimate was available for land water, and therefore, no standard deviation is shown. GRD estimates are expressed as the ratio of local MSL to GMSL per unit rise/fall with the 1:1 and zero contours indicated by the solid and dotted gray lines, respectively. The spatial patterns of GRD can have an important impact on projections of local MSL change. Depending on the geographic location, components of GMSL change can be greatly attenuated (if the location is close to where the GRD pattern is zero) and even result in a change of sign of one or more components (where the GRD pattern has negative values).	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-3	Figure 3: Estimates of the combined effect of mass changes on Earth’s gravity, rotation, and solid earth deformation (GRD) on local relative sea level. Panels (a), (b), (c), (g) and (h) show the mean of three sets of estimates with corresponding standard deviations across estimates shown in (d), (e), (f), (j) and (k). only a single estimate was available for land water, and therefore, no standard deviation is shown. GRD estimates are expressed as the ratio of local MSL to GMSL per unit rise/fall with the 1:1 and zero contours indicated by the solid and dotted gray lines, respectively. The spatial patterns of GRD can have an important impact on projections of local MSL change. Depending on the geographic location, components of GMSL change can be greatly attenuated (if the location is close to where the GRD pattern is zero) and even result in a change of sign of one or more components (where the GRD pattern has negative values). Computing the standard deviations across the three GRD estimates shows that differences are largest in the regions of ice/water mass loss (Figures 3d–3f, 3j, and 3k), corresponding to the negative value regions seen in the mean GRD patterns (Figures 3a–3c and 3g–3i). Away from these areas, the agreement among GRD estimates is high, with the standard deviation representing only a few % of the local mean signal. The circular spatial structures seen in the panels of standard deviation for the Greenland components (Figures 3j and 3k) and Antarctic ice dynamics (Figure 3e) resemble a 2-1 pattern of spherical harmonics and are indicative of slight differences in the rotational effects among the three estimates. Although all three estimates are based on the same well-understood physics, differences arise from the methods used to compute the Love numbers, as well as different grid formulations and spatial resolutions to solve the convolution integral (Table S3; see Martinec et al., 2018, for a discussion). From a practical standpoint, we find that the small differences among estimates lead to a negligible uncertainty for the tide gauge locations considered here, compared to the other factors (see section 4.3). For future studies that consider regions in closer proximity to the ice mass changes, increasing the spatial resolution would promote greater consistency among the GRD estimates. Glacial Isostatic Adjustment Similar to the effects of GRD discussed in the previous section, ongoing glacial isostatic adjustment (GIA) also leaves its imprint in the spatial pattern of MSL change. GIA is associated with the adjustment of Earth’s lithosphere and viscous mantle material to past changes in ice loading since the last glaciation (e.g., Tamisiea & Mitrovica, 2011). This adjustment process gives rise to areas of upward and downward vertical land motion, and the associated mass redistribution also influences Earth’s rotation and gravity field with additional impacts on local MSL. It is well known that GIA leads to substantial spatial variations in the rates of MSL change observed at tide gauges and, such as the lower rate of sea-level rise seen for the north of the United Kingdom compared to the south (Howard et al., 2019; Palmer, Harris, et al., 2018). Since the adjustment time scales of GIA are thousands of years, we make the approximation that the contemporary rates of its effect on local MSL change are valid for the projections (i.e., the rates are assumed to be time constant). We use three global GIA estimates in this study. The first is based on the ICE-5G (VM2 L90) model (Peltier, 2004). The second is based on ICE-6G_C (VM5a) (Argus et al., 2014; Peltier et al., 2015). ICE-6G_C is a refinement of the ICE-5G model, based on a wider range of observational constraints, including new data from Global Positioning System (GPS) receivers and as time-dependent gravity observations from both surface measurements and the satellite-based Gravity Recovery and Climate Experiment GRACE (Peltier et al., 2015). Peltier et al. (2015) state that the GIA solution from ICE-6G_C uses an improved ice loading history compared to ICE5G. Both of these data sets were sourced from http://www.atmosp.physics.utoronto.ca/~peltier/data.php. The final global GIA data product represents an independent estimate from the Australian National University based on an update of Nakada and Lambeck (1988) in 2004–2005. This final GIA estimate is identical to that used by Slangen et al. (2014). All three GIA data sets are provided on a 1 × 1° latitude-longitude grid. There are substantial differences among all three GIA estimates, despite ICE5G and ICE6G originating from the same modeling group. The overall spread in GIA estimates is largest for areas of North America, the Arctic, and Antarctica, that is, the regions of large ice mass changes during the last deglaciation. A detailed comparison and explanation for the differences is beyond the scope of this paper. A major limitation in GIA modeling is the lack of 3-D earth structures together with glaciation histories which in combination can be constrained locally against observational data. However, the optimized global 1-D estimates presented here represent a compromise, and therefore, our study may tend to overestimate the GIA uncertainty compared to locally optimized solutions. For example, UKCP18 used a regional, observationally constrained GIA solution with substantially smaller estimated uncertainties reported here (Howard et al., 2019; Palmer, Howard, et al., 2018). Methods Global-Mean Sea-Level Projections The local MSL projections presented here are based on 21st century process-based projections of GMSL presented in IPCC AR5 (Church et al., 2013). The GMSL projections are composed of seven components: (i) global-mean thermosteric sea level; and barystatic sea level due to (ii) Antarctic surface mass balance; (iii) Antarctic ice dynamics; (iv) Greenland surface mass balance; (v) Greenland ice dynamics; (vi) worldwide glaciers; and (vii) net changes in land water storage. The first component is also referred to as “global thermal expansion” and is the only term that does not constitute a change in ocean mass following Gregory et al. (2019). For the period out to 2100, the GMSL projections are underpinned by 21 CMIP5 climate model simulations (Taylor et al., 2012) of global thermal expansion (GTE) and global-mean surface temperature (GMST) change under the Representative Concentration Pathway (RCP) climate change scenarios (Meinshausen et al., 2011). For the extended period out to 2300 we use projections of GTE and GMST change from a physically based emulator that has been tuned to 16 CMIP5 models (Palmer, Harris, et al., 2018) under the RCP extensions (Meinshausen et al., 2011). Of the two sets of CMIP5 models, 11 are common across both the 21st century projections and the extended 2300 projections (Table S2). Note that the extended projections were not included in AR5 and represent one of the novel aspects of this study. We stress here that there is a much greater degree of uncertainty associated with the extended projections to 2300 than for the 21st century projections. For example, the RCP extensions make very simple assumptions about emissions trajectories, and there is deep uncertainty associated with the response of ice sheets on multicentury time scales (e.g., Edwards et al., 2019). While we present the two time horizons alongside each other for reader convenience, the extended 2300 projections should be regarded with a lower degree of confidence and treated as illustrative of the potential changes.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-4	Methods Global-Mean Sea-Level Projections The local MSL projections presented here are based on 21st century process-based projections of GMSL presented in IPCC AR5 (Church et al., 2013). The GMSL projections are composed of seven components: (i) global-mean thermosteric sea level; and barystatic sea level due to (ii) Antarctic surface mass balance; (iii) Antarctic ice dynamics; (iv) Greenland surface mass balance; (v) Greenland ice dynamics; (vi) worldwide glaciers; and (vii) net changes in land water storage. The first component is also referred to as “global thermal expansion” and is the only term that does not constitute a change in ocean mass following Gregory et al. (2019). For the period out to 2100, the GMSL projections are underpinned by 21 CMIP5 climate model simulations (Taylor et al., 2012) of global thermal expansion (GTE) and global-mean surface temperature (GMST) change under the Representative Concentration Pathway (RCP) climate change scenarios (Meinshausen et al., 2011). For the extended period out to 2300 we use projections of GTE and GMST change from a physically based emulator that has been tuned to 16 CMIP5 models (Palmer, Harris, et al., 2018) under the RCP extensions (Meinshausen et al., 2011). Of the two sets of CMIP5 models, 11 are common across both the 21st century projections and the extended 2300 projections (Table S2). Note that the extended projections were not included in AR5 and represent one of the novel aspects of this study. We stress here that there is a much greater degree of uncertainty associated with the extended projections to 2300 than for the 21st century projections. For example, the RCP extensions make very simple assumptions about emissions trajectories, and there is deep uncertainty associated with the response of ice sheets on multicentury time scales (e.g., Edwards et al., 2019). While we present the two time horizons alongside each other for reader convenience, the extended 2300 projections should be regarded with a lower degree of confidence and treated as illustrative of the potential changes. While AR5 included scenario-independent projections of Antarctic ice dynamics based on the assessed literature, we use a parameterization of scenario-dependent projections presented by Levermann et al. (2014). This procedure is based on temperature-dependent log-normal fits to the percentiles from probability distribution functions for the sea-level contribution at 2100 for each scenario (Levermann et al., 2014; Table 6, “shelf models” with time delay). All percentiles are reproduced to within ±0.01 m by our fits, except that the 95th percentile for RCP2.6 is slightly too high (0.26 m for the fit compared to 0.23 m in their table). We use the parameterized 5th to 95th percentile ranges at 2100 with the time dependence obtained as in the AR5 (Church et al., 2013; 13.SM1.6). Recent work has highlighted the potential importance of self-sustaining dynamic ice feedbacks (DeConto & Pollard, 2016), which are not explicitly accounted for in Levermann et al. (2014). However, the Levermann et al. (2014) study yields a similar projected range to other recent studies that do include these effects (Edwards et al., 2019). In addition, a recent analysis suggests that the likelihood of rapid acceleration of dynamic ice loss from West Antarctica simulated by DeConto and Pollard (2016) was overestimated (Edwards et al., 2019). We follow the same approach as AR5 in constructing a 450,000-member Monte Carlo simulation for each RCP scenario that forms the basis of both the GMSL and local MSL projections. The methods used for each component and for our two different time horizons are summarized in Table 1. With the exception of changes in Greenland ice dynamics and land water storage, all GMSL components are dependent on the climate change scenario. The scenario-independent projections have ranges based on the literature assessed in AR5. Table 1: The Methods Used for Each Component of Global-Mean Sea Level (GMSL) Change According to Time Horizon Sea-level component 21st century method Extended 2300 method Global thermal expansion (GTE) Projections are based on simulations with an ensemble of 21 CMIP5 models (Table S2) as described by Church et al. (2013) and in the text above. Any scenarios not available for a given model were estimated by the method of Good et al. (2013) from its other RCP and abrupt 4 × CO2 experiments. Projections are based on the CMIP5-based emulator ensemble of Palmer, Harris, et al. (2018) and the corresponding CMIP5 model expansion efficiencies documented by Lorbacher et al. (2015). Antarctica: surface mass balance GMSL rise is projected from global-mean surface temperature (GMST) change T(t) (as described by Church et al. (2013), as the time-integral of APR(1–S) T(t), where A is the time-mean snowfall accumulation during 1985–2005, P = 5.1 ± 1.5% °C −1 is the rate of increase of snowfall with Antarctic warming, R = 1.1 ± 0.2 is the ratio of Antarctic to global warming, and S is a number in the range 0.00–0.035 that quantifies the increase in ice discharge due to increased accumulation. The Monte Carlo chooses P, R, and S independently; P and R are normally distributed and S uniformly. Projections of GMST change come from the same ensemble of 21 CMIP5 models as for GTE. The same relationship with global surface temperature change is applied out to 2300 (Church et al., 2013). Projections of time-integral global surface temperature change come from the CMIP5-based 16-member emulator ensemble of Palmer, Harris, et al. (2018; Table S2). Antarctica: ice dynamics A scenario-dependent projection based on the results of Levermann et al. (2014). GMSL rise is modeled as a quadratic function of time, beginning with the observational rate of dynamic mass loss in 2006 and reaching Lex at 2100, where x is chosen by the Monte Carlo from a normal distribution with zero mean and standard deviation λ. The parameters L and λ are scenario-dependent; for instance, RCP2.6 has L = 56 mm and λ = 0.92, RCP8.5 91 mm and 0.86. The 2100 rate is held constant between 2100 and 2300. Greenland: surface mass balance GMSL rise is projected from GMST change (Church et al., 2013) as the time integral of EFG(T(t)), where G gives the change in Greenland SMB as a cubic function of GMST change according to Equation (2) of Fettweis et al. (2013), derived from regional climate model projections. F is a factor representing systematic uncertainty in G, and E is a factor in the range 1.00–1.15 representing the enhancement of mass loss due to reduction of surface elevation. The Monte Carlo chooses E and F independently; E is uniformly distributed, and F = eN, where N is normally distributed with zero mean and standard deviation of 0.4. Projections of GMST change are the same as for Antarctic surface mass balance. The 2100 rate is held constant between 2100 and 2300. Greenland: ice dynamics Scenario-dependent projection based on the literature at the time of AR5 (Church et al., 2013). GMSL rise is modeled as a quadratic function of time, beginning with the observational rate of dynamic mass loss in 2006 and reaching 0.020–0.085 m for RCP8.5 and 0.014–0.063 m for the other RCPs at 2100. The Monte Carlo chooses the final amount uniformly within the ranges given. The 2100 rate is held constant between 2100 and 2300.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-5	Antarctica: ice dynamics A scenario-dependent projection based on the results of Levermann et al. (2014). GMSL rise is modeled as a quadratic function of time, beginning with the observational rate of dynamic mass loss in 2006 and reaching Lex at 2100, where x is chosen by the Monte Carlo from a normal distribution with zero mean and standard deviation λ. The parameters L and λ are scenario-dependent; for instance, RCP2.6 has L = 56 mm and λ = 0.92, RCP8.5 91 mm and 0.86. The 2100 rate is held constant between 2100 and 2300. Greenland: surface mass balance GMSL rise is projected from GMST change (Church et al., 2013) as the time integral of EFG(T(t)), where G gives the change in Greenland SMB as a cubic function of GMST change according to Equation (2) of Fettweis et al. (2013), derived from regional climate model projections. F is a factor representing systematic uncertainty in G, and E is a factor in the range 1.00–1.15 representing the enhancement of mass loss due to reduction of surface elevation. The Monte Carlo chooses E and F independently; E is uniformly distributed, and F = eN, where N is normally distributed with zero mean and standard deviation of 0.4. Projections of GMST change are the same as for Antarctic surface mass balance. The 2100 rate is held constant between 2100 and 2300. Greenland: ice dynamics Scenario-dependent projection based on the literature at the time of AR5 (Church et al., 2013). GMSL rise is modeled as a quadratic function of time, beginning with the observational rate of dynamic mass loss in 2006 and reaching 0.020–0.085 m for RCP8.5 and 0.014–0.063 m for the other RCPs at 2100. The Monte Carlo chooses the final amount uniformly within the ranges given. The 2100 rate is held constant between 2100 and 2300. Glaciers GMSL rise is projected from GMST change (Church et al., 2013) as mfI(t)p, where I(t) is the time integral of GMST change (in °C yr) since 2006. Four glacier models are represented by different f,p pairs, with f in the range 3–5 mm and p ≈ 0.7. The Monte Carlo gives equal probability to the four glacier models and chooses the random normally distributed factor m with a standard deviation of 20% representing systematic uncertainty. Projections of GMST change are the same as for Antarctic surface mass balance. The same relationship with global surface temperature change is applied out to 2300 (Church et al., 2013) with a cap on the total sea level equivalent of 0.32 m to reflect current estimates of global glacier volume (Farinotti et al., 2019; Grinsted, 2013). Projections of global surface temperature change come from the CMIP5-based 16-member emulator ensemble of Palmer, Howard, et al. (2018; Table S2). Land water storage Scenario-independent projection based on the literature at the time of AR5 (Church et al., 2013). GMSL rise is modeled as a quadratic function of time, beginning with the estimated rate for 2006 and having its time-mean for 2081–2100 uniformly distributed within the range − 10 to + 90 mm by the Monte Carlo. The 2100 rate is held constant between 2100 and 2300. For the scenario-dependent terms, the ensemble spread arises from differences among the underlying CMIP5 (or emulator) simulations of GTE and GMST change and from any additional methodological uncertainties (Church et al., 2013). For each scenario, the climate model ensemble (CMIP5 or emulator) was treated as a normal distribution, with time-dependent ensemble mean QM(t) and standard deviation QS(t), where Q is GTE or GMST, both with respect to the time mean of 1986–2005, and t is time. Larger Monte Carlo ensembles were constructed with members Qi(t) = QM(t) + riQS(t), where {ri} is a set of normal random numbers (with zero mean and unit standard deviation). The {ri} are time-independent, and the same {ri} were used for GTE and GMST, so that variations within the ensemble were correlated over time and between the two quantities. The glacier contribution to GMSL is based on a relationship between the global glacier contribution and GMST change (Church et al., 2013), which is also applied post-2100. The total contribution is capped at 0.32 m, based on current estimates of total glacier mass (Farinotti et al., 2019; Grinsted, 2013). However, we note that this is a simplistic assumption. It is possible that remaining glaciers might reach a new steady state under a stable future climate following preferential loss of low-altitude ablation areas, a possibility that was not accounted for in the AR5 projections, or here. The different GMSL components are combined using a 450,000-member Monte Carlo simulation that samples from the underlying distributions. The procedure preserves the correlation between GTE and GMST change in the underlying CMIP5 model simulations (or the emulator ensemble for the period post-2100). As a result, many of the GMSL components are correlated, as discussed further in section 4. In addition, the effect of increased accumulation on the dynamics of the Antarctic ice sheet is represented in the same way as described in AR5 (Church et al., 2013; SM1.5), resulting in these terms also being weakly correlated. The sampled distributions are based on the 5th to 95th percentile ranges of the climate model simulations and literature-based assessed ranges for the scenario-independent terms. Each member of the Monte Carlo simulation is composed of a time series for each of the seven GMSL contributions listed in Table 1 with the correlations between terms preserved. Local Sea-Level Projections As we move to local MSL projections, a number of additional processes are taken into account. First, the spatial patterns of MSL change associated with each of the barystatic GMSL contributions (Table 1, ii–vii) are incorporated using estimates of the effects on Earth’s gravity, rotation, and solid earth deformation (GRD, Figure 3). Following previous studies (Bilbao et al., 2015; Palmer, Howard, et al., 2018; Perrette et al., 2013), the effects of local changes in ocean density and circulation are included by establishing regression relationships between global thermal expansion and local sterodynamic sea-level change in CMIP5 climate model simulations (see supporting information, Figures S1–S4). Finally, the spatial pattern of local MSL change from ongoing glacial isostatic adjustment (GIA, Figure 4) is included in our local MSL projections. Figure 4: (a)–(c) three estimates of the effect of glacial isostatic adjustment (GIA) on sea-level change. The zero line is indicated by the dotted contours. (d) the standard deviation of the three GIA estimates. Units for all panels are mm yr^{−1}. The projections of local MSL change for specific tide gauge locations (Figure 1, Table S1) are derived directly from the GMSL Monte Carlo projections described in the previous section. This represents an advance over the local MSL projections presented in AR5 (Church et al., 2013), which combined the different components post hoc using statistical approximations (see supplementary materials of Cannaby et al., 2016; Church et al., 2013). These approximations break the correlation structure among sea-level components and compromise the traceability of the local projections, including our understanding of how the different variances combine for total sea-level change locally.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-6	Local Sea-Level Projections As we move to local MSL projections, a number of additional processes are taken into account. First, the spatial patterns of MSL change associated with each of the barystatic GMSL contributions (Table 1, ii–vii) are incorporated using estimates of the effects on Earth’s gravity, rotation, and solid earth deformation (GRD, Figure 3). Following previous studies (Bilbao et al., 2015; Palmer, Howard, et al., 2018; Perrette et al., 2013), the effects of local changes in ocean density and circulation are included by establishing regression relationships between global thermal expansion and local sterodynamic sea-level change in CMIP5 climate model simulations (see supporting information, Figures S1–S4). Finally, the spatial pattern of local MSL change from ongoing glacial isostatic adjustment (GIA, Figure 4) is included in our local MSL projections. Figure 4: (a)–(c) three estimates of the effect of glacial isostatic adjustment (GIA) on sea-level change. The zero line is indicated by the dotted contours. (d) the standard deviation of the three GIA estimates. Units for all panels are mm yr^{−1}. The projections of local MSL change for specific tide gauge locations (Figure 1, Table S1) are derived directly from the GMSL Monte Carlo projections described in the previous section. This represents an advance over the local MSL projections presented in AR5 (Church et al., 2013), which combined the different components post hoc using statistical approximations (see supplementary materials of Cannaby et al., 2016; Church et al., 2013). These approximations break the correlation structure among sea-level components and compromise the traceability of the local projections, including our understanding of how the different variances combine for total sea-level change locally. The local MSL projection Monte Carlo simulations presented here are computed as follows. For a given RCP scenario, a single instance of the 450,000-member Monte Carlo of GMSL is randomly drawn. Each instance includes a time series for the seven GMSL components that preserves the underlying correlations among them. The barystatic timeseries (Table 1, ii–vii) are combined with the corresponding GRD estimates (Figure 3) from one of the three sets at the tide gauge latitude and longitude. This selection is made at random with all GRD patterns based on the same model, in order to preserve any correlated errors. The only exception is for land water, for which only a single GRD estimate is available (Slangen et al., 2014). The timeseries of global thermal expansion is combined with a randomly drawn regression coefficient from one of the 21 CMIP5 models in order to estimate the sterodynamic sea-level change at the tide gauge location. The resulting seven timeseries of local MSL change are then combined with an estimate of the rate of MSL change associated with GIA using one of the three estimates (Figure 4) drawn at random. This procedure (shown schematically in Figure 5) is repeated 100,000 times for each tide gauge location to build up a distribution of MSL projections under each RCP scenario. Following the approach of AR5, we take the 5th and 95th percentiles of this distribution to indicate the spread of projections for individual components and the total MSL change. Figure 5: A schematic representation of the Monte Carlo simulation performed for the local mean sea level (MSL) projections. The above process is repeated 100,000 times to build up a distribution of sea level projections for each tide gauge location for each RCP scenario. Results Global-Mean Sea-Level Projections Our projections of GMSL change show good agreement with recent observations based on satellite altimeter measurements (Figure 6). For the overlapping period of 2007–2015 the 50th percentile of the RCP4.5 projection gives the same rate as the altimeter observations of 3.8 mm per year. The observed rate of GMSL for the entire 1993–2015 period is 3.0 mm per year, indicating an acceleration over time (Nerem et al., 2018) that is also seen in the projections. For the period out to 2030 there is little difference among the projected rates of across the three RCP scenarios. Figure 6: Comparison of satellite altimeter observations (black) with our projections of global-mean sea-level change for the period 1993 to 2030. The 50th percentile and 5th to 95th percentile range for RCP4.5 are shown by the solid line and shaded region, respectively. Also shown are the 5th and 95th percentile projections for RCP2.6 and RCP8.5 (dotted lines) as indicated in the figure legend. The satellite altimeter timeseries has been adjusted so that the mean value matches the 50th percentile of the RCP4.5 projections over the period 2007–2015. Our projections of GMSL change over the 21st century (Figure 7; Table 2) yield similar ranges to those presented in AR5 (Church et al., 2013) and SROCC (Oppenheimer et al., 2019). The inclusion of an updated Antarctic ice dynamics component following Levermann et al. (2014) in the present study increases the overall uncertainties and the skewness of the distribution and results in a slightly higher central estimate for RCP8.5 compared to AR5. The SROCC projections were also based on AR5, but with an updated estimate of the contribution from Antarctica based on several process-based studies (including Levermann et al., 2014). The SROCC projected ranges at 2100 are very similar to AR5, except for the RCP8.5 scenario, which is systematically higher and shows a larger uncertainty. Our extended GMSL projections show a high degree of consistency with the CMIP5-based 21st century projections evaluated at 2100 (Table 2), with all ranges agreeing to within a few centimeters. Figure 7: Projections of global-mean sea-level change for RCP2.6 (left), RCP4.5 (middle), and RCP8.5 (right) based on the 21st century methods (a–c) and the extended 2300 methods (d–f) (Table 1). Sea-level components are shown as indicated in the figure legend. The shaded regions show the 5th to 95th percentile range from the 450,000-member Monte Carlo simulation for global thermal expansion (red) and the total (gray). The dashed and dotted lines indicate the 50th percentile and 5th to 95th percentile range from the Monte Carlo simulation presented in IPCC AR5 (Church et al., 2013). The gray shaded bars on the right-hand side of each plot indicates the 5th to 95th percentile range at 2100 or 2300 from the IPCC SROCC (Oppenheimer et al., 2019). All projections are plotted relative to a baseline period of 1986–2005. Note the change of y-axis scale for for panel (f). Table 2: Comparison of Projected Ranges of Global-Mean Sea-Level Rise Projection Year RCP2.6 RCP4.5 RCP8.5 IPCC AR5 2100 0.28–0.61 m 0.36–0.71 m 0.52–0.98 m IPCC SROCC 2100 0.28–0.59 m 0.38–0.72 m 0.61–1.11 m This study (21st century) 2100 0.28–0.66 m 0.37–0.78 m 0.55–1.11 m This study (extended 2300) 2100 0.28–0.67 m 0.35–0.78 m 0.52–1.11 m This study (extended 2300) 2200a 0.5–1.5 m 0.7–1.8 m 1.3–2.9 m This study (extended 2300) 2300a 0.6–2.2 m 0.9–2.6 m 1.7–4.5 m IPCC SROCC 2300a 0.6–1.1 m — 2.3–5.4 m SROCC expert elicitation 2300a 0.5–2.3 m — 2.0–5.4 m Nauels et al. (2017) 2300a 0.8–1.4 m 1.3–2.3 m 3.4–6.8 m Note. All projections are expressed relative to a baseline period of 1986–2005. ^a Due to large uncertainties associated with post-2100 projections, these values are reported to the nearest 0.1 m.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-7	Table 2: Comparison of Projected Ranges of Global-Mean Sea-Level Rise Projection Year RCP2.6 RCP4.5 RCP8.5 IPCC AR5 2100 0.28–0.61 m 0.36–0.71 m 0.52–0.98 m IPCC SROCC 2100 0.28–0.59 m 0.38–0.72 m 0.61–1.11 m This study (21st century) 2100 0.28–0.66 m 0.37–0.78 m 0.55–1.11 m This study (extended 2300) 2100 0.28–0.67 m 0.35–0.78 m 0.52–1.11 m This study (extended 2300) 2200a 0.5–1.5 m 0.7–1.8 m 1.3–2.9 m This study (extended 2300) 2300a 0.6–2.2 m 0.9–2.6 m 1.7–4.5 m IPCC SROCC 2300a 0.6–1.1 m — 2.3–5.4 m SROCC expert elicitation 2300a 0.5–2.3 m — 2.0–5.4 m Nauels et al. (2017) 2300a 0.8–1.4 m 1.3–2.3 m 3.4–6.8 m Note. All projections are expressed relative to a baseline period of 1986–2005. ^a Due to large uncertainties associated with post-2100 projections, these values are reported to the nearest 0.1 m. The extended 2300 projections illustrate the long-term committed sea-level rise under all RCP scenarios and the large uncertainties associated with these time horizons. At these extended time horizons, there is a greater distinction between scenarios than for the 21st century, and the benefits of reduced greenhouse gas emissions on the potential magnitude of committed future sea-level rise are clear (cf. RCP2.6 and RCP8.5 at 2300, Table 2). For the extended 2300 projections, the total glacier ice mass becomes exhausted between 2200 and 2300 under RCP4.5 and between 2100 and 2300 under RCP8.5 (Figure 7). Given the different methods, and the inherently large uncertainty associated with projections on multicentury time horizons, our projected values at 2300 are broadly consistent with the estimates presented in IPCC SROCC (Oppenheimer et al., 2019) and Nauels et al. (2017) (Table 2). Our results show a substantially larger projected range for RCP2.6 (0.6–2.2 m) than the SROCC likely range (0.6–1.1 m) and Nauels et al. (2017; 0.8–1.4 m). This larger range arises primarily from the Antarctica ice dynamics term (Figure 8; Figure S5) and may have important implications for adaptation planning. Both SROCC (2.3–5.4 m) and Nauels et al. (2017; 3.4–6.8 m) show higher projected ranges under RCP8.5 than the present study (1.7–4.5 m). For SROCC, these larger values arise primarily from the Antarctic component (Figure 8). For Nauels et al. (2017) the difference seems to arise from larger contributions and greater uncertainties in both global thermal expansion and Greenland surface mass balance (Figure S5). Figure 8: Components of projected global-mean sea-level (GMSL) change at 2100 (a–c, based on 21st century methods) and 2300 (d–f, based on extended 2300 methods) for RCP2.6 (left), RCP4.5 (middle), and RCP8.5 (right). The horizontal lines and shaded regions indicate the 50th percentile and the 5th to 95th percentile range, respectively, from the 450,000-member Monte Carlo simulation. Scenario-independent projections are shown in gray. For reference, the corresponding global-mean surface temperature (GMST) change is shown in the final column of each panel, with secondary y-axis on the right-hand side. All projections are expressed relative to the 1986–2005 average. Corresponding projected ranges from IPCC SROCC (Oppenheimer et al., 2019) are indicated by the dashed rectangles, based on the supplementary data files (GMSL and Antarctica) and table 13.SM.1/table 13.8 (other components) of Church et al. (2013). In order to gain some initial insights into the drivers of GMSL change, we present the breakdown of components at 2100 and 2300 based on the 5th to 95th percentile range (Figure 8). For all scenarios and both time horizons, the single largest component of uncertainty is that associated with the contribution from Antarctica (combined effects of changes in surface mass balance and ice dynamics). The 5th to 95th percentile range for Antarctica includes negative values, which arises from positive surface mass balance owing to a warmer atmosphere transporting more moisture. The components and their uncertainties generally increase under the higher emissions scenarios for both time horizons. At 2100, the RCP8.5 scenario induces substantial increases in the contribution ranges for Greenland and worldwide glaciers. The exhaustion of glacier mass for the extended projections under RCP4.5 and RCP8.5 results in reduced uncertainty for this term at 2300. Since the 21st century projections in both SROCC and the current study use AR5 methods with updates only for Antarctica, the GMSL-component projections at 2100 are identical to SROCC except for that term (Figures 8a–8c). For all three RCP scenarios our projections show substantially larger uncertainties in the Antarctica component with higher 95th percentiles that translate into more modest differences in GMSL. For the projections on extended time horizons, the methods differ to a greater extent. The SROCC 2300 projections are based on Table 13.8 of AR5 (Church et al., 2013), which drew upon a diverse set of model simulations that were broadly categorized as “Low,” “Medium,” and “High” scenarios. The extended 2300 projections presented here are based on the RCP scenarios, using a physical framework that is consistent with the 21st century projections and traceable to CMIP5 climate model simulations. For 2300, we see substantial differences between SROCC and the present study for the available GMSL components (Figures 8d–8f). No estimate of post-2100 land water changes were made for AR5/SROCC, and our methods use a simple assumption of applying the 2100 rates over the period 2100–2300 (Table 1). The magnitude and relative importance of GMSL components at 2300 show strong scenario dependence. For RCP8.5 the dominant terms become thermal expansion, Greenland and Antarctica with the scenario-independent land water changes and mass-limited glacier contribution becoming less important compared to RCP2.6 or RCP4.5. RCP8.5 also shows the largest difference between the projected ranges for the present study and SROCC, with substantial differences for all three of the leading component terms. The 5th to 95th percentile component ranges combine nonlinearly to the overall projected ranges for GMSL (Figure 8). The reason for this is illustrated in Figure 9, which shows the correlation between components evaluated across the 450,000-member Monte Carlo set at 2100. Global thermal expansion, Greenland surface mass balance, and worldwide glaciers are all positively correlated: Stronger warming promotes an increased contribution to GMSL from all of these terms. Figure 9: Correlation matrices of the different GMSL components for each RCP scenario based on the Monte Carlo spread at 2100. The matrices illustrate the relationships between GMSL components and explain why the total variance is not identical to the sum of the variances of the components.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-8	For 2300, we see substantial differences between SROCC and the present study for the available GMSL components (Figures 8d–8f). No estimate of post-2100 land water changes were made for AR5/SROCC, and our methods use a simple assumption of applying the 2100 rates over the period 2100–2300 (Table 1). The magnitude and relative importance of GMSL components at 2300 show strong scenario dependence. For RCP8.5 the dominant terms become thermal expansion, Greenland and Antarctica with the scenario-independent land water changes and mass-limited glacier contribution becoming less important compared to RCP2.6 or RCP4.5. RCP8.5 also shows the largest difference between the projected ranges for the present study and SROCC, with substantial differences for all three of the leading component terms. The 5th to 95th percentile component ranges combine nonlinearly to the overall projected ranges for GMSL (Figure 8). The reason for this is illustrated in Figure 9, which shows the correlation between components evaluated across the 450,000-member Monte Carlo set at 2100. Global thermal expansion, Greenland surface mass balance, and worldwide glaciers are all positively correlated: Stronger warming promotes an increased contribution to GMSL from all of these terms. Figure 9: Correlation matrices of the different GMSL components for each RCP scenario based on the Monte Carlo spread at 2100. The matrices illustrate the relationships between GMSL components and explain why the total variance is not identical to the sum of the variances of the components. Conversely, Antarctic surface mass balance is strongly anticorrelated with these terms because a warmer atmosphere tends to promote greater snowfall on Antarctica and reduce GMSL. As discussed in section 3.1, the AR5 methods resulted in a weak correlation between the surface mass balance and ice dynamics terms for Antarctica, which is also included here. We find similar correlations among components for all RCPs (Figure 9), although these tend to be slightly reduced for the higher emissions scenarios. Analysis of correlations at 2300 in the extended projections yields similar results (Figure S6), except for the glacier term that shows weaker correlations for RCP4.5 and RCP8.5 owing to the cap on total ice mass (illustrated in Figures 7e and 7f). Local Sea-Level Projections In this section we present our local MSL projections for 16 example tide gauge sites (Figure 1; Table S1). We focus our presentation on the highest (RCP8.5) and lowest (RCP2.6) emissions scenarios and include annual-mean tide gauge and satellite altimeter timeseries to illustrate the observed trends and local sea-level variability (Figure 10). In general, there is good agreement between the observed decadal rates of MSL change and the early part of the projections, noting that the satellite altimeter timeseries do not account for vertical land motion processes associated with, for example, GIA, local subsidence, or tectonic activity. Locations of poorer agreement between observed and projected MSL trends include Lima and Port Louis. However, the high degree of consistency between altimeter and tide gauge observations at these locations suggests the discrepancy arises from climatic variability rather than non-GIA vertical land-motion processes. There is an apparent jump in the Pago Pago tide gauge timeseries towards the end of the record that could be related to a nearby earthquake in 2009 that resulted in several tsunami waves hitting the island. This jump is not seen in the satellite altimeter timeseries, confirming the likely role of substantial vertical land motion at this location. The observed interannual sea-level variability varies considerably by location and demonstrates that the reality of future sea-level change will be a combination of the climate response and unforced variability (e.g., Roberts et al., 2016). Figure 10: Local sea-level projections for RCP2.6 (blue) and RCP8.5 (red). Shaded regions indicate the 5th to 95th percentile range of the 100,000-member Monte Carlo simulation. The dotted lines indicate the 5th and 95th percentile projections from the IPCC SROCC (Oppenheimer et al., 2019). Local annual-mean tide gauge data are indicated by the solid black line. Local annual-mean satellite altimeter data are indicated by the solid gray line. All timeseries are shown relative to the 1986–2005 average. Note the different y-axis for Barentsburg. As with GMSL, local MSL projections for the 21st century are similar to those reported in the IPCC SROCC (Figure 10) with agreement varying somewhat across tide gauge sites. Differences are thought to arise primarily from (i) the methods for estimating sterodynamic sea level and/or (ii) the methods used to combine sea-level components. It is apparent that the SROCC projections include some residual variability that originates from the underlying CMIP5 climate model simulations of sterodynamic sea-level change, which may also be present in the 1986–2005 reference state. Our regression-based approach to local sterodynamic sea-level change is designed to better isolate the climate change signal, resulting in smoother projections that do not include this simulated variability. This regression approach makes the approximation of a linear relationship between local sterodynamic sea-level change and global thermal expansion, which may also result in some differences with SROCC projections. Statistical approximations were used to combine the different local sea-level components for AR5 outside of the GMSL Monte Carlo framework that assumed terms were either perfectly correlated or perfectly uncorrelated, as described in equation 13.SM.1 of Church et al. (2013). This breaks the correlation structure among the GMSL components (Figure 9) and likely results in differences in the SROCC projected ranges for some locations. Re-running our local projections using only the GIA estimates used for AR5/SROCC (i.e., Lambeck and ICE5G, Figure 4) makes a negligible difference to the results shown in Figure 10. Analysis of the differences among our GRD fingerprints suggests that any differences in this regard are also likely to be negligible (see section 4.3). Most tide gauge locations show that MSL is currently rising and that this rise will accelerate over the 21st century under the RCP8.5 scenario. The 21st century rates of sea-level change under RCP2.6 are relatively stable and most locations show the scenarios diverging from the mid-21st century. For most locations, the change in sea level over the 21st century is large compared to the tide gauge variability and implies that adaptation measures will be necessary to preserve current levels of coastal flood protection. Barentsburg (Svalbard) and Reykjavik (Iceland) show atypical MSL projections. In both cases, the proximity to Greenland results in negative sea-level rise from this component (see Figure 3), which largely cancels out the positive contributions from the other climatic components, resulting in small scenario dependency at these locations. Barentsburg has a substantial rate of MSL fall associated with GIA, which accounts for the more negative values seen at this location compared to Reykjavik. While Oslo retains substantive scenario dependency, the negative GIA signal results in a much-reduced rates of rise under RCP8.5 and the expectation of a sea-level fall under RCP2.6. Barentsburg, Reykjavik, and Oslo clearly illustrate that projections of GMSL cannot necessarily be taken as indicative of local MSL change. Excluding these atypical tide gauge locations, we still see substantive variations in future sea-level rise across the remaining tide gauge sites. The range of behavior is spanned by New York and Stanley II with ranges at 2100 under RCP2.6 (RCP8.5) of 0.27–0.84 m (0.57–1.35 m) and 0.21–0.51 m (0.45–0.91 m), respectively. New York has a large spread in sterodynamic sea-level change and also a substantial positive contribution from GIA. The relative proximity of Stanley II (the Falkland Islands) to Antarctica results in a strong attenuation of the MSL change associated with Antarctic ice dynamics, which reduces both the overall magnitude and the spread of uncertainty in future projections.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-9	Most tide gauge locations show that MSL is currently rising and that this rise will accelerate over the 21st century under the RCP8.5 scenario. The 21st century rates of sea-level change under RCP2.6 are relatively stable and most locations show the scenarios diverging from the mid-21st century. For most locations, the change in sea level over the 21st century is large compared to the tide gauge variability and implies that adaptation measures will be necessary to preserve current levels of coastal flood protection. Barentsburg (Svalbard) and Reykjavik (Iceland) show atypical MSL projections. In both cases, the proximity to Greenland results in negative sea-level rise from this component (see Figure 3), which largely cancels out the positive contributions from the other climatic components, resulting in small scenario dependency at these locations. Barentsburg has a substantial rate of MSL fall associated with GIA, which accounts for the more negative values seen at this location compared to Reykjavik. While Oslo retains substantive scenario dependency, the negative GIA signal results in a much-reduced rates of rise under RCP8.5 and the expectation of a sea-level fall under RCP2.6. Barentsburg, Reykjavik, and Oslo clearly illustrate that projections of GMSL cannot necessarily be taken as indicative of local MSL change. Excluding these atypical tide gauge locations, we still see substantive variations in future sea-level rise across the remaining tide gauge sites. The range of behavior is spanned by New York and Stanley II with ranges at 2100 under RCP2.6 (RCP8.5) of 0.27–0.84 m (0.57–1.35 m) and 0.21–0.51 m (0.45–0.91 m), respectively. New York has a large spread in sterodynamic sea-level change and also a substantial positive contribution from GIA. The relative proximity of Stanley II (the Falkland Islands) to Antarctica results in a strong attenuation of the MSL change associated with Antarctic ice dynamics, which reduces both the overall magnitude and the spread of uncertainty in future projections. We combine the tide gauge data with our local MSL projections to explore the relative importance of variability, scenario, and model uncertainty over the 21st century following Hawkins and Sutton (2009). Local sea-level variability is estimated by de-trending the tide gauge records and computing the standard deviation of the residual timeseries. The scenario standard deviation is estimated using the central estimates under RCP2.6, RCP4.5, and RCP8.5. Finally, the model uncertainty is estimated by computing the average standard deviation of the Monte Carlo simulation across the three RCP scenarios. Our analysis suggests that sea-level variability is likely to be a key driver of MSL change for the coming decades at all tide gauge locations (Figure 11). Conversely, the impact of RCP scenario will only make a substantive contribution towards the end of the 21st century. At Barentsburg and Reykjavik, differences across scenarios explain 10–20% of the projected variance, which is related to the negative contribution from Greenland canceling out other terms (as discussed above). At all locations, model uncertainty explains a large share of the overall variance and is particularly important for Barentsburg and Reykjavik. Figure 11: Assessment of the fraction of total variance of sea-level change explained by model, scenario, and variability, following Hawkins and Sutton (2009) as indicated in the figure legend for Auckland. The extended 2300 projections again illustrate the large levels of committed sea-level change associated with both RCP2.6 and RCP8.5 for the coming centuries (Figure 12). The projections show greater separation between these two scenarios post-2100 and the large degree of uncertainty on these time horizons. For several sites, the projected range at 2300 for RCP8.5 exceeds 5 m. Even under RCP2.6 central estimates of sea-level rise are in excess of 1 m for most locations, and the projected range exceeds 2 m at many locations. The geographical variations in projections seen over the 21st century (Figure 10) are essentially preserved (as a proportion of the signal size), resulting in differences in the projected ranges up to several meters. At these extended time horizons, the projected sea-level changes are an order of magnitude greater than the interannual tide gauge variability. At most locations, the magnitude of MSL rise and the projected range is larger than for the corresponding projection of GMSL (Figure 7). A large part of the increased spread comes from the amplification of the Antarctic ice sheet signals (the greatest source of uncertainty, Figure 8) by the GRD patterns, which have local values greater than unity for most tide gauge sites. Figure 12: Local sea-level projections for RCP2.6 (blue) and RCP8.5 (red). Shaded regions indicate the 5th to 95th percentile range of the 100,000-member Monte Carlo simulation. Annual tide gauge data are indicated by the black line. All timeseries are shown relative to the 1986–2005 average. Note the different y-axis for Barentsburg, Oslo, and Reykjavik. Analysis of Model Uncertainty In this section, we further explore the contributions to the model uncertainty that is represented by the projected ranges of sea-level change for a given scenario. In particular, we consider which MSL components are dominant in determining the total variance in projected ranges as a function of geographic location, time horizon, and RCP scenario. We compute the total variance for the GMSL timeseries and several example tide gauge locations for both RCP2.6 and RCP8.5. We also compute the covariance matrix across the Monte Carlo ensembles as a function of time. The off-diagonal elements of the matrix are combined into an additional term that we label “interactions”—with this contribution arising from correlations among the components. While this analysis was performed on both, the 21st century and extended 2300 projections, the results up to 2100 are similar (Figure S7). For this reason, we focus our presentation on the extended 2300 results so that we can look across all relevant time scales. In some instances, the anticorrelation between terms leads to a reduction of total variance. For simplicity of the graphical representation and to focus discussion on the relative importance of contributions to variance in general, our analysis is based on the absolute variances. The total variance at 2300 under RCP8.5 is more than double that for RCP2.6, both globally and at all tide gauge locations (Figures 13 and 14; left column), indicating the inherently larger uncertainties under high emissions scenarios, related to the uncertainty in model climate sensitivity. For GMSL (Figure 13, top row) the ensemble spread is initially dominated by global thermal expansion, but uncertainty in Antarctic ice dynamics becomes the dominant term from the latter half of the 21st century. Prior to 2100, there is little difference in the breakdown of variance by RCP scenario. Post-2100 we see a much larger contribution from Greenland surface mass balance under RCP8.5, becoming the second largest source of variance after Antarctic ice dynamics. Figure 13: Time evolution of variance associated with model uncertainty for GMSL and three example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. For GMSL, “exp” refers to global thermal expansion. For tide gauge sites, “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.” Figure 14: Time evolution of variance associated with model uncertainty for four example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. Note that “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.” The breakdown of variance for Barentsburg (Svalbard; Figure 13) shows a dominant contribution from glaciers over the 21st century, and this term remains important out to 2300. This feature is due to the large glacier mass on Svalbard, which leads to large negative values in the GRD estimates associated primarily with vertical land uplift. We see strong scenario dependence on the contributions to variance post-2100, with Greenland surface mass balance and sterodynamic sea-level change becoming much more important under RCP8.5 than RCP2.6. Variance arising from GIA estimates makes only a minor contribution at this location.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-10	Figure 13: Time evolution of variance associated with model uncertainty for GMSL and three example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. For GMSL, “exp” refers to global thermal expansion. For tide gauge sites, “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.” Figure 14: Time evolution of variance associated with model uncertainty for four example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. Note that “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.” The breakdown of variance for Barentsburg (Svalbard; Figure 13) shows a dominant contribution from glaciers over the 21st century, and this term remains important out to 2300. This feature is due to the large glacier mass on Svalbard, which leads to large negative values in the GRD estimates associated primarily with vertical land uplift. We see strong scenario dependence on the contributions to variance post-2100, with Greenland surface mass balance and sterodynamic sea-level change becoming much more important under RCP8.5 than RCP2.6. Variance arising from GIA estimates makes only a minor contribution at this location. In contrast, Reykjavik (Iceland; Figure 13) has a large contribution to variance from the different GIA estimates, and this is the dominant source of variance over most of the 21st century. Post-2100, Antarctic ice dynamics dominates the variance under RCP2.6, with GIA remaining an important contribution for this scenario. For RCP8.5, Greenland surface mass balance and Antarctic ice dynamics contribute similarly to total variance post-2100. As discussed in the previous section, New York (USA; Figure 13) has a particularly large uncertainty associated with future changes in ocean circulation, and this is reflected in the dominance of sterodynamic sea-level change over the 21st century. GIA is also relatively important at this location, particularly under RCP2.6. Again, it is ultimately the Antarctic ice dynamics that becomes the dominant source of variance post-2100, but sterodynamic sea-level change remains a sizeable contribution under RCP8.5. The breakdown of variance for Mera (Japan; Figure 14) and Diamond Harbour (India; Figure 14) is typical of many lower latitude locations with characteristics that are very similar to that of GMSL (Figure 13, top row). Mera has a substantive contribution from land water under RCP2.6 that is absent from Diamond Harbour. At both locations, the contribution from GIA is very small, indicating that future changes will be dominated by contemporary climate-driven changes. As with all tide gauge locations, we cannot rule out the potential importance of nonclimatic processes, particularly those associated with vertical motion (e.g., subsidence and tectonic activity). Palermo (Argentina; Figure 14) shows a marked reduction in the contribution from Antarctic ice dynamics, which is associated with its proximity to West Antarctica (Figure 3b). This is also reflected in the reduced total variance seen at this tide gauge location for both RCP scenarios compared to GMSL. This results in a larger relative importance of many of the other MSL components. Variance arising from different scenarios of land water change is the second largest post-2100 term (after Antarctic ice dynamics) under RCP2.6. For RCP8.5, post-2100 variance is dominated by Greenland surface mass balance. Like Palermo, Stanley II (Falkland Islands; Figure 14) also sees a reduced total variance compared to GMSL resulting from the proximity of Antarctica. In this case, Antarctic ice dynamics makes a negligible contribution to total variance because the tide gauge location is close to the zero contour of the associated GRD pattern of MSL change (Figure 3b). The relative importance of the other terms over the 21st century is similar to Palermo. Post-2100, land water dominates the variance under the RCP2.6 scenario, and Greenland surface mass balance dominates for the RCP8.5 scenario. In the absence of substantive signals from Antarctic ice dynamics, the anticorrelated contributions of Greenland and Antarctic surface mass balance lead to a particularly large “interactions” term that reduces the overall variance compared to GMSL. The importance of the land water contribution for both Palermo and Stanley II under RCP2.6 arises for a number of reasons. First, the land water projections are scenario-independent, so its relative importance increases under RCP2.6 compared to RCP8.5. Second, the relative importance of land water is further increased at these locations due to the strong attenuation of the Antarctica signals, in relation to GRD (Figure 3), as discussed above. Third, both Palermo and Stanley II are in a region where the global land water contribution (and its uncertainty) is amplified at regional scales by the GRD patterns (Figure 3). Overall, we see a strong geographic and time dependence of the contribution to total variance. There is little scenario dependence until after 2100, and this is typified by a substantially increased contribution from Greenland surface mass balance under RCP8.5. Antarctic ice dynamics tend to dominate the total variance from the latter half of the 21st century out to 2300 under both RCP2.6 and RCP8.5. However, this is not the case for locations in proximity to West Antarctica (e.g., southern South America), where the GRD pattern of MSL results in greatly attenuated signals for Antarctic ice dynamics. The contrasting results presented here make a clear case for the need for site-specific local sea-level projections. In addition, the reduction of variance (uncertainty) in future projections implies different research priorities, depending on geographic location and time horizon. As part of our analysis of variance, we also investigate the contribution from uncertainty in the GRD estimates presented in section 2.4 (Figure 3). We choose the three tide gauge sites with the largest spread in one or more GRD components, that is, Barentsburg, Reykjavik, and Stanley II, and conduct the following simple analysis. We run an additional instance of the MSL Monte Carlo for each location using the average value of the GRD fingerprints at each site for RCP8.5 (i.e., the scenario with the largest signal). The projected MSL and component ranges are then compared to the full Monte Carlo simulations that include a random choice of GRD estimate. The results demonstrate that using multiple GRD estimates has an essentially negligible contribution to the total variance—the differences at 2100 for RCP8.5 are just about perceptible for Barentsburg (Figure 15). It is therefore reasonable to use only a single set of GRD estimates when computing local MSL projections. However, our analysis does not account for uncertainty in the associated space-time mass distributions that is used to compute the GRD patterns. This is an area that may benefit from further research. Figure 15: Projected ranges of MSL change at 2100 for three example tide gauge locations under RCP8.5. The dashed lines indicate the results when the average of three GRD estimates (rather than random selection) is used in the Monte Carlo simulations. All projections are expressed relative to a baseline period of 1986–2005. Scenario-independent components are indicated in gray. Summary and Conclusions	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
83d9565fb013-11	As part of our analysis of variance, we also investigate the contribution from uncertainty in the GRD estimates presented in section 2.4 (Figure 3). We choose the three tide gauge sites with the largest spread in one or more GRD components, that is, Barentsburg, Reykjavik, and Stanley II, and conduct the following simple analysis. We run an additional instance of the MSL Monte Carlo for each location using the average value of the GRD fingerprints at each site for RCP8.5 (i.e., the scenario with the largest signal). The projected MSL and component ranges are then compared to the full Monte Carlo simulations that include a random choice of GRD estimate. The results demonstrate that using multiple GRD estimates has an essentially negligible contribution to the total variance—the differences at 2100 for RCP8.5 are just about perceptible for Barentsburg (Figure 15). It is therefore reasonable to use only a single set of GRD estimates when computing local MSL projections. However, our analysis does not account for uncertainty in the associated space-time mass distributions that is used to compute the GRD patterns. This is an area that may benefit from further research. Figure 15: Projected ranges of MSL change at 2100 for three example tide gauge locations under RCP8.5. The dashed lines indicate the results when the average of three GRD estimates (rather than random selection) is used in the Monte Carlo simulations. All projections are expressed relative to a baseline period of 1986–2005. Scenario-independent components are indicated in gray. Summary and Conclusions We have presented MSL projections under the RCP2.6, RCP4.5, and RCP8.5 climate change scenarios for both global-mean sea level and for a number of example tide gauge locations around the world. Our 21st century projections are directly traceable to the CMIP5-based sea-level projections presented in AR5 (Church et al., 2013) with updated treatment of the contribution from Antarctic ice dynamics following Levermann et al. (2014) and show similar results to projections presented in the SROCC (Oppenheimer et al., 2019). Our regression approach for sterodynamic sea-level change more cleanly isolates the forced signal and enables us to characterize the relative importance of variance arising from scenario, model spread, and observed sea-level variability over the 21st century, following Hawkins and Sutton (2009). A key aspect of the study is the use of the same Monte Carlo framework for both global and local sea-level projections. This means that global and local projections are entirely consistent, preserving the correlations among components and allowing us to quantify the contributions to total variance (uncertainty) by geographic location, time horizon, and scenario. We introduced a new set of exploratory extended projections to 2300 that are rooted in CMIP5 projections through the use of a physically based emulator to extend individual CMIP5 climate model simulations (Palmer, Harris, et al., 2018). These emulator-based projections are designed for maximum consistency with the 21st century projections and show a high level of agreement at 2100. Our main findings are summarized as follows: We have developed a consistent set of local sea-level projections that are directly traceable to the GMSL projections developed for AR5 and preserve the relationships between components (an important factor in determining the fraction of variance explained). Our projections of GMSL at both 2100 and 2300 yield similar numbers to those recently reported in the IPCC Special Report on the Ocean and Cryosphere in a Changing Climate (IPCC, 2019), noting the large inherent uncertainties associated with multicentury time horizons. For RCP2.6, our projected ranges at 2300 are larger than some recent studies (including the SROCC likely range) and may have important implications for long-term adaptation planning. Combined analysis of local MSL projections and tide gauge records suggests that sea-level variability dominates the total variance in the coming decades, with climate change scenario becoming increasingly important over the 21st century. Model uncertainty tends to be the largest source of variance from the mid-21st century. Local MSL projections can show large departures from the GMSL response and are typically associated with substantially larger uncertainties. A few locations will see MSL decrease in future owing to spatial patterns of GRD and the local contribution from GIA. On century time scales, the projected MSL changes are large compared to observations of local sea-level variability. This indicates that many places will be exposed to greater coastal flood risk unless effective adaptation measures are taken. The extended projections to 2300 illustrate the large degree of committed sea-level rise even in strong mitigation scenarios. They also illustrate the substantially increased risk for the highest emissions scenarios, which are associated with several meters of MSL rise at most tide gauge locations. Correlations between component terms mean that the variance of the total MSL change is not identical to the sum of the variances of the components. Moreover, the breakdown of variance depends on both, geographic location and time horizon, with differences in scenario post-2100. Antarctic ice dynamics dominate the total variance post-2100 except at locations where GRD patterns strongly attenuate this signal. This study highlights the need for development of site-specific MSL projections for effective planning (GMSL projections of cannot be reliably used to indicate the local changes). The time-space-scenario dependence in the contributions to total variance suggests that research priorities for reducing uncertainty in sea-level projections are likely to vary by geographic location and planning time horizon.	https://sealeveldocs.readthedocs.io/en/latest/palmer20.html
4418cc28ef3a-0	Hay et al. (2015) Title: Probabilistic reanalysis of twentieth-century sea-level rise Key Points: Reconstruction of the global mean sea-level changes since 1900, from combining the probability distributions of a set of tide gauge records, refines estimates and is more consitent with estimates based on the sum of contributions The global mean sea level rose by about 1.2 ± 0.2 mm/year between 1901 and 1990 and by 3.0 ± 0.7 mm/year between 1993 and 2010. The rate of sea-level rise has been accelerating Keywords: sea level reconstruction, tide gauge, global mean sea level rise, probabilistic framework Corresponding author: Carling C. Hay. Citation: Hay, C. C., Morrow, E., Kopp, R. E., & Mitrovica, J. X. (2015). Probabilistic reanalysis of twentieth-century sea-level rise. Nature, 517(7535), 481–484. doi:10.1038/nature14093 URL: https://www.nature.com/articles/nature14093 Abstract Estimating and accounting for twentieth-century global mean sea-level (GMSL) rise is critical to characterizing current and future human-induced sea-level change. Several previous analyses of tide gauge records [1,2,3,4,5,6] — employing different methods to accommodate the spatial sparsity and temporal incompleteness of the data and to constrain the geometry of long-term sea-level change — have concluded that GMSL rose over the twentieth century at a mean rate of 1.6 to 1.9 millimetres per year. Efforts to account for this rate by summing estimates of individual contributions from glacier and ice-sheet mass loss, ocean thermal expansion, and changes in land water storage fall significantly short in the period before 1990 [7]. The failure to close the budget of GMSL during this period has led to suggestions that several contributions may have been systematically underestimated [8]. However, the extent to which the limitations of tide gauge analyses have affected estimates of the GMSL rate of change is unclear. Here we revisit estimates of twentieth-century GMSL rise using probabilistic techniques [9,10] and find a rate of GMSL rise from 1901 to 1990 of 1.2 ± 0.2 millimetres per year (90% confidence interval). Based on individual contributions tabulated in the Fifth Assessment Report [7] of the Intergovernmental Panel on Climate Change, this estimate closes the twentieth-century sea-level budget. Our analysis, which combines tide gauge records with physics-based and model-derived geometries of the various contributing signals, also indicates that GMSL rose at a rate of 3.0 ± 0.7 millimetres per year between 1993 and 2010, consistent with prior estimates from tide gauge records4. The increase in rate relative to the 1901–90 trend is accordingly larger than previously thought; this revision may affect some projections11 of future sea-level rise. Editorial Summary: Twentieth century sea levels revisited Rates of sea-level rise calculated from tide gauge data tend to exceed bottom-up estimates derived from summing loss of ice mass, thermal expansion and changes in land storage. Carling Hay et al. provide a statistical reassessment of the tide gauge record — which is subject to bias due to sparse and non-uniform geographic coverage and other uncertainties — and conclude that sea-level rose by about 1.2 millimetres per year from 1901 to 1990. This is slightly lower than prior estimates and is consistent with the bottom-up estimates. The same analysis applied to the period 1993–2010, however, indicates a sea-level rise of about three millimetres per year, consistent with other work and suggesting that the recent acceleration in sea-level rise has been greater than previously thought. Main Tide gauges provide records of local sea-level changes that, in the case of some sites, extend back to the eighteenth century [12,13,14]. However, using the database of tide gauge records [15] to estimate historical GMSL rise (defined as the increase in ocean volume normalized by ocean area) is challenging. Tide gauges sample the ocean sparsely and non-uniformly, with a bias towards coastal sites and the Northern Hemisphere, and with few sites at latitudes greater than 60° (see, for example, refs 4, 9). In addition, tide gauge time series show significant inter-annual to decadal variability, and they are characterized by missing data (that is, intervals without observations at the start, middle or end of a time series). From the perspective of estimating GMSL changes, the data are contaminated by local and regional signals due to ongoing glacial isostatic adjustment (GIA) associated with past ice ages [16,17], the spatially non-uniform pattern of sea-level rise associated with changes in contemporary land ice sources[18,19,20,21], ocean/atmosphere dynamics [22], and other local factors including tectonics, sediment compaction, groundwater pumping and harbour development. Different approaches have been used to address these complexities in efforts to estimate twentieth-century GMSL rise [23]. These include averaging rates at sites with the longest records [1,2], averaging rates determined from regional binning of records [3], incorporating shorter records into the analysis to distinguish between secular trends and decadal-scale variability [3], and using altimetry records to determine dominant sea-level geometries and then using tide gauge records to estimate the time-varying amplitudes of these geometries [4,5]. In most cases, other criteria were applied to cull the tide gauge sites adopted in the analysis (for example, excluding sites near tectonic activity or major urban centres). Estimates of twentieth-century GMSL rise from these previous analyses range from 1.6 to 1.9 mm yr−1 (refs 1, 2, 3, 4, 5, 6) and define an important enigma. Independent model- and data-based estimates of the individual sources of GMSL, including mass flux from glaciers and ice sheets, thermal expansion of oceans, and changes in land water storage, are insufficient to account for the GMSL rise estimated from tide gauge records8, particularly before 19907. For example, a tabulation of contributions to GMSL rise from 1901 to 1990 in the Fifth Assessment Report (AR5; ref. 7) of the Intergovernmental Panel of Climate Change (IPCC) total 0.5 ± 0.4 mm yr−1 (90% confidence interval, CI) less than a recent tide gauge derived rate of 1.5 ± 0.2 mm yr−1 (90% CI) estimated by Church and White4 for the same period (the confidence range for this estimate is taken from AR5; refs 7 and 23). Using IPCC terminology, the latter suggests that it is ‘extremely likely’ (probability P = 95%) that GMSL rise from 1901 to 1990 was greater than 1.3 mm yr−1, although the bottom-up sum of contributions is ‘likely’ (P > 67%) below this level. The above discrepancy has been attributed to underestimation of almost all possible sources: thermal expansion, glacier mass balance, and Greenland or Antarctic ice sheet mass balance7,8. In this Letter, we revisit the analysis of GMSL since the start of the twentieth century using Kalman smoothing9 (KS; see Methods). This statistical technique naturally accommodates spatially sparse and temporally incomplete sampling of a global sea-level field, provides a rigorous, probabilistic framework for uncertainty propagation, and can correct for a distribution of GIA and ocean models. We applied the approach to analyse annual records from 622 tide gauges included in the Permanent Service for Mean Sea Level (PSMSL) Revised Local Reference database15,24 and reconstruct the global field of sea-level change for each year from 1900 to 2010. To examine the skill with which the KS reconstruction reproduces the tide gauge observations, we compute the time series of residuals at each tide gauge site and examine the distribution of the mean residual (that is, bias) for each site (Fig. 1a). The mean of the mean residuals across all 622 observations is 0.3 mm, with a standard deviation of 5.1 mm, indicating minimal systemic bias. Figure 1: Fit of the KS-based reconstruction of sea level to the tide gauge record. a, Histogram of mean residuals (mm) between the sea-level reconstruction and the tide gauge observations at all 622 sites. The mean of all mean residuals is 0.3 ± 5.1 mm (±1 s.d.). b–f, Time series of reconstructed annual sea level (black lines, KS mean estimate; grey shading, 1σ uncertainty) at New York, USA (b), Fremantle, Australia (c), Zemlia Bunge, Russia (d), Vaasa, Finland (e), and Champlain, Canada (f), together with the associated annual mean tide gauge observations (red lines).	https://sealeveldocs.readthedocs.io/en/latest/hay15.html
4418cc28ef3a-1	In this Letter, we revisit the analysis of GMSL since the start of the twentieth century using Kalman smoothing9 (KS; see Methods). This statistical technique naturally accommodates spatially sparse and temporally incomplete sampling of a global sea-level field, provides a rigorous, probabilistic framework for uncertainty propagation, and can correct for a distribution of GIA and ocean models. We applied the approach to analyse annual records from 622 tide gauges included in the Permanent Service for Mean Sea Level (PSMSL) Revised Local Reference database15,24 and reconstruct the global field of sea-level change for each year from 1900 to 2010. To examine the skill with which the KS reconstruction reproduces the tide gauge observations, we compute the time series of residuals at each tide gauge site and examine the distribution of the mean residual (that is, bias) for each site (Fig. 1a). The mean of the mean residuals across all 622 observations is 0.3 mm, with a standard deviation of 5.1 mm, indicating minimal systemic bias. Figure 1: Fit of the KS-based reconstruction of sea level to the tide gauge record. a, Histogram of mean residuals (mm) between the sea-level reconstruction and the tide gauge observations at all 622 sites. The mean of all mean residuals is 0.3 ± 5.1 mm (±1 s.d.). b–f, Time series of reconstructed annual sea level (black lines, KS mean estimate; grey shading, 1σ uncertainty) at New York, USA (b), Fremantle, Australia (c), Zemlia Bunge, Russia (d), Vaasa, Finland (e), and Champlain, Canada (f), together with the associated annual mean tide gauge observations (red lines). Comparing reconstructions and tide gauge observations at a selection of individual sites (Fig. 1b–f) shows generally excellent agreement, although there are a small number of outliers. An example outlier is the Champlain tide gauge (Fig. 1f), which has a mean residual of 52 mm. This particular misfit (also evident at other sites in the vicinity) can be attributed to the St Lawrence being a regulated water system where flow is dominated by anthropogenic control rather than global-scale climate dynamics25. The eight sites that have mean residuals greater than ±3σ (15 mm) from the mean exhibit an average interannual sea-level variability (estimated as the standard deviation after detrending the tide gauge observations) of ±130 mm, more than triple the mean inter-annual variability of ±40 mm across all sites. Although these outliers have large inter-annual variability, the site-specific variability is incorporated into the covariances computed in the probabilistic reconstruction, and the uncertainties in the estimated sea-level trends at these sites reflect this. The sum of the KS-estimated GMSL changes associated with the mass balance of the Greenland and Antarctic ice sheets, the mass balance of 18 mountain glacier regions, and thermal expansion (Fig. 2, blue line and shading; see Methods) is characterized by an average GMSL rate of 1.2 ± 0.2 mm yr−1 (90% CI) for 1901–90. As shown in Fig. 3, this is significantly lower than the estimates of 1.5 ± 0.2 mm yr−1 from Church and White4 (magenta line in Fig. 2) and 1.9 mm yr−1 from Jevrejeva et al.3 (red line in Fig. 2). The KS-estimated acceleration is 0.017 ± 0.003 mm yr−2, larger than our estimates based on the Church and White4 (0.009 ± 0.002 mm yr−2) and Jevrejeva et al.3 (0.011 ± 0.006 mm yr−2) time series (see Methods). Figure 2: Time series of GMSL for the period 1900–2010. Shown are estimates of GMSL based on KS (blue line), GPR (black line), Church and White4 (magenta line) and Jevrejeva et al.3 (red line). Shaded regions show ±1σ pointwise uncertainty. Inset, trends for 1901–90 and 1993–2010, and accelerations, all with 90% CI. Confidence intervals for Church and White4 are from refs 7 and 23. Confidence intervals were not available for Jevrejeva et al.3; data in this reference ends in 2002, so the rate quoted here for 1993–2010 is actually for 1993–2002. Since the GPR methodology outputs decadal sea level, no trend is estimated for 1993–2010. Accelerations are consistently estimated from the KS, GPR, and GMSL time series in refs 3 and 4 (see Methods) from 1901 to the end of each reconstruction. Figure 3: Comparison of mean GMSL rates for 1901–90. Shown are estimates of GMSL rise for the period 1901–90 obtained from six different sampling methods along with previously published rates (see main text for description of each). The box covers the 1σ uncertainty range, while the bars represent the 90% CI. In the case of the Jevrejeva et al.3 estimate, uncertainties and confidence intervals were not available. Church and White [4] combined stationary empirical orthogonal functions (EOFs), computed from ∼20 years of satellite altimetry data spanning latitudes up to about ±60°, with amplitudes estimated from sparse tide gauge observations. Given the relatively short duration of the altimeter record, the EOFs may be dominated by patterns due to interannual variability rather than the geometry associated with long-term sea-level change26,27. Jevrejeva et al.3 used tide gauge records to compute regional sea-level means and from these computed a global average. Both methodologies involve spatially sparse, temporally incomplete sampling of the global sea-level field, which introduces a potentially significant bias into estimates of GMSL. The KS technique differs from these approaches by using the spatial information inherent in the observations to infer the weights associated with the individual, underlying contributions to the sea-level change. The method extracts global information from the sparse field by taking advantage of the physics-based and model-derived geometry of the contributing processes, thereby reducing the potential for sampling bias. To understand the origin of the differences between the KS estimate and the higher values of refs 3 and 4, and in particular to quantify the impact of regional binning, spatial sparsity and missing data, we performed several tests. First, we applied to the KS global sea-level reconstruction a regional binning algorithm similar to that of Jevrejeva et al.3. In particular, we sampled the reconstruction at the locations of the 622 tide gauge sites, imposed sections of missing data consistent with the PSMSL data availability15, binned the tide gauges into 12 ocean regions, and averaged across these regions to compute a GMSL curve. The resulting estimate of the mean GMSL rate from 1901 to 1990 (Fig. 3; ‘KS PSMSL sampling’), 1.6 ± 0.4 mm yr−1 (90% CI), is significantly closer to the estimate of Jevrejeva et al.3, indicating that combined spatial sparsity and missing data generate an upward bias in estimates of GMSL rates (Fig. 3). Second, we performed a bootstrapping test that repeated the above algorithm for tide gauge subsets ranging from 25 to 600 sites that confirmed this result (see Methods and Extended Data Fig. 3). We also implemented a test to estimate the possible bias in the estimate of GMSL rate introduced in the EOF analysis of Church and White4 (see Methods; Fig. 3; ‘KS EOF’); the result was consistent with the difference between the KS and Church and White4 results in Fig. 2. We performed several other tests to explore the impact of sparsity and missing data on the estimates. Specifically, we applied the binning algorithm as described above but without imposing sections of missing data. The resulting mean GMSL rate estimate for 1901–90 was 1.0 ± 0.4 mm yr−1, close to the KS result (Fig. 3; ‘KS 622 sites, no missing data’). Third, we sampled the full reconstruction at a large number of globally distributed sites—that is, the sampling was not confined to the tide gauge sites and no sections of missing data were imposed on the time series—and performed the same regional binning and averaging (‘KS global reconstruction’). The resulting rate estimate, 1.2 ± 0.1 mm yr−1, was identical to the KS result (Fig. 3). This indicates that regional binning of estimates, in the absence of sparsity and missing data, does not introduce a significant bias.	https://sealeveldocs.readthedocs.io/en/latest/hay15.html
4418cc28ef3a-2	First, we applied to the KS global sea-level reconstruction a regional binning algorithm similar to that of Jevrejeva et al.3. In particular, we sampled the reconstruction at the locations of the 622 tide gauge sites, imposed sections of missing data consistent with the PSMSL data availability15, binned the tide gauges into 12 ocean regions, and averaged across these regions to compute a GMSL curve. The resulting estimate of the mean GMSL rate from 1901 to 1990 (Fig. 3; ‘KS PSMSL sampling’), 1.6 ± 0.4 mm yr−1 (90% CI), is significantly closer to the estimate of Jevrejeva et al.3, indicating that combined spatial sparsity and missing data generate an upward bias in estimates of GMSL rates (Fig. 3). Second, we performed a bootstrapping test that repeated the above algorithm for tide gauge subsets ranging from 25 to 600 sites that confirmed this result (see Methods and Extended Data Fig. 3). We also implemented a test to estimate the possible bias in the estimate of GMSL rate introduced in the EOF analysis of Church and White4 (see Methods; Fig. 3; ‘KS EOF’); the result was consistent with the difference between the KS and Church and White4 results in Fig. 2. We performed several other tests to explore the impact of sparsity and missing data on the estimates. Specifically, we applied the binning algorithm as described above but without imposing sections of missing data. The resulting mean GMSL rate estimate for 1901–90 was 1.0 ± 0.4 mm yr−1, close to the KS result (Fig. 3; ‘KS 622 sites, no missing data’). Third, we sampled the full reconstruction at a large number of globally distributed sites—that is, the sampling was not confined to the tide gauge sites and no sections of missing data were imposed on the time series—and performed the same regional binning and averaging (‘KS global reconstruction’). The resulting rate estimate, 1.2 ± 0.1 mm yr−1, was identical to the KS result (Fig. 3). This indicates that regional binning of estimates, in the absence of sparsity and missing data, does not introduce a significant bias. To assess the robustness of our probabilistic reanalysis, we also performed a second, independent statistical analysis based on Gaussian process regression28 (GPR), a technique that also naturally accommodates data sparsity and gaps, and incorporates a suite of GIA and ocean models (see Methods; black line in Fig. 2). The mean GMSL rate for 1901–90 estimated from the GPR analysis, 1.1 ± 0.4 mm yr−1, is consistent with the results of the KS analysis (Fig. 3). Previous analyses appear to have overestimated the mean GMSL rate over the twentieth century. The KS estimate for the period 1901–90 indicates that it is ‘very likely’ (probability P = 90%) that the rate of GMSL rise during this period was between 1.0 and 1.4 mm yr−1. This estimate closes the sea-level budget for 1901–90 estimated in AR5 (ref. 7) without appealing to an underestimation of individual contributions from ocean thermal expansion, glacier melting, or ice sheet mass balance. Moreover, it may contribute to the ultimate resolution of Munk’s sea-level enigma28 (defined by the argument that Earth rotation measurements and bounds on ocean warming are inconsistent with a rate of sea-level rise beginning in the late nineteenth century of 1.5–2.0 mm yr−1), since it may lower the signal of twentieth century ice melting in Earth rotation measurements. In contrast, for the period 1993–2010—which coincides with the era of satellite altimetry measurements of sea surface height changes29—the KS estimate is consistent with previous results (Fig. 2). The KS estimate, 3.0 ± 0.7 mm yr−1 (90% CI), is essentially identical to the tide gauge analysis of Church and White4 (2.8 ± 0.5 mm yr−1; ref. 23). It is also consistent with the estimate based on TOPEX and Jason altimeter measurements (3.2 ± 0.4 mm yr−1; ref. 29 as cited by ref. 23 for the period 1993–2010, see also ref. 7). To assess the anomalous nature of recent sea-level change, we compute 15-year rates through the KS-derived GMSL time series in Fig. 2 from 1901 to 2010. Figure 4 shows both the time series and distribution of these 96 rates, where the 5 most recent time windows are shown in red. The former is in qualitative agreement with a previous inference of multi-decadal trends in acceleration during the twentieth century30. While the rates show significant variability, the rate for the 1996–2010 time window, 3.1 mm yr−1, is the largest of all computed 15-year rates. Figure 4: Moving 15-year averages of GMSL rate estimated using the KS reconstruction of sea level across the entire interval 1901–2010. The x-axis represents the mid-point of each 15-year averaging window, and the shading gives the 1σ uncertainty range. Inset, histogram of 15-year mean GMSL rate estimates (mm yr−1) for all time windows. The five most recent windows are shown in red. We have revisited twentieth century GMSL rise using probabilistic techniques that combine sea-level records with physics-based and model-derived geometries of the contributing processes. Our estimated GMSL trend for the period 1901–90 (1.2 ± 0.2 mm yr−1) is lower than previous estimates, indicating that the rate of GMSL rise during the last two decades represents a more significant increase than previously recognized. Projections of future sea-level rise based on the time series of historical GMSL, notably semi-empirical approaches11, should accordingly be revisited. Methods Probabilistic estimation methods Kalman smoothing (KS) and Gaussian process regression (GPR), both discussed in detail below, share three advantages over the approaches taken in traditional tide gauge analyses. First, the Bayesian nature of both approaches naturally accommodates the spatiotemporal changes in the availability of the sea-level records (that is, sparsity and missing data). Second, the probabilistic approaches correct for a distribution of GIA and ocean models rather than adopting a specific model for each process, and they thus reduce a potentially important bias in previous estimates of the GMSL change17,31. Last, as both methods are fully probabilistic, they allow for the propagation of measurement and inferential uncertainties and correlations throughout the complete analysis time period. Despite these commonalities, the implementations of KS and GPR differ significantly. Kalman smoother The KS methodology is divided into four steps9, the first three of which are repeated by employing the spatial fields of GIA and ocean dynamic models from all possible combinations of 161 different Earth rheological models and 6 global climate model (GCM) simulations from CMIP5 (ref. 32) (see below for details of the rheological and climate models). First, a priori model estimates of both local sea level and the individual mass contributions from the Greenland, West Antarctic and East Antarctic ice sheets, as well as 18 major mountain glacier regions, are recursively corrected by tide gauge observations as the estimates are propagated forward through time. The local sea level is linked to the individual mass contributions through the unique spatial patterns, or ‘fingerprints,’ of sea-level change associated with rapid mass loss from land-based ice18,19,20,21. The forward step yields an estimate of local sea level and land ice contributions at each time slice, conditional on all earlier observations and a particular combination of GIA and GCM models. Second, the procedure is run backward in time, with the initial state estimate being the last estimate from the first step. The third, smoothing step optimally combines the results of the first two passes based upon the uncertainties of the respective estimates. The result is an estimate of local sea levels and land ice contributions conditional upon the entire set of observations and specific pairings of GIA and GCM models. Finally, the results from different GIA/GCM combinations are linearly combined, weighted by their likelihood, to yield an a posteriori probability distribution for local sea levels and land ice contributions, conditional upon the tide gauge observations. A comprehensive discussion of our application of the KS technique to the analysis of tide gauge measurements is given in ref. 9, which also includes synthetic tests to assess the performance of the procedure. Several subsequent refinements of this approach are summarized below. Reference 9 defined the state vector to include estimates of sea level at every tide gauge site, the mass loss rates of three ice sheets, and the temporally correlated noise in the sea-level observations. Using only tide gauge observations limits our ability to separate estimates of sea level from estimates of the temporally correlated noise. This led us to modify the KS approach in two ways. First, the state vector includes only an estimate of total sea level at every tide gauge site in addition to the desired mass loss rates. This yields the following state vector, xk, at every time step, k:	https://sealeveldocs.readthedocs.io/en/latest/hay15.html
4418cc28ef3a-3	A comprehensive discussion of our application of the KS technique to the analysis of tide gauge measurements is given in ref. 9, which also includes synthetic tests to assess the performance of the procedure. Several subsequent refinements of this approach are summarized below. Reference 9 defined the state vector to include estimates of sea level at every tide gauge site, the mass loss rates of three ice sheets, and the temporally correlated noise in the sea-level observations. Using only tide gauge observations limits our ability to separate estimates of sea level from estimates of the temporally correlated noise. This led us to modify the KS approach in two ways. First, the state vector includes only an estimate of total sea level at every tide gauge site in addition to the desired mass loss rates. This yields the following state vector, xk, at every time step, k: xk = [hk Bk]^T where hk is a vector of sea level at the 622 tide gauge sites, and Bk is a vector containing the scalar weightings of 3 ice sheets and 18 mountain glacier regions (see below), as well as a uniform component that accounts for global mean thermal expansion and any additional mass contributions from smaller mountain glaciers. Second, while in ref. 9 the observation model consisted of the sum of the estimated sea level, correlated noise, and white noise, here, the observation model consists only of the estimated sea level and white noise at each tide gauge site. Temporal correlations due to ocean dynamics are now modelled by the annual, spatial, CMIP5 ocean model fields (see below for a more detailed description of the CMIP5 model fields). Sea level is modelled as the Euler integration of the contributions from melt sources, Bk-1y, (with y being the matrix of sea-level fingerprints associated with rapid land-ice mass loss), the ongoing rate of sea-level change due to GIA, G, and the rate of change of sea level due to ocean dynamics, Sdot_{k-1} , from the spatial fields in the CMIP5 model outputs: hk = h_{k-1} + ∆t(B_{k-1} y + G + Sdot_{k-1}) + wh where wh represents a zero-mean, white noise term associated with sea level. The scalar weightings of the fingerprints are modelled as a random walk: Bk = B_{k-1} + wB where wB represents a zero-mean, white noise term associated with the melt contributions. The forward filtering pass of the Kalman smoother follows the steps outlined in ref. 9. A final departure from the methodology presented in ref. 9 is that we implemented a three-pass fixed-interval smoother33 in place of a Rauch-Tung-Stiebel two-pass smoother [34]. Gaussian process regression The GPR approach, in contrast, models sea level as a multivariate Gaussian field defined by spatiotemporal mean and covariance functions that describe the underlying processes responsible for sea-level variability. Specifically, Gaussian process priors describing the contributions from land ice, GIA, and ocean models are conditioned simultaneously upon the available observations to produce the conditional, posterior distribution of sea level at decadal intervals throughout the twentieth century. In contrast to the KS, the GPR approach directly estimates the intertemporal covariance of the posterior; the associated computational demands require the use of decadal rather than annual means. Rather than being based upon discrete GIA and GCM models as in the KS approach, the GPR approach employs Gaussian process priors for the GIA and ocean dynamics contributions that are estimated, respectively, from the 161 GIA model predictions and 6 GCM outputs (see below). The distribution describing each land ice mass contribution is modelled assuming a prior spatio-temporal covariance, with the temporal component estimated from previous, non-sea level based estimates of land ice melt and the spatial component from the sea-level fingerprints associated with the melt source. We model decadal-average sea level as a spatiotemporal field: f(x,t) = fGIA(x,t) + fM(x,t) + fLSL(x,t) where fGIA, fM, and fLSL are respectively the components of sea level due to ongoing GIA, land ice mass loss, and local effects associated with ocean dynamics, tectonics and other non-climatic factors, each as a function of location, x, and time, t. Each sea-level component is modelled as a Gaussian process with a prior mean function, μi(x,t), and covariance function, Ki(x,t,x′,t′). The total field can be partitioned into observed sites, f1, and unobserved sites, f2, and subsequently written as a joint, multivariate distribution, such that: [f1, f2] ~ N([μ1, μ2],[K11 K12, K^T12 K22]) Observations, y, are modelled as the underlying sea-level field with additive white noise characterized by zero mean and a covariance Σp, such that the joint distribution becomes: [y, f2] ~ N([μ1, f2],[K11+Σp K12, K^T12 K22]) Using standard statistical results (see, for example, ref. 35), the posterior mean and covariance, f2 and V2, of the unobserved field conditioned upon the observations are: f2 = f2 + K^T12 [K11+Σp]^{-1} y and V2 = K22 - K^T12 [K11+Σp]^{-1} K12 To estimate the underlying constituents of the total sea-level field, the prior mean and covariance of the unobserved field (that is, μ2, K12, K22) are set to the distribution of the desired quantity alone. For example, setting μ2, K12, and K22 equal to μ2M, K12M, K22M, returns the posterior mean and covariance of sea-level change due to the melt contributions. Once all the underlying constituent sea-level fields are determined, the global mean of those components can be computed and added to estimate GMSL. The elements of the prior covariance matrix of the melt contribution, KM, are defined as: K^Mi,j = Σ^n_{a=1}(A^{M,L}_{i,j,a} + A^{M,RQ}_{i,j,a})(B^M_{i,j,α}) where the subscripts indicate the ith row and jth column element of the ath ice sheet or mountain glacier. The time dependence of the covariance matrix is taken to be the sum of a linear component, AM,L, which accounts for secular changes in the melt contributions, and a rational quadratic term, AM,RQ, that represents a smoothly-varying function of variability: A^{M,L}(tq,tp) = k1 tq tp and A^{M,RQ}(tq,tp) = k2 (1 + ∆t^2q,p / 2 α τs^2)^{-α} Here, tq and tp represent the time at the qth and pth time step, Δtq,p represents the time difference between these steps, and k1, k2, α, and τs are hyperparameters that define the linear amplitude, rational quadratic amplitude, roughness, and characteristic timescale of the covariance functions35. To estimate the hyperparameters we adopt an empirical Bayesian approach where we compute the parameters that maximize the likelihood of reconstructed time series of previous mountain glacier estimates36 and ice sheet estimates37. The spatial weighting of the prior covariance, BM, is computed as the outer product of the unique fingerprint associated with melt from the corresponding land-based ice source. The prior spatiotemporal mean and covariance for the GIA contribution to sea-level change, μGIA(x,t) and KGIA(x,t), respectively, are taken as the sample mean and covariance of the 161 predictions of sea-level change described below. The distribution of the contribution to sea-level changes from thermosteric and ocean dynamic effects is partially modelled as the sample mean and covariance of the CMIP5 model outputs32. However, since a small number of models are used to compute the distribution statistics, the estimated distribution may not be representative of the parent distribution. Consequently, we augment the sample covariance with a space-time separable covariance structure consisting of the product of two Matérn functions [35], C: one representing the temporal distribution and the other representing the spatial, such that the total prior covariance describing local sea-level change is given by: KLSL(x,t) = KCMIP5(x,t) + C(t,ν1,τ) C(x,ν2,L)	https://sealeveldocs.readthedocs.io/en/latest/hay15.html
4418cc28ef3a-4	where the subscripts indicate the ith row and jth column element of the ath ice sheet or mountain glacier. The time dependence of the covariance matrix is taken to be the sum of a linear component, AM,L, which accounts for secular changes in the melt contributions, and a rational quadratic term, AM,RQ, that represents a smoothly-varying function of variability: A^{M,L}(tq,tp) = k1 tq tp and A^{M,RQ}(tq,tp) = k2 (1 + ∆t^2q,p / 2 α τs^2)^{-α} Here, tq and tp represent the time at the qth and pth time step, Δtq,p represents the time difference between these steps, and k1, k2, α, and τs are hyperparameters that define the linear amplitude, rational quadratic amplitude, roughness, and characteristic timescale of the covariance functions35. To estimate the hyperparameters we adopt an empirical Bayesian approach where we compute the parameters that maximize the likelihood of reconstructed time series of previous mountain glacier estimates36 and ice sheet estimates37. The spatial weighting of the prior covariance, BM, is computed as the outer product of the unique fingerprint associated with melt from the corresponding land-based ice source. The prior spatiotemporal mean and covariance for the GIA contribution to sea-level change, μGIA(x,t) and KGIA(x,t), respectively, are taken as the sample mean and covariance of the 161 predictions of sea-level change described below. The distribution of the contribution to sea-level changes from thermosteric and ocean dynamic effects is partially modelled as the sample mean and covariance of the CMIP5 model outputs32. However, since a small number of models are used to compute the distribution statistics, the estimated distribution may not be representative of the parent distribution. Consequently, we augment the sample covariance with a space-time separable covariance structure consisting of the product of two Matérn functions [35], C: one representing the temporal distribution and the other representing the spatial, such that the total prior covariance describing local sea-level change is given by: KLSL(x,t) = KCMIP5(x,t) + C(t,ν1,τ) C(x,ν2,L) where KCMIP5 is the sample covariance of the CMIP5 model outputs, ν1 and τ are the smoothness parameter and characteristic timescale of the temporal Matérn function, respectively, and ν2 and L are the smoothness parameter and characteristic length scale of the spatial Matérn function, respectively. For the exponents within the Matérn functions we follow ref. 10 and set the exponent on the spatial component to ν2 = 5/2 (reflecting a relatively smooth, twice-differentiable field) and the exponent on the temporal component to ν1 = 3/2 (reflecting a once-differentiable time series, in which rate is always defined but can change abruptly). As with the melt covariance hyperparameters, we use an empirical Bayesian approach to estimate the maximum-likelihood time and length scales of the Matérn functions to be 46 years and 90 km, respectively. Note that there is some trade-off between the Matérn exponent values and the hyperparameter characteristic scales: the selection of, say, a lower exponent (giving rise to a less smooth functional form) would result in a longer length scale. In addition to capturing the inaccuracies of the ocean dynamics distribution, the Matérn functions also model local tectonic, geomorphological and other non-climatic contributions to local sea-level change. These hyperparameters, and a white-noise variance, are computed by finding the parameters that maximize the likelihood of the available tide gauge observations given the complete sea-level model. Sea-level fingerprints Extended Data Fig. 1a and b shows global maps of sea-level change, known as sea-level fingerprints, associated with rapid, uniform mass loss across the Greenland Ice Sheet (GIS) and the West Antarctic Ice Sheet (WAIS), respectively. The sea-level changes are normalized by the equivalent GMSL change. Both fingerprints are characterized by a large amplitude sea-level fall in the region adjacent to the melting ice sheet with a gradual rise in sea level moving away from the ice sheet. The computation of the fingerprints is based upon a gravitationally self-consistent sea-level theory that takes into account shoreline migration and changes in grounded, marine-based ice cover as well as the impact on sea level of perturbations in the Earth’s rotation axis38,39,40. In addition to the GIS and WAIS, fingerprints were computed for the East Antarctic Ice Sheet (EAIS) and glaciers of Alaska, the Alps, Baffin Island, the Caucasus, Ellesmere Island, Franz Josef Land, High Mountain Asia, Altai, Iceland, Kamchatka, the low-latitude Andes, New Zealand, Novaya Zemlya, Patagonia, Scandinavia, Severnaya Zemlya, Svalbard, and Western Canada/US. We also include a spatially uniform pattern to account for changes in GMSL due to land ice sources not included in the above set of glaciers. In the Kalman smoother, this uniform ‘fingerprint’ also captures changes in GMSL due to globally uniform thermal expansion and terrestrial water storage variations9. GIA models The first step when analysing tide gauge records is to correct for sea-level contributions due to the ongoing GIA of the Earth in response to the ice age cycles. Predictions of GIA are dependent on the geometry and deglaciation history of the Late Pleistocene ice sheets and the Earth’s viscoelastic structure. In this study, we computed 160 different GIA predictions distinguished on the basis of the adopted lower-mantle viscosity, upper-mantle viscosity, and thickness of a high-viscosity (effectively elastic) lithosphere. Additionally, we computed a GIA prediction using the VM2 viscosity profile41. These were combined with the ICE-5G (Ref. 41) global ice sheet reconstruction for the last glacial cycle. A detailed description of physical processes that contribute to the total GIA signal can be found in ref. 42. We adopted values for the three rheological model parameters that encompass all recent estimates of the Earth’s structure. The lower-mantle viscosity was varied in the range (2–100) × 1021 Pa s, upper-mantle viscosity in the range (0.3–1) × 1021 Pa s, and lithospheric thickness in the range 72–150 km. Extended Data Fig. 2a and b shows the mean and standard deviation of the model predictions. The largest variance is seen in the region within the near field of the former ice sheets, including areas of ancient ice cover and the so-called peripheral bulges. Ocean dynamics models We treat the thermosteric and ocean dynamic contributions to sea level using the historical experiment output from 6 global climate models of the World Climate Research Programme’s (WCRP) Coupled Model Intercomparison Project phase 5 (CMIP5) data set32. Following ref. 9, the models we use are: bcc-csm1-1 from the Beijing Climate Center, CanESM2 from Environment Canada, the NOAA-GFDL model GFSL-ESM2M, the Institut Pierre Simone Laplace IPSL-CM5A-LR model, MRI-CGCM3 from the Japanese Meteorological Institute, and NorESM1-M from the Norwegian Climate Centre. For the KS methodology, we use the zero-mean spatial field ‘zos’ that is supplied by all the models. In the GPR, we add to ‘zos’ each model’s estimated globally averaged sea-level change due to thermal expansion: ‘zossga’. While the CMIP5 model outputs are provided as global ocean grids, the field values at the specific locations of tide gauges are required, as input, to both the KS and GPR analyses. Where the tide gauges are coincident with model grid points, the associated value of the model output is used. Otherwise, an inverse distance weighting interpolation scheme is used to estimate the field at the desired location. We examined three alternative interpolation schemes to assess the sensitivity of the KS GMSL estimate to this choice: (1) a nearest-neighbour approach, selecting the value on the CMIP5 grid that is closest to the tide gauge site; (2) a Delaunay interpolant, computing a linear interpolation between the irregularly spaced model cells along the coastlines; and (3) a Gaussian process (or simple kriging) methodology. For the Gaussian process interpolation, we employed a Gaussian process prior with a mean equal to the mean of the model grid values within a 200 km radius of the tide gauge location and a Matérn covariance function with smoothness parameter equal to 5/2. Since we are interested in the variability of the ocean models immediately surrounding each tide gauge site, the length scale of the Matérn covariance function was set to 1° (∼110 km). Neither the nearest-neighbour approach nor the Delaunay interpolated altered the estimate of the GMSL rate over the time period 1901–90. The Gaussian process interpolation scheme changed the GMSL estimate by less than 2%, significantly smaller than the estimated ±0.2 mm yr−1 90% CI on the estimate. Computation of GMSL rates and accelerations	https://sealeveldocs.readthedocs.io/en/latest/hay15.html
4418cc28ef3a-5	While the CMIP5 model outputs are provided as global ocean grids, the field values at the specific locations of tide gauges are required, as input, to both the KS and GPR analyses. Where the tide gauges are coincident with model grid points, the associated value of the model output is used. Otherwise, an inverse distance weighting interpolation scheme is used to estimate the field at the desired location. We examined three alternative interpolation schemes to assess the sensitivity of the KS GMSL estimate to this choice: (1) a nearest-neighbour approach, selecting the value on the CMIP5 grid that is closest to the tide gauge site; (2) a Delaunay interpolant, computing a linear interpolation between the irregularly spaced model cells along the coastlines; and (3) a Gaussian process (or simple kriging) methodology. For the Gaussian process interpolation, we employed a Gaussian process prior with a mean equal to the mean of the model grid values within a 200 km radius of the tide gauge location and a Matérn covariance function with smoothness parameter equal to 5/2. Since we are interested in the variability of the ocean models immediately surrounding each tide gauge site, the length scale of the Matérn covariance function was set to 1° (∼110 km). Neither the nearest-neighbour approach nor the Delaunay interpolated altered the estimate of the GMSL rate over the time period 1901–90. The Gaussian process interpolation scheme changed the GMSL estimate by less than 2%, significantly smaller than the estimated ±0.2 mm yr−1 90% CI on the estimate. Computation of GMSL rates and accelerations The mean and uncertainty of GMSL rates are estimated using a generalized least squares regression of a linear trend to the reconstructed GMSL time series. While the GPR methodology outputs a full temporal covariance matrix, the KS methodology does not. For this purpose, we adopt a temporal covariance matrix Σ with elements having the form: Σ_{i,j} = σi σj exp(-(tj - ti)/τ where σi and σj are the instantaneous uncertainties in GMSL at time i and j, respectively, derived in the multi-model KS analysis. To estimate the decorrelation timescale, τ, we examined the annual PSMSL tide gauge data and computed the mean temporal correlation coefficient across all tide gauges. This coefficient approaches zero after 2 years, and we set τ to 3 years. Estimates of acceleration in GMSL cited in the main text for the two probabilistic analyses are computed using a generalized least squares fit of a quadratic through the associated GMSL time series. Estimates of acceleration for the Church and White4 and Jevrejeva et al.3 time series listed in Fig. 2 are based on a weighted least squares regression through the published time series (see figure legend). Analysis of bias introduced by using a subset of tide gauges We used a bootstrapping technique to assess the potential biases introduced in estimates of GMSL rates when only a subset of tide gauge records is used. We randomly sampled our global sea-level reconstruction based on the Kalman smoother at a specific number of tide gauge sites in the database of 622 sites, computed the associated GMSL curve by binning the sites into 12 regions and averaging the result, and then used this curve to determine the rate of sea-level change over the time period 1901–90. The time series of the sea-level reconstruction at any given tide gauge site were sampled to match any missing data at that site in the PSMSL database. We repeated the analysis 100 times for subsets ranging in size from 25 to 600 sites. The mean sea-level rate we computed in this exercise and its associated uncertainty are shown in Extended Data Fig. 3 as a function of the number of sites. The horizontal blue line and shading is the mean rate of sea-level rise from 1901 to 1990, and its associated uncertainty, respectively, obtained from the KS-derived time series (1.2 ± 0.2 mm yr−1; Figs 2, 3). The mean sea-level rate obtained from this analysis asymptotes towards its final value and the spread in rates decreases monotonically as the number of tide gauges used in the analysis increases. The asymptote lies ∼0.4 mm yr−1 above the KS estimate, which is consistent with the difference between the KS and ‘KS PSMSL sampling’ rate estimates for 1901–90 shown in Fig. 3. This result suggests that the combined effects of data sparsity and missing data introduce an upward bias into the estimate of GMSL. This bias is reduced in the KS (and GPR) methodologies because these techniques extract global information by using the observations, together with model-based geometries (or covariances) associated with the underlying contributions, to estimate (and sum) these contributions. Analysis of bias introduced by an EOF analysis of altimetry records To compare our results with EOF-based reconstructions of sea level4,43, we computed the GMSL time series following the approach adopted by Church and White4, but replacing altimetry and tide gauge observations with our KS reconstruction. The EOFs were computed using the KS sea-level reconstruction from 1993 to 2010, limited to the latitudinal observation range of satellite altimetry (65° N to 65° S). As in ref. 43, a spatially uniform EOF was added to the basis set to account for changes in mean sea level within the altimetry data (here the KS reconstruction), while the weights of the EOFs were computed using the first differences of the KS reconstruction at the tide gauge locations (sampled to reflect missing data in the PSMSL database) in order to eliminate dependence on a consistent datum. The GMSL time series was computed using an area-weighted mean of the EOF-reconstruction. To compute the uncertainty in our estimated GMSL, we sampled our distribution for each KS reconstructed tide gauge 1,000 times and computed the corresponding EOF-derived GMSL time series. We used this distribution of GMSL curves with a generalized least squares regression to compute a trend and uncertainty. This analysis yielded a linear trend of 1.4 ± 0.4 mm yr−1, demonstrating the existence of a bias since the ‘true’ underlying reconstruction has a trend of 1.2 ± 0.2 mm yr−1 (see Fig. 3, ‘KS EOF’). Inverted barometer correction The results in the manuscript were obtained using tide gauge observations that were not corrected for the inverse barometer (IB) effect. Previous studies (for example, refs 44 and 45) have shown that the sea-level response to atmospheric pressure changes can be non-negligible on regional scales. In order to investigate the potential effect that atmospheric pressure changes have on our probabilistic estimate of GMSL, we repeated the KS analysis on the full tide gauge data set after we corrected these records for the IB effect. Specifically, we used the HadSLP2 global reconstructed atmospheric pressure data set46 to compute the IB correction. We next applied the correction to the observations at the 622 tide gauge sites and then re-ran the KS analysis. The 1901–90 GMSL rate of change associated with this analysis is 1.2 ± 0.2 mm yr−1, consistent with the value cited in the main text. We conclude that while the IB effect can impact regional sea-level histories, it has a negligible effect on our probabilistic estimates of GMSL. Optimality of the Kalman smoother Local sea levels observed by tide gauges reveal significant interannual and decadal variability. This variability can lead to temporal correlation in the sea-level time series that needs to be considered if one seeks to obtain optimal estimates of the underlying GMSL contributions. In order to test the optimality of the Kalman smoother, we investigated the properties of the innovation sequence by computing the residuals between the observations and the KS model estimate of sea level at every tide gauge site. Since every KS estimate of sea level is accompanied by its associated uncertainty, we randomly sampled from each sea-level distribution to obtain 100 time series of residuals for every site. Following the optimality test described in ref. 47, we computed the mean AR(1) coefficient across the 100 samples at each tide gauge site. An optimal Kalman smoother is characterized by a white noise innovation sequence. In practice, this means that, within uncertainty, the AR(1) coefficients of the innovation sequences will be close to zero. In the exercise above, we obtained a mean AR(1) coefficient of 0.2 ± 0.3 (90% confidence). This indicates that our innovation sequence is (within uncertainty) white noise and that the smoother is, or is close to, optimal. Sensitivity of GMSL estimates to limitations of the CMIP5 climate simulations The presence of unmodelled ocean dynamics can also affect the smoother performance. As described above, the limitations of the CMIP5 simulations as models for the true dynamic variability of the oceans is addressed in the GPR analysis by augmenting the covariance computed from the climate runs with two additional terms: a covariance modelled with two Matérn functions, and a white noise variance.	https://sealeveldocs.readthedocs.io/en/latest/hay15.html
4418cc28ef3a-6	Optimality of the Kalman smoother Local sea levels observed by tide gauges reveal significant interannual and decadal variability. This variability can lead to temporal correlation in the sea-level time series that needs to be considered if one seeks to obtain optimal estimates of the underlying GMSL contributions. In order to test the optimality of the Kalman smoother, we investigated the properties of the innovation sequence by computing the residuals between the observations and the KS model estimate of sea level at every tide gauge site. Since every KS estimate of sea level is accompanied by its associated uncertainty, we randomly sampled from each sea-level distribution to obtain 100 time series of residuals for every site. Following the optimality test described in ref. 47, we computed the mean AR(1) coefficient across the 100 samples at each tide gauge site. An optimal Kalman smoother is characterized by a white noise innovation sequence. In practice, this means that, within uncertainty, the AR(1) coefficients of the innovation sequences will be close to zero. In the exercise above, we obtained a mean AR(1) coefficient of 0.2 ± 0.3 (90% confidence). This indicates that our innovation sequence is (within uncertainty) white noise and that the smoother is, or is close to, optimal. Sensitivity of GMSL estimates to limitations of the CMIP5 climate simulations The presence of unmodelled ocean dynamics can also affect the smoother performance. As described above, the limitations of the CMIP5 simulations as models for the true dynamic variability of the oceans is addressed in the GPR analysis by augmenting the covariance computed from the climate runs with two additional terms: a covariance modelled with two Matérn functions, and a white noise variance. To assess the sensitivity of the KS analysis to unmodelled ocean dynamics, we examined its response to (1) a known synthetic ocean dynamic signal and (2) the inclusion of the dynamic response to freshwater hosing of the North Atlantic. We used the mean KS estimates of the ice sheet melt rates and uniform sea-level contribution, as well as the multi-model estimate of the GIA contribution, to construct synthetic sea-level observations at the 622 tide gauge sites. We then added the dynamic sea-level change associated with one of the six CMIP5 climate models and ran the multi-model KS using the five remaining climate models to obtain an estimate of the GMSL rate. We repeated this analysis for each of the CMIP5 simulations. By not including the climate model used in constructing the synthetics in the multi-model component of the KS methodology, we tested the ability of the smoother to account for unmodelled dynamics. The 1901–90 GMSL rates determined from the complete set of 6 analyses ranged from 1.1 to 1.3 mm yr−1. Five of these analyses yielded a 90% CI of 0.2 mm yr−1, while the sixth yielded a 90% CI of 0.3 mm yr−1. These values are consistent with the results for the KS analysis cited in the main text (1.2 ± 0.2 mm yr−1). To assess the sensitivity of our GMSL results to ocean dynamic effects due to freshwater input (‘hosing’) from GIS melt, we used the results of a previous study48 to investigate the dynamic sea-level signal arising from North Atlantic freshwater ‘hosing’ simulations. Specifically, we computed the difference between the results of the 0.1 Sv hosing run and the control (no-hosing) simulation described in ref. 48 and scaled this difference by 0.05 to approximate a synthetic dynamic signal for a GIS melt rate equivalent to 0.5 mm yr−1 GMSL rise over the twentieth century. After subtracting a uniform 0.5 mm yr−1 from the spatial pattern, we calculated time series of this signal at all 622 tide gauge sites, added these to the observed record, and repeated the KS analysis. The presence of these unmodelled dynamics has negligible effect on our estimate of GMSL. The 1901–90 rate estimated in the above test agrees with the value presented in the manuscript (1.2 ± 0.2 mm yr−1). While the above sensitivity tests indicate that the probabilistic analyses have quantified, with reasonable accuracy, the impact of uncertainties in CMIP5 models of ocean dynamic variability, improving such models is an important requirement in any effort to further refine estimates of GMSL rates. Kalman smoother reconstruction of sea level at sites with no observations To investigate how well the KS is able to estimate sea level at sites without observations, we ran the Kalman smoother using data from 450 randomly chosen tide gauge sites and estimated the sea level at the remaining 172 sites. Extended Data Fig. 4 shows the GMSL time series estimated in this new analysis as well as a comparison of the estimated and observed sea level at a representative subset of 5 of these 172 sites (the remaining sites show similar fits). We calculate a 1901–90 GMSL rate of 1.2 ± 0.2 mm yr−1, consistent with the results presented in the manuscript when all 622 tide gauge sites are used in the analysis. The consistency between the estimated and observed values at the 172 tide gauge sites also indicates that limitations of the CMIP5 simulations in modelling ocean dynamics are not degrading the ability to predict sea-level trends at sites without observations. More generally, the analysis demonstrates the power of the KS method in reconstructing sea level when the method is applied with physics-based and model-derived geometries of the underlying physical processes. Extended data figures and tables Extended Data Figure 1: Illustrative sea-level fingerprints. a, b, Normalized sea-level changes due to rapid melting of the Greenland Ice Sheet (a) and the West Antarctic Ice Sheet (b). The variable ‘normalized sea-level change’ on the colour scale is formally dimensionless, but may be interpreted as having the unit of metres of sea-level change per metre of the equivalent GMSL change associated with the melt event. Extended Data Figure 2: The present-day rate of change of sea level in mm yr−1 due to GIA for a suite of Earth models. a, b, Mean sea-level change (a) and standard deviation (b) computed from the output of 161 GIA model simulations (see text). In both frames, the colour scale saturates in the near field, which includes areas of post-glacial rebound and peripheral subsidence. Extended Data Figure 3: Bootstrapping analysis of GMSL rate for 1901–90 obtained by sampling the global reconstruction of sea level. Data points show the mean computed from a bootstrapping analysis of the 1901–90 GMSL rate as a function of the number of geographic sites used in the analysis (ranging from 25 to 600). Error bars, ±1s.d. Sites are obtained by randomly sampling the global KS reconstruction at a subset of tide gauge sites and introducing data gaps that are consistent with those that exist in the PSMSL database15. The analysis was repeated 100 times for each choice of the number of sites. Also shown (horizontal blue line and shading) is the 1901–90 rate and its 90% CI computed from the KS GMSL curve in Fig. 2 (1.2 ± 0.2 mm yr−1; Figs 2 and 3). Extended Data Figure 4: Results of the KS analysis performed using a random subset of 450 tide gauges. a, KS-estimated GMSL curve derived using a subset of 450 of the 622 tide gauge records discussed in the main text (blue line) and the reconstruction of Church and White4 (magenta line) and Jevrejeva et al.3 (red line). The shaded regions represent the 1σ certainty range. Panels b–f show the KS reconstructions (black lines) at a representative set of 5 of the 122 sites that were not used in the estimation procedure. The observations are shown in red.	https://sealeveldocs.readthedocs.io/en/latest/hay15.html
5a019e65abdb-0	Suzuki et al. (2005) Title: Projection of future sea level and its variability in a high-resolution climate model: Ocean processes and Greenland and Antarctic ice-melt contributions Corresponding author: Tatsuo Suzuki Citation: Suzuki, T., Hasumi, H., Sakamoto, T. T., Nishimura, T., Abe‐Ouchi, A., Segawa, T., et al. (2005). Projection of future sea level and its variability in a high‐resolution climate model: Ocean processes and Greenland and Antarctic ice‐melt contributions. Geophysical Research Letters, 32(19), n/a-n/a. https://doi.org/10.1029/2005gl023677 URL: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2005GL023677 Abstract Using a high-resolution climate model, we projected future sea level and its variability based on two scenarios for 21st century greenhouse gas emission. The globally averaged sea level rise attributable to the steric contribution was 23 and 30 cm for the two scenarios. The results of the high-resolution model and a medium-resolution version of the same model for global and local sea level change agreed well. However, the high-resolution model represented more detailed ocean structure changes under global warming. The changes affected not only the spatial distribution of sea level rise, but also the changes in local sea level variability associated with ocean eddies. The enhanced eddy activity was responsible for extreme sea level events. Introduction Sea level change is an important aspect of future climate change for human societies and the environment. Estimates of the rate of globally averaged sea level change during the 20th century are in the range of 1 to 2 mm yr−1 [Church et al., 2001]. This globally averaged rise in sea level is mainly the result of the thermal expansion of seawater and land-ice melt. Future projections of sea level change have been calculated using various coupled atmosphere–ocean general circulation models (CGCMs [e.g., Gregory and Lowe, 2000; Gregory et al., 2001; Meehl et al., 2005]). Gregory et al. [2001] compared the results of the projections that followed the IS92a scenario for greenhouse gas (GHG) emissions. The rates of globally averaged sea level rise predicted by the models for the 21st century were in the range of 2.0–3.7 mm yr−1, but regional sea level change was not spatially uniform and some regions experienced more than twice the global average rate of rise. However, different models, while sharing some features, predicted different distributions. The geographical distribution of sea level change is principally determined by changes in density structure and wind stress forcing, both of which affect ocean circulation [Church et al., 2001]. Reproducing ocean structures is important for estimating the distribution of future sea level changes. However, the details of ocean structures, such as western boundary currents and fronts with pronounced horizontal gradients of water properties, have not been reproduced by existing coarse-resolution CGCMs. In this study, we compared sea level projections made by CGCMs with differing resolutions, focusing particularly on detailed features of the spatial distribution of sea level changes as predicted by a high-resolution model. We also present global and local changes in 21st century sea level variability. Extreme changes in sea levels could severely affect human activities, and storm surges and coastal wave height have been identified in previous studies as sources of extreme sea levels [Church et al., 2001]. Unfortunately, the resolution of CGCMs has not been adequate to resolve these events. Extreme sea levels associated with ocean eddies, however, are represented by high-resolution models. Warm eddies increase the flooding risk in coastal areas. For example, Okinawa Island flooded on 22 July 2001 without the passage of an atmospheric low; rather, a warm eddy was responsible for increasing the sea level by more than 15 cm [Tokeshi and Yanagi, 2003]. Therefore, it would be advantageous to project changes in sea level variability using a high-resolution climate model that included eddies. Models and Experiments The CGCM used in this study was the Model for Interdisciplinary Research on Climate, version 3.2 (MIROC3.2), developed at the Center for Climate System Research, University of Tokyo, National Institute for Environmental Studies and Frontier Research Center for Global Change [K-1 Model Developers, 2004]. The ocean component explicitly represented sea surface elevation. The higher-resolution version of MIROC3.2 (MIROC3.2_hi) consisted of a T106 global atmospheric spectral model with 56 vertical levels, an eddy-permitting global ocean model in which horizontal resolution was zonally 0.28° and meridionally 0.19° with 48 vertical levels, and other components (i.e., land, river, and sea ice). In the medium-resolution version of MIROC3.2 (MIROC3.2_med), a T42 global atmospheric spectral model with 20 vertical levels was coupled with a medium-resolution OGCM in which horizontal resolution was zonally 1.4° and meridionally 0.56° around the equator, with 44 vertical levels. The same physical parameterizations were applied to these models, but resolution-dependent parameters were adjusted per model. Both models were integrated by prescribing external conditions, including solar and volcanic forcing, GHG concentrations, aerosol emissions, and land use. MIROC3.2_hi was spun up for 109 years under the fixed external conditions of the year 1900 after coupling the atmosphere and the ocean. MIROC3.2_med was spun up for 560 years under the fixed conditions of the year 1850. For control runs, we continued the integration under the same conditions for 100 years in MIROC3.2_hi and 400 years in MIROC3.2_med. Twentieth century experiments (20C3M) were performed after the spin-up. These runs were forced by the historical external conditions from 1900 to 2000 in MIROC3.2_hi and from 1850 to 2000 in MIROC3.2_med. Future projections for the 21st century were initiated by the final states of the 20C3M runs and performed by prescribing the external conditions according to the Intergovernmental Panel on Climate Change (IPCC) special report for emission scenarios. CO2 concentrations at the end of the 21st century were 720 and 550 ppm under the A1B scenario and the B1 scenario, respectively [Intergovernmental Panel on Climate Change, 2000]. For MIROC3.2_med, three 20th century runs were performed using different initial conditions (1st, 101st, and 201st years for the control run). We used the ensemble mean of the three members in this study. The model integration was restricted by computing resource limitations, especially for the high-resolution model, and the control runs showed some trends. Therefore, we subtracted these trends from the results. Globally Averaged Sea Level Rise The steric contribution (thermal expansion and haline contraction) to sea level rise was estimated from the model ocean temperature and salinity. As the Boussinesq approximation was adopted in the ocean model, the globally averaged sea level rise attributable to steric factors was diagnosed indirectly from density changes as the equivalent volume change under mass conservation: urn:x-wiley:00948276:media:grl20251:grl20251-math-0001 where ΔH is the globally averaged sea level rise, S is the surface area of the ocean, Z is the ocean depth, ρ is the in situ density, and Δρ is its difference from the reference state. Because the haline contribution was small, the globally averaged change in sea level was caused mainly by changes in ocean heat content. To validate the model, the upper ocean heat content was compared with observations [Levitus et al., 2005]. Decadal variations were not in phase, but long-term trends and the changes over the last decade for both models were similar to observations (data not shown). The 21st century steric contribution was projected to be about 30 cm for the A1B scenario and 23 cm for the B1 scenario in MIROC3.2_hi (Figure 1a). This result was similar to that for MIROC3.2_med (Figure 1b) and within the range of estimations of previous CGCMs [Gregory et al., 2001]. Both models had linear trends in the control runs: about 0.15 mm yr−1 in MIROC3.2_hi and about 0.14 mm yr−1 in MIROC3.2_med. These linear trends were subtracted from the projections. Figure 1: A time series of globally averaged sea level change in (a) MIROC3.2_hi and (b) MIROC3.2_med (ensemble mean). Solid lines indicate the steric contribution. Broken lines represent the Greenland ice sheet and dotted lines indicate the Antarctica ice sheet.	https://sealeveldocs.readthedocs.io/en/latest/suzuki05.html
5a019e65abdb-1	Globally Averaged Sea Level Rise The steric contribution (thermal expansion and haline contraction) to sea level rise was estimated from the model ocean temperature and salinity. As the Boussinesq approximation was adopted in the ocean model, the globally averaged sea level rise attributable to steric factors was diagnosed indirectly from density changes as the equivalent volume change under mass conservation: urn:x-wiley:00948276:media:grl20251:grl20251-math-0001 where ΔH is the globally averaged sea level rise, S is the surface area of the ocean, Z is the ocean depth, ρ is the in situ density, and Δρ is its difference from the reference state. Because the haline contribution was small, the globally averaged change in sea level was caused mainly by changes in ocean heat content. To validate the model, the upper ocean heat content was compared with observations [Levitus et al., 2005]. Decadal variations were not in phase, but long-term trends and the changes over the last decade for both models were similar to observations (data not shown). The 21st century steric contribution was projected to be about 30 cm for the A1B scenario and 23 cm for the B1 scenario in MIROC3.2_hi (Figure 1a). This result was similar to that for MIROC3.2_med (Figure 1b) and within the range of estimations of previous CGCMs [Gregory et al., 2001]. Both models had linear trends in the control runs: about 0.15 mm yr−1 in MIROC3.2_hi and about 0.14 mm yr−1 in MIROC3.2_med. These linear trends were subtracted from the projections. Figure 1: A time series of globally averaged sea level change in (a) MIROC3.2_hi and (b) MIROC3.2_med (ensemble mean). Solid lines indicate the steric contribution. Broken lines represent the Greenland ice sheet and dotted lines indicate the Antarctica ice sheet. The contributions of ice-sheet melt were estimated using the methods of Wild et al. [2003]. The contributions of the Greenland and Antarctic ice-sheet melts exhibited opposite tendencies in both models, as in previous estimations [Church et al., 2001]. However, the amplitude in MIROC3.2_hi was larger than that in MIROC3.2_med (Figure 1). The difference between the two models appeared to be a result of the differences in projected temperature rises in the Greenland and snowfall increases in the Antarctica. Both of these differences in arctic temperature and snowfalls are strongly related to the difference in the climate sensitivity of the MIROC3.2_hi and MIROC3.2_med, in which the A1B run induced global warming of about 4.0°C in MIROC3.2_hi and 3.4°C in MIROC3.2_med at the end of the 21st century, respectively. This different sensitivity may partly due to the difference in the control SST and sea ice distribution in the two models. The globally averaged sea level rise in MIROC3.2_hi is similar to that in MIROC3.2_med in spite of the different sensitivity. It is because the total heat flux into the ocean for the 21st century is similar in the both models, though the net heat flux into ocean in MIROC3.2_hi is larger than that in MIROC3.2_med during the early 21st century. The upper ocean in MIROC3.2_hi also warms up more than that in MIROC3.2_med. The reasons for these differences are currently under investigation. Regional Sea Level Change The sea level patterns corresponding to major ocean gyres were well represented by both models (data not shown). In particular, narrow western boundary currents, such as the Kuroshio, were reproduced realistically in MIROC3.2_hi [Sakamoto et al., 2005]. The globally averaged sea level rise estimated in the previous section was added to the local sea levels calculated by each model to obtain sea level projections. Mass balance was ensured in a globally averaged sense by this procedure [Greatbatch, 1994]. Some regions experienced substantially higher sea level rise in both models than the 21st century global average (Figure 2). Figure 2: The changes in mean sea level between 1980 and 2000 (20C3M) and between 2080 and 2100 (A1B) in (a) MIROC3.2_hi and (b) MIROC3.2_med (ensemble mean). To distinguish sea level changes caused by global warming from background variability, such as decadal variations, we estimated area-weighted spatial standard deviations of the local sea level change with respect to the control climate. Gregory et al. [2001] assumed that sea level changes associated with global warming and background variability were not spatially correlated. Under this assumption, increased spatial standard deviation would be attributable to global warming if changes in background variability were relatively small. The spatial standard deviation increased with time for both models and reached 7–13 cm by the end of the 21st century (Figure 3), indicating that the projected sea level changes were sufficiently significant relative to background variability. There was a conspicuous signal from the last decade of the 20th century in MIROC3.2_hi. Figure 3: The spatial standard deviation of the decadal mean field of local sea level change with respect to the control climate. Solid lines indicate MIROC3.2_hi and broken lines represent MIROC3.2_med (ensemble mean). The regions with large sea level changes were more restricted to specific areas and the magnitudes of change were more pronounced in MIROC3.2_hi than in MIROC3.2_med (Figure 2). These results were consistent with the fact that spatial variability in MIROC3.2_hi was larger than in MIROC3.2_med at the end of the 21st century (Figure 3). The sea level variability associated with eddies was also shown in MIROC3.2_hi (Figure 4a), and this spatial distribution was consistent with satellite obsersvations. These changes were closely related to regional sea level changes and were as large as several centimeters for some regions during the 21st century (Figure 4b). Figure 4: (a) The root-mean-square (rms) of the sea level anomaly from the 3-month running mean for the control run in MIROC3.2_hi. (b) Changes in the rms between 1980 and 2000 (20C3M) and between 2080 and 2100 (A1B) in MIROC3.2_hi. Both models exhibited a region of large sea level rises in the North Pacific. These sea level changes were also shown in the Hadley Centre coupled atmosphere-ocean general circulation model (HadCM3), which has a horizontal resolution similar to that of MIROC3.2_med [Gregory and Lowe, 2000]. This feature has not been represented in previous coarser-resolution models [Gregory et al., 2001]. With higher resolution, as in MIROC3.2_med and HadCM3, fronts at the western boundary currents and their extensions were more sharply reproduced, so sea level changes associated with their shifting or intensification were better captured. Such features became further differentiated at higher resolution (Figure 2). There was a reduced sea level rise north of the Kuroshio Current at approximately 150°E and an enhanced sea level rise to the south in MIROC3.2_hi. This sea level change was associated with the acceleration of the Kuroshio caused by changes in wind stress and the consequential spin-up of the Kuroshio recirculation [Sakamoto et al., 2005]. In contrast, the Kuroshio in MIROC3.2_med overshot to the north in comparison with that in MIROC3.2_hi. Therefore, the region of large sea level rises in MIROC3.2_med extended northward relative to that in MIROC3.2_hi. MIROC3.2_hi also exhibited a region of reduced sea level rises in the North Pacific subpolar gyre. We believe that this feature was related to the intensification of the Aleutian Low, which is also considered to be the cause of the Kuroshio acceleration [Sakamoto et al., 2005]. This acceleration was associated with the enhanced eddy activity in the Kuroshio and the Kuroshio extension under global warming (Figure 4b). These features were not represented in MIROC3.2_med, partly because the subpolar gyre was not well represented due to the overshooting of the Kuroshio. Similar acceleration of a western boundary current and enhanced eddy activity were also detected east of Australia.	https://sealeveldocs.readthedocs.io/en/latest/suzuki05.html
5a019e65abdb-2	Both models exhibited a region of large sea level rises in the North Pacific. These sea level changes were also shown in the Hadley Centre coupled atmosphere-ocean general circulation model (HadCM3), which has a horizontal resolution similar to that of MIROC3.2_med [Gregory and Lowe, 2000]. This feature has not been represented in previous coarser-resolution models [Gregory et al., 2001]. With higher resolution, as in MIROC3.2_med and HadCM3, fronts at the western boundary currents and their extensions were more sharply reproduced, so sea level changes associated with their shifting or intensification were better captured. Such features became further differentiated at higher resolution (Figure 2). There was a reduced sea level rise north of the Kuroshio Current at approximately 150°E and an enhanced sea level rise to the south in MIROC3.2_hi. This sea level change was associated with the acceleration of the Kuroshio caused by changes in wind stress and the consequential spin-up of the Kuroshio recirculation [Sakamoto et al., 2005]. In contrast, the Kuroshio in MIROC3.2_med overshot to the north in comparison with that in MIROC3.2_hi. Therefore, the region of large sea level rises in MIROC3.2_med extended northward relative to that in MIROC3.2_hi. MIROC3.2_hi also exhibited a region of reduced sea level rises in the North Pacific subpolar gyre. We believe that this feature was related to the intensification of the Aleutian Low, which is also considered to be the cause of the Kuroshio acceleration [Sakamoto et al., 2005]. This acceleration was associated with the enhanced eddy activity in the Kuroshio and the Kuroshio extension under global warming (Figure 4b). These features were not represented in MIROC3.2_med, partly because the subpolar gyre was not well represented due to the overshooting of the Kuroshio. Similar acceleration of a western boundary current and enhanced eddy activity were also detected east of Australia. Another important difference between the two models was found in the western tropical Pacific. In MIROC3.2_hi, there was a reduced sea level rise east of Mindanao Island that spread to the eastern tropical Pacific (Figure 2a). This feature was caused by intensification of the wind-induced Ekman upwelling under global warming. This wind-induced Ekman upwelling in this region was not well resolved in MIROC3.2_med [Suzuki et al., 2005]. The reduced sea level rise was associated with the acceleration of the North Equatorial Current (NEC) and the North Equatorial Counter Current (NECC). The acceleration of these currents and the Subtropical Counter Current increased the meridional gradient of zonal velocity, which was associated with an enhanced eddy (Figure 4b). The region stretched zonally to 120°W. These responses of the zonal flows to global warming will be investigated in future studies. Both models showed a narrow band of enhanced sea level rise in the Southern Ocean. Under global warming in our models, the westerlies shifted southward and strengthened. These changes in the wind field contributed to the southward shift and the intensification of the circumpolar fronts, which are linked to sea level change. These changes were also connected with a couple of narrow bands of enhanced and reduced eddy activity that stretched east from Argentina to the south of Australia in MIROC3.2_hi. A dipole pattern of sea level change in the North Atlantic Ocean, i.e., an enhanced rise north of the Gulf Stream extension and a reduced rise to the south, was recognized in both models. Bryan [1996] suggested that this pattern was consistent with weakening of the upper branch of the Atlantic Meridional Overturning Circulation (AMOC). The AMOC was weakened from 14 Sv (1 Sv = 106 m3 s−1) to 9 Sv in MIROC3.2_hi and from 19.5 Sv to 12.5 Sv in MIROC3.2_med during the 21st century. Discussion and Conclusions The dynamic effect of sea level pressure was not included in either of the ocean components. The change in spatial standard deviation estimated from model sea level pressure during the 21st century was less than 2 cm. These changes, while not negligible, were small in comparison to the spatial variability caused by ocean structure changes (Figure 3). Therefore, we did not indicate the contribution of sea level pressure in this study. The strengthening of eddy activity was recognized in a globally averaged sense. The global average of the root-mean-square (rms) increased from 4.8 to 5.1 cm in the A1B scenario and from 4.8 to 5.0 cm in the B1 scenario during the 21st century. These changes were small compared to levels of globally averaged sea level rise. However, enhanced eddy activity was confined to specific areas, and those areas overlapped with the areas of enhanced sea level rise around some coastal regions and islands, suggesting that the frequency of extreme sea levels may increase in those regions during the 21st century. We have described future sea level changes as projected by MIROC3.2_hi according to the 21st century scenarios for GHG emissions and compared them with the results of MIROC3.2_med. The globally averaged sea level rise during the 21st century predicted by the two models was similar. The distribution of sea level changes in MIROC3.2_hi also resembled that in MIROC3.2_med on a large scale. However, MIROC3.2_hi presented more detailed ocean structure changes under global warming. The changes in the ocean structure affected not only the spatial distribution of sea level rise, but also changes in local sea level variability. Therefore, it is critical to consider changes in sea level variability when assessing the possible effects on human activities.	https://sealeveldocs.readthedocs.io/en/latest/suzuki05.html
f3e8df801819-0	Welcome to SeaLevelDocs’s documentation! Contents: Suzuki et al. (2005) Hughes et al. (2010) Church and White (2011) Kuhlbrodt and Gregory (2012) Church et al. (2013) Hallberg et al. (2013) Kopp et al. (2014) Hay et al. (2015) DeConto and Pollard (2016) Chen et al. (2017) Chen and Tung (2018) Horton et al. (2018) Little et al. (2019) Bamber et al. (2019) Gregory et al. (2019) Hamlington et al. (JGR, 2020) Sadai et al. (2020) Palmer et al. (2020) Yin et al. (2020) Yuan and Kopp (2021) Couldrey et al. (2021) Dangendorf et al. (2021) Zika et al. (2021) DeConto et al. (2021) Li et al. (2022) Slangen et al. (2022) van de Wal et al. (2022) Bamber et al. (2022) Wickramage et al. (2023) Li et al. (2023) Park et al. (2023) Indices and tables Index Module Index Search Page	https://sealeveldocs.readthedocs.io/en/latest/index.html
88add8ede049-0	Hamlington et al. (JGR, 2020) Title: Understanding of contemporary regional sea-level change and the implications for the future Key Points: An overview of the current state of understanding of the processes that cause regional sea-level change is provided Areas where the lack of understanding or gaps in knowledge inhibit the ability to assess future sea-level change are discussed The role of the expanded sea-level observation network in improving our understanding of sea-level change is highlighted Corresponding author: Hamlington Citation: Hamlington, B. D., Gardner, A. S., Ivins, E., Lenaerts, J. T. M., Reager, J. T., Trossman, D. S., et al. (2020). Understanding of contemporary regional sea-level change and the implications for the future. Reviews of Geophysics, 58, e2019RG000672. https://doi.org/10.1029/2019RG000672 Abstract Global sea level provides an important indicator of the state of the warming climate, but changes in regional sea level are most relevant for coastal communities around the world. With improvements to the sea-level observing system, the knowledge of regional sea-level change has advanced dramatically in recent years. Satellite measurements coupled with in situ observations have allowed for comprehensive study and improved understanding of the diverse set of drivers that lead to variations in sea level in space and time. Despite the advances, gaps in the understanding of contemporary sea-level change remain and inhibit the ability to predict how the relevant processes may lead to future change. These gaps arise in part due to the complexity of the linkages between the drivers of sea-level change. Here we review the individual processes which lead to sea-level change and then describe how they combine and vary regionally. The intent of the paper is to provide an overview of the current state of understanding of the processes that cause regional sea-level change and to identify and discuss limitations and uncertainty in our understanding of these processes. Areas where the lack of understanding or gaps in knowledge inhibit the ability to provide the needed information for comprehensive planning efforts are of particular focus. Finally, a goal of this paper is to highlight the role of the expanded sea-level observation network - particularly as related to satellite observations - in the improved scientific understanding of the contributors to regional sea-level change. Plain Language Summary This review paper addresses three important questions: (1) What do we currently know about the processes contributing to sea level change? (2) What observations do we use to gain this knowledge? and (3) Where are these gaps in our knowledge and the need for further improvement in our understanding of the drivers of regional sea level? By answering these specific questions in a focused manner, this paper should be a useful resource for other scientists, sea-level stakeholders, and a broader audience of those interested in sea level and our changing climate. Introduction Global mean sea level (GMSL) is an important indicator of a warming climate (Church & White, 2011; Milne et al., 2009; Stammer et al., 2013), but changes in regional sea level are most relevant to coastal communities around the world (Kopp et al., 2015; Nicholls, 2011; Woodworth et al., 2019). The regional variability of the processes driving sea-level change (SLC), along with their uncertainties and relative importance over different time scales, pose challenges to planning efforts. Available observations of sea level show clear spatial and temporal inhomogeneity. From satellite altimeter observations covering the time period from 1993 to present, regional rates of rise can be more than double the global average in some locations while near zero at other locations (Cazenave & Llovel, 2010). Furthermore, as a result of internal variability, the pattern of linear trends in regional sea level has shifted or reversed in many regions from the first half of the altimeter record to the second (e.g., Han et al., 2017; Peyser et al., 2016; Figure 1). Over longer time periods (i.e., hundreds of years), tide gauge records also show regional differences in the rates of SLC, owing in part to the vertical motion of the land upon which the gauges sit (e.g., Church et al., 2004; Church & White, 2006; Hay et al., 2015; Kleinherenbrink et al., 2018; Santamaroa-Gomez et al., 2014, 2017; Thompson et al., 2016). Understanding and accounting for these regional differences are critical first steps in providing information that is useful for planning efforts at the coast. Figure 1: Satellite altimeter-measured regional sea-level trend patterns from (top) 1993–2005, (middle) 2006–2018, and (bottom) 1993–2018. Black contours and gray shading denote areas where the estimated trend is not significant at the 95% confidence level. Due in large part to improvements in the sea.level observing system, the processes contributing to recent SLC are now well known. The uncertainty in the budget of GMSL rise over the last decade has been reduced (Cazenave et al., 2018), allowing for an assessment of the relative contributions of different processes that are important on global scales. While more challenging on regional levels, satellite observations, along with in situ measurements, have also led to a dramatically improved understanding of the processes causing regional differences in SLC. Fundamentally, the drivers that dominantly impact GMSL have a regional signature, and no process will result in a change that is uniform across the ocean (Milne et al., 2009; Stammer et al., 2013). Similarly, no contributor to SLC is constant in time, and the time scales upon which the processes vary can differ dramatically. Separating the contributors temporally and geographically can be useful when considering a particular planning horizon, although the range of variability inherent to the individual contributors can make this difficult. Additionally, it is the combined impact of several factors operating on these different scales that is of direct importance. The causes of global and regional SLC have been the focus of recent review papers, with regional change most comprehensively discussed and summarized in Stammer et al. (2013), Kopp et al. (2015), and Slangen et al. (2017). The understanding of these processes has progressed in recent years, and the outstanding gaps in knowledge and remaining uncertainties have shifted accordingly. The intent of the present paper is to provide an overview and update of the current state of understanding of the processes that cause regional SLC and to identify and discuss limitations and uncertainty in our understanding of these processes. Although the focus is on contemporary SLC, we do include discussion of projections of future SLC. In particular, we are concerned with areas where lack of understanding or gaps in knowledge inhibit comprehensive planning efforts at the regional level. While we do not make explicit connections to planning efforts, we expect that a detailed discussion of uncertainties could be useful to those translating science into actionable plans (e.g., Horton et al., 2018). This paper is a resource for those interested in particular aspects of regional SLC by giving a detailed presentation of the most recent estimates of their contributions and a discussion of where improvement may be made in the coming years. Finally, a goal of this paper is to highlight the potential role of the expanded sea-level observation network - particularly as related to satellites - to understanding the contributors to regional SLC. This paper is organized according to the individual processes of regional relative SLC, with each process covered in a section. In section 2, we provide a brief summary of how the contributors to regional sea level are separated and we present definitions for the terminology adopted in the remainder of the paper. Sections 3 through 8 discuss the individual processes contributing to regional SLC, with each section broken into two components: (1) a summary of the current state of knowledge, and (2) an overview of current limitations or areas of uncertainty and a discussion of where progress will likely be made in the coming years. In section 9, we summarize advances toward overcoming these limitations or reducing uncertainties that may be expected through recent and future additions to the sea-level observational network, with particular emphasis on satellite-based observations. Processes Contributing to Regional SLC	https://sealeveldocs.readthedocs.io/en/latest/hamlington20jgr.html
88add8ede049-1	The causes of global and regional SLC have been the focus of recent review papers, with regional change most comprehensively discussed and summarized in Stammer et al. (2013), Kopp et al. (2015), and Slangen et al. (2017). The understanding of these processes has progressed in recent years, and the outstanding gaps in knowledge and remaining uncertainties have shifted accordingly. The intent of the present paper is to provide an overview and update of the current state of understanding of the processes that cause regional SLC and to identify and discuss limitations and uncertainty in our understanding of these processes. Although the focus is on contemporary SLC, we do include discussion of projections of future SLC. In particular, we are concerned with areas where lack of understanding or gaps in knowledge inhibit comprehensive planning efforts at the regional level. While we do not make explicit connections to planning efforts, we expect that a detailed discussion of uncertainties could be useful to those translating science into actionable plans (e.g., Horton et al., 2018). This paper is a resource for those interested in particular aspects of regional SLC by giving a detailed presentation of the most recent estimates of their contributions and a discussion of where improvement may be made in the coming years. Finally, a goal of this paper is to highlight the potential role of the expanded sea-level observation network - particularly as related to satellites - to understanding the contributors to regional SLC. This paper is organized according to the individual processes of regional relative SLC, with each process covered in a section. In section 2, we provide a brief summary of how the contributors to regional sea level are separated and we present definitions for the terminology adopted in the remainder of the paper. Sections 3 through 8 discuss the individual processes contributing to regional SLC, with each section broken into two components: (1) a summary of the current state of knowledge, and (2) an overview of current limitations or areas of uncertainty and a discussion of where progress will likely be made in the coming years. In section 9, we summarize advances toward overcoming these limitations or reducing uncertainties that may be expected through recent and future additions to the sea-level observational network, with particular emphasis on satellite-based observations. Processes Contributing to Regional SLC As we discuss in the sections to follow, changes in sea level arise from a diverse set of physical processes. As a result, scientists from a range of disciplines are working on different questions related to sea level. The need to address the impacts of ongoing and future SLC, along with associated policy considerations, further increases the breadth of those studying or interested in SLC. This diversity and broad interest have led to inconsistency in sea-level terminology that can hinder progress in research, communication, and policy. To address this issue, Gregory et al. (2019) have provided guidelines and clearly defined terminology for discussing SLC. In Gregory et al. (2019), SLC refers to the geocentric SLC, specifically the change in the height of sea level with respect to the terrestrial reference frame. When including the movement of the land at the coast, the phrase relative SLC is used, which is the change in the height of the mean sea surface relative to the solid surface, and thus includes the effects of vertical land motion (VLM). Given that relative SLC encompasses both geocentric SLC and VLM, and to simplify the discussion in this paper, we have chosen to use SLC to refer to changes in relative sea level for the remainder of this paper. The definition of spatial scales is separated by regional and global. The term “regional” is used to refer to processes that are considered properties of regions, with spatial of hundreds of kilometers and less. Unless specified, this includes local changes that occur at a specific geographic location. Processes are said to be of “global” scale if they contribute to variability in GMSL. The global mean refers specifically to the area-weighted mean of SLC for the entire connected surface of the ocean. There are several ways to separate and distinguish between the different processes contributing to regional SLC. Here, we separate the contributors into six different sections. Regional and global SLC associated with ice mass changes is divided into contributions from ice sheets (section 3) and contributions from glaciers (section 4), recognizing that the observational and measurement considerations can differ between the two. Further changes arising from variability in land water storage are presented in a separate section (section 5). Each of these three contributors are discussed first in terms of their impact on GMSL, and then in terms of their regional signature through changes in Earth Gravitation, Rotation, and Deformation (GRD), caused by redistributions of land ice and water (discussed in more detail below). The primary intent of this paper is to discuss regional SLC, but the magnitude of the regional contributions of these factors is related to the size of their GMSL contribution. These three contributors are also intentionally covered first due to the similarity of the mechanism that impacts regional SLC. Regional SLC associated with steric variability and ocean dynamics (also referred to as sterodynamic SLC) is combined into a single discussion (section 6), which includes both natural and anthropogenic contributions. This section also covers dynamic SLC that may occur as a result of freshwater input into the ocean associated with the contributors in sections 3 through 5. Given its large contribution to the SLC at the coast, a section is included on VLM, covering a range of temporal and spatial scales (section 7). Finally, as the goal here is to cover a wide range of time scales that impact regional and local SLC, a section on higher-frequency variability is provided that includes variations in sea level associated with astronomic tides, storm surges, ocean swell, wave setup, and wave run-up (section 8). We use the term sea level in this paper to refer to both the lower-frequency variations described in sections 3 through 6, and the higher-frequency variations in section 8. Pugh and Woodworth (2014) define sea level as the sum of four main components: mean sea level, astronomical tides, a meteorological component, and waves. Using this description, sections 3 through 6 largely discuss changes in mean sea level, while section 8 covers the other higher-frequency components. As a summary of the contributing factors covered in this paper, Table 1 provides an overview of the relevant time scales of each process in addition to the magnitude of its associated contribution on a yearly basis. One of the main takeaways from this breakdown is the wide range of time scales and subcomponents associated with each factor, and the degree to which each needs to be accounted for within any particular time frame of interest. Table 1: Components of Regional Sea-Level Rise Covered in This Paper, Along With Their Relevant Time Scales and Potential Magnitude Component Dominant temporal scales Potential magnitude (yearly) Ice sheets years to centuries millimeters to centimeters Glaciers (outside of ice sheets) months to centuries millimeters to centimeters Steric and dynamic sea-level change months to decades millimeters to meters Land water storage months to decades millimeters to centimeters High-frequency water level variability minutes to years centimeters to meters Solid earth deformation/vertical land motion years to centuries millimeters to meters Contributions From Ice Sheets Current State of Knowledge	https://sealeveldocs.readthedocs.io/en/latest/hamlington20jgr.html
88add8ede049-2	We use the term sea level in this paper to refer to both the lower-frequency variations described in sections 3 through 6, and the higher-frequency variations in section 8. Pugh and Woodworth (2014) define sea level as the sum of four main components: mean sea level, astronomical tides, a meteorological component, and waves. Using this description, sections 3 through 6 largely discuss changes in mean sea level, while section 8 covers the other higher-frequency components. As a summary of the contributing factors covered in this paper, Table 1 provides an overview of the relevant time scales of each process in addition to the magnitude of its associated contribution on a yearly basis. One of the main takeaways from this breakdown is the wide range of time scales and subcomponents associated with each factor, and the degree to which each needs to be accounted for within any particular time frame of interest. Table 1: Components of Regional Sea-Level Rise Covered in This Paper, Along With Their Relevant Time Scales and Potential Magnitude Component Dominant temporal scales Potential magnitude (yearly) Ice sheets years to centuries millimeters to centimeters Glaciers (outside of ice sheets) months to centuries millimeters to centimeters Steric and dynamic sea-level change months to decades millimeters to meters Land water storage months to decades millimeters to centimeters High-frequency water level variability minutes to years centimeters to meters Solid earth deformation/vertical land motion years to centuries millimeters to meters Contributions From Ice Sheets Current State of Knowledge Using measurements from the joint NASA (US) /DLR (Germany) Gravity Recovery and Climate Experiment (GRACE) twin satellite mission, the Greenland and Antarctic Ice Sheets lost mass and collectively contributed around 1.17 ± 0.17 mm yr^{-1} to GMSL (Figure 2) from 2002 to 2017, about one third of the total GMSL rise (Dieng et al., 2017). This rate has been steadily increasing since the 1990s (Bamber et al., 2018). The Greenland Ice Sheet holds enough water to raise GMSL by 7.4 m, while the Antarctic Ice Sheet has the potential to increase GMSL by 58 m (Fretwell et al., 2013; Morlighem et al., 2017). Although both ice sheets are currently losing mass, they do so at different rates via different mechanisms. The Antarctic Ice Sheet mass loss has increased threefold from 2002-2007 (0.2 ± 0.1 mm yr^{-1} sea-level equivalent) to 2012-2017 (0.6 ± 0.1 mm yr^{-1}) (Shepherd et al., 2018) and is mostly attributed to an increase in ice sheet discharge from glacier acceleration in West Antarctica (Gardner et al., 2018; Mouginot et al., 2014; Rignot et al., 2011). This increase is driven by a combination of an intrinsic geometric instability associated with marine-based ice sheets grounded on bedrock that deepens toward the center of the ice sheet and changes in the availability of warm, circumpolar deep water under floating ice shelves due to decadal atmospheric variability (Jenkins et al., 2016). Warm ocean water acts in tandem with atmospheric warming to thin and break up foating ice shelves (Khazendar et al., 2016; Liu et al., 2015; Paolo et al., 2015), leading to acceleration and retreat of the glaciers they buttress (Shepherd et al., 2018; Wouters et al., 2015). In contrast, the Greenland Ice Sheet mass loss is dominated by changes in surface mass balance (SMB, precipitation minus sublimation and meltwater runoff), with a smaller contribution caused by increased discharge from marine terminating outlet glaciers (Enderlin et al., 2014; Shepherd et al., 2018). Increase in runoff along the entire Greenland Ice Sheet margin is predominantly caused by atmospheric warming which promotes the intensification of ice sheet surface melt (Van den Broeke et al., 2016) and in turn rates of frontal (ocean) melting (Carroll et al., 2016). Figure 2: Time series and spatial patterns of ice sheet mass changes as measured by GRACE (2002-2017, Wise et al., 2018). In the upper plot, the solid lines show the GRACE mass balance from Antarctica (blue) and Greenland (red), with uncertainties contoured in the same color, and the three dotted lines show the lower, middle, and upper estimates of ice sheet mass loss in the business-as-usual, high-emissions RCP8.5 future scenario (IPCC, 2013). The numbers in the upper plot give the best linear Þt for each ice sheet. The lower plots show the linear trend in units of cm water equivalent per year squared over the 2002-2017 period. Three independent observational methods are used to calculate current ice sheet mass loss rates: gravimetry, altimetry, and the inputÐoutput method (Shepherd et al., 2018). Each method has various strengths and weaknesses, with differing sensitivities to necessary corrections. Mass loss estimates from gravimetry (Velicogna & Wahr, 2006) provide the only direct measure of mass change of the ice sheets, but require a correction due to glacial isostatic adjustment (GIA) processes, which dominates the uncertainty in derived mass-loss rates. GIA uncertainties are largest for Antarctic Ice Sheet, and while estimates vary among studies, a recent study (Caron et al., 2018) estimates Antarctic Ice Sheet GIA uncertainty to be ~40 Gt (Gigaton = 10^{12} kg) per year, which is approximately 30% of the mass trend. Greenland, on the other hand, has a GIA uncertainty of ~13 Gt/yr, which is less than 5% of the Greenland Ice Sheet mass loss trend. Repeated satellite and airborne laser and radar altimetry provide detailed surface height change observations over ice sheets, but conversion from surface height to mass loss requires knowledge of spatial and temporal variability in firn density, a parameter that is poorly constrained due to sparse observations within the ice sheet interior (Pritchard et al., 2009). The input-output method (Gardner et al., 2018; Rignot et al., 2011, 2019; Shirzaei & Bgmann, 2012, 2018) - the only method that gives a longer time series of ice sheet mass balance (Kjeldsen et al., 2015; Rignot et al., 2019; Mouginot et al., 2019) - combines observations of ice flux across the grounding line from satellite remote sensing with modeled SMB estimates. In general, most observational time series are less than 20 years old, making the detection of mass loss acceleration in the presence of large natural variability challenging, especially in ice sheet SMB (Wouters et al., 2013). Radar altimetry from CryoSat-2 (launched in 2010), as well as new gravimetry (GRACE Follow-On, GRACE-FO) and laser altimeter (ICESat-2) missions launched in 2018, will extend the time series and provide continuous monitoring of ice sheet changes in the coming years. We depend on a suite of numerical models to project future ice sheet changes, and these models also contribute to constraining past and present behavior. These models are traditionally used in a stand-alone framework but are increasingly “coupled” to represent the full spectrum of ice sheet-climate interactions. Atmospheric (surface climate and SMB) and oceanic (e.g., temperature, salinity, circulation, sea ice) forcings to the ice sheet are supplied by a variety of climate models, which are either produced for the full globe (global circulation models and climate reanalysis) or spatially limited to one particular ice sheet and surroundings (regional climate models). While circulation models historically focused on coupled ocean-land-atmosphere processes, modern earth system models also include the carbon cycle through dynamic atmospheric chemistry, as well as forcing of the ocean and atmosphere by the ice sheets. Regional climate models have become a preferred tool in representing ice sheet surface climate and SMB because they incorporate surface energy and snow hydrology processes and have the spatial resolutions (~5 km) necessary to accurately model the Greenland Ice Sheet and individual Antarctic Ice Sheet basins (Agosta et al., 2019; Lenaerts et al., 2017; Noel et al., 2018; Van Wessem et al., 2018), often with steep topographic slopes around ice sheet margins. However, the accuracy of any regional climate model depends on the quality of the atmospheric forcing at the model domain boundaries, and observations necessary to evaluate climate and SMB over extensive areas of northern Greenland and Antarctica are lacking.	https://sealeveldocs.readthedocs.io/en/latest/hamlington20jgr.html
88add8ede049-3	We depend on a suite of numerical models to project future ice sheet changes, and these models also contribute to constraining past and present behavior. These models are traditionally used in a stand-alone framework but are increasingly “coupled” to represent the full spectrum of ice sheet-climate interactions. Atmospheric (surface climate and SMB) and oceanic (e.g., temperature, salinity, circulation, sea ice) forcings to the ice sheet are supplied by a variety of climate models, which are either produced for the full globe (global circulation models and climate reanalysis) or spatially limited to one particular ice sheet and surroundings (regional climate models). While circulation models historically focused on coupled ocean-land-atmosphere processes, modern earth system models also include the carbon cycle through dynamic atmospheric chemistry, as well as forcing of the ocean and atmosphere by the ice sheets. Regional climate models have become a preferred tool in representing ice sheet surface climate and SMB because they incorporate surface energy and snow hydrology processes and have the spatial resolutions (~5 km) necessary to accurately model the Greenland Ice Sheet and individual Antarctic Ice Sheet basins (Agosta et al., 2019; Lenaerts et al., 2017; Noel et al., 2018; Van Wessem et al., 2018), often with steep topographic slopes around ice sheet margins. However, the accuracy of any regional climate model depends on the quality of the atmospheric forcing at the model domain boundaries, and observations necessary to evaluate climate and SMB over extensive areas of northern Greenland and Antarctica are lacking. The relation between ice sheets and climate is defined by a two-way connection: While ice sheets respond to atmospheric and oceanic conditions, they also influence the surrounding climate, for example, via the discharge of freshwater into oceans (Bronselaer et al., 2018; Schloesser et al., 2019) and changes in topographic geometries (e.g., Fyke et al., 2018). To this end, the ice sheet modeling community has increasingly focused on simulations that are fully coupled to climate models. The ongoing intercomparison of climate models (Sixth Coupled Model Intercomparison Project; CMIP6) includes several models that couple to dynamical ice sheet models for the first time (Nowicki et al., 2016). The initial development has been associated with atmosphere/ice sheet coupling over the Greenland Ice Sheet (e.g., Lipscomb et al., 2013). Major, ongoing challenges of such models include matching the temporal and spatial scales of the ice sheet model with the global models, providing accurate initial conditions for the ice sheet model, and allowing for the variable extent of the ice-covered surface. Initial improvements have been made in the representation of SMB in earth system models guided by lessons from regional climate models (e.g., Vizcaino et al., 2013). Advances in the coupling of ocean and ice sheet models (e.g., Goldberg et al., 2018) will continue to improve our ability to model the Antarctic Ice Sheet, particularly in West Antarctica, where oceanic forcings are likely to play a pivotal role in future ice sheet mass loss. Recent studies have demonstrated the impact of ice-ocean coupling on such sub-ice-shelf melt rates and grounding line migration (Golledge et al., 2019; Jordan et al., 2017; Seroussi et al., 2017). The ice-sheet mass loss to the ocean strongly influences regional sea level, as associated changes in Earth’s GRD responses dictate the spatial distribution of water across the global ocean (Farrell & Clark, 1976; Milne & Mitrovica, 1998; Mitrovica et al., 2001). These so-called “sea-level fingerprints” are crucial to determining regional SLC (Figures 3a and 3b). In general, mass loss causes a sea level fall in the near field, a reduced sea-level rise at intermediate distances, and a greater-than-global-mean sea-level rise at larger distances. Sea-level fingerprints can be computed for specific portions of ice sheets, enabling accurately quantified sensitivities of basin-scale ice mass loss to local sea-level rise at any coastal cities. The collapse of Petermann Glacier in Greenland, for example, would lead to 38% lower sea-level rise at New York and 20% higher sea-level rise at Tokyo relative to the global mean (Larour et al., 2017; Mitrovica et al., 2018). Estimating the current and projecting future contributions from the two ice sheets - including spatial variability in the contribution across each ice sheet - is thus critical to understanding regional SLC. Updated assessments of the regional impact on coastal cities will continue to be made as our understanding of mass loss from ice sheets advances and projections are improved. Uncertainties and Future Outlook While significant progress has been made in recent years as described above, estimating future ice sheet contributions to sea level relies on models, which contain large uncertainties. These uncertainties exist in every stage of modeling ice sheets in future climates, from fundamental understanding of ice sheet physical processes (e.g., DeConto & Pollard, 2016), initialization (Goelzer et al., 2018), parameter, and boundary condition choices (e.g., Larour et al., 2012; Nias et al., 2016; Schlegel et al., 2015), to the quality of atmospheric and ocean forcings, which in turn rely on climate models with all their associated uncertainties (Nowicki & Seroussi, 2018; Robel et al., 2019); all of these uncertainties can limit the quality of model projections. For example, climate model-driven projections reported in the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC) underestimated mass loss from 2006 to present, especially in the case of the Greenland Ice Sheet, including in the strongest warming (business-as-usual) RCP8.5 scenarios (Figure 2). This example highlights the need for extensive evaluation of present-day model performance, careful selection of model forcing, and, on the longer term, a focus on earth system model development to improve high-latitude atmospheric (e.g., clouds, radiation, precipitation) and oceanic processes, horizontal resolution and/or statistical downscaling (Lenaerts et al., 2019). Multimodel ensembles and intercomparisons (e.g., the Ice Sheet Model Intercomparison Project, ISMIP6; Nowicki et al., 2016) will also provide critical contributions to uncertainty quantifications. Figure 3: Contribution to relative sea.level rise (mm/year) from 2002 to 2015 from (a) Antarctica Ice Sheet mass loss, (b) Greenland Ice Sheet mass loss, (c) terrestrial water storage variability, and (d) glacier mass loss. Adapted from Adhikari and Ivins (2016). Ice sheet contributions are especially important when planning for future SLC (e.g., Garner & Keller, 2018; Oppenheimer & Alley, 2016; Sriver et al., 2018; Sweet et al., 2017). The research community is increasingly employing probabilistic approaches when making projections of future sea-level contributions from ice sheets (Edwards et al., 2019; Little et al., 2013; Ritz et al., 2015; Schlegel et al., 2018), which are necessary for holistic probabilistic projections of sea-level rise (e.g., Kopp et al., 2014, 2017; Perrette et al., 2013, Slangen et al., 2014). Probabilistic projections, however, are subject to the same limitations as the models or structured expert judgements (e.g., Bamber & Aspinall, 2013; Bamber et al., 2019) used to construct them. There is some utility in turning to past analogs of high sea-level contributions from ice sheets (e.g., the last interglacial or Pliocene) to calibrate ice sheet models and improve probabilistic projections (e.g., Edwards et al., 2019), but these too are impacted by prior model uncertainties, as well as by the uncertainties in paleo-reconstructions. Furthermore, the efficacy of using modern ice sheet trends for constraining future contributions remains an active area of research (Kopp et al., 2017). As these deep uncertainties in ice sheet contributions are elucidated and probabilistic projections continue to improve, they will inform policy decisions that are based on projected probabilities that regional- and global-scale sea levels will exceed critical levels (e.g., Bakker et al., 2017; Buchanan et al., 2016; Rasmussen et al., 2018; Sweet et al., 2017).	https://sealeveldocs.readthedocs.io/en/latest/hamlington20jgr.html
88add8ede049-4	Figure 3: Contribution to relative sea.level rise (mm/year) from 2002 to 2015 from (a) Antarctica Ice Sheet mass loss, (b) Greenland Ice Sheet mass loss, (c) terrestrial water storage variability, and (d) glacier mass loss. Adapted from Adhikari and Ivins (2016). Ice sheet contributions are especially important when planning for future SLC (e.g., Garner & Keller, 2018; Oppenheimer & Alley, 2016; Sriver et al., 2018; Sweet et al., 2017). The research community is increasingly employing probabilistic approaches when making projections of future sea-level contributions from ice sheets (Edwards et al., 2019; Little et al., 2013; Ritz et al., 2015; Schlegel et al., 2018), which are necessary for holistic probabilistic projections of sea-level rise (e.g., Kopp et al., 2014, 2017; Perrette et al., 2013, Slangen et al., 2014). Probabilistic projections, however, are subject to the same limitations as the models or structured expert judgements (e.g., Bamber & Aspinall, 2013; Bamber et al., 2019) used to construct them. There is some utility in turning to past analogs of high sea-level contributions from ice sheets (e.g., the last interglacial or Pliocene) to calibrate ice sheet models and improve probabilistic projections (e.g., Edwards et al., 2019), but these too are impacted by prior model uncertainties, as well as by the uncertainties in paleo-reconstructions. Furthermore, the efficacy of using modern ice sheet trends for constraining future contributions remains an active area of research (Kopp et al., 2017). As these deep uncertainties in ice sheet contributions are elucidated and probabilistic projections continue to improve, they will inform policy decisions that are based on projected probabilities that regional- and global-scale sea levels will exceed critical levels (e.g., Bakker et al., 2017; Buchanan et al., 2016; Rasmussen et al., 2018; Sweet et al., 2017). As ice sheet models improve in their resolution, initialization procedures, and process implementation, they become increasingly reliant on observations to both force their behavior and validate their performance. Accurate reproduction of ice sheet dynamics, especially near grounding lines, requires high-resolution surface and bed topography (Aschwanden et al., 2016, 2019; Morlighem et al., 2014; Nias et al., 2018), estimates of basal shear stress (Parizek et al., 2013), and sub-ice shelf bathymetry (Schodlok et al., 2012). Geometric constraints on outlet glacier dynamics have improved dramatically in recent years (e.g., Greenbaum et al., 2015; Morlighem et al., 2017; Vaughan et al., 2012), but technological advancements (e.g., radar tomo.graphy; Al-Ibadi et al., 2018) and geophysical methods development (toward observational validation of sub-surface model parameters such as basal shear stress (Brisbourne et al., 2017), temperature (MacGregor et al., 2015; Schroeder et al., 2016), and englacial velocity (Holschuh et al., 2017, 2019; Leysinger Vieli et al., 2007) could drive significant improvement in model projections. Importantly, new aerogeophysical campaigns and satellite missions will be required to collect data optimized for these new techniques, as well as to fill gaps in existing subsurface observations. Ice sheet model development should focus on including geophysical observations directly, and extending the data assimilation capabilities from the inclusion of snapshot surface observations to the inclusion of time series data (Goldberg & Heimbach, 2013; Larour et al., 2014) to take full advantage of the abundance of remote sensing observations now available. Contributions From Glaciers Current State of Knowledge […] Figure 4: Time series of cumulative mass anomalies from GRACE for all primary glacier regions of the Randolph Glacier Inventory, except the Greenland and Antarctic periphery, covering the time period from 2002 to 2017. From Wouters et al. (2019). […] Uncertainties and Future Outlook […] Contributions From Changes in Land Water Storage Current State of Knowledge […] Uncertainties and Future Outlook […] Steric Sea-Level and Ocean Dynamics Current State of Knowledge […] Uncertainties and Future Outlook […] VLM/Solid Earth Deformation Current State of Knowledge […] Uncertainties and Future Outlook […] Contributions From High-Frequency Water Level Variability Current State of Knowledge […] Uncertainties and Future Outlook […] Near-Term Outlook of Regional Relative Sea-Level Understanding […] Summary Changes in GMSL provide an integrative measure of the state of the climate system, encompassing both the ocean and cryosphere and may be viewed as an important indicator of what is currently happening to the climate in the present and what may happen in the future. While an increase in GMSL portends an increase in sea level of some magnitude along the world’s coastlines, the response on regional levels is not uniform. Water that is added to the ocean from land will not be distributed evenly everywhere (sections 3-5) and changes in ocean dynamics add to the regional variability in sea-level rise (section 6). Using observations from tide gauges and satellite altimetry, the extent of the spatial variability in the rate of sea-level rise can be understood. With the recent improved understanding of GRD effects on sea level, and the suite of VLM effects outlined here, it has become clear that the use of a single global rate to describe sea level around the globe is problematic, and improved assessment of sea-level rise on regional levels is required from a planning perspective. Over the past century, coastal sea levels have risen over the majority of the globe. The effect of increasing sea level relative to land is a significant reduction in the elevation gap between typical high tides and a threshold elevation at which flooding begins (Sweet et al., 2018). Coastal communities were established with this gap in mind, recognizing flooding might occur under the most extreme of conditions. Recent reports (e.g., Sweet et al., 2017, 2018) have detailed the rapidly declining gap along the coastlines of the world, and the accelerated effect this has had on flood frequencies in many coastal locations. One important implication of these analyses is that the narrowed gap between high tide and flooding conditions can now be overcome by sea-level variability on a range of time scales. Subsequently, from a decision-making perspective, improved projections of future regional SLC are needed over a variety of time horizons, not simply the longest. As discussed here, sea level varies on time scales from short-term (section 8), to seasonal-to-decadal (sections 5 and 6) or longer (sections 3-5). When considered in tandem with the movement of land relative to the ocean (section 7), contributions to sea level at each of these time scales can combine constructively, increasing sea levels and high-tide flood frequencies over both the short and long terms. The gap described above is known in many locations, and time horizons can be generated over which high-tide flood frequency will begin to increase rapidly. When considering only the long-term trend, this time horizon is usually found to be on the order of decades. However, when combined with the other contributors to sea-level variability, it is highly likely that in the short term (on the order of years) the cumulative effect of high-tide flooding will extend beyond “nuisance” levels and becomes too frequent for business as usual in coastal areas. As such, there is a strong need for improved information regarding future sea-level rise across a range of time scales. While understanding the long-term contributions from melting glaciers and ice-sheets is essential, so too is understanding, quantifying, and possibly predicting the variability that will occur on seasonal to decadal time scales. Recent studies suggest that these contributors are becoming distinguishable with the records available (e.g., Fasullo & Nerem, 2018; Nerem et al., 2018). With the observations that are available - or will become available soon - coupled with improved data analysis and modeling efforts, our understanding of future regional SLC will continue to advance in the coming years. Knowing that planning efforts are underway and that sea-level rise is already impacting many parts of the world’s coastlines, it is worth taking inventory of the current state of understanding and clearly identifying areas of uncertainty that are impacting our ability to provide complete, accurate, and actionable information at the coast. Such assessments should be undertaken frequently to update relevant information in light of new science results, and to assist those tasked with translating current scientific understanding into plans that can be put into action at the coast.	https://sealeveldocs.readthedocs.io/en/latest/hamlington20jgr.html
bc7116e222f0-0	Wickramage et al. (2023) Title: Sensitivity of MPI-ESM Sea Level Projections to Its Ocean Spatial Resolution Corresponding author: Chathurika Wickramage Keywords: Ocean, Sea level, Climate models, Mesoscale models, Model comparison Citation: Wickramage, C., Köhl, A., Jungclaus, J., & Stammer, D. (2023). Sensitivity of MPI-ESM Sea Level Projections to Its Ocean Spatial Resolution, Journal of Climate, 36(6), 1957-1980. doi:10.1175/JCLI-D-22-0418.1 URL: https://journals.ametsoc.org/view/journals/clim/36/6/JCLI-D-22-0418.1.xml Abstract The dependence of future regional sea level changes on ocean model resolution is investigated based on Max Planck Institute Earth System Model (MPI-ESM) simulations with varying spatial resolution, ranging from low resolution (LR), high resolution (HR), to eddy-rich (ER) resolution. Each run was driven by the shared socioeconomic pathway (SSP) 5-8.5 (fossil-fueled development) forcing. For each run the dynamic sea level (DSL) changes are evaluated by comparing the time mean of the SSP5-8.5 climate change scenario for the years 2080–99 to the time mean of the historical simulation for the years 1995–2014. Respective results indicate that each run reproduces previously identified large-scale DSL change patterns. However, substantial sensitivity of the projected DSL changes can be found on a regional to local scale with respect to model resolution. In comparison to models with parameterized eddies (HR and LR), enhanced sea level changes are found in the North Atlantic subtropical region, the Kuroshio region, and the Arctic Ocean in the model version capturing mesoscale processes (ER). Smaller yet still significant sea level changes can be found in the Southern Ocean and the North Atlantic subpolar region. These sea level changes are associated with changes in the regional circulation. Our study suggests that low-resolution sea level projections should be interpreted with care in regions where major differences are revealed here, particularly in eddy active regions such as the Kuroshio, Antarctic Circumpolar Current, Gulf Stream, and East Australian Current. Significance Statement Sea level change is expected to be more realistic when mesoscale processes are explicitly resolved in climate models. However, century-long simulations with eddy-resolving models are computationally expensive. Therefore, current sea level projections are based on climate models in which ocean eddies are parameterized. The representation of sea level by these models considerably differs from actual observations, particularly in the eddy-rich regions such as the Southern Ocean and the western boundary currents, implying erroneous ocean circulation that affects the sea level projections. Taking this into account, we review the sea level change pattern in a climate model with featuring an eddy-rich ocean model and compare the results to state-of-the-art coarser-resolution versions of the same model. We found substantial DSL differences in the global ocean between the different resolutions. Relatively small-scale ocean eddies can hence have profound large-scale effects on the projected sea level which may affect our understanding of future sea level change as well as the planning of future investments to adapt to climate change around the world. Introduction Rising sea levels are among the largest threats of anthropogenic global warming to society, with far-reaching consequences for many coastal and island populations around the globe (Church et al. 2011; Fox-Kemper et al. 2021). Our understanding of the global mean sea level trends has improved significantly over the past decades based on coordinated modeling efforts, such as the Coupled Model Intercomparison Project (CMIP) and the analysis of past observations (e.g., Church et al. 2011; Fox-Kemper et al. 2021). However, a quantitative understanding of processes that lead to regional to local-scale sea level changes is still pending, which affects the ability to accurately forecast future coastal changes (Church et al. 2013a). Although the quantitative budget of the global sea level rise is understood, insufficient data hampers the understanding on regional and coastal scale, which makes improving modeling efforts essential for closing the gaps. Physically, regional to local-scale sea level change is a global problem as it depends on processes taking place locally as well as remotely. As such, it is directly or indirectly affected by all components of the climate system as well as contributions originating from the solid Earth such as vertical seafloor movement or changes in gravity (Stammer et al. 2013). However, the relative contribution of individual processes or forcing components to the regional or local sea level changes is strongly dependent on the spatial and temporal scales under consideration and might also change under global warming conditions. In this context a caveat of previous CMIP projections is that underlying climate models are of relatively low spatial resolution, thereby intrinsically excluding the impact of resolved ocean eddies on the solution. To what extent existing sea level projections are thus biased toward large-scale climate mode responses as opposed to regional dynamical balances remains therefore to be understood, making it difficult to provide quantitative information about future sea level projections in specific regions, such as coastlines. This situation now gradually changes as climate projections from eddy-resolving models are becoming available accounting for the dynamical processes associated with boundary or coastal current dynamics, which are important for accurate small-scale sea level change information (e.g., van Westen et al. 2020; van Westen and Dijkstra 2021; Li et al. 2022). Given these new opportunities, it is now timely to test previous CMIP sea level change projections against novel eddy-rich climate projections. Against this background, the aim of this paper is to provide an understanding of the sensitivity of CMIP type sea level projections to model resolution and, thus, to define an uncertainty level in CMIP previous projections resulting from the lack of small-scale processes and eddy mechanisms. Specifically, we aim to quantify differences in the resulting sea level projections obtained under shared socioeconomic pathway (SSP) 5-8.5 forcing (O’Neill et al. 2017) with respect to lower-resolution simulations using a hierarchy of spatial resolutions. The respective work will be based on the eddy-rich Max Planck Institute Earth System Model (MPI-ESM-ER) experiments (Gutjahr et al. 2019), which have been performed as part of the CMIP6-endorsed HighResMIP (Haarsma et al. 2016). At the same time, we aim to identify causes for those changes in terms of mechanisms, which can result from differences in the ocean dynamics and differences in the air–sea exchange of heat, freshwater, or momentum. Geographically, our work focuses on the North Atlantic, the North Pacific, and the Southern Ocean, where large-scale effects from mesoscale eddies can be expected, given the presence of strong and eddy-rich western boundary currents. The structure of the remaining paper is as follows: Section 2 describes the model and the evaluation methods used. Common characteristics of sea level change in the models are described in detail in section 3. In section 4, we compare the ocean model projection in each basin. We conclude and summarize with section 5. Materials and methods Climate model simulations with MPI-ESM Our study is based on climate projections obtained with the MPI-ESM1.2 and performed under the protocol of the CMIP phase 6 (CMIP6; Eyring et al. 2016). The MPI-ESM is a fully coupled climate model, using ECHAM6 (Stevens et al. 2013) for the atmosphere and MPIOM (Jungclaus et al. 2013) for the ocean. Details on the CMIP6 version in comparison with its predecessor can be found in Mauritsen et al. (2019). Our study considers three versions of the same model configuration, including the low-resolution (LR), high-resolution (HR), and eddy-rich (ER) models, all driven by the SSP5-8.5 (fossil-fueled development) (O’Neill et al. 2017) forcing. From each model simulation monthly mean fields are available. Configurations for all experiments evaluated here are summarized in Table 1. Table 1: Summary of MPI-ESM projections used in this study.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-1	The structure of the remaining paper is as follows: Section 2 describes the model and the evaluation methods used. Common characteristics of sea level change in the models are described in detail in section 3. In section 4, we compare the ocean model projection in each basin. We conclude and summarize with section 5. Materials and methods Climate model simulations with MPI-ESM Our study is based on climate projections obtained with the MPI-ESM1.2 and performed under the protocol of the CMIP phase 6 (CMIP6; Eyring et al. 2016). The MPI-ESM is a fully coupled climate model, using ECHAM6 (Stevens et al. 2013) for the atmosphere and MPIOM (Jungclaus et al. 2013) for the ocean. Details on the CMIP6 version in comparison with its predecessor can be found in Mauritsen et al. (2019). Our study considers three versions of the same model configuration, including the low-resolution (LR), high-resolution (HR), and eddy-rich (ER) models, all driven by the SSP5-8.5 (fossil-fueled development) (O’Neill et al. 2017) forcing. From each model simulation monthly mean fields are available. Configurations for all experiments evaluated here are summarized in Table 1. Table 1: Summary of MPI-ESM projections used in this study. The coupled control simulations were initialized and forced following the CMIP6 protocol for HR and LR. The reference year for the preindustrial control simulation (piControl) is 1850, and it is conducted under conditions that have been selected to be typical of the time before the start of large-scale industrialization. A control simulation typically begins after an initial spinup phase, during which the climate system reaches a state close to an equilibrium (Eyring et al. 2016). For ER, the coupled control simulation was however initialized following the CMIP6-HighResMIP protocol (Haarsma et al. 2016). The ER control run was initialized after 30 years of spinup initialized from the averaged state of the Met Office Hadley Centre EN4 observational dataset from 1950 to 1954 (Good et al. 2013) for the ocean and ER atmosphere initialized from HR atmospheric state (Gutjahr et al. 2019). The length of control run is 1000 years for LR, 500 years for HR, and 200 years for ER. The low-resolution version of MPI-ESM (MPI-ESM-LR) approximately has a 1.9° horizontal resolution for the atmosphere (spectral truncation at T63; 210 km at the equator; 192 × 96 longitude/latitude) and 47 hybrid sigma pressure level extending to a 0.01 hPa top level. The ocean component has a bipolar 1.5° horizontal resolution (GR1.5; approximately 150 km near the equator; 256 × 220 longitude/latitude) and 40 vertical levels with layer thickness ranging from 12 m near the surface to several hundred meters at depth. The horizontal grid spacing varies from 185 km in the tropical Pacific to 22 km around Greenland. The poles of the ocean model are over Greenland and Antarctica (coast of the Weddell Sea). The LR version cannot capture mesoscale ocean processes and dynamics (for more details, Mauritsen et al. 2019). The high-resolution configuration, MPI-ESM-HR (Müller et al. 2018), uses a 0.9° horizontal resolution (T127; 384 × 192 longitude/latitude) for the atmosphere, which is approximately 100 km around the equator. HR has a relatively highly resolved stratosphere extending to a 0.01 hPa top level with 95 vertical levels (L95). A tripolar grid 0.4° horizontal resolution (TP04; 802 × 404 longitude/latitude) is used for the ocean component. Two poles are placed in the Northern Hemisphere over central Asia (Siberia) and Canada, providing quasi-homogeneous resolution of a approximately 40 km in the Arctic Ocean. In the Southern Hemisphere, grid distances decrease with increasing latitude. South of the Antarctic Circumpolar Current (ACC) at around 60°S the resolution is 20 km. HR comprises 40 unevenly spaced vertical levels, allocating 20 levels within the upper 700 m. HR is permitting eddies in the tropics but not resolving the Rossby radius in the higher latitudes. Even though HR fails to resolve the Rossby radius length scales, key for the representation of boundary currents and fronts, it still can capture reasonable eddy-like structures (see Jungclaus et al. 2013; Müller et al. 2018). The Gent–McWilliams (GM) parameterization (Gent and McWilliams 1990) of mesoscale eddies is used in LR and HR. The GM coefficients in HR and LR are constant and quite small. They are scaled with the grid spacing. The GM parameterization decreases linearly with increasing resolution, and a value of 250 m−2 s−1 was chosen for a grid cell that is 400 km wide (Gutjahr et al. 2019). The eddy-rich MPI-ESM-ER (Gutjahr et al. 2019) has the same T127/L95 atmospheric component as HR. However, the horizontal resolution of the ocean component is on a tripolar 6-min (TP6M) horizontal grid (approximately 0.1° or 10 km) in both latitude and longitude, and has 80 vertical levels. ER has three poles over North America, Russia, and Antarctica. In the eddy-rich-resolution model simulations, the GM parameterization for mesoscale eddies is disabled, and eddy effects are resolved according to the ratio of the first baroclinic deformation radius to the horizontal grid spacing. Eddies are not resolved at higher latitudes and over shallow/shelf regions. The grid resolution is smaller south of 50°S (Table 1; for more details, check Mauritsen et al. 2019; Putrasahan et al. 2021). The ER model has nominal horizontal resolution of ∼10 km which means that the large-scale (order of 1000 km) and oceanic mesoscale eddies (order of 10 and larger) are resolved almost everywhere; however, the ocean submesoscale eddies are typically less than 10 km are not included in the ER simulation. In addition, the air–sea interactions from processes such as mesoscale storms are not resolved by the atmospheric component of ER model. As part of our analysis, we compare results from all model version described above under the SSP5-8.5 climate forcing scenario covering 2080–99, to their historical simulations during 1995–2014. In all cases we consider ensemble means using all available members, which are 10 members in LR, 2 in HR, and 3 in ER to minimize the impact of climate variability. Prior to analyzing the model output, we interpolated it onto the same grid of 1° horizontal resolution. As the development of ER was computationally expensive, it has not been tuned and spun up according to the standard of HR and LR (Mauritsen et al. 2019). Therefore, the linear trend obtained from the only member of the control run was removed from the historical and scenario data. While we focus mostly on effects of ocean resolution, we note that LR features also considerably lower resolution in the atmosphere. Therefore, we put particular emphasis in the discussion on changes we diagnose in the ER configuration, which was run with the same atmosphere as the HR model. Analyzing model output This study considers the dynamic sea level (DSL), which is defined as the mean sea level above the geoid due to ocean dynamics (Gregory et al. 2019): ζ = η − η′. (1) Here ζ is the variable “zos” according to the CMIP terminology (Griffies et al. 2016), η, which is named “sterodynamic sea level,” is the sea surface height relative to a reference geopotential surface, and η′ denotes a global mean (Gregory et al. 2019). Hence, DSL change (Δζ) should have a zero global mean by definition. We therefore subtracted the global mean from each input field. As we are interested in future sea level change, our work focuses on the dynamic sea level change (Δζ) pattern, which is calculated from the difference between the DSL change in SSP5-8.5 forcing scenario (Δηs) relative to the DSL change in the historical simulation (Δηh): Δζ = Δηs − Δηh. (2) Considering that changes in circulation and changes in wind stress in principle are the key drivers of these changes, we also calculated changes in the barotropic streamfunction ψ, changes in surface wind stress, and variation in the meridional overturning circulation and analyzed their differences as function of model resolution.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-2	Analyzing model output This study considers the dynamic sea level (DSL), which is defined as the mean sea level above the geoid due to ocean dynamics (Gregory et al. 2019): ζ = η − η′. (1) Here ζ is the variable “zos” according to the CMIP terminology (Griffies et al. 2016), η, which is named “sterodynamic sea level,” is the sea surface height relative to a reference geopotential surface, and η′ denotes a global mean (Gregory et al. 2019). Hence, DSL change (Δζ) should have a zero global mean by definition. We therefore subtracted the global mean from each input field. As we are interested in future sea level change, our work focuses on the dynamic sea level change (Δζ) pattern, which is calculated from the difference between the DSL change in SSP5-8.5 forcing scenario (Δηs) relative to the DSL change in the historical simulation (Δηh): Δζ = Δηs − Δηh. (2) Considering that changes in circulation and changes in wind stress in principle are the key drivers of these changes, we also calculated changes in the barotropic streamfunction ψ, changes in surface wind stress, and variation in the meridional overturning circulation and analyzed their differences as function of model resolution. The Sverdrup streamfunction was estimated using wind stress data based on the Sverdrup relation (Sverdrup 1947). The Sverdrup transport was integrated zonally along a latitude (y) from the eastern boundary (xe) to each zonal location (x) of the basin as follows: ψSv = 1/(βρ) ∫^x_{xe} curl(τ) dx′, (3) where β denotes the meridional derivative of the Coriolis parameter, ρ is the mean density of the ocean, and curl(τ) denotes the wind stress curl. To analyze the meridional displacement of gyres, the mean latitude y of the barotropic streamfunction is calculated according to θ = ∫ψ × y dx dy / ∫ψ dx dy. (4) We consider the zero contour as the boundary of each gyre and consider positive (negative) barotropic streamfunction values for subtropical (subpolar) gyre. The contours between the minimum and maximum transport (positive contours) in the Drake Passage were considered to calculate the mean central latitude of ACC transport. Significance and trend Assuming that the variance remains unchanged under climate forcing conditions, the 95% significance of the difference of changes between the resolutions ER and HR was determined according to the formula, √(2σ^2_{ER}/N_{ER} + 2σ^2_{HR}/N_{HR}) × t_{95%}, with t_{95%} the Student t value, N_{ER} and N_{HR} are the respective numbers of members, and σ^2_{ER} and σ^2_{HR} are the variances of the ER and HR control simulations, respectively. The factor 2 accounts for the fact that the changes have twice the variance of the fields they are calculated from. A similar method was applied to calculate the error bar or envelope, (1/√N)√2σ^2_{ER} × t_{95%} for the ensemble mean changes [see von Storch and Zwiers (2002) for more details]. The linear least squares fitting was used to calculate the yearly average time series trend. The analyses were performed using CDO and NCL software (NCL 2019). Commonalities of sea level changes in MPI-ESM Over the past years, the global ocean has accounted for around 91% of anthropogenically induced Earth’s heat content increase, resulting in an observed thermal expansion and associated sea level rise of about 0.54 (0.40–0.68) mm yr−1 over the years from 1901 to 2018 (Fox-Kemper et al. 2021). In contrast, the simulated thermal expansion in IPCC AR6 leads to sea level rise of 30 (24–36) cm under SSP5-8.5 for the year 2100 relative to a baseline of 1995–2014. From the MPI- ESM model simulations we can infer a comparable global mean thermosteric sea level (GMTSL) rise (Fig. 1) of 30.30, 30.06, and 31.95 cm at the end of the twenty-first century relative to the 1950s for ER, HR, and LR, respectively. Over the period 1901–2018, changes are around 60 mm and compare well with the observed change of 63.2 mm due to thermal expansion (Fox-Kemper et al. 2021). The average simulated rate of thermosteric sea level rise due to global ocean heating for the SPP5-8.5 scenario is 3.31 mm yr−1 in ER, 3.34 mm yr−1 in HR, and 3.51 mm yr−1 in LR between 2030 and 2099. Figure 1: Global 12-month running mean (a) sea surface temperature and (b) thermosteric sea level change for MPI-ESM-ER, MPI-ESM-HR, and MPI-ESM-LR, relative to the 1950s. The purple line in (a) represents a 12-month running mean of respective global mean HadISST values. Figure 1 also compares the increase in thermosteric sea level rise with the respective increases in global SST. On global average, both curves suggest an equivalent increase in thermosteric sea level rise of 0.11 m per 1°C of SST increase. However, SST curves are considerable noisier and, in that sense, can only be considered a very crude proxy for thermosteric sea level rise. This holds especially for individual ensemble members and should be true also for the real world. We note that a respective correspondence cannot be expected to hold on regional scale due to the temperature dependence of the thermosteric expansion coefficient and the influence of salinity. The common global pattern of LR, HR, and ER for the changes in DSL (in m), the barotropic streamfunction (BSF; Sv; 1 Sv ≡ 106 m3 s−1), and the wind stress (N m−2) are discussed in this section. These changes between the patterns over the SSP5-8.5 years 2080–99 relative to the historical simulation (years 1995–2014) are shown in Fig. 2. We will discuss how the models differ from one another on regional scale in the following section. Figure 2: Anomalies of (a)–(c) dynamic sea level (m), (d)–(f) barotropic streamfunction (Sv), (g)–(i) sea surface temperature (°C), and (j)–(l) wind stress (N m−2), for (left) MPI-ESM-LR, (center) MPI-ESM-HR, and (right) MPI-ESM-ER between the SSP5-8.5 averaged over the period 2080–99 and the historical period averaged over the years 1995–2014. The DSL is a helpful tool for analyzing the ocean processes contributing to sea level changes due to the close link between the DSL and the ocean circulation through the geostrophic relation. According to Figs. 2a–c, sea level changes are not homogeneous in the global ocean but show diverse regional patterns. At the end of the century, the respective sea level change leads to a dipole pattern in the North Atlantic with generally increasing sea level north of the Gulf Stream (in the southern part of the subpolar gyre) and a decrease in the subtropical gyre. An opposite dipole pattern exists in the North Pacific, where sea level is higher south of the Kuroshio (in the subtropical region) and lower farther to the north (in the subpolar region). In the Southern Ocean, the ridge-like pattern is associated with a sea level increase north of ∼50°S and decrease south of ∼50°S. The aforementioned sea level change patterns have also been reported previously, such as Chen et al. (2019), Church et al. (2013a,b), Couldrey et al. (2021), Fox-Kemper et al. (2021), Gregory et al. (2016), and Lyu et al. (2020), and common to all the models. As in previous studies (Prandi et al. 2012; Rose et al. 2019; Xiao et al. 2020), the highest sea level rise is also found in our models in the Arctic Ocean.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-3	Figure 2: Anomalies of (a)–(c) dynamic sea level (m), (d)–(f) barotropic streamfunction (Sv), (g)–(i) sea surface temperature (°C), and (j)–(l) wind stress (N m−2), for (left) MPI-ESM-LR, (center) MPI-ESM-HR, and (right) MPI-ESM-ER between the SSP5-8.5 averaged over the period 2080–99 and the historical period averaged over the years 1995–2014. The DSL is a helpful tool for analyzing the ocean processes contributing to sea level changes due to the close link between the DSL and the ocean circulation through the geostrophic relation. According to Figs. 2a–c, sea level changes are not homogeneous in the global ocean but show diverse regional patterns. At the end of the century, the respective sea level change leads to a dipole pattern in the North Atlantic with generally increasing sea level north of the Gulf Stream (in the southern part of the subpolar gyre) and a decrease in the subtropical gyre. An opposite dipole pattern exists in the North Pacific, where sea level is higher south of the Kuroshio (in the subtropical region) and lower farther to the north (in the subpolar region). In the Southern Ocean, the ridge-like pattern is associated with a sea level increase north of ∼50°S and decrease south of ∼50°S. The aforementioned sea level change patterns have also been reported previously, such as Chen et al. (2019), Church et al. (2013a,b), Couldrey et al. (2021), Fox-Kemper et al. (2021), Gregory et al. (2016), and Lyu et al. (2020), and common to all the models. As in previous studies (Prandi et al. 2012; Rose et al. 2019; Xiao et al. 2020), the highest sea level rise is also found in our models in the Arctic Ocean. Many changes in DSL displayed in Figs. 2a–c can be associated with changes in the vertically integrated large-scale circulation as depicted by BSF (Figs. 2d–f). In the North Atlantic, circulation in subpolar regions north of the Gulf Stream and Labrador Sea and the subtropical gyre weaken, whereas the circulation southeast of Greenland strengthens (negative BSF anomalies in high latitude). The latter is accompanied by a smaller DSL. However, the opposite happens in the North Pacific, where subpolar gyre and the Kuroshio region (northern part of subtropical gyre) strengthens, while the southern part of subtropical gyre weakens. A band of positive streamfunction north of Gulf Stream could possibly indicate the poleward shift of the North Atlantic Subtropical Gyre. Moreover, Southern Ocean circulation strengthens between 40° and 50°S in all models (Figs. 2d–f). While almost every corner in the world is heating up, a cooling temperature patch, known as warming hole (aka cold blob), is identified in the vicinity of southeast Greenland (Figs. 2g–i). This warming hole feature, however, appears to be north of the warming hole stated in earlier studies in response to warming (e.g., Chemke et al. 2020; Drijfhout et al. 2012; Gervais et al. 2018; Menary and Wood 2018). Together with the warming hole, a patch of DSL decline (Figs. 2a–c), and a reinforced high latitude circulation (Figs. 2d–f) are identified in the same subpolar region in all configurations. Numerous previous studies have discussed the occurrence of the warming hole as a result of a weakening Atlantic meridional overturning circulation (AMOC) (such as Caesar et al. 2018; Drijfhout et al. 2012; Gervais et al. 2018; Keil et al. 2020; Menary and Wood 2018; Rahmstorf et al. 2015). The variation in the BSF field is linked to the changes in the wind stress curl field by the Sverdrup relation. However, it might also reflect the changes in the interior density gradients and/or interactions with sloping bottom bathymetry (e.g., Yeager 2015). We examine the Sverdrup relation in the North Atlantic and North Pacific Oceans later in section 4. A distinct feature of wind stress changes in the models is the strengthening of the westerly wind in the Southern Ocean at the end of the twenty-first century (Figs. 2j–l). Over most of the Pacific Ocean except the tropical North Pacific wind stress strengthens. Changes over the Atlantic are less clear, leading mostly to weakened wind stress, except for the eastern subtropical North Atlantic. Discrepancies of regional sea level change in MPI-ESM In the following we will discuss the detailed resolution dependence of the time mean changes from historical to SSP5-8.5 on the model resolution separately for the North Atlantic, the North Pacific, and the Southern Ocean. North Atlantic DSL change in the subpolar gyre region is characterized by two distinct features: a decrease in the basins southeast of Greenland and an increase in the rest of the subpolar gyre (Figs. 3a–c). Statistically significant differences at 95% are marked by dots in the spatial pattern difference and by nonoverlapping error bars in the zonal averages. This DSL decrease over the basins southeast of Greenland (the Irminger Sea and Icelandic Basin) is smallest in ER and largest in LR (by the magnitude of ∼0.2 m see in Figs. 3a,c,d,e). The increase in the Labrador Sea and farther south until north of the Gulf Stream is also smaller in ER than HR and LR (by the magnitude of ∼0.16 m see in Figs. 3d,e). The changes southeast of Greenland dominate the zonal mean (north of 50°N) of DSL, indicating a prominent decrease in LR and almost no change in HR (Fig. 4a). In contrast, ER shows an overall sea level increase in the northern part of the subpolar gyre, pointing to a smaller decline in the basins southeast of Greenland (Figs. 3c,d, 4a). In the southern part of the subpolar gyre (between 40° and 50°N), the MPI-ESM models show a DSL increase with the largest rise in HR (Fig. 4a). The increase in HR can be noticed in the spatial pattern by a negative and positive sign of the difference north of the Gulf Stream (north of 40°N) in Figs. 3e and 3f, respectively. Figure 3: North Atlantic differences of (a)–(f) dynamic sea level (m) and (g)–(l) wind stress (N m−2); panels (a)–(c) and (g)–(i) illustrate the anomalies of the SSP5-8.5 (2080–99) average relative to the historical simulation, averaged over 1995–2014 for MPI-ESM-LR in (a) and (g), MPI-ESM-HR in (b) and (h), and MPI-ESM-ER in (c) and (i); panels (d)–(f) and (j)–(l) illustrate the differences of the anomalies between the models [ER minus LR in (d) and (j), ER minus HR in (e) and (k), and HR minus LR in (f) and (l)]. The contours represent the historical mean (1995–2014; contour interval is 0.1 m). The contour colors denote solid red for the positive values, green line for the zero contour, and dash blue for the negative values. In all panels stippling indicates statistically significant differences (95% confidence level). All projections were interpolated onto the same grid prior to computing differences of the anomalies. Figure 4: North Atlantic (a) zonal mean sea surface height (m); (b) zonal mean barotropic streamfunction (Sv); (c) zonal mean zonal wind stress (N m−2) anomalies, each for the time mean differences over historical simulation and SSP5-8.5 for ER, HR, and LR, respectively. Error bar envelopes represent two standard deviations.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-4	Figure 3: North Atlantic differences of (a)–(f) dynamic sea level (m) and (g)–(l) wind stress (N m−2); panels (a)–(c) and (g)–(i) illustrate the anomalies of the SSP5-8.5 (2080–99) average relative to the historical simulation, averaged over 1995–2014 for MPI-ESM-LR in (a) and (g), MPI-ESM-HR in (b) and (h), and MPI-ESM-ER in (c) and (i); panels (d)–(f) and (j)–(l) illustrate the differences of the anomalies between the models [ER minus LR in (d) and (j), ER minus HR in (e) and (k), and HR minus LR in (f) and (l)]. The contours represent the historical mean (1995–2014; contour interval is 0.1 m). The contour colors denote solid red for the positive values, green line for the zero contour, and dash blue for the negative values. In all panels stippling indicates statistically significant differences (95% confidence level). All projections were interpolated onto the same grid prior to computing differences of the anomalies. Figure 4: North Atlantic (a) zonal mean sea surface height (m); (b) zonal mean barotropic streamfunction (Sv); (c) zonal mean zonal wind stress (N m−2) anomalies, each for the time mean differences over historical simulation and SSP5-8.5 for ER, HR, and LR, respectively. Error bar envelopes represent two standard deviations. These changes in sea level also reflect changes in circulation indicated by the vertically integrated flow (displayed in Figs. 6a–c). The cyclonic circulation in southeast of Greenland strengthens in LR and somewhat less so in HR. It is even lesser in ER (Figs. 6a–c), as does decline in sea level. Subpolar gyre circulation weakens in the Labrador Sea and the southern part of the gyre. In contrast, the weakening is considerably larger in the ER model. A weakening subpolar gyre as seen in ER was also reported for the circulation changes from the 1990s to the 2000s observed by altimetry data (Lee et al. 2010; Häkkinen and Rhines 2004, 2009), with inconclusive attribution to whether these changes remain part of natural climate variability or are already a sign of a long-term trend. Nevertheless, the circulation changes can have important consequences for the distribution of water masses as a weakening subpolar gyre can lead to an increasing transport of warm and salty Atlantic water into the Nordic seas during the historical period (Hátún et al. 2005) and significant reduction in Labrador Sea deep convection. Many previous studies have also highlighted the relation between weakening AMOC and DSL changes in the North Atlantic (Bouttes et al. 2014; Chen et al. 2019; Fox-Kemper et al. 2021; Hu et al. 2011; Levermann et al. 2005; Lyu et al. 2020; Pardaens et al. 2011; Yin et al. 2009, 2010). The projected change of the AMOC is likely to depend on the model resolution and therefore impact DSL differently. The pattern of the AMOC change is nearly identical to its mean AMOC (Figs. 5a–c), indicating a consistent weakening of all branches from historical to SSP5-8.5 by about one-third of its strength. The overlaid contours of historical mean in Figs. 5a–c provide a comparison with its anomalies. Similar to the DSL change, the warming hole feature becomes smaller in size with improved horizontal resolution, especially in ER compared to HR (Figs. 2h,i and 3b,c), whereas the AMOC slowing (Fig. 5g) does not differ much in the final years despite the smaller trend in LR at 26°N. Although, the AMOC weakening is larger in LR than in HR and ER between 30° and 60°N centered around 1500-m depth (Figs. 5a,d,f). This puts a question mark to studies linking the strength of the warming hole directly to a slowing down of the AMOC (Caesar et al. 2018, using observations; Menary and Wood 2018; Rahmstorf et al. 2015). In this sense, Keil et al. (2020) have argued for multiple drives of the warming hole feature. During the historical period, the anthropogenically forced changes of both the gyre and overturning circulation induce heat transport increase out of the subpolar region to the Greenland–Iceland–Norwegian (GIN) Seas and then farther to the Arctic, contributing to the warming hole feature in the North Atlantic (Keil et al. 2020). Figure 5: Anomalies of the Atlantic meridional overturning circulation streamfunction (Sv) for (a) MPI-ESM-LR, (b) MPI-ESM-HR, and (c) MPI-ESM-LR, each for the time mean differences over historical simulation and SSP5-8.5. Differences of the anomalies between the models: (d) ER minus LR, (e) ER minus HR, and (f) HR minus LR. The contours represent the historical mean (1995–2014; contour interval is 3 Sv). The contour colors denote solid red for the positive values, green line for the zero contour, and dash blue for the negative values. In all panels stippling indicates statistically significant differences (95% confidence level). (g) Time series of 3-yr running mean AMOC streamfunction (Sv) relative to 1950s at 1000 m and 26°N. Dashed lines represent a linear trend over the period 2030–99 for LR, HR, and ER, plotted in green, red, and blue, respectively. In the subpolar gyre, heat transport is driven by both gyre and overturning circulation. The strengthened barotropic circulation in high latitudes, where the northern part of the subpolar region and GIN Seas (Figs. 6a–c) can similarly contribute to the development of the warming hole and the associated sea level decline. Conversely, the AMOC effect is mitigated by a weaker strengthening of the circulation in vicinity of Greenland in ER (Figs. 6c,e) leading to a smaller DSL decrease and a smaller-scale warming hole in comparison to HR. The DSL decline and strengthening subpolar circulation are larger in LR than in HR and ER (Figs. 6d,f), despite the warming hole in LR being smaller (Fig. 2g and Fig. S1 in the online supplemental material). This could be because LR has a weaker GIN Seas circulation (Figs. 6a,d,f), which could indicate smaller heat transport out of the subpolar region. Future DSL change in the subpolar North Atlantic and the formation of the warming hole are hence resolution dependent. Figure 6: Anomalies of (a)–(c) barotropic streamfunction and (g)–(h) Sverdrup streamfunctions for the SSP5-8.5 (2080–99) average relative to the historical simulation, averaged over 1995–2014 in North Atlantic for MPI-ESM-LR in (a) and (g), MPI-ESM-HR in (b) and (h), and MPI-ESM-ER in (c) and (i); the differences of the anomalies between the models: (d) ER minus LR, (e) ER minus HR, and (f) HR minus LR. The contours represent the historical mean [1995–2014; contour intervals are 2 Sv in (a)–(c) and 5 Sv in (g)–(h)]. The contour colors denote solid red for the positive values, green line for the zero contour, and blue dashed for the negative values. The stippling indicates the statistically significant regions at the 95% confidence level. Curry et al. (1998), Böning et al. (2006), and Häkkinen et al. (2011) show the impact of the surface wind stress on both subpolar gyre variability and the strength. The decline of the surface wind stress over the subpolar gyre region could result in the spindown of the subpolar gyre circulation, leading to sea level increase (Chafik et al. 2019; Putrasahan et al. 2019). A noticeable difference between eddy-rich and lower resolutions is that wind stress reduces in ER over the subpolar North Atlantic (50°–65°N) in contrast to LR and HR (Figs. 3j,k), for which wind stress intensify (Fig. 4c). These changes agree with the noted spindown of the subpolar gyre in ER and the strengthened gyre circulation in LR and HR (Figs. 6a–c). However, the extensive spread in zonal mean zonal wind stress anomalies implies a minor resolution dependency in subpolar gyre region (Fig. 4c) at the 95% significant level.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-5	Figure 6: Anomalies of (a)–(c) barotropic streamfunction and (g)–(h) Sverdrup streamfunctions for the SSP5-8.5 (2080–99) average relative to the historical simulation, averaged over 1995–2014 in North Atlantic for MPI-ESM-LR in (a) and (g), MPI-ESM-HR in (b) and (h), and MPI-ESM-ER in (c) and (i); the differences of the anomalies between the models: (d) ER minus LR, (e) ER minus HR, and (f) HR minus LR. The contours represent the historical mean [1995–2014; contour intervals are 2 Sv in (a)–(c) and 5 Sv in (g)–(h)]. The contour colors denote solid red for the positive values, green line for the zero contour, and blue dashed for the negative values. The stippling indicates the statistically significant regions at the 95% confidence level. Curry et al. (1998), Böning et al. (2006), and Häkkinen et al. (2011) show the impact of the surface wind stress on both subpolar gyre variability and the strength. The decline of the surface wind stress over the subpolar gyre region could result in the spindown of the subpolar gyre circulation, leading to sea level increase (Chafik et al. 2019; Putrasahan et al. 2019). A noticeable difference between eddy-rich and lower resolutions is that wind stress reduces in ER over the subpolar North Atlantic (50°–65°N) in contrast to LR and HR (Figs. 3j,k), for which wind stress intensify (Fig. 4c). These changes agree with the noted spindown of the subpolar gyre in ER and the strengthened gyre circulation in LR and HR (Figs. 6a–c). However, the extensive spread in zonal mean zonal wind stress anomalies implies a minor resolution dependency in subpolar gyre region (Fig. 4c) at the 95% significant level. Dynamically, the simplest concept for the barotropic circulation is the balance between wind stress curl and the advection of vorticity which is described by the Sverdrup streamfunction. For the time-mean circulation the validity of the concept has been approved for interior of the subtropical circulation (e.g., Sonnewald et al. 2019; Wunsch and Roemmich 1985; Wunsch 2011). The degree of the Sverdrup concept’s validity to describe temporal changing circulation changes was previously demonstrated (e.g., Willebrand et al. 1980; Hautala et al. 1994; Stammer 1997; Morris et al. 1996, Thomas et al. 2014) and an agreement was found even outside of the subtropics. The Sverdrup streamfunction was calculated in the North Atlantic and North Pacific (Figs. 6, 9) to quantify the impact of the wind stress curl. The weakening of the North Atlantic Subpolar Gyre between 50° and 60°N can be explained by the weakening of surface wind stress curl in ER (Fig. 6i). Despite the strengthened circulation (Figs. 6a,b, 4b) and slightly intensified surface wind stress (Fig. 4c) in LR and HR, their curl weakens in both models (Figs. 6g,h). The changes in subpolar gyre circulation do not entirely concur with the Sverdrup dynamics in HR, although HR and ER have the same T127/L95 atmospheric component. All three resolutions show a weakening subtropical gyre (Figs. 6a–c), associated with a negative DSL, and just north of the Gulf Stream a band of positive DSL (Figs. 3a–c) that signifies the reduction of Gulf Stream transport associated with the weakening subtropical gyre and the declining AMOC (Fig. 5g). The weakening of the Gulf Stream under warming conditions during the twenty-first century (Yang et al. 2016) could be related to the sea level changes in the North Atlantic associated with higher sea level rise north of the Gulf Stream (Bouttes et al. 2014; Chen et al. 2019; Levermann et al. 2005; Yin et al. 2009). The argument requires that the DSL is lower than the global mean such that a relaxation locally leads to a DSL rise; the same applies to the barotropic streamfunction. ER shows the largest DSL reduction in the subtropical gyre (Figs. 3d,e), although a weakening of wind stress is smaller south of 40°N (Figs. 3j,k, 4c). However, the spindown of the subtropical gyre is larger in the ER compared to HR and LR (Figs. 6d,e, 4b). There is no considerable difference in the AMOC slowdown (Figs. 5e,g) between HR and ER. The AMOC deceleration is larger by 0.25 Sv in ER than in HR and 1.25 Sv than in LR (Fig. 5g) by 2100 at 26°N. The AMOC linear trend, calculated over 1995–2099, amounts to −0.061 Sv yr−1 for ER, −0.059 Sv yr−1 for HR, and −0.052 Sv yr−1 for LR. Despite the similar magnitude of AMOC decline at 26°N in HR and ER, spindown of the subtropical gyre and the DSL reduction is larger in ER than in HR. In addition to the effect of AMOC slowdown, changes in wind stress could also be responsible for the sea level reduction in the subtropical gyre south of the Gulf Stream (40°N), as suggested by Bouttes et al. (2012). The Sverdrup circulation of the North Atlantic Subtropical Gyre weakens dramatically in all three models. Moreover, the wind stress weakens south of 40°N in the zonal mean, with a stronger decline in HR and LR relative to the ER (Fig. 4c). The associated Sverdrup circulation is illustrated in Figs. 6g–i for comparison with the gyre changes as illustrated by the barotropic streamfunction (Figs. 6a–c, 4b). Additionally, the contours representing the present-day state are superimposed over the streamfunction anomalies to better visualize the shifting and changing of gyres. The wind stress curl changes tend to weaken the gyre circulation in the southern part of the North Atlantic Subtropical Gyre, whereas the northern part experiences a strengthening in all model versions (Figs. 6g–i). This suggests a poleward shift of the weakening subtropical gyre and Gulf Stream (Figs. 6a–c). North Pacific In contrast to the North Atlantic, in ER the DSL increases south of the Kuroshio Extension (over the subtropical gyre) and decreases farther to the north (in the subpolar gyre, Fig. 7a). This characteristic North Pacific dipole pattern is opposite to that in the North Atlantic and its axis is located along the steep DSL gradient associated with the Kuroshio causing further steepening of the frontal zone. In the southern part of the subpolar gyre, the DSL reduction is much greater in ER and HR than in LR (Figs. 7d,f), though there is no substantial difference between HR and ER (Fig. 7e). The DSL decreases in the lower resolutions along the eastern boundary of the North Pacific (Figs. 7a,b), while ER shows an increase in the Gulf of Alaska (Figs. 7c). The discrepancies between the models are negligible in the northern part of the subpolar gyre (Figs. 7d–f). In the subtropical gyre, the DSL increase is larger in ER compared to the LR and HR (Figs. 7d,e), and in HR than in LR (Fig. 7f), along with a significant increase east of Japan in the eddy-rich model (Fig. 8a). Figure 7: As in Fig. 3, but for the North Pacific. Figure 8: As in Fig. 4, but for the North Pacific.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-6	North Pacific In contrast to the North Atlantic, in ER the DSL increases south of the Kuroshio Extension (over the subtropical gyre) and decreases farther to the north (in the subpolar gyre, Fig. 7a). This characteristic North Pacific dipole pattern is opposite to that in the North Atlantic and its axis is located along the steep DSL gradient associated with the Kuroshio causing further steepening of the frontal zone. In the southern part of the subpolar gyre, the DSL reduction is much greater in ER and HR than in LR (Figs. 7d,f), though there is no substantial difference between HR and ER (Fig. 7e). The DSL decreases in the lower resolutions along the eastern boundary of the North Pacific (Figs. 7a,b), while ER shows an increase in the Gulf of Alaska (Figs. 7c). The discrepancies between the models are negligible in the northern part of the subpolar gyre (Figs. 7d–f). In the subtropical gyre, the DSL increase is larger in ER compared to the LR and HR (Figs. 7d,e), and in HR than in LR (Fig. 7f), along with a significant increase east of Japan in the eddy-rich model (Fig. 8a). Figure 7: As in Fig. 3, but for the North Pacific. Figure 8: As in Fig. 4, but for the North Pacific. In HR and ER, the subpolar gyre strengthens (Figs. 9a,b), but in LR, the barotropic streamfunction field exhibits only minor change (Fig. 9c). Although the subpolar gyre is stronger in ER than in LR, the differences in subpolar gyre circulation between the models are relatively insignificant (Figs. 9d–f). The changes in the subtropical gyre indicate a pattern with a positive north and negative south parts of the gyre (Figs. 9a–c). The northern part of subtropical gyre (the Kuroshio region) strengthens; here ER displays a considerable amplification (30°–35°N; Fig. 8b). In the same region, we also noticed the high rate of DSL changes (Fig. 8a). Although the streamfunction differences are significant between the eddy-rich and eddy parameterized models in the northern part of the subtropical gyre (Figs. 9d,e), weakening southern part shows minor changes (Figs. 9d,e). The differences between LR and HR are significant (insignificant) in the southern (northern) part of the subtropical gyre circulation (Fig. 9f). Figure 9: As in Fig. 6, but for the North Pacific. In comparison to the DSL change, changes of the barotropic circulation (Figs. 9a–c) show some distinctively different patterns in the tropical regions. Although, the DSL changes across the Kuroshio Extension front are still reflected by similar albeit much weaker streamfunction changes, the strengthening of the subtropical gyre encompasses only the region 30°–40°N, while for DSL it reaches down to 20°N. The negative DSL change south of 20°N covers the entire tropical region, while for the streamfunction it reaches down to only 15°N, where the pattern continues southward by a positive signal. Streamfunction and DSL are therefore inconsistent in the region north of the equator until 30°N, a region where the Pacific and the Indian Ocean are connected by the Indonesian Throughflow passages (Zhang et al. 2014). Because of the absence of deep-water formation and deep convection in the North Pacific, similar changes seen in CMIP5 models were mainly attributed to changes in the wind field (Sakamoto et al. 2005; Yin et al. 2010). Merrifield (2011) has shown the relevance of off-equatorial wind changes for explaining the features of the large observed sea level trend in the western tropical Pacific during the 1990s and early 2000s. Figure 8c depicts the zonally averaged zonal wind stress changes for three models, showing an increase north of about 38°–45°N and a decrease to the south. Although differences in wind stress change between the models are not significant at the 95% confidence interval of the multimember mean, ER shows less weakening in the subtropical and more strengthening in the subpolar in comparison to HR and LR (Figs. 7j,k, 8c). Changes in wind stress curl were found to explain the intensification of the subtropical gyre in the South Pacific since the early 1990s (Roemmich et al. 2007, 2016), which were also shown by Köhl and Stammer (2008) to explain the sea level trends during the longer time scale 1960–2001 over much of the Pacific Ocean. We will therefore again examine the wind-driven component of the circulation by the Sverdrup streamfunction (Figs. 9g–i). Consistent with the barotropic streamfunction, the Sverdrup circulation changes show that the southern part of the North Pacific Subtropical Gyre weakens in all models, while the northern part of the gyre (north of 30°N) indicates a strengthening (Figs. 9g–i). Cheon et al. (2012) argue that this pattern (positive north part and negative south part) observed in the subtropical gyre indicates a poleward shift rather than a strengthening of the gyre under a warming future. Yin et al. (2010) further corroborate this hypothesis of the subtropical gyre poleward shift, resulting from the poleward shift of subtropical high in the western Pacific and the associated wind system. These characteristic patterns of DSL change in the western North Pacific because of the poleward shift and intensification of the Kuroshio have also been studied earlier in earlier CMIP models (e.g., Church et al. 2013a,b; Fox-Kemper et al. 2021; Sueyoshi and Yasuda 2012; Suzuki and Tatebe 2020; Terada and Minobe 2018; Yin 2012; Zhang et al. 2014). However, whether this pattern in the Sverdrup and barotropic streamfunction indicates a poleward shift of the subtropical gyre or a strengthening of the Kuroshio is debatable (see section 4d). Different from the other resolutions, in LR, Sverdrup circulation weakens in the southern part of the North Pacific Subpolar Gyre, while the northern part shows a strengthening (Fig. 9g). It significantly strengthens in ER, and crosses the present-day zero contour (Fig. 9i), while HR shows no considerable change (Fig. 9h). Previous studies have emphasized that increased model resolution is necessary for the representation of accurate western boundary currents such as Gulf Stream, Kuroshio, and East Australian Currents (e.g., Chassignet and Xu 2017; Chassignet et al. 2020; Griffies et al. 2015; Hewitt et al. 2017, 2020; Roberts et al. 2018; Small et al. 2014). Similarly, Hurlburt et al. (1996) and Nishikawa et al. (2020) demonstrate that eddy-rich horizontal resolution can realistically represent the Oyashio–Kuroshio fronts. Therefore, the strong, narrow current, noted in the DSL and barotropic streamfunction fields (Figs. 7, 9) east of Japan, denotes the more accurately represented Kuroshio in the eddy-rich model, which alters the characteristics of the Kuroshio–Oyashio front by representing a significant number of mesoscale activities in comparison to eddy parameterized models. The Kuroshio, between 30° and 35°N, is significantly intensified in the ER model but slightly strengthens in the high-resolution and the low-resolution climate models. The Southern Ocean Like in the North Atlantic and North Pacific, a north–south gradient of DSL change is found in the Southern Ocean for all resolutions (Figs. 10a–c), with increasing sea level north to ∼50°S and decreasing south of ∼50°S, which has been described as a belt-like pattern (Yin et al. 2010). The increase (decrease) of DSL north (south) of the ACC is smaller in ER than in HR and LR at the end of the twenty-first century (Figs. 11a,b, 12a). In earlier studies, the strengthening and poleward shift of Southern Hemisphere westerlies have been shown to induce such a pattern of DSL changes, although it was also noted that it is not sufficient to explain all of the projected changes (Thompson and Solomon 2002; Bouttes et al. 2012; Frankcombe et al. 2013).	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-7	Previous studies have emphasized that increased model resolution is necessary for the representation of accurate western boundary currents such as Gulf Stream, Kuroshio, and East Australian Currents (e.g., Chassignet and Xu 2017; Chassignet et al. 2020; Griffies et al. 2015; Hewitt et al. 2017, 2020; Roberts et al. 2018; Small et al. 2014). Similarly, Hurlburt et al. (1996) and Nishikawa et al. (2020) demonstrate that eddy-rich horizontal resolution can realistically represent the Oyashio–Kuroshio fronts. Therefore, the strong, narrow current, noted in the DSL and barotropic streamfunction fields (Figs. 7, 9) east of Japan, denotes the more accurately represented Kuroshio in the eddy-rich model, which alters the characteristics of the Kuroshio–Oyashio front by representing a significant number of mesoscale activities in comparison to eddy parameterized models. The Kuroshio, between 30° and 35°N, is significantly intensified in the ER model but slightly strengthens in the high-resolution and the low-resolution climate models. The Southern Ocean Like in the North Atlantic and North Pacific, a north–south gradient of DSL change is found in the Southern Ocean for all resolutions (Figs. 10a–c), with increasing sea level north to ∼50°S and decreasing south of ∼50°S, which has been described as a belt-like pattern (Yin et al. 2010). The increase (decrease) of DSL north (south) of the ACC is smaller in ER than in HR and LR at the end of the twenty-first century (Figs. 11a,b, 12a). In earlier studies, the strengthening and poleward shift of Southern Hemisphere westerlies have been shown to induce such a pattern of DSL changes, although it was also noted that it is not sufficient to explain all of the projected changes (Thompson and Solomon 2002; Bouttes et al. 2012; Frankcombe et al. 2013). Figure 10: Anomalies of (a)–(c) dynamic sea level, (d)–(f) barotropic streamfunction, and (g)–(i) wind stress for the SSP5-8.5 (2080–99) average relative to the historical simulation, averaged over 1995–2014 in Southern Ocean for (left) MPI-ESM-LR, (center) MPI-ESM-HR, and (right) MPI-ESM-ER. The contours represent the historical mean [1995–2014; contour intervals are 0.1 m in (a)–(c) and 2 Sv in (d)–(f)]. The contour colors denote solid red for the positive values, green line for the zero contour, and blue dashed for the negative values. The stippling indicates the statistically significant regions at the 95% confidence level. In (a), W stands for Weddell Sea and R stands for Ross Sea. Figure 11: Differences of the anomalies between the models [(left) ER minus LR, (center) ER minus HR, and (right) HR minus LR] for (a)–(c) dynamic sea level, (d)–(f) barotropic streamfunction, and (g)–(i) wind stress in Southern Ocean. The stippling indicates the statistically significant regions at the 95% confidence level. Figure 12: As in Fig. 4, but for the Southern Ocean. The projected wind stress change shows a decrease north of the ACC, with a peak around 38°S, and an increase to the south centered around 58°S (Figs. 10g–i, 12c). This dipole-type pattern in the zonal component of the wind stress is interpreted by Fyfe and Saenko (2006) as the strengthening and poleward shift. The differences in wind stress between ER and HR are minor (Figs. 11h, 12c). The changes in LR are considerably larger than both ER and HR (Figs. 11g,i, 12c). The pattern of DSL reflects circulation changes characterized by similar patterns of barotropic streamfunction change (Figs. 10d–f). The intriguing feature of the projected circulation change is an intensifying region centered around 45°S (Fig. 12b). This strengthening could be caused by a potential southerly shift of subtropical gyres, and as horizontal resolution improves, the magnitude of the strengthening decreases. The poleward shift in sea surface height contours is consistent with regional sea level rise patterns (Gille 2014). Therefore, understanding gyre shift is crucial for sea level change studies. The historical mean contours overlaid over the anomalies can further explain this poleward movement (Figs. 10a–e). Positive dynamic sea level anomalies and negative anomalies in the barotropic streamfunction field both cross the present-day zero contour, which indicates a poleward shift. This poleward shift of subtropical gyres, interpreted as the belt-like pattern, is less pronounced in ER than in HR and LR (Figs. 10d,e). Similarly, ER reflects a thinner belt of DSL increase compared to HR and LR (Figs. 10a,b). A dipole-like pattern of changes in the southern Indian Ocean and South Atlantic Ocean is revealed by changes in streamfunction for the Southern Ocean (Figs. 10d–f). The southern (northern) part of the Indian Ocean Gyre is shown to be strengthening (weakening) by around 18 Sv (10 Sv) in the ER. The southern part of the South Atlantic Gyre is also strengthening by about 13 Sv (Fig. 10f). However, the South Pacific Gyre weakens in all the projections by about 6 Sv. When the changes in subpolar gyres are taken into account, we found that all projections show a weakening Weddell Gyre (Fig. 10d–f), which is more pronounced in HR than in LR and ER (Figs. 11e,f). With a larger acceleration in ER (Fig. 11d), the Ross Gyre strengthens in LR and ER (Figs. 10d,f), whereas it weakens in HR (Fig. 10e). Eddies are omnipresent in the Southern Ocean, especially along the ACC (e.g., Constantinou and Hogg 2019; Frenger et al. 2015). These eddies are crucial for establishing the stratification in the presence of wind and buoyancy forcing (Karsten et al. 2002). The intensified Southern Hemisphere westerlies enhance the Southern Ocean eddy activity, leading to the phenomenon known as eddy saturation (Straub 1993) and eddy compensation. As a result, the strength of the ACC, the isopycnal slope, and the meridional circulation of the Southern Ocean become less sensitive to the enhanced wind forcing. Studies using higher-resolution ocean models (e.g., Farneti et al. 2010; Hallberg and Gnanadesikan 2006; Meredith and Hogg 2006) or observations (Böning et al. 2008; Chidichimo et al. 2014; Firing et al. 2011) have shown the insensitivity of ACC or the Southern Ocean meridional overturning circulation to the enhanced westerlies. They stated that the non-eddy-resolving models respond with an accelerated ACC, steeper isopycnals, and robust meridional overturning circulation to wind intensification forcing changes. To evaluate the response of ACC to the intensified westerly wind stress, we investigated the Drake Passage transport independently for the two time periods (Fig. 13a). Even though the studies cited above oppose the ACC’s sensitivity to changing westerly winds, we discovered an accelerating ACC particularly in our eddy-rich model (Fig. 13b). With a substantial increase in ER, the strength of the ACC increases in ER and LR (Fig. 13a). The LR reveals increased transport between 60° and 68°S, with no changes north of 60°S. The transport in HR remains unchanged until 64°S, south of which it begins to weaken (Fig. 13a), reflecting its insensitivities to accelerated southern westerlies. Furthermore, Shi et al. (2021) and Swart et al. (2018) show that factors other than wind influence the Southern Ocean circulation. Warming in the upper ocean generates a density change, accelerating the ACC. Heat and freshwater fluxes at the surface could also cause changes observed in ER. Figure 13: (a) Time mean transport through Drake Passage; the dashed line for the time mean future (2080–99) transport and the solid line indicates the historical (1995–2014) transport (Sv). (b) Time series of ACC transport through Drake Passage relative to 1950s mean at 65°W. The green, red, and blue lines represent MPI-ESM-LR, MPI-ESM-HR, and MPI-ESM-ER, respectively. The positive barotropic streamfunction values were considered for the calculation.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-8	To evaluate the response of ACC to the intensified westerly wind stress, we investigated the Drake Passage transport independently for the two time periods (Fig. 13a). Even though the studies cited above oppose the ACC’s sensitivity to changing westerly winds, we discovered an accelerating ACC particularly in our eddy-rich model (Fig. 13b). With a substantial increase in ER, the strength of the ACC increases in ER and LR (Fig. 13a). The LR reveals increased transport between 60° and 68°S, with no changes north of 60°S. The transport in HR remains unchanged until 64°S, south of which it begins to weaken (Fig. 13a), reflecting its insensitivities to accelerated southern westerlies. Furthermore, Shi et al. (2021) and Swart et al. (2018) show that factors other than wind influence the Southern Ocean circulation. Warming in the upper ocean generates a density change, accelerating the ACC. Heat and freshwater fluxes at the surface could also cause changes observed in ER. Figure 13: (a) Time mean transport through Drake Passage; the dashed line for the time mean future (2080–99) transport and the solid line indicates the historical (1995–2014) transport (Sv). (b) Time series of ACC transport through Drake Passage relative to 1950s mean at 65°W. The green, red, and blue lines represent MPI-ESM-LR, MPI-ESM-HR, and MPI-ESM-ER, respectively. The positive barotropic streamfunction values were considered for the calculation. The stronger DSL increase relative to HR and LR is significant in ER in the South Pacific (25°–60°S), especially east of Australia (Figs. 11a,b). The East Australian Current system strengthens as DSL rises in the ER model (Figs. 11d,e). This intensified circulation represented by the barotropic streamfunction field is more significant in ER than in HR and LR. Lower-resolution models poorly simulate the western boundary current system due to unresolved mesoscale processes, whereas the East Australian Current is adequately simulated in ER. Increasing southward transport of the East Australia Current in a warming climate was shown to be a response to the intensified South Pacific wind stress curl (Goyal et al. 2021; Roemmich et al. 2007), which can be seen in ER. We also found the enormously increased sea surface temperature in southeastern Australia in ER as indicated in previous studies (Wu et al. 2012; Hobday and Pecl 2014), which causes this region to be a global warming hotspot. The DSL decreases in southern Indian Ocean and the Pacific (north of 30°S), though it increases in the Atlantic (Figs. 10a–c). The increase in the South Atlantic is larger in LR than in HR as well as ER (Figs. 11a,c). In the south Indian Ocean, the DSL decreases, in order of decreasing the reduction, in HR, LR, and ER (Figs. 11a–c), similarly in the South Pacific. Displacement of major ocean gyres The poleward shift of major ocean gyres, which secondarily drives sea level change, was previously discussed by Yang et al. (2020). To comprehend how differences in poleward shift affect different DSL responses, we have calculated the change in position of major ocean gyres in the three versions of MPI-ESM and quantified the linear trend from 2030 to 2100 in Fig. 14. Most gyres shift toward the poles (except for the North Pacific), indicating a statistically significant poleward gyre displacement as a response to a warmer climate. Figure 14: Time series of 3 yr running mean latitudinal variations of the ocean gyres relative to the 1950s mean (°). The position is calculated based on the barotropic streamfunction weighted center of each gyre, and the contours between the minimum and maximum eastward transport in the Drake passage were considered to calculate the mean central latitude of ACC transport. Abbreviations in each panel are North Atlantic Subpolar Gyre (NASPG), North Atlantic Subtropical Gyre (NASTG), North Pacific Subpolar Gyre (NPSPG), North Pacific Subtropical Gyre (NPSTG), and Antarctic Circumpolar Current (ACC). The linear trend is indicated by the dashed lines between 2030 and 2099 for LR, HR, and ER in green, red, and blue, respectively. The related linear trends are presented in the box (° yr−1). The poleward shift of the North Atlantic Subtropical Gyre is almost identical in all projections, showing a magnitude of 0.0138°, 0.0112°, and 0.0161° per year in ER, HR, and LR, respectively (Fig. 14b). The North Atlantic Subpolar Gyre, on the other hand, responds differently in each projection, with the highest displacement in ER and trends of 0.0129°, 0.0125°, and 0.0085° per year in ER, HR, and LR, respectively (Fig. 14a), from 2030 to 2099. Periods over which trends occur are also not very consistent. While HR shows no trend until the last few decades, poleward trends start in the early to mid-twenty-first century for LR and ER, respectively, but it seems to cease after 2070 for ER pointing to considerable influence of climate variability. The North Pacific Subpolar Gyre behaves contrarily to the North Atlantic in ER, which shows a southward shift, indicated by a negative trend of 0.0088° per year, while northward shifts are small with 0.0067 per year in HR and 0.0011° per year in LR (Fig. 14c). Similarly, North Pacific Subtropical Gyre displacements are also relatively small in HR and LR, with linear trends of 0.0028° and 0.0010° yr−1, respectively (Fig. 14d). Both poleward shifts of the ocean gyres in the North Pacific are not statistically significant in LR, and in ER, the subtropical gyre is experiencing a statistically insignificant downward trend of 0.0026° year−1. The displacements of the North Pacific Subpolar and Subtropical Gyres are negligible when compared to interannual variability in all configurations (Figs. 14c,d). Consistent with the changes of the barotropic streamfunction (Figs. 9c), the boundary of the negative circulation anomaly crossing the zero contour of present day (solid green line) in the Sverdrup circulation implies that the North Pacific Subpolar Gyre strengthens and moves southward in ER (Fig. 9i). Thus, ER shows a strengthening Kuroshio due to the stronger wind rather than a poleward displacement (Fig. 14d), despite the interpretation of earlier CMIP5 model investigations that the changes in the Sverdrup streamfunction indicates a northward movement of the subtropical gyre (Cheon et al. 2012). It is worth mentioning that the latitudinal variations observed in the North Atlantic Subtropical Gyre are consistent with a pattern of positive north and negative south parts of the gyre in the Sverdrup streamfunction field (Figs. 6g–i) and barotropic circulation (Figs. 6a–c). Our results indicate a distinct poleward shift of the ACC (Fig. 14e), which has been linked to climate change in many earlier studies (e.g., Morrow et al. 2008; Yang et al. 2020). Interestingly, the eddy-rich model shows less poleward migration of the ACC with induced transport increase in comparison to the high- and low-resolution models. Between 2030 and 2100, the HR has the highest trend of 0.0044° yr−1, whereas the LR has the lowest trend of 0.0024° yr−1, while ER’s linear trend lies with 0.0028° yr−1 in between. In ER, the latitudinal displacement of the ACC is less than in HR and LR (Fig. 14e), and we also observed the smallest poleward shift of the ACC in the barotropic streamfunction (Figs. 11c,d) and sea surface height field (Figs. 11a,b).	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-9	The displacements of the North Pacific Subpolar and Subtropical Gyres are negligible when compared to interannual variability in all configurations (Figs. 14c,d). Consistent with the changes of the barotropic streamfunction (Figs. 9c), the boundary of the negative circulation anomaly crossing the zero contour of present day (solid green line) in the Sverdrup circulation implies that the North Pacific Subpolar Gyre strengthens and moves southward in ER (Fig. 9i). Thus, ER shows a strengthening Kuroshio due to the stronger wind rather than a poleward displacement (Fig. 14d), despite the interpretation of earlier CMIP5 model investigations that the changes in the Sverdrup streamfunction indicates a northward movement of the subtropical gyre (Cheon et al. 2012). It is worth mentioning that the latitudinal variations observed in the North Atlantic Subtropical Gyre are consistent with a pattern of positive north and negative south parts of the gyre in the Sverdrup streamfunction field (Figs. 6g–i) and barotropic circulation (Figs. 6a–c). Our results indicate a distinct poleward shift of the ACC (Fig. 14e), which has been linked to climate change in many earlier studies (e.g., Morrow et al. 2008; Yang et al. 2020). Interestingly, the eddy-rich model shows less poleward migration of the ACC with induced transport increase in comparison to the high- and low-resolution models. Between 2030 and 2100, the HR has the highest trend of 0.0044° yr−1, whereas the LR has the lowest trend of 0.0024° yr−1, while ER’s linear trend lies with 0.0028° yr−1 in between. In ER, the latitudinal displacement of the ACC is less than in HR and LR (Fig. 14e), and we also observed the smallest poleward shift of the ACC in the barotropic streamfunction (Figs. 11c,d) and sea surface height field (Figs. 11a,b). The differences between the anomalies seen in the Arctic Ocean are not thoroughly covered in a separate section. Thus, the discrepancies found are outlined here at the end of the section 4. We noted a dipole pattern of the difference in the DSL changes between ER and the lower-resolution models in the Arctic Ocean (Fig. S5). The ER reveals no changes to the north of Greenland, where we diagnose strong sea level increase in HR and LR (Fig. 2c and Fig. S2i). As expected, the model differences of the freshwater content change show a similar behavior to the differences of sea level change (Fig. S6). Although the Beaufort Gyre weakens (Figs. 2d,e and Figs. S3d,e) at the end of the twenty-first century, our models, HR and LR, show a DSL increase in the Canada Basin (Figs. 2a,b) associated with freshwater accumulation (Fig. S7). Furthermore, we do not find any considerable changes in the wind stress field, except for its increase in the Chukchi Sea in ER (Fig. 2j and Fig. S4). The induced anticyclonic circulation in ER (Fig. 2f) causes the increase in DSL (Fig. 2c) and freshwater content (Fig. S7i) in the Canada Basin. Summary and concluding remarks By analyzing model simulations from eddy-rich (ER), high-resolution (HR), and low-resolution (LR) versions of MPI-ESM run under the SSP5-8.5 scenario forcing, we found substantial DSL change differences in the global ocean among the different resolutions. We note that HR and ER have the same atmospheric component, whereas LR also has a lower-resolution atmosphere. However, when comparing LR and HR, we cannot fully discriminate between the effects of resolution changes versus intrinsic changes of surface forcing in either simulation. This is because the response to changes in external forcing is a complex coupled phenomenon that depends on details of the surface boundary condition formulation. Because of this, even ocean models coupled to the same atmosphere (Semmler et al. 2021) show different regional or global expressions of such change as soon as ocean surface fields are different. Nevertheless, in many cases we can link the responses in sea level change to the different ocean resolution, in particular comparing the HR and ER versions of the model. All models simulate a meridional dipole pattern of sea level change in the North Atlantic. This dipole pattern is identified by a larger sea level rise relative to the global mean in the subpolar region and sea level decrease in the subtropical region. However, southeast of Greenland, we identify a patch of sea level decline, which shrinks with the enhanced horizontal resolution. The warming hole feature is also located in the same area as the sea level decline, indicating a similar pattern of behavior, particularly in HR and ER. We have mainly focused on the changes in the circulation to examine the causes of these differences, since the long-term changes in sea level are linked to the changes in the circulation. In HR and LR, the subpolar gyre strengthens in the barotropic streamfunction field, although its component driven by the wind stress curl weakens. The strengthening circulation in the subpolar region, on the one hand, can induce the heat transport out of the region into the GIN Seas and then farther north to the Arctic, forming NA warming hole. However, in this region the changes in wind stress curl in the subpolar region do not agree with the change in circulation in the eddy permitting models. In the ER model, the subpolar gyre weakens in both barotropic and Sverdrup fields, as well as the region of the sea level decline becomes small. Furthermore, the magnitude of the sea level decline is also smaller in ER compared to HR and LR. The sea level increase over the rest of the subpolar gyre has been simulated by all the models. This increase becomes smaller in the eddy-rich model, indicating lower sea level in the Labrador Sea and north of Gulf Stream. This lower increase is caused by lower freshwater content in ER than in HR and LR. Previous studies have widely reported the relation between the uncertainty in twenty-first century sea level change in the North Atlantic and the variability in projected weakening of AMOC (Yin et al. 2009; Bouttes et al. 2014). We observed a significant AMOC decline in all the versions. This decrease is very similar in HR and ER at 26°N. However, the DSL changes in the North Atlantic are considerably different between these models. These results indicate that the North Atlantic DSL change is not responding as we anticipated to the weakening AMOC. The sea level decline in the subtropical region is larger in ER compared to HR and LR, caused by the larger weakening of the circulation in ER. The poleward shift of North Atlantic Subtropical Gyre, which is also observed in the Sverdrup field as a pattern of positive north and negative south parts of the gyre, is considerable in all the models. However, the differences between models are not significant. Interestingly, a pattern of positive north and negative south parts of the gyre identified in the Sverdrup streamfunction most likely indicates the strengthening of Kuroshio rather than the poleward shift of the subtropical gyre in the ER simulation, because the gyres in the North Pacific show a negligible poleward displacement. It is further explained by identifying a significant DSL change, robust circulation, and less reduction of wind stress in the Kuroshio region (30°–35°N). These robust changes are identified only in the eddy-rich model, because of the realistically represented western boundary currents. It is well known that in the Southern Ocean changes in sea level correspond to changes in ACC and barotropic circulation (e.g., van Westen and Dijkstra 2021). Southern Ocean sea level change is smaller in ER, indicating a minor poleward shift of the ACC in comparison to HR and LR. The ACC, as in earlier studies, is known to be insensitive to the strengthening westerlies in the higher-resolution models that explicitly resolve eddies. One interesting result is that ACC strengthens in ER, but remains unchanged or slightly weakened in HR. The findings indicate that ER, as opposed to HR, appears to be more sensitive to strong westerlies. The general understanding is that many low-lying coastal areas experience substantial threats from sea level rise due to their relatively low elevation above sea level. Figure 1 by Magnan et al. (2022) provides a detailed overview of the low-lying islands and coasts of the world. However, we have not included in-depth discussion about low lying coastal areas that are located in the Indian Ocean and southwest Pacific due to small model differences of DSL anomalies (maximum around ±40 mm; see supplemental Figs. S8–S13). Interesting to note is that the differences of DSL change between HR and ER is significant in these regions and larger than the difference between LR and ER.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
bc7116e222f0-10	The poleward shift of North Atlantic Subtropical Gyre, which is also observed in the Sverdrup field as a pattern of positive north and negative south parts of the gyre, is considerable in all the models. However, the differences between models are not significant. Interestingly, a pattern of positive north and negative south parts of the gyre identified in the Sverdrup streamfunction most likely indicates the strengthening of Kuroshio rather than the poleward shift of the subtropical gyre in the ER simulation, because the gyres in the North Pacific show a negligible poleward displacement. It is further explained by identifying a significant DSL change, robust circulation, and less reduction of wind stress in the Kuroshio region (30°–35°N). These robust changes are identified only in the eddy-rich model, because of the realistically represented western boundary currents. It is well known that in the Southern Ocean changes in sea level correspond to changes in ACC and barotropic circulation (e.g., van Westen and Dijkstra 2021). Southern Ocean sea level change is smaller in ER, indicating a minor poleward shift of the ACC in comparison to HR and LR. The ACC, as in earlier studies, is known to be insensitive to the strengthening westerlies in the higher-resolution models that explicitly resolve eddies. One interesting result is that ACC strengthens in ER, but remains unchanged or slightly weakened in HR. The findings indicate that ER, as opposed to HR, appears to be more sensitive to strong westerlies. The general understanding is that many low-lying coastal areas experience substantial threats from sea level rise due to their relatively low elevation above sea level. Figure 1 by Magnan et al. (2022) provides a detailed overview of the low-lying islands and coasts of the world. However, we have not included in-depth discussion about low lying coastal areas that are located in the Indian Ocean and southwest Pacific due to small model differences of DSL anomalies (maximum around ±40 mm; see supplemental Figs. S8–S13). Interesting to note is that the differences of DSL change between HR and ER is significant in these regions and larger than the difference between LR and ER. It is expected that the eddy-rich models improve the representation of the eddy activities, providing more accurate and informative sea level change patterns over the following decades. In general, the DSL change pattern and dynamics are similar in eddy-rich compared to the coarser-resolution climate models in each ocean basin, suggesting that the coarser-resolution models will remain valid in understanding the sea level change patterns. On the other hand, the detailed, quantitative responses depend on the resolution. The robust changes found in MPI-ESM-ER suggest that improved resolution will have an impact on the interpretation of regional sea level change in the following decades. Therefore, the sea level projections of coarse-resolution models should be interpreted with caution, predominantly in the eddy active regions such as Kuroshio, ACC, Gulf Stream, and East Australian Current, and one should consider restrictions associated with limited climate model horizontal resolutions, when planning future adaptation and mitigation investments.	https://sealeveldocs.readthedocs.io/en/latest/wickramage23.html
a9cea5dae349-0	DeConto et al. (2021) Title: The Paris Climate Agreement and future sea-level rise from Antarctica Corresponding author: Robert M. DeConto Citation: DeConto, R. M., Pollard, D., Alley, R. B., Velicogna, I., Gasson, E., Gomez, N., et al. (2021). The Paris Climate Agreement and future sea-level rise from Antarctica. Nature, 593(7857), 83–89. doi: 10.1038/s41586-021-03427-0 URL: https://www.nature.com/articles/s41586-021-03427-0 Abstract The Paris Agreement aims to limit global mean warming in the twenty-first century to less than 2 degrees Celsius above preindustrial levels, and to promote further efforts to limit warming to 1.5 degrees Celsius [1]. The amount of greenhouse gas emissions in coming decades will be consequential for global mean sea level (GMSL) on century and longer timescales through a combination of ocean thermal expansion and loss of land ice [2]. The Antarctic Ice Sheet (AIS) is Earth’s largest land ice reservoir (equivalent to 57.9 metres of GMSL) [3], and its ice loss is accelerating [4]. Extensive regions of the AIS are grounded below sea level and susceptible to dynamical instabilities [5,6,7,8] that are capable of producing very rapid retreat [8]. Yet the potential for the implementation of the Paris Agreement temperature targets to slow or stop the onset of these instabilities has not been directly tested with physics-based models. Here we use an observationally calibrated ice sheet–shelf model to show that with global warming limited to 2 degrees Celsius or less, Antarctic ice loss will continue at a pace similar to today’s throughout the twenty-first century. However, scenarios more consistent with current policies (allowing 3 degrees Celsius of warming) give an abrupt jump in the pace of Antarctic ice loss after around 2060, contributing about 0.5 centimetres GMSL rise per year by 2100—an order of magnitude faster than today [4]. More fossil-fuel-intensive scenarios [9] result in even greater acceleration. Ice-sheet retreat initiated by the thinning and loss of buttressing ice shelves continues for centuries, regardless of bedrock and sea-level feedback mechanisms [10,11,12] or geoengineered carbon dioxide reduction. These results demonstrate the possibility that rapid and unstoppable sea-level rise from Antarctica will be triggered if Paris Agreement targets are exceeded. Introduction Most of the AIS terminates in the ocean, with massive ice shelves (floating extensions of glacial ice) providing resistance (buttressing) to the seaward flow of the grounded ice upstream [13]. About a third of the AIS rests on bedrock hundreds to thousands of metres below sea level [3], and in places where subglacial bedrock slopes downwards away from the ocean (reverse-sloped), the ice margin is susceptible to a marine ice-sheet instability (MISI) [5,6] and possibly a marine ice-cliff instability (MICI) [7,8]. The West Antarctic Ice Sheet (WAIS), which has the potential to cause about 5 m of GMSL rise, is particularly vulnerable. The WAIS is losing ice faster than other Antarctic sectors [4] and it sits in a deep basin >2.5 km below sea level in places [3]. Marine ice instabilities MISI and MICI can be triggered by the thinning or loss of buttressing ice shelves in response to a warming ocean, atmosphere or both [14]. MISI is related to a self-sustaining positive feedback between seaward ice flux across the grounding line (the boundary between grounded and floating ice) and ice thickness [5,6]. If buttressing is lost and retreat is initiated on a reverse-sloped bed, the retreating grounding line will encounter thicker ice, strongly increasing ice flow. Retreat will continue until the grounding line reaches forward-sloping bedrock, or sufficient resistive stress is restored by the regrowth of a buttressing ice shelf that is confined within coastal embayments or is thick enough to ‘pin’ on shallow bedrock features. Grounding lines on reverse-sloped bedrock are conditionally unstable15 and retreat at a rate determined by the complex interplay between ice flow and stress fields, bedrock conditions, surface mass balance and other factors that make model-ling these dynamics difficult. MICI is also theorized to be triggered where buttressing ice shelves disappear or become too small to provide substantial back stress [7,8]. At unsupported grounding lines where ice thickness exceeds a critical value, the weight of ice above sea level can produce deviatoric stresses that exceed the material yield strength of the ice. This causes structural failure [16,17], possibly manifest as repeated slumping and calving events17. Once initiated, failure can continue until the collapsing ice front backs into shallow water, where subaerial cliff heights and the associated stresses drop below their critical values, viscous deformation lowers the cliff, or sufficient buttressing is restored by an ice shelf. In undamaged ice, with small grain sizes and without large bubbles or pre-existing weaknesses, slowly emerging subaerial ice cliffs could exceed 500 m in height before failing [18]. However, natural glacial ice is typically damaged, especially near crevassed calving fronts and in fast-flowing ice upstream [19]. Assuming ice properties representative of glaciers, stress-balance calculations [16] point to maximum sustainable cliff heights of around 200 m. This value is reduced to about 100 m or less [8,16] where deep surface and basal crevasses effectively thin the supportive ice column (increasing the stress), possibly explaining why the tallest ice cliffs observed today are about 100 m tall. Recent modelling [18] using values of fracture toughness and pre-existing flaw size appropriate for damaged ice fronts [17] and consistent with field observations [19] indicates that tensile fracturing can occur at cliffs as low as 60 m. This reinforces the argument for including ice-cliff calving in ice-sheet models [20], despite current uncertainties in ice properties and the lack of observations which make ice-cliff calving laws difficult to formulate. Thick, marine-terminating glaciers such as Jakobshavn Isbræ in Greenland demonstrate how efficient calving can deliver ice to the ocean. The terminus of Jakobshavn is about 10 km wide, 1,000 m thick and flows seawards at about 12 km yr−1 (ref. 21). Since the glacier lost its ice shelf in the late 1990s, the ice front (with an intermittent ~100-m ice cliff) has retreated >12 km into the thicker ice upstream, albeit with a recent re-advance coincident with regional ocean cooling [22]. The average effective calving rate (flow speed + retreat) between 2002 and 2015 is estimated at 13.2 ± 0.9 km yr−1 (1σ; ref. 21). Calving in narrow fjord settings such as Jakobshavn is controlled by a complex combination of ductile and brittle processes, as well as buoyancy. After calving, subsequent fracture-driven failure is delayed until accelerated flow thins the terminus to near-flotation, allowing tidal flexure, basal crevassing, slumping or other processes to initiate the next event17,23. Resistive stresses from lateral shear along fjord walls, as well as thick mélange strengthened by sea ice, slow calving in winter, but annual ice loss remains high. Jakobshavn-style calving is not widespread in Antarctica today, because most marine-terminating grounding lines with comparable ice thickness are supported by the resistive back stress of ice shelves. Crane Glacier, previously buttressed by the Larsen B ice shelf on the Antarctic Peninsula, is an exception. When the ice shelf suddenly collapsed in 2002 after becoming covered in meltwater, the glacier sped up by a factor of three24. A persistent 100-m ice cliff formed at the terminus25 and the calving front retreated into its narrow fjord. The drainage of Crane Glacier was too small to contribute substantially to sea level, but similar events could become widespread in Antarctica if temperatures continue to rise. Importantly, some Antarctic glaciers are vastly larger than their Greenland counterparts. For example, Thwaites Glacier in West Antarctica terminates in the open Amundsen Sea rather than in a narrow fjord. The main trunk of Thwaites Glacier is about 120 km wide, widening upstream into the heart of the WAIS. Today, the heavily crevassed grounding zone of Thwaites Glacier is minimally buttressed and retreating on reverse-sloped bedrock at >1 km yr−1 in places26, possibly owing to MISI. The terminus currently sits in water too shallow (about 600 m deep) to produce an unstable cliff face, but if retreat continues into deeper bedrock and thicker ice, a calving face taller than that of Jakobshavn could appear, with stresses and strain rates exceeding thresholds for brittle failure16,17,18. Similar vulnerabilities exist at other Antarctic glaciers, particularly where buttressing ice shelves are already thinning from contact with warm sub-surface waters14.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-1	Calving in narrow fjord settings such as Jakobshavn is controlled by a complex combination of ductile and brittle processes, as well as buoyancy. After calving, subsequent fracture-driven failure is delayed until accelerated flow thins the terminus to near-flotation, allowing tidal flexure, basal crevassing, slumping or other processes to initiate the next event17,23. Resistive stresses from lateral shear along fjord walls, as well as thick mélange strengthened by sea ice, slow calving in winter, but annual ice loss remains high. Jakobshavn-style calving is not widespread in Antarctica today, because most marine-terminating grounding lines with comparable ice thickness are supported by the resistive back stress of ice shelves. Crane Glacier, previously buttressed by the Larsen B ice shelf on the Antarctic Peninsula, is an exception. When the ice shelf suddenly collapsed in 2002 after becoming covered in meltwater, the glacier sped up by a factor of three24. A persistent 100-m ice cliff formed at the terminus25 and the calving front retreated into its narrow fjord. The drainage of Crane Glacier was too small to contribute substantially to sea level, but similar events could become widespread in Antarctica if temperatures continue to rise. Importantly, some Antarctic glaciers are vastly larger than their Greenland counterparts. For example, Thwaites Glacier in West Antarctica terminates in the open Amundsen Sea rather than in a narrow fjord. The main trunk of Thwaites Glacier is about 120 km wide, widening upstream into the heart of the WAIS. Today, the heavily crevassed grounding zone of Thwaites Glacier is minimally buttressed and retreating on reverse-sloped bedrock at >1 km yr−1 in places26, possibly owing to MISI. The terminus currently sits in water too shallow (about 600 m deep) to produce an unstable cliff face, but if retreat continues into deeper bedrock and thicker ice, a calving face taller than that of Jakobshavn could appear, with stresses and strain rates exceeding thresholds for brittle failure16,17,18. Similar vulnerabilities exist at other Antarctic glaciers, particularly where buttressing ice shelves are already thinning from contact with warm sub-surface waters14. Because of the very strong dependency of crack growth with increasing stress17,27, a previously unseen style of calving and ice failure might emerge at unbuttressed Antarctic ice fronts with higher freeboard than glaciers on Greenland7,8. The potential pace of fracturing in such settings remains uncertain20, but once a calving front backs into thicker ice upstream, brittle failure could outpace viscous flow, inhibiting the growth of a new shelf. Complete, sustained loss of an ice shelf is not required for structural failure16. If a small floating shelf survives or reforms without providing substantial buttressing, the grounding zone would remain under sufficient stress for collapse. Re-emerging ice shelves would remain vulnerable to warm ocean waters and surface meltwater, as evidenced at Jakobshavn and Crane glaciers; despite fast flow and mélange buttressing, persistent ice tongues have not reformed and calving continues. Extensive loss of buttressing ice shelves (prerequisite for both MISI and MICI) represents a possible tipping point in Antarctica’s future. This is concerning, because ice shelves are vulnerable to both oceanic melt from below14 and surface warming above28. Rain and meltwater can deepen crevasses28 and cause flexural stresses29, leading to hydrofracturing and ice-shelf collapse. Vulnerability to surface meltwater is enhanced where firn (the transitional layer between surface snow and underlying ice) becomes saturated and where ocean-driven thinning is already underway28. Air temperatures above Antarctica’s largest ice shelves remain too cold to produce sustained meltwater rates associated with collapse30,31; however, given sufficient future warming, this could change. Modelling the AIS response to warming We build on previous work [8] by improving a hybrid ice sheet–shelf model that includes viscous ice processes related to MISI and brittle processes related to MICI. The model allows conditionally unstable grounding lines (MISI) on reverse-sloped bedrock in response to flow and stress fields, bed conditions and surface mass balance. The model accounts for oceanic sub-ice melt and meltwater-driven hydrofracturing of ice shelves, leading to ice-cliff calving at thick, marine-terminating ice fronts where stresses exceed ice strength (MICI). Model improvements and extensions described in Methods and Supplementary Information include new formulations of ice-shelf buttressing, hydrofracturing and coupling with a comprehensive Earth–sea-level model, as well as ice–climate (meltwater) feedback mechanisms using the NCAR Community Earth System Model. Parametric uncertainty is assessed using modern and geological observations and statistical emulation. Regional climate model (RCM) forcing used in future ice-sheet ensembles is substantially improved relative to ref. 8, with the trajectory of warming being comparable to that of other studies30 (Supplementary Information). We test the future response of the AIS to scenarios representing +1.5 °C and +2 °C global warming limits, a +3 °C scenario representing current policies [32] and extended RCP emissions scenarios9. We consider recently proposed negative feedback mechanisms that could slow the pace of future ice loss, and emissions scenarios allowing a temporary overshoot of Paris Agreement temperature targets followed by rapid carbon dioxide reduction (CDR), assuming that such geoengineering is possible. The results identify emissions-forced climatic thresholds capable of triggering rapid retreat of the AIS. Calibrated model ensembles To account for the current uncertainty in key parameters controlling (1) the sensitivity of crevasse penetration to surface melt and rainwater (hydrofracturing) and (2) the ice-cliff calving rate, we run 196 ice-sheet simulations for each climate scenario described below. Each ensemble member uses a unique combination of parameter values (Extended Data Table 1), scored using a binary history-matching approach [8,33]. Scoring is based on the model’s ability to simulate the observed ice loss, d𝑀¯/d𝑡, between 1992 and 2017 (IMBIE) [4], and Antarctica’s contribution to sea level in the last interglacial period (LIG) [34] and the mid-Pliocene epoch [35,36] (Methods). Ensemble members falling outside the likely range of observations are discarded, and only parameter combinations within the bounds of all three constraints are included in future projections. Both modern and geological constraints contain considerable uncertainty with poorly known sample distributions, so weighting of individual model outcomes is avoided. In Supplementary Information we compare our ensemble scoring to a more rigorous Gaussian process emulation approach [33,37] to verify that the central estimates of our calibrated ensembles are robust. Comparing simulated and IMBIE estimates of d𝑀¯/d𝑡 (Extended Data Fig. 1a) eliminates 33 ensemble members (n = 163). The effect of replacing IMBIE with alternative (narrower) ranges of d𝑀¯/d𝑡 on the basis of solely GRACE data from 2002–2017 [38] (Methods) is shown in Extended Data Fig. 2. The model performs well over the IMBIE interval with and without hydrofracturing and ice-cliff calving enabled. Although IMBIE provides guidance on processes that cause contemporary mass change (surface mass balance, oceanic-shelf thinning and grounding-line dynamics), it does not sufficiently test the brittle-ice processes theorized to become important in a warmer climate [7,8]. Furthermore, the 25-year IMBIE record is very short relative to the dynamical response time of an ice sheet, and interdecadal and longer variability is not captured. Collectively, these issues motivate our use of geological records from past warm periods as additional training constraints. Adding the LIG constraint (3.1–6.1 m; 129–128 kyr ago) to IMBIE eliminates 44 additional parameter combinations (n = 119), but only at the lower bound of the parameter range. Without MICI, the model is incapable of simulating realistic LIG ice loss. Even at the top of the parameter range, simulated rates of GMSL rise remain below 1 cm yr−1, slower than indicated by some LIG proxy records39 (Extended Data Fig. 1b, c). Adding a warm mid-Pliocene test (11–21 m; 3.26–3.03 Ma) further reduces the ensemble to n = 109 by eliminating some of the highest-valued parameter combinations. Similar to the LIG, hydrofracturing and ice-cliff calving must be included to satisfy Pliocene geological observations, including regional retreat into East Antarctic basins40 (Extended Data Figs. 1d, 3). The model’s ability to simulate current rates of ice loss without ice-cliff calving, while failing to simulate past retreat under warmer climate conditions (Extended Data Figs. 1, 3), is at odds with the findings of ref. 33, which assumed a lower range of Pliocene sea-level constraints than the more recent data35,36 used here.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-2	Adding the LIG constraint (3.1–6.1 m; 129–128 kyr ago) to IMBIE eliminates 44 additional parameter combinations (n = 119), but only at the lower bound of the parameter range. Without MICI, the model is incapable of simulating realistic LIG ice loss. Even at the top of the parameter range, simulated rates of GMSL rise remain below 1 cm yr−1, slower than indicated by some LIG proxy records39 (Extended Data Fig. 1b, c). Adding a warm mid-Pliocene test (11–21 m; 3.26–3.03 Ma) further reduces the ensemble to n = 109 by eliminating some of the highest-valued parameter combinations. Similar to the LIG, hydrofracturing and ice-cliff calving must be included to satisfy Pliocene geological observations, including regional retreat into East Antarctic basins40 (Extended Data Figs. 1d, 3). The model’s ability to simulate current rates of ice loss without ice-cliff calving, while failing to simulate past retreat under warmer climate conditions (Extended Data Figs. 1, 3), is at odds with the findings of ref. 33, which assumed a lower range of Pliocene sea-level constraints than the more recent data35,36 used here. Processes other than ice-cliff calving could be invoked to improve geological model–data comparisons. For example, Pliocene retreat in East Antarctica has been simulated in a model without MICI, using a sub-ice melt scheme allowing the presence of melt beneath grounded ice upstream of the grounding line41. Tidally driven seawater intrusion and non-zero melt beneath discontinuous sectors of grounding zones has been observed26; however, model treatments used so far41 have been questioned on physical grounds42. Alternative (Coulomb) sub-glacial sliding laws have been proposed43 that can substantially increase the rate of ice loss in models with ice shelves removed44, but they have not been tested with realistic palaeoclimate forcing. We stress that hydrofracturing and ice-cliff calving processes incorporated here are observed phenomena, tested under both modern and geological settings. Ice loss in both LIG and Pliocene ensembles saturates at the upper range of parameter values (Extended Data Fig. 1). The LIG is sufficiently warm to cause complete WAIS retreat, but not warm enough to trigger retreat into East Antarctic basins, even if our nominal ice-cliff calving limit (13,000 m yr−1) is doubled. Similarly, maximum ice loss in the Pliocene ensemble reflects the loss of most marine-based ice, as supported by observations35, but not more. As such, the geological constraints do not rule out the possibility of faster Antarctic ice-cliff calving rates than those observed on Greenland today, which would substantially increase our future projections while remaining consistent with geological observations. Implications of the Paris Agreement We run ensembles of the transient response of the AIS to future greenhouse gas emissions scenarios (Methods) representing global mean warming limits of +1.5 °C, +2 °C and +3 °C (similar to current policies and Nationally Determined Contributions, NDCs32), as well as RCP2.6, RCP4.5 and RCP8.59. Only simulations with validated parameter combinations (Extended Data Fig. 4d) are included in the analysis. The +1.5 °C, +2 °C and +3 °C scenarios assume that there is no overshoot in temperature; once these global mean temperatures are reached in 2040, 2060 and 2070, respectively, atmosphere and ocean forcings are held constant. In the +1.5 °C and +2 °C ensembles, Antarctic ice loss continues at a pace similar to today’s throughout the 21st century (Fig. 1, Table 1). The median contribution to sea level in 2100 is 8 cm with +1.5 °C warming and 9 cm with +2 °C. By contrast, about 10% of the ensemble members in the +3 °C scenario produce onset of major WAIS retreat before 2100. This skews the upper bound of the +3 °C distribution (33 cm at the 90th percentile), substantially increasing the ensemble median (15 cm in 2100) relative to the +1.5 °C and +2 °C scenarios. The jump in late 21st century ice loss at +3 °C is mainly caused by retreat of Thwaites Glacier (Fig. 2; Extended Data Fig. 5), which destabilizes the entire WAIS in some ensemble members. Figure 1: Antarctic contribution to future GMSL rise. a–h, The fan charts show the time-evolving uncertainty and range around the median ensemble value (black line) in 10% increments. a, c, e, g, Ensemble results from 2000–2100, including median rates of GMSL rise (red line). b, d, f, h The same as a, c, e, g, extended to 2300. a, b, Emissions consistent with a +1.5.°C global mean warming scenario. c, d, Emissions consistent with +2.0.°C. e, f, Emissions consistent with +3.0.°C. g, h, RCP8.5. In h, two additional RCP8.5 simulations are shown with average calibrated parameter values (Methods) but with atmosphere and ocean forcing provided by the NCAR CESM1.2.2 GCM with (blue line) and without (red line) Antarctic meltwater feedback46. Note the expanded vertical axes in g and h. Table 1: Antarctic sea-level contributions. Ensemble medians (top six rows) using IMBIE, LIG and Pliocene observational constraints, reported in metres, relative to 2000. Values in parentheses are the 17th–83rd percentiles (likely range). Scenarios refer to the maximum global mean temperature reached relative to pre-industrial (1850) or following RCPs. Alternative ensemble outcomes using more restrictive ranges of ice-cliff calving parameters are provided in Extended Data Table 2. Model simulations corresponding to Fig. 3 (bottom 12 rows) use average calibrated parameter values (Extended Data Table 1). NDC simulations follow the standard +3 °C emissions scenario or consider CDR beginning in 2200, 2150, 2100, 2090, 2080, 2070, 2060, 2050, 2040 or 2030. An alternative scenario maintains the atmosphere and ocean climate forcing at 2020 (with no additional future warming). Note that the +3.0 °C ensemble median (third row) differs from the corresponding +3.0 °C (NDCs) simulation using average model parameter values (seventh row) owing to the skewness of the ensembles (Fig. 1). Figure 2: Ice-sheet evolution following the +3 °C global warming emissions trajectory. A single +3 °C ensemble member with average hydrofracturing and ice-cliff calving parameters. Transient atmosphere and ocean forcing follows the +3 °C scenario, roughly consistent with current policies (NDCs). Floating and grounded ice thickness is shown in blue. The grounding line position is shown with a black line. The red square over the Thwaites Glacier (TG) and Pine Island Glacier (PIG) sector of West Antarctica corresponds to the high-resolution (1,000 m) nested model domain in Extended Data Fig. 5. a, Initial ice-sheet conditions. b, Model ice sheet in 2100, showing the onset of major retreat of Thwaites Glacier. c, Change in ice thickness in 2100. d, The ice sheet in 2300, with Thwaites Glacier retreat leading to the loss of the WAIS. e, Change in ice thickness in 2300. With more extreme RCP8.5 warming, thinning and hydrofracturing of buttressing ice shelves becomes widespread, triggering marine ice instabilities in both West and East Antarctica. The RCP8.5 median contribution to GMSL is 34 cm by 2100. This is substantially less than reported by ref. 8 (64–105 cm), owing to a combination of improved model physics and revised atmospheric forcing (Methods) that delays the onset of surface melt by about 25 years. Nonetheless, the median contribution to GMSL reaches 1 m by 2125 and rates exceed 6 cm yr−1 by 2150 (Extended Data Figs. 6, 7). By 2300, Antarctica contributes 9.6 m of GMSL rise under RCP8.5, almost 10 times more than simulations limiting warming to +1.5 °C.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-3	Figure 2: Ice-sheet evolution following the +3 °C global warming emissions trajectory. A single +3 °C ensemble member with average hydrofracturing and ice-cliff calving parameters. Transient atmosphere and ocean forcing follows the +3 °C scenario, roughly consistent with current policies (NDCs). Floating and grounded ice thickness is shown in blue. The grounding line position is shown with a black line. The red square over the Thwaites Glacier (TG) and Pine Island Glacier (PIG) sector of West Antarctica corresponds to the high-resolution (1,000 m) nested model domain in Extended Data Fig. 5. a, Initial ice-sheet conditions. b, Model ice sheet in 2100, showing the onset of major retreat of Thwaites Glacier. c, Change in ice thickness in 2100. d, The ice sheet in 2300, with Thwaites Glacier retreat leading to the loss of the WAIS. e, Change in ice thickness in 2300. With more extreme RCP8.5 warming, thinning and hydrofracturing of buttressing ice shelves becomes widespread, triggering marine ice instabilities in both West and East Antarctica. The RCP8.5 median contribution to GMSL is 34 cm by 2100. This is substantially less than reported by ref. 8 (64–105 cm), owing to a combination of improved model physics and revised atmospheric forcing (Methods) that delays the onset of surface melt by about 25 years. Nonetheless, the median contribution to GMSL reaches 1 m by 2125 and rates exceed 6 cm yr−1 by 2150 (Extended Data Figs. 6, 7). By 2300, Antarctica contributes 9.6 m of GMSL rise under RCP8.5, almost 10 times more than simulations limiting warming to +1.5 °C. In alternative ensembles, the upper bound of the maximum calving rate (VCLIFF) is reduced from 13 km yr−1 to 11 km yr−1 or 8 km yr−1 to reflect Jakobshavn’s recent slowdown22, but the effect on the calibrated ensemble medians is small (Extended Data Table 2). The main ensembles (Fig. 1, Table 1) use 13 km yr−1 as the upper bound because LIG and Pliocene responses saturate above these values and observations at Jakobshavn demonstrate that such rates are indeed possible. Future simulations excluding hydrofracturing and ice-cliff calving produce less GMSL rise than our ensemble medians (Extended Data Fig. 6). Similar to other models without ice-cliff calving45, enhanced precipitation in East Antarctica partially compensates for MISI-driven retreat in West Antarctica, but these simulations are excluded from the future projections because they fail to reproduce the LIG and Pliocene. Negative feedback mechanisms slowing ice loss Because our model includes hydrofracturing, the onset of major retreat is sensitive to the pace of future atmospheric warming. We compare our RCM/CCSM4-driven RCP8.5 ensemble to two alternative simulations, with atmosphere and ocean forcing supplied by the NCAR CESM1.2.2 GCM. Both CESM-forced simulations follow RCP8.5, but one includes Antarctic meltwater feedback (Methods) by adding time-evolving liquid-water and solid-ice discharge at the appropriate ocean grid cells in the GCM46. Including meltwater discharge in CESM expands Southern Ocean sea ice, stratifies the upper ocean, and warms the subsurface (400 m water depth) by 2–4 °C around most of the Antarctic margin in the early 22nd century46. Conversely, expanded sea ice suppresses surface atmospheric warming by more than 5 °C, slowing the onset of surface melt and hydrofracturing in the ice-sheet model. The net result of competing sub-surface ocean warming (enhanced sub-shelf melt) and atmospheric cooling (reduced surface melt) produces a substantial negative feedback on the pace of ice-sheet retreat (Fig. 1h). This contrasts ref. 41, which found a net positive (ocean-driven) meltwater feedback using an ice-sheet model without hydrofracturing. The CESM-driven simulations bracket our RCM/CCSM4-driven ensembles, supporting the timing of retreat in our main ensembles. Our RCM and CESM1.2.2 climate forcings are evaluated relative to independent CMIP5 and CMIP6 GCMs in Supplementary Information. We test two additional negative-feedback mechanisms proposed to provide a stabilizing influence on marine ice-sheet retreat. First, the potential for channelized supraglacial runoff to delay or stop ice-shelf hydrofracturing47 is examined by reducing meltwater-enhanced surface crevassing in regions of compressional ice-shelf flow (Supplementary Information). Despite a reduced influence of meltwater, we find that hydrofracturing in a warming climate still occurs near ice-shelf calving fronts, where the ice is thinnest, convergence and buttressing are minimal13 and air temperatures (melt rates) are highest. Once initiated, meltwater-enhanced calving near the shelf edge reduces compressional flow in ice upstream and calving propagates. As a result, reduced wet crevassing in compressional flow does little to protect buttressing ice shelves48 and the impact on our simulations is minimal (Supplementary Fig. 3). Second, we examine the potential for rapid bedrock uplift and ice–ocean gravitational effects to lower relative sea level and reduce ice loss at retreating grounding lines12. Exceptionally fast uplift rates due to low mantle viscosities in the Amundsen Sea sector of West Antarctica have been invoked to slow future retreat of the WAIS10. This is tested by replacing the model’s standard Elastic Lithosphere/Relaxed Asthenosphere representation of deforming bedrock with a more complete viscoelastic (Maxwell) Earth model combining a radially varying, depth-dependent lithosphere and viscosity structure with gravitationally self-consistent sea-level calculations (Methods)12. Simulations assuming the lowest upper mantle viscosity10 with rapid bedrock uplift under all of West Antarctica show limited potential for ice–Earth feedback mechanisms to slow retreat over the next approximately two centuries (Extended Data Fig. 8). This finding is consistent with other recent studies11,12,49, although future work should explore these effects at higher resolution and with a three-dimensional Earth structure50 including lateral heterogeneity of viscoelastic properties under West and East Antarctica. Implications of delayed mitigation An additional set of simulations was run using a single combination of ice-model parameters representing calibrated ensemble averages (Extended Data Table 1). The simulations either maintain current (2020) atmosphere and ocean conditions without any future warming, or begin to follow the +3 °C emissions pathway, except assuming that CDR mitigation is initiated at different times in the future, beginning in 2030, 2040, 2050, 2060, 2070, 2080, 2090, 2100, 2150 or 2200. We optimistically assume that CDR technologies will be capable of reducing CO2 atmospheric mixing ratios with an e-folding time of one century (Fig. 3a). Figure 3: AIS thresholds and commitments to GMSL rise with delayed mitigation. a, Greenhouse gas (GHG) emissions scenarios that initially follow the +3 °C (NDCs) scenario, followed by CDR (carbon dioxide reduction/negative emissions), assuming relaxation towards preindustrial levels with an e-folding time of 100 years. The timing when CDR commences is shown in b. The solid black line is the same +3 °C simulation shown in Fig. 2 and Extended Data Fig. 5. The dashed black line assumes there is no additional GHG increase or warming after 2020. GHG concentrations are shown in CO2 equivalent, in units of preindustrial atmospheric level (PAL; 280 ppm). b, GMSL contributions from Antarctica, corresponding to the scenarios in a, over the 21st century. All simulations use identical model physics and average hydrofracturing and ice-cliff calving parameters. Note the sharp increase in late-21st-century ice loss when CDR is delayed until 2070. c, The same as b, but extended to 2500 (see Table 1). Note the long-term dependence of GMSL rise on the timing when mitigation begins. All scenarios exceed 1 m by 2500, and no scenario shows recovery of the ice sheet, including those returning to near-preindustrial levels of GHGs by about 2300.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-4	Implications of delayed mitigation An additional set of simulations was run using a single combination of ice-model parameters representing calibrated ensemble averages (Extended Data Table 1). The simulations either maintain current (2020) atmosphere and ocean conditions without any future warming, or begin to follow the +3 °C emissions pathway, except assuming that CDR mitigation is initiated at different times in the future, beginning in 2030, 2040, 2050, 2060, 2070, 2080, 2090, 2100, 2150 or 2200. We optimistically assume that CDR technologies will be capable of reducing CO2 atmospheric mixing ratios with an e-folding time of one century (Fig. 3a). Figure 3: AIS thresholds and commitments to GMSL rise with delayed mitigation. a, Greenhouse gas (GHG) emissions scenarios that initially follow the +3 °C (NDCs) scenario, followed by CDR (carbon dioxide reduction/negative emissions), assuming relaxation towards preindustrial levels with an e-folding time of 100 years. The timing when CDR commences is shown in b. The solid black line is the same +3 °C simulation shown in Fig. 2 and Extended Data Fig. 5. The dashed black line assumes there is no additional GHG increase or warming after 2020. GHG concentrations are shown in CO2 equivalent, in units of preindustrial atmospheric level (PAL; 280 ppm). b, GMSL contributions from Antarctica, corresponding to the scenarios in a, over the 21st century. All simulations use identical model physics and average hydrofracturing and ice-cliff calving parameters. Note the sharp increase in late-21st-century ice loss when CDR is delayed until 2070. c, The same as b, but extended to 2500 (see Table 1). Note the long-term dependence of GMSL rise on the timing when mitigation begins. All scenarios exceed 1 m by 2500, and no scenario shows recovery of the ice sheet, including those returning to near-preindustrial levels of GHGs by about 2300. We find that without future warming beyond 2020, Antarctica continues to contribute to 21st-century sea-level rise at a rate roughly comparable to today’s, producing 5 cm of GMSL rise by 2100 and 1.34 m by 2500 (Fig. 3, Table 1). Simulations initially following the +3 °C pathway, but with subsequent CDR delayed until after 2060, show a sharp jump in the pace of 21st-century sea-level rise (Fig. 3b). Every decade that CDR mitigation is delayed has a substantial long-term consequence on sea level, despite the fast decline in CO2 and return to cooler temperatures (Fig. 3c). Once initiated, marine-based ice loss is found to be unstoppable on these timescales in all mitigation scenarios (Fig. 3). The commitment to sustained ice loss is caused mainly by the onset of marine ice instabilities triggered by the loss of ice shelves that cannot recover in a warmer ocean with long thermal memory (Fig. 3c). In summary, these results demonstrate that current policies allowing +3 °C or more of future warming could exceed a threshold, triggering extensive thinning and loss of vulnerable Antarctic ice shelves and ensuing marine ice instabilities starting within this century. Resulting ice loss would be irreversible on multi-century timescales, even if atmospheric temperatures return to preindustrial-like values (Fig. 3). Relative to the +3 °C scenario, sea-level rise resulting from the +1.5 °C and +2 °C aspirations of the Paris Agreement (Fig. 1) would have much less impact on low-lying coastlines, islands and population centres, pointing to the importance of ambitious mitigation. Strong circum-Antarctic atmospheric cooling feedback caused by meltwater discharge [46] slows the pace of retreat under RCP8.5 (Fig. 1h). However, other proposed negative feedback mechanisms associated with ice–Earth–sea level interactions and reduced hydrofracturing through surface runoff do little to slow ice loss on 21st- to 22nd-century timescales. Although we attempt to constrain parametric uncertainty, this study uses a single ice-sheet model, and structural uncertainty is accounted for only in the model improvements described herein. Similarly, our main ensembles use a single method of climate forcing, although with future warming comparable to other state-of-the-art climate models (Supplementary Figs. 1, 2), and alternative simulations driven by CESM1.2.2 produce similar results (Fig. 1). More work is clearly needed to further explore this uncertainty, using multiple ice-sheet models accounting for processes associated with MISI and MICI, and with future climate forcing that includes interactive climate–ice sheet coupling. Ice-cliff calving remains a key wild card. Although founded on basic physical principles and observations, its potential to produce even faster rates of ice loss than those simulated here remains largely untested with process-based models of mechanical ice failure. Here we find that limiting rates of ice-cliff calving to those observed on Greenland can still drive multi-metre-per-century rates of sea-level rise from Antarctica (Extended Data Fig. 7). Given the bedrock geography of the much larger and thicker AIS, the possibility of even faster mechanical ice loss should be a top priority for further investigation. Methods Ice-sheet modelling framework The ice sheet–shelf model uses hybrid ice dynamics [51] with an internal boundary condition on ice velocity at the grounding line [6]. Grounding lines can migrate freely, and the model accounts for the buttressing effects of ice shelves with pinning points and side shear (see Supplementary Information). In our solution of the dynamical shallow shelf (SSA) equations, ice velocities across grounding lines are imposed as a function of local sub-grid ice thickness, with the sub-grid interpolation accurate to the limit of the resolved bathymetry. This is also true for diagnosed stresses and ice-cliff failure rates, which makes the model largely independent of grid resolution (Extended Data Fig. 5). A resolution of 10 km is used for continental simulations used in our main ensembles (Figs. 1–3). A nested 1-km grid is used for a select simulation over West Antarctica (Extended Data Fig. 5). The model uses a standard Weertman-type basal sliding law51, with basal sliding coefficients determined by an inverse method that iteratively matches model ice-surface elevations to observations under modern climate conditions52. We use Bedmap253 bathymetric boundary conditions. Using alternative BedMachine3 bathymetry is found to have only a small effect on continental-scale sea-level projections (<1.5% difference under RCP8.5 in 2300). Several advances relative to previous versions of the model7,8,51 are described below and in Supplementary Information. Sub-ice melt rates The model used here includes an updated treatment of sub-ice oceanic melting. Oceanic melt rates are calculated at each floating ice grid cell as a quadratic function of the difference between nearest sub-surface ocean temperatures at 400-m water depth and the pressure melting point of ice51,54. The model accounts for evolving connectivity between a given ice model grid cell and the open ocean, and elevated plume melt on subsurface vertical ice faces51. All melt calculations are performed with spatially uniform physics, including a single, uniform coefficient in the ocean melt relation based on a 625-member ensemble of simulations of WAIS retreat through the last deglaciation55. Although it would be possible to perform inverse calculations for a distribution of coefficients within each basin based on modern ice-shelf melt observations41, their patterns are likely to change substantially within the timescales of our simulations as ocean circulation, grounding-line extents and cavity geometries evolve. A 1.5 °C sub-surface ocean temperature adjustment is used in the Amundsen Sea sector to bring ocean melt rates closer to observations56 when using CCSM4 ocean model temperatures that underestimate observed shelf-bottom water temperatures57. This is a substantial improvement relative to the 3 °C temperature adjustment required previously8. Ice-shelf hydrofracturing In the model, surface crevasses deepen as a function of the stress field and local meltwater and rainfall availability [7,8,58], leading to hydrofracturing when surface and basal crevasses penetrate 75% or more of the total ice thickness. With greatly increased surface melt, model ice shelves can be completely lost. In the standard wet crevassing scheme, we assume a quadratic relationship between surface crevasse penetration depth dw (in metres) and total meltwater production R (rain plus surface melt minus refreezing; in m yr−1). A tunable prefactor, CALVLIQ, is varied between zero (no meltwater influence on crevassing) and 195 m−1 yr2 in the ensembles presented in the main text. dw = CALVLIQ R^2. Calving occurs in places where the sum of the surface and basal crevasse penetration caused by extensional stresses, accumulated strain (damage), thinning and meltwater (dw), exceeds the critical fraction (0.75) of total ice thickness (see appendix B of ref. 7).	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-5	The model used here includes an updated treatment of sub-ice oceanic melting. Oceanic melt rates are calculated at each floating ice grid cell as a quadratic function of the difference between nearest sub-surface ocean temperatures at 400-m water depth and the pressure melting point of ice51,54. The model accounts for evolving connectivity between a given ice model grid cell and the open ocean, and elevated plume melt on subsurface vertical ice faces51. All melt calculations are performed with spatially uniform physics, including a single, uniform coefficient in the ocean melt relation based on a 625-member ensemble of simulations of WAIS retreat through the last deglaciation55. Although it would be possible to perform inverse calculations for a distribution of coefficients within each basin based on modern ice-shelf melt observations41, their patterns are likely to change substantially within the timescales of our simulations as ocean circulation, grounding-line extents and cavity geometries evolve. A 1.5 °C sub-surface ocean temperature adjustment is used in the Amundsen Sea sector to bring ocean melt rates closer to observations56 when using CCSM4 ocean model temperatures that underestimate observed shelf-bottom water temperatures57. This is a substantial improvement relative to the 3 °C temperature adjustment required previously8. Ice-shelf hydrofracturing In the model, surface crevasses deepen as a function of the stress field and local meltwater and rainfall availability [7,8,58], leading to hydrofracturing when surface and basal crevasses penetrate 75% or more of the total ice thickness. With greatly increased surface melt, model ice shelves can be completely lost. In the standard wet crevassing scheme, we assume a quadratic relationship between surface crevasse penetration depth dw (in metres) and total meltwater production R (rain plus surface melt minus refreezing; in m yr−1). A tunable prefactor, CALVLIQ, is varied between zero (no meltwater influence on crevassing) and 195 m−1 yr2 in the ensembles presented in the main text. dw = CALVLIQ R^2. Calving occurs in places where the sum of the surface and basal crevasse penetration caused by extensional stresses, accumulated strain (damage), thinning and meltwater (dw), exceeds the critical fraction (0.75) of total ice thickness (see appendix B of ref. 7). The crevassing scheme is modified here relative to previous model versions7,8,51, by reducing wet crevassing in areas of low-to-moderate meltwater production (<1,500 mm yr−1), ramping linearly from zero, where no meltwater is present, to dw, where R = 1,500 mm yr−1. This small modification improves performance by maintaining more realistic ice-shelf calving fronts under present climate conditions, although it conservatively precludes the loss of ice shelves with thicknesses comparable to the Larsen B until R approaches ~1,400 mm yr−1, which is more than that observed before the actual collapse (~750 mm yr−1)30. Whereas liquid water embedded in firn and partial refreezing of meltwater are accounted for8,59, the detailed evolution of firn density and development of internal ice lenses are not, which could affect the timing at which hydrodrofacturing is simulated to begin. A modification to hydrofracturing described in Supplementary Information tests the possible influence of channelized meltwater flow and supraglacial runoff in compressional ice-shelf regimes. Calving and ice-cliff failure Two modes of brittle fracturing causing ice loss are represented in the model: (1) ‘standard’ calving of ice bergs from floating ice, and (2) structural failure of tall ice cliffs at the grounding line. Similar to other models, standard calving depends mainly on the grid-scale divergence of ice flow, producing crevasses to depths at which the extensional stress is equal to the hydrostatic imbalance58. Crevasse penetration is further increased as a function of surface meltwater and rain availability (see above). Unlike most continental-scale models, we also account for ice-cliff calving at thick, marine-terminating grounding lines. Such calving is a complex product of forces related to glacier speed, thickness, longitudinal stress gradients, bed conditions, side shear, pre-existing crevasses, mélange and other factors60. Determining the precise mode and rate of failure is the focus of ongoing work17,18,20,61; at present, a suitable physically based calving model has yet to be developed. In our model7,8, ice-cliff calving occurs where static stresses at the calving front (assumed to be exactly at floatation) begin to exceed the depth-averaged yield strength of glacial ice, assumed here to be 0.5 MPa (ref. 16). We account for crevassing near the cliff face (influenced by the stress regime and the presence of meltwater7), which thins the supportive ice column and increases the stress at the ice front. Where the critical stress threshold is exceeded, ice-cliff calving is applied as a horizontal wastage rate, ramping linearly from zero up to a maximum rate as effective cliff heights (adjusted for buttressing and crevassing) increase from ~80 to 100 m and above. This maximum calving rate is treated as a tunable model parameter (VCLIFF), replacing the arbitrary default value of 3 km yr−1 in equation A.4 of ref. 7. In this formulation, ice-cliff calving rates in places diagnosed to be undergoing structural failure are generally much smaller than VCLIFF (Extended Data Fig. 5). We note that the linear cliff height–calving relationship with an imposed calving limit (VCLIFF) used here is conservative relative to another proposed calving law20 that assumes a power-law dependence on cliff height and no upper bound on the calving rate. Furthermore, our model numerics preclude regular calving in places undergoing ice-cliff failure, so the computed ice-cliff calving rate can be considered as the sum of all calving processes at thick marine-terminating ice fronts. This allows direct comparison of model calving (Extended Data Fig. 5) with observations. Mélange can slow calving by providing some back stress at confined calving fronts [62,63], but it has limited effect on the large unconfined widths of Antarctic outlets [64], so it is ignored here. Ensemble parameters Our primary perturbed physics ensembles use a 14 × 14 matrix (n = 196) of CREVLIQ and VCLIFF in the hydrofracturing and ice-cliff calving parameterizations described above (Extended Data Table 1). The 14 values of CREVLIQ vary between 0 and 195 m−1 yr2 in evenly spaced increments. VCLIFF varies between 0 and 13 km yr−1. Previous studies7,8 considered a smaller, arbitrary range of VCLIFF values of up to 5 km yr−1; however, observed rates of horizontal ice loss through ice-cliff calving can reach 13 km yr−1 at the terminus of Jakobshavn Isbræ in West Greenland21, so we limit the top of our parameter range in our main ensembles to this observationally justifiable value. As discussed in the main text, this upper bound might be too small for Antarctic settings with thicker ice margins, taller unconfined ice fronts and higher deviatoric stresses at unbuttressed grounding lines. Select simulations extending the upper bounds of CALVLIQ and VCLIFF above 195 m−1 yr2 and 13 km yr−1, respectively, are shown in Extended Data Fig. 1. Setting these parameter values to zero (Extended Data Figs. 1, 6) effectively eliminates hydrofracturing and ice-cliff calving, limiting rates of ice loss to processes associated with standard calving, surface mass balance, sub-ice melt and MISI, as in most other continental-scale ice-sheet models. Ensemble scoring based on recent observations Future ice-sheet simulations begin in 1950 to allow comparisons with observations over the satellite era. For consistency, initial ice-sheet conditions (ice thickness, bed elevation, velocity, basal sliding coefficients and internal ice and bed temperatures) follow the same procedure as in ref. 8 and are identical in all simulations. Initialization involves a 100,000-kyr spin-up using observed mean annual ocean climatology65 and standard SeaRISE66 atmospheric temperature and precipitation fields [67].	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-6	Ensemble parameters Our primary perturbed physics ensembles use a 14 × 14 matrix (n = 196) of CREVLIQ and VCLIFF in the hydrofracturing and ice-cliff calving parameterizations described above (Extended Data Table 1). The 14 values of CREVLIQ vary between 0 and 195 m−1 yr2 in evenly spaced increments. VCLIFF varies between 0 and 13 km yr−1. Previous studies7,8 considered a smaller, arbitrary range of VCLIFF values of up to 5 km yr−1; however, observed rates of horizontal ice loss through ice-cliff calving can reach 13 km yr−1 at the terminus of Jakobshavn Isbræ in West Greenland21, so we limit the top of our parameter range in our main ensembles to this observationally justifiable value. As discussed in the main text, this upper bound might be too small for Antarctic settings with thicker ice margins, taller unconfined ice fronts and higher deviatoric stresses at unbuttressed grounding lines. Select simulations extending the upper bounds of CALVLIQ and VCLIFF above 195 m−1 yr2 and 13 km yr−1, respectively, are shown in Extended Data Fig. 1. Setting these parameter values to zero (Extended Data Figs. 1, 6) effectively eliminates hydrofracturing and ice-cliff calving, limiting rates of ice loss to processes associated with standard calving, surface mass balance, sub-ice melt and MISI, as in most other continental-scale ice-sheet models. Ensemble scoring based on recent observations Future ice-sheet simulations begin in 1950 to allow comparisons with observations over the satellite era. For consistency, initial ice-sheet conditions (ice thickness, bed elevation, velocity, basal sliding coefficients and internal ice and bed temperatures) follow the same procedure as in ref. 8 and are identical in all simulations. Initialization involves a 100,000-kyr spin-up using observed mean annual ocean climatology65 and standard SeaRISE66 atmospheric temperature and precipitation fields [67]. We consider three different estimates of recent changes in Antarctic ice mass to test the performance of each ensemble member with a unique combination of model physical parameters (Extended Data Table 1). We use the average annual mass change d𝑀¯/d𝑡 from 1992–2017 (equivalent to a GMSL change of 0.15–0.46 mm yr−1) provided by the IMBIE assessment4, which is based on a combination of satellite altimetry, gravimetry and surface mass balance estimates. We use the 25-year average to minimize the influence of simulated and observed interannual variability (Extended Data Fig. 1a) on ensemble scoring, although decadal and longer variability68 are not fully captured. Alternative target ranges use mass change calculations based solely on the Gravity Recovery and Climate Experiment (GRACE), following the methodology described in ref. 38 and updated from April 2002 to June 2017. The glacial isostatic adjustment (GIA) component of the GRACE estimates represents the largest source of uncertainty. We use three GIA models69,70,71. For each model we use a range of GIA corrections generated by the authors of69,70,71, assuming a range of viscosities and lithospheric thicknesses69,70,71. The lower bound of our mass change estimates is calculated using the minimum GIA correction from the three models69,70,71 and the upper bound is calculated using the maximum GIA correction. This yields a 2002–2017 average estimate of 0.2–0.54 mm yr−1, close to the central estimate from IMBIE over the same interval. Alternatively, we consider viscosity profiles from each of these studies that have been reported to provide the best fit to observations69,70,71. This substantially narrows and shifts the 2002–2017 range towards higher values (0.39 to 0.53 mm yr−1), which is impactful on our ensemble scoring and future projections, highlighting the need for more precise modern observations. Although the uncertainty range of estimates based solely on GRACE is smaller, the longer IMBIE record is used as our default training constraint over the modern era. LIG ensemble LIG simulations use model physics, parameter values and initial conditions identical to those used in our Pliocene and future simulations. The ice-driving atmospheric and oceanic climatology representing conditions between 130 and 125 kyr ago is the same as that used in ref. 8, and is based on a combination of regional atmospheric modelling and proxy-based reconstructions of air and ocean temperatures72. Differences in the timing and magnitude of our modelled Antarctic ice-sheet retreat relative to independent LIG simulations73 reflect the different approaches to LIG climate forcing and structural differences in our ice-sheet models, including the inclusion of hydrofracturing and ice-cliff calving in this study. Our ensemble scoring uses a LIG target range of Antarctic ice loss equivalent to 3.1–6.1 m, which is assumed to have occurred early in the interglacial between 129 and 128 kyr ago (Extended Data Fig. 1). The range used here is based on a prior estimate of GMSL of 5.9 ± 1.7 m by 128.6 ± 0.8 kyr ago35 (2σ uncertainty), rounded to the nearest half metre (4.5–7.5 m) to reflect the current uncertainty in the magnitude (due to GIA effects and dynamic topography) and timing of LIG sea-level estimates35,74. The Antarctic component is deconvolved from the GMSL value by assuming that Greenland contributed no more than 1 m before 128 kyr ago75,76,77, with an additional 0.4 m contributed by thermosteric effects75. Contributions from mountain glaciers in the early LIG are not known and are not included in our simple accounting. We find that rounding the exact GMSL values from ref. 35 (5.9 ± 1.7 m or 2.8–6.2 m after accounting for Greenland and thermosteric components) has no appreciable effect on the outcome of the calibrated ensembles. The target range of 3.1–6.1 m used here is lower than the 3.6–7.4 m range used in ref. 8, but we emphasize that it is based on a coral record from a single location (Seychelles), and ongoing work may further refine this range. For example, a recent study73 attempting to simultaneously fit relative sea-level data at several locations was able to reproduce early LIG changes observed in the Seychelles without a substantial contribution from Antarctica, but it required a thin lithosphere in the Earth model used to correct for GIA. Conversely, another study78 indicated that a North American ice sheet may have persisted until ~126 kyr ago or later. If true, this would require a substantial Antarctic contribution to GMSL to offset remaining North American ice in the early LIG. These alternative scenarios remain speculative, but they highlight the ongoing uncertainty in the palaeo sea-level records. Our LIG and Pliocene ensemble data (Extended Data Figure 1) are provided as source data to allow others to test the impact of alternative palaeo sea-level interpretations on the future projections. Pliocene ensemble Mid-Pliocene simulations also use consistent ice model physics and the same RCM climate forcing described in ref. 8, assuming 400 ppm CO2, an extreme warm austral summer orbit and 2 °C of ocean warming to represent maximum mid-Pliocene warmth in Antarctica. The ice-sheet simulations are run for 5,000 model years, the approximate duration that the warm orbital parameters are valid (Extended Data Fig. 1). The Pliocene maximum GMSL target range of 11–21 m is based on two recent, independent estimates of warm mid-Pliocene (3.26–3.03 Myr ago) sea level36,37. In ref. 36, shallow marine sediments are used to estimate the glacial–interglacial range of GMSL variability over this interval. Assuming ±5 m of uncertainty in the sea-level reconstructions and up to 5 m of GMSL change contributed by Greenland, at times orbitally out of phase with the timing of Antarctic ice loss36, the central estimate of Antarctica’s contribution to GMSL is 17.8 ± 5 m. This value is adjusted downwards to 16 m, according to an independent estimate derived from Mediterranean cave deposits corrected for geodynamical processes37. Combining the lower central estimate of ref. 37 and the uncertainty range of ref. 36 provides an Antarctic GMSL target range of 11–21 m, close to the range of 10–20 m used in ref. 8, albeit with considerable uncertainty. Future ensembles	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-7	Pliocene ensemble Mid-Pliocene simulations also use consistent ice model physics and the same RCM climate forcing described in ref. 8, assuming 400 ppm CO2, an extreme warm austral summer orbit and 2 °C of ocean warming to represent maximum mid-Pliocene warmth in Antarctica. The ice-sheet simulations are run for 5,000 model years, the approximate duration that the warm orbital parameters are valid (Extended Data Fig. 1). The Pliocene maximum GMSL target range of 11–21 m is based on two recent, independent estimates of warm mid-Pliocene (3.26–3.03 Myr ago) sea level36,37. In ref. 36, shallow marine sediments are used to estimate the glacial–interglacial range of GMSL variability over this interval. Assuming ±5 m of uncertainty in the sea-level reconstructions and up to 5 m of GMSL change contributed by Greenland, at times orbitally out of phase with the timing of Antarctic ice loss36, the central estimate of Antarctica’s contribution to GMSL is 17.8 ± 5 m. This value is adjusted downwards to 16 m, according to an independent estimate derived from Mediterranean cave deposits corrected for geodynamical processes37. Combining the lower central estimate of ref. 37 and the uncertainty range of ref. 36 provides an Antarctic GMSL target range of 11–21 m, close to the range of 10–20 m used in ref. 8, albeit with considerable uncertainty. Future ensembles We improve on previous work8 with new atmospheric climatologies used to run future ice-sheet simulations using dynamically downscaled meteorological fields of temperature and precipitation provided by an RCM79 adapted to Antarctica. RCM snapshots are run at 1950 and with increasing levels of effective CO2 (2, 4 and 8 times the preindustrial level) while accounting for topographic changes in the underlying ice sheet as described in ref. 8. The resulting meteorological fields are then time-interpolated and log-weighted to match transient CO2 concentrations following the emissions scenarios simulated here. This technique is computationally efficient and flexible, allowing a number of multi-century emissions scenarios to be explored, including non-standard RCP scenarios (Fig. 1) and those including CDR mitigation (Fig. 3). Unlike in ref. 8, sea surface temperatures and sea ice boundary conditions in the nested RCM come from the same transient NCAR CCSM480 runs that provide the time-evolving sub-surface ocean temperatures used in our sub-ice melt rate calculations. This eliminates the need for an imposed lag between transient greenhouse gas concentrations and equilibrated RCM climates as done previously8. Our revised approach delays the future timing at which surface meltwater begins to appear on ice-shelf surfaces, and the resulting atmospheric temperatures compare favourably with independent CMIP5 and CMIP6 GCMs (Supplementary Figs. 1, 2) and NCAR CESM1.2.2 (Fig. 1h). Monthly mean surface air temperatures and precipitation from the RCM are used to calculate the net annual surface mass balance on the ice sheet. These fields are bilinearly interpolated to the relatively fine ice-sheet grid, and temperatures are adjusted for the vertical difference between RCM and ice-sheet elevations using a simple lapse-rate correction. The lapse-rate correction is also applied to precipitation on the basis of a Clausius–Clapeyron-like relation. A two-step zero-dimensional box model using positive-degree days for snow and ice melt captures the basic physical processes of refreezing versus runoff in the snow–firn column8,59. The total surface melt available to influence surface crevassing (Supplementary Fig. 1) is the fraction of meltwater that is not refrozen near the surface, plus any rainwater. A spatially dependent bias correction based on reanalysis (Supplementary Fig. 2) could be applied to the RCM forcing, but such corrections are unlikely to remain stationary. Instead, we apply a uniform 2.9 °C temperature correction, reflecting the austral summer cold bias in the RCM over ice surface elevations lower than 200 m, where surface melt is most likely to begin. The cold bias, caused by an underestimate of net long-wave radiation, is observed in other Antarctic RCMs and GCMs81,82. Correcting for the cold bias and accounting for rainwater increases the total available surface meltwater in our RCP8.5 simulations relative to other studies31 (see Supplementary Information). The +1.5 °C simulations initially follow a RCP4.5 emission trajectory9, with time-evolving atmospheric fields provided by the RCM and matching sub-surface ocean temperatures from an RCP4.5 CCSM4 simulation80. The ice-driving climatology evolves freely until 2040, when decadal global mean surface air temperatures first reach +1.5 °C relative to 1850. Once the +1.5 °C temperature target is reached, the atmosphere and ocean forcings are fixed (maintained) at their 2040 levels for the duration of the simulations. The +2 °C scenario is also based on RCP4.5, but warming is allowed to evolve until 2060. 21st-century warming does not reach +3 °C under RCP4.5, so our +3 °C scenario (roughly representing the NDCs) is based on RCP8.5, with atmospheric and oceanic forcing fixed beyond 2070. Warming trajectories over major Antarctic ice shelves are shown in Supplementary Figs. 1, 2. Ice-sheet ensembles following extended RCP2.6, RCP4.5 and RCP8.5 scenarios9 are shown in Extended Data Fig. 6 for comparison with ref. 8. Alternative future ensembles (Extended Data Table 2) truncate the upper bound of the VCLIFF calving parameter from 13 km yr−1 (Table 1) to either 11 km yr−1 or 8 km yr−1, to account for the possibility that 13 km yr−1 calving rates observed at Jakobshavn between 2002 and 201521 are not representative of the glacier’s long-term behaviour. This reduces the raw ensembles from n = 196 to n = 168 and n = 126, respectively. An upper bound of 8 km yr−1 is difficult to justify because higher values cannot be excluded by the modern, LIG and Pliocene history matching. Furthermore, 8 km yr−1 is very close to the validated average value of 7.7 km yr−1 in the main ensemble. Using an upper bound of 11 km yr−1 instead of 13 km yr−1 has only a small effect on future projections (Extended Data Table 2). We consider 13 km yr−1 to be a reasonable upper bound for our main ensembles (Fig. 1) because this rate has been observed in nature21 and because ensemble members using this value cannot be excluded on the basis of model performance (Extended Data Fig. 1). Coupled ice–Earth–sea level model Most simulations use a standard Elastic Lithosphere/Relaxed Asthenosphere (ELRA) representation of vertical bedrock motion [51]. The ELRA model accounts for time-evolving bedrock deformation under changing ice loads, assuming an elastic lithospheric plate above local isostatic relaxation. Alternative simulations (Extended Data Fig. 8) account for full Earth–ice coupling using a viscoelastic (Maxwell) Earth model, combining a radially varying, depth-dependent lithosphere and mantle structure and gravitationally self-consistent sea-level calculations following the methodology described in ref. 12. Seismic [83,84] and geodetic [85,86] observations suggest substantial lateral variability in a viscoelastic Earth structure, with lower-than-average viscosities in parts of West Antarctica leading to faster uplift where ice mass is lost at the grounding line. Owing to the current uncertainties in Earth’s viscoelastic properties, we test a broad range of viscosity profiles. These include two end-member profiles described in refs. 12,49; one with a relatively high viscosity profile (HV) consistent with standard, globally tuned profiles; and one with a thinned lithosphere and a low-viscosity zone of 1,019 Pa s in the uppermost upper mantle (LVZ) that is broadly representative of West Antarctica. Here, we test a new profile (BLVZ) similar to LVZ, but assuming a vertical profile with the upper zone one order of magnitude less viscous than in LVZ, as recently proposed for the Amundsen Sea region10. The BLVZ model is consistent with the best-fitting radial Earth model in ref. 10, and uses a lithospheric thickness of 60 km, a shallow upper mantle from 60 km to 200 km depth with a viscosity of 3.98 × 1018 Pa s, a deep upper mantle from 200 km to 400 km with a viscosity of 1.59 × 1019 Pa s, a transition zone from 400 km to 670 km depth with a viscosity of 2.51 × 1019 Pa s, and a lower mantle viscosity of 1 × 1019 Pa s.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-8	Coupled ice–Earth–sea level model Most simulations use a standard Elastic Lithosphere/Relaxed Asthenosphere (ELRA) representation of vertical bedrock motion [51]. The ELRA model accounts for time-evolving bedrock deformation under changing ice loads, assuming an elastic lithospheric plate above local isostatic relaxation. Alternative simulations (Extended Data Fig. 8) account for full Earth–ice coupling using a viscoelastic (Maxwell) Earth model, combining a radially varying, depth-dependent lithosphere and mantle structure and gravitationally self-consistent sea-level calculations following the methodology described in ref. 12. Seismic [83,84] and geodetic [85,86] observations suggest substantial lateral variability in a viscoelastic Earth structure, with lower-than-average viscosities in parts of West Antarctica leading to faster uplift where ice mass is lost at the grounding line. Owing to the current uncertainties in Earth’s viscoelastic properties, we test a broad range of viscosity profiles. These include two end-member profiles described in refs. 12,49; one with a relatively high viscosity profile (HV) consistent with standard, globally tuned profiles; and one with a thinned lithosphere and a low-viscosity zone of 1,019 Pa s in the uppermost upper mantle (LVZ) that is broadly representative of West Antarctica. Here, we test a new profile (BLVZ) similar to LVZ, but assuming a vertical profile with the upper zone one order of magnitude less viscous than in LVZ, as recently proposed for the Amundsen Sea region10. The BLVZ model is consistent with the best-fitting radial Earth model in ref. 10, and uses a lithospheric thickness of 60 km, a shallow upper mantle from 60 km to 200 km depth with a viscosity of 3.98 × 1018 Pa s, a deep upper mantle from 200 km to 400 km with a viscosity of 1.59 × 1019 Pa s, a transition zone from 400 km to 670 km depth with a viscosity of 2.51 × 1019 Pa s, and a lower mantle viscosity of 1 × 1019 Pa s. Two sets of coupled ice–Earth–sea level simulations are run for each viscosity profile, with and without hydrofracturing and ice-cliff calving enabled (Extended Data Fig. 8). Simulations with the brittle processes enabled use values of CALVLIQ (105 m−1 yr2) and VCLIFF (6 km yr−1) close to the ensemble averages. The simulations follow our standard RCP forcing to test the effect of ice–Earth–sea level feedback on future projections. We find that the effects on equivalent sea-level rise are quite small on timescales of a few centuries and similar to those using the ELRA bed model, confirming that the use of the latter in our main ensembles (Fig. 1) is adequate. CESM ice-sheet simulations Two additional ice-sheet simulations are run using future atmospheric and oceanic forcing provided by two different RCP8.5 simulations described in ref. 46 and using the NCAR CESM 1.2.2 GCM with CAM5 atmospheric physics87. Ice-sheet model physics and parameter values are identical in both simulations. Hydrofracturing (CALVLIQ) and cliff calving (VCLIFF) parameters use calibrated ensemble averages of 107 m−1 yr2 and 7.7 km yr−1, consistent with the RCM-driven simulations shown in Figs. 2, 3. The standard RCP8.5 simulation ignores future Antarctic meltwater and dynamic discharge, whereas an alternative simulation accounts for time-evolving and spatially resolved liquid-water and solid-ice inputs around the Antarctic margin (peaking at >2 Sv in the early 22nd century), provided by an offline RCP8.5 ice-sheet simulation including hydrofracturing and ice-cliff calving46. The evolving temperature and precipitation fields from CESM are spatially interpolated and lapse-rate-adjusted to the ice-sheet model grid, using the same surface mass balance scheme used in our main RCM-forced ensembles. Similarly, sub-ice melt rates from CESM are calculated in exactly the same way as those provided by CCSM4 in our main ensembles. Although this discrete two-step coupling between CESM and the ice-sheet model does not account for time-continuous, fully coupled ice–ocean–climate feedback mechanisms, the two simulations (with and without ice-sheet discharge) span the envelope of possible outcomes when two-way meltwater feedback is fully accounted for. The two simulations using CESM with and without meltwater feedback are shown in Fig. 1h for comparison with our main RCM/CCSM4-forced ensembles. Extended data figures and tables Extended Data Fig. 1 Ensemble observational targets. 196 simulations (grey lines), each using a unique combination of hydrofracturing and ice-cliff calving parameters (Extended Data Table 1) compared with observations (blue dashed boxes). Solid blue lines show simulations without hydrofracturing and ice-cliff calving. Red lines show simulations with maximum parameter values in our main ensemble. Additional simulations (black lines) allow ice-cliff calving rates of up to 26 km yr−1, twice the maximum value used in our main ensembles. The vertical heights of the blue boxes represent the likely range of observations. Changes in ice mass above floatation are shown in equivalent GMSL. a, Simulated annual contributions to GMSL in the RCP8.5 ensemble compared with the 1992–2017 IMBIE4 observational average (0.15–0.46 mm yr−1; dashed blue box). b, LIG ensemble simulations from 130 to 125 kyr ago. The height of the dashed blue box shows the LIG target range (3.1–6.1 m), the width represents ~1,000-yr age uncertainty34. c, The same LIG simulations as in b, showing the rate of GMSL change contributed by Antarctica, smoothed over a 25-yr window. The peak in the early LIG is mainly caused by marine-based ice loss in West Antarctica. d, The same as b, except for warmer mid-Pliocene conditions. Maximum ice loss is compared with observational estimates of 11–21 m (refs. 35,36; blue dashed lines). Note the saturation of the simulated GMSL values near the top of the LIG and Pliocene ensemble range, and the failure of the model to produce realistic LIG or Pliocene sea levels without hydrofracturing and ice-cliff calving enabled (blue lines). Extended Data Fig. 2 RCP8.5 ensembles calibrated with alternative GRACE estimates. a, b, The fan charts show the time-evolving uncertainty and range around the median ensemble value (black line) in 10% increments. RCP8.5 ice-sheet model ensembles calibrated with GRACE estimates of annual mass change averaged from 2002–2017 using alternative GIA corrections (Methods). Use of GIA corrections produces estimates of mass loss between 2002 and 2017 of 0.2–0.54 mm yr−1 (a) and 0.39–0.53 mm yr−1 (b). The more restrictive and higher range of GRACE estimates in b skews the distribution and shifts the ensemble median values of GMSL upwards from 27 cm to 30 cm in 2100 and from 4.44 m to 4.94 m in 2200. Extended Data Fig. 3 Last Interglacial and Pliocene ice-sheet simulations. a–e, Ice-sheet simulations with the updated model physics used in our future ensembles and driven with the same LIG and Pliocene climate forcing used in ref. 8. Simulations without hydrofracturing and ice-cliff calving (a, b, d) correspond to blue lines in Extended Data Fig. 1. Simulations using maximum hydrofracturing and ice-cliff calving parameters (c, e) correspond to red lines in Extended Data Fig. 1. a, Modern (1950) ice-sheet simulation. b, c, LIG simulations run from 130 to 125 kyr ago are shown at 125 kyr ago. Values at the top of each panel are the maximum GMSL contribution between 129 and 128 kyr ago. Values in parentheses are the GMSL contribution at 125 kyr ago. d, e, Warm Pliocene simulations. Values shown are the maximum GMSL achieved during the simulations. Smaller values in parentheses show GMSL contributions after 5,000 model years (Extended Data Fig. 2d). Ice mass gain after peak retreat is caused by post-retreat bedrock rebound and enhanced precipitation in the warm Pliocene atmosphere.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-9	Extended Data Fig. 2 RCP8.5 ensembles calibrated with alternative GRACE estimates. a, b, The fan charts show the time-evolving uncertainty and range around the median ensemble value (black line) in 10% increments. RCP8.5 ice-sheet model ensembles calibrated with GRACE estimates of annual mass change averaged from 2002–2017 using alternative GIA corrections (Methods). Use of GIA corrections produces estimates of mass loss between 2002 and 2017 of 0.2–0.54 mm yr−1 (a) and 0.39–0.53 mm yr−1 (b). The more restrictive and higher range of GRACE estimates in b skews the distribution and shifts the ensemble median values of GMSL upwards from 27 cm to 30 cm in 2100 and from 4.44 m to 4.94 m in 2200. Extended Data Fig. 3 Last Interglacial and Pliocene ice-sheet simulations. a–e, Ice-sheet simulations with the updated model physics used in our future ensembles and driven with the same LIG and Pliocene climate forcing used in ref. 8. Simulations without hydrofracturing and ice-cliff calving (a, b, d) correspond to blue lines in Extended Data Fig. 1. Simulations using maximum hydrofracturing and ice-cliff calving parameters (c, e) correspond to red lines in Extended Data Fig. 1. a, Modern (1950) ice-sheet simulation. b, c, LIG simulations run from 130 to 125 kyr ago are shown at 125 kyr ago. Values at the top of each panel are the maximum GMSL contribution between 129 and 128 kyr ago. Values in parentheses are the GMSL contribution at 125 kyr ago. d, e, Warm Pliocene simulations. Values shown are the maximum GMSL achieved during the simulations. Smaller values in parentheses show GMSL contributions after 5,000 model years (Extended Data Fig. 2d). Ice mass gain after peak retreat is caused by post-retreat bedrock rebound and enhanced precipitation in the warm Pliocene atmosphere. Extended Data Fig. 4 RCP8.5 ensembles calibrated with modern and palaeo observations. The fan charts show the time-evolving uncertainty and range around the median ensemble value (black line) in 10% increments. Mean and median ensemble values are shown at 2100. a, Raw ensemble with a range of plausible model parameters based on glaciological observations (Extended Data Table 1). b, The ensemble trimmed with IMBIE4 (1992–2017) estimates of ice mass change. c, The ensemble trimmed with IMBIE rates of ice mass change plus LIG sea-level constraints between 129 and 128 kyr ago34. d, The same as c, except with the addition of maximum mid-Pliocene sea-level constraints35,36 (Extended Data Fig. 1). Future ensembles in the main text (Fig. 1, Table 1) use the combined IMBIE + LIG + Pliocene history matching constraints as shown in d. Extended Data Fig. 5 Future retreat of Thwaites Glacier (TG) and Pine Island Glacier (PIG) with +3 °C global warming. The Amundsen Sea sector of the ice sheet in a nested, high-resolution (1 km) simulation using average calibrated values of hydrofracturing and ice-cliff calving parameters (CALVLIQ = 107 m−1 yr2; VCLIF = 7.7 km yr−1), consistent with those used in CESM1.2.2-forced simulations (Fig. 1h) and CDR simulations (Fig. 3, Table 1). a–c, The ice sheet in 2050. d–f, The ice sheet in 2100. a, d, Ice-sheet geometry and annually averaged ice-cliff calving rates at thick, weakly buttressed grounding lines. The solid line in all panels is the grounding line and the dashed line is its initial position. Note that simulated ice-cliff calving rates are generally much slower than the maximum allowable value of 7.7 km yr−1. Ice shelves downstream of calving ice cliffs are the equivalent of weak mélange, incapable of stopping calving64. b, e, Ice surface speed showing streaming and fast flow just upstream of calving ice cliffs where driving stresses are greatest. c, f, Change in ice thickness relative to the initial state. g, GMSL contributions within the nested domain at model spatial resolutions spanning 1–10 km. Extended Data Fig. 6 Antarctic contribution to sea level under standard RCP forcing. a–c, The fan charts show the time-evolving uncertainty and range around the median ensemble value (thick black line) in 10% increments. The RCP ensembles use the same IMBIE, LIG and Pliocene observational constraints applied to the simulations in Fig. 1. GMSL contributions in simulations without hydrofracturing or ice-cliff calving (excluded from the validated ensembles) are shown for East Antarctica (thin blue line), West Antarctica (thin red line) and the total Antarctic contribution (thin black line). a, RCP2.6; b, RCP4.5; and c, RCP8.5. Extended Data Fig. 7 Long-term magnitudes and rates of GMSL rise contributed by Antarctica. a, Ensemble median (50th percentile) projections of GMSL rise contributed by Antarctica with emissions forcing consistent with the +1.5 °C and +2.0 °C Paris Agreement ambitions, versus a +3.0 °C scenario closer to current NDCs. b, Median (50th percentile) rates of GMSL rise in the same emissions scenarios as in a, illustrating a sharp jump in ice loss in the warmer +3.0 °C scenario after 2060 (also see Fig. 1), and reduced net ice loss before 2060 (black line) caused by increased snowfall. c, Ensemble median (50th percentile) projections of GMSL rise contributed by Antarctica with emissions forcing consistent with standard RCP scenarios, highlighting the potential for extreme GMSL rise under high (RCP8.5) emissions. d, Ensemble median (50th percentile) rates of GMSL rise in the same RCP scenarios as shown in c. Note the much larger vertical-axis scales in c and d relative to a and b. Extended Data Fig. 8 Coupled ice–Earth–sea level model simulations. a–c, Simulations without hydrofracturing and ice-cliff calving processes. d–f, Simulations with hydrofracturing and ice-cliff calving enabled (Methods). GMSL contributions are from the WAIS only. Various Earth viscosity profiles (coloured lines) are compared with the ice-sheet model’s standard ELRA formulation (black line). The most extreme viscosity profile (blue line) assumes a thin lithosphere and very weak underlying mantle, like that observed in the Amundsen sea10, but extended continent-wide. a, RCP2.6 without hydrofracturing or ice-cliff calving. b, RCP2.6 with hydrofracturing and ice-cliff calving. c, RCP4.5 without hydrofracturing or ice-cliff calving. d, RCP4.5 with hydrofracturing and ice-cliff calving. e, RCP8.5 without hydrofracturing or ice-cliff calving. f, RCP8.5 with hydrofracturing and ice-cliff calving. Extended Data Table 1 Model ensemble parameter values. Parameter values used in unique combinations to generate 196 model ensemble members. Blue and red values correspond to the simulations shown by blue and red lines in Extended Data Fig. 1. Thirteen additional combinations extending CALVLIQ to 390 m−1 yr2 and VCLIFF to 26 km yr−1 are shown in black in Extended Data Fig. 1. The average calibrated parameter values based on IMBIE, LIG and Pliocene history matching (Extended Data Fig. 1) are CALVLIQ = 107 m−1 yr2 and VCLIFF = 7.7 km yr−1. The corresponding median values are 105 m−1 yr2 and 7 km yr−1.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-10	Extended Data Fig. 8 Coupled ice–Earth–sea level model simulations. a–c, Simulations without hydrofracturing and ice-cliff calving processes. d–f, Simulations with hydrofracturing and ice-cliff calving enabled (Methods). GMSL contributions are from the WAIS only. Various Earth viscosity profiles (coloured lines) are compared with the ice-sheet model’s standard ELRA formulation (black line). The most extreme viscosity profile (blue line) assumes a thin lithosphere and very weak underlying mantle, like that observed in the Amundsen sea10, but extended continent-wide. a, RCP2.6 without hydrofracturing or ice-cliff calving. b, RCP2.6 with hydrofracturing and ice-cliff calving. c, RCP4.5 without hydrofracturing or ice-cliff calving. d, RCP4.5 with hydrofracturing and ice-cliff calving. e, RCP8.5 without hydrofracturing or ice-cliff calving. f, RCP8.5 with hydrofracturing and ice-cliff calving. Extended Data Table 1 Model ensemble parameter values. Parameter values used in unique combinations to generate 196 model ensemble members. Blue and red values correspond to the simulations shown by blue and red lines in Extended Data Fig. 1. Thirteen additional combinations extending CALVLIQ to 390 m−1 yr2 and VCLIFF to 26 km yr−1 are shown in black in Extended Data Fig. 1. The average calibrated parameter values based on IMBIE, LIG and Pliocene history matching (Extended Data Fig. 1) are CALVLIQ = 107 m−1 yr2 and VCLIFF = 7.7 km yr−1. The corresponding median values are 105 m−1 yr2 and 7 km yr−1. Extended Data Table 2 Antarctic sea-level contributions with alternative maximum ice-cliff calving rates. Ensemble median GMSL contributions using IMBIE, LIG and Pliocene observational constraints (in metres) relative to 2000. Values in parentheses are the 17th–83rd percentiles (likely range). Scenarios refer to the maximum global mean temperature reached relative to pre-industrial (1850) or following extended RCPs, and with the upper bound of the ice-cliff calving parameter (VCLIFF) set at the maximum observed value of 13 km yr−1 (n = 196; Table 1), or alternatively at 11 km yr−1 (n = 168) or 8 km yr−1 (n = 126). Reducing the upper bound of the ice-cliff calving parameter has a relatively small impact on ensemble medians, especially in the near term. The average calibrated value of VCLIFF constrained by observational constraints is 7.7 km yr−1, which severely truncates the upper tail of the distributions when using 8 km yr−1 as the sampling limit. Supplementary Information Uncertainty in surface melt rates and climate forcing As discussed in the main text and Methods, our ice sheet model accounts for rain and meltwater- induced wet crevassing and hydrofracturing that can trigger the sudden loss of buttressing ice shelves, as mean summer temperatures approach and exceed -1oC. As a result, our future simulations (Fig. 1) are sensitive to the timing when substantial quantities of liquid water appear on vulnerable ice shelf surfaces. In our prior work1, RCP8.5 climate forcing used to run future ice sheet simulations produced substantially more melt than indicated by an independent study2, using different regional and global climate models. Here, we compare the updated climate forcing used in this study with those produced by the CMIP5 GCMs used in ref-2 and 22 state-of-the-art CMIP6 GCMs3. Surface melt rates produced by the climate models used in this study (Supplementary Figure 1) are only ~25% as high as those in our previous modeling1, but they remain somewhat higher (especially around the East Antarctic Margin) than those calculated by the empirical temperature- melt relationship used ref.-2. These differences are mainly due to atmospheric temperatures in our model being corrected to account for a cold bias of ~2.9 oC in low elevations over ice surfaces relative to observations4. Similar cold biases of ~2.3 and ~2.4 oC, caused by a deficit of net longwave radiation, are found in the RACMO2 RCM forced by ERA-Interim reanalysis5 and the CESM GCM6. Given the exponential relationship between melt and summer mean (DJF) surface temperature2, our bias-corrected temperatures increase our future melt rates relative to those using uncorrected climate model temperatures, or those using RACMO2 as the bias-correction benchmark2. Additional relatively minor departures from ref-2 are caused by different approaches used to calculate total surface melt from air temperatures. Here, melt rates are calculated by a box model7, using positive degree days for snow and ice melt with standard coefficients8, and accounting for partial refreezing of meltwater1. In our ice sheet model, total surface melt available to influence surface crevassing (Supplementary Figure 1) is the fraction of meltwater not refrozen in the near- surface, plus any rainwater. Under RCP8.5, rainwater in our calculations adds ~10% to total meltwater production in areas of high melt at the end of the 21st century. Supplementary Figure 1: Comparison of surface melt and rainwater production rates. Surface water production rates (rain plus meltwater not refrozen in the near surface, m yr-1) in the last decade of the 21st century under RCP8.5 emissions calculated by the surface mass balance scheme in our ice sheet model. a-f, Melt rates from six global climate models (GCMs)9-13 used in a previous assessment2 are compared with the climate models used in this study (g-i). g-i, Surface melt and rainwater rates produced by the regional climate model (RCM) and GCM used in this study. Spatial patterns differ among the climate models. There is more melt water produced on the Ross and Filchner-Ronne ice shelves in the RCM relative to the other models, but the RCM shows less warming over the Amundsen Sea and most of the East Antarctic margin. The two CESM1.2.2 simulations either ignore (h) or include (i) meltwater (freshwater and iceberg discharge) feedbacks between the GCM and ice sheet model (Fig. 1f). As discussed in the main text, the smaller melt rates in i are the result of a strong negative atmospheric warming feedback caused by sea ice expansion when ice sheet discharge is accounted for in the GCM14. The blue to yellow transition in the color bar (750 mm yr-1) is the approximate meltwater production rate preceding the breakup of the Larsen B ice shelf in 20022. Melt and rainwater required to break up thick (>600 m) ice shelves in our hydrofracturing model is closer to 1,400 mm yr-1. Here, we compare the timing of future summer warming over four regions of the Antarctic margin (Supplementary Figure 2) simulated by the RCM used to force our main ice sheet model ensembles under RCP8.5 (Fig. 1g,h) relative to ERA5 reanalysis15, five CMIP5 climate models following RCP8.5 used in a previous assessment of future surface melt trajectories2, and 22 CMIP6 GCMs3 following SSP5-8516. The regions include three major buttressing ice shelves (Larsen, Ross, Filcher-Ronne), and the Amundsen Sea, where weakly buttressed outlet glaciers, including Thwaites Glacier, are currently thinning and retreating17. The CMIP6 models sampled here include ACCESS-CM2, ACCESS-ESM1-5, BCC-CSM2-MR, CAMS-CSM1-0, CanESM5, CESM2, CESM2-WACCM, EC-Earth3, EC-Earth3-Veg, FGOALS-f3-L, FIO-ESM-2-0, GFDL-CM4, GFDL-ESM4, INM-CM4-8, INM-CM5-0, IPSL-CM6A-LR, MIROC6, MPI-ESM1-2-HR, MPI- ESM1-2-LR, MRI-ESM2-0, NESM3, NorESM2-LM. This comparison places the climate forcing used in our ice sheet simulations within the context of other state-of-the-art climate models, including a variant of CESM (CESM1.2.2-CAM5) used to test the importance of climate-ice sheet feedbacks in Figure 1h. We focus on the summer melt season, because of its connection to ice- shelf breakup.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-11	Here, we compare the timing of future summer warming over four regions of the Antarctic margin (Supplementary Figure 2) simulated by the RCM used to force our main ice sheet model ensembles under RCP8.5 (Fig. 1g,h) relative to ERA5 reanalysis15, five CMIP5 climate models following RCP8.5 used in a previous assessment of future surface melt trajectories2, and 22 CMIP6 GCMs3 following SSP5-8516. The regions include three major buttressing ice shelves (Larsen, Ross, Filcher-Ronne), and the Amundsen Sea, where weakly buttressed outlet glaciers, including Thwaites Glacier, are currently thinning and retreating17. The CMIP6 models sampled here include ACCESS-CM2, ACCESS-ESM1-5, BCC-CSM2-MR, CAMS-CSM1-0, CanESM5, CESM2, CESM2-WACCM, EC-Earth3, EC-Earth3-Veg, FGOALS-f3-L, FIO-ESM-2-0, GFDL-CM4, GFDL-ESM4, INM-CM4-8, INM-CM5-0, IPSL-CM6A-LR, MIROC6, MPI-ESM1-2-HR, MPI- ESM1-2-LR, MRI-ESM2-0, NESM3, NorESM2-LM. This comparison places the climate forcing used in our ice sheet simulations within the context of other state-of-the-art climate models, including a variant of CESM (CESM1.2.2-CAM5) used to test the importance of climate-ice sheet feedbacks in Figure 1h. We focus on the summer melt season, because of its connection to ice- shelf breakup. The evolution of atmospheric warming in the RCM used in our main ensembles (using CCSM4 ocean boundary conditions) is comparable to the subset of CMIP5 GCMs2. When global mean temperatures reach +1.5 oC, +2.0 oC, and +3.0 oC, warming averaged over Antarctica is slightly lagged, reaching +1.48, oC, +1.50 oC, and +1.82 oC, respectively. Both the RCM and CESM1.2.2 used in our study are considerably colder than ERA5 and most CMIP6 GCMs over the main ice shelves. Summer temperatures over the sensitive Larsen and Amundsen Sea regions approach the threshold for producing extensive rain and surface meltwater faster in almost all of the CMIP6 GCMs than either the RCM or CESM1.2.2 (Supplementary Figure 2a-b). Bias correcting the summer temperatures (TDJF) in the climate models relative to the 40-year average of summer temperatures in ERA5 (𝑇 (𝑡) = 𝑇 (𝑡) − 𝑇 + 𝑇 ), substantially reduces the range of simulated temperatures among the climate models, especially in the late 20th and early 21st centuries (Supplementary Figure 2e-h). However, we note that the range of bias-corrected temperatures among the models still expands markedly toward the end of the 21st century. Because of the strong cold bias around the periphery of Antarctica in CESM relative to both observations6 and ERA5 (red vs. orange lines in Supplementary Figure 2), corrected temperatures in CESM (Supplementary Figure 2e-h) show more warming in 2100 than the median of the bias-corrected CMIP6 GCMs. Clearly the wide range of warming rates simulated by these climate models, particularly among CMIP6 GCMs, represents considerable uncertainty in the timing when surface meltwater production and ice shelf loss might begin in the future. The quantified impact of this climatic uncertainty on our ice sheet projections should be explored in future work. Supplementary Figure 2: Future atmospheric warming over Antarctic ice shelves. Summer (DJF) surface (2- meter) air temperature (oC) simulated by five CMIP5 and 22 CMIP6 global climate models (GCMs) over the period 1940-2100. CMIP5 models follow RCP8.5 emissions and CMIP6 models follow SSP5-85. GCM temperatures (averaged over 10-year intervals) are compared with ERA5 reanalysis (orange line), the RCM (RCP8.5) used in our main ensembles (blue crosses) and CESM1.2.2 (RCP8.5; red dashed line) used in ice sheet simulations shown in Figure 1h. The inset shows the model domains corresponding to the Larsen, Ross, and Filchner-Ronne ice shelves, and the Amundsen Sea sector of West Antarctica. a-d, Uncorrected, raw model temperatures averaged over the individual model domains. e-h, Bias corrected temperatures using ERA5. Blue crosses show the RCM temperatures at specific times (1950, 2000, and when effective atmospheric CO2 reaches 2 and 4 times preindustrial levels). Ice shelf hydrofracturing in compressional flow regimes It is conceivable that in regions of compressional ice-shelf flow, liquid water flowing on the surface might tend to reach the margins and run off, instead of penetrating into crevasses and causing hydrofracture. This potential influence of compressional ice flow on hydrofracturing is tested by modifying the model’s wet crevassing (hydrofracturing) scheme (see Methods). In this case, the total meltwater production rate R is reduced by ×0.1 as a function of the local ice convergence rate (yr-1) at convergences >0.01, ramping to ×1 where convergence is zero. We find that reducing wet crevasse penetration in regions of convergent flow has little influence on our continental-scale results (Supplementary Figure 3). In climate scenarios with minimal surface melt (RCP2.6), Antarctic ice loss is dominated by WAIS retreat in response to ocean- driven thinning of ice shelves and the associated reduction in buttressing. In such instances, the influence of hydrofracturing is minimal and modifications to our wet crevassing scheme are inconsequential. Under more extreme future warming scenarios (RCP8.5), shelf loss is largely driven by massive meltwater production and the sudden onset of widespread meltwater-enhanced calving (hydrofracturing). In the model, this hydrofracturing begins near the calving fronts where the ice is thinnest, convergence and buttressing are minimal18, and air temperatures (melt rates) are highest. Once initiated, meltwater-induced calving reduces convergence and compressional flow in the ice upstream and the meltwater enhanced calving propagates, resulting in the complete loss of major ice shelves, despite the reduction of 𝑑4 in convergent flow regimes. Extending these results with a more sophisticated, physically based, time-dependent19 hydrofracturing scheme is the subject of ongoing work. However, these results combined with the relatively high melt rates required to trigger destruction of ice shelves like the Larsen B, add confidence that the model formulation used in our main ensembles is reasonable. Supplementary Figure 3: Global mean sea level contributions from Antarctica with a modified hydrofracturing scheme. Simulations follow two future greenhouse gas emissions scenarios, using our nominal model formulation of hydrofracturing used throughout the main text (solid lines), compared with an alternative formulation reducing meltwater influence on crevasse penetration in convergent (compressive) flow regimes (dashed lines). Reformulation of buttressing at grounding lines The hybrid ice sheet model used here heuristically blends vertically integrated shallow ice/shallow shelf approximations (SIA/SSA)20, with the seaward ice flux at grounding lines imposed as a boundary condition according to an analytical expression relating ice flux to ice thickness21. This expression includes a term θ representing buttressing by ice shelves, i.e., the amount of back stress caused by pinning points or lateral forces on the ice shelf further downstream. The buttressing factor θ is defined as the ratio of vertically averaged horizontal deviatoric stress normal to the grounding line, relative to its value if the ice shelf was freely floating with no back stress. The analysis for grounding-line flux and buttressing in ref.6 is limited to one-dimensional flowline geometry. In our standard model20, the expression is applied across individual one-grid-cell-wide segments separating pairs of grounded and floating grid cells, so that the orientation of each single- cell “grounding-line” segment is parallel to either the x or the y axis. Although this is consistent with the one-dimensional character of the formulation in ref.21, it neglects the actual orientation of the real, slightly wider-scale grounding line, and results in non-isotropic θ values for u and v staggered-grid velocities.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-12	Supplementary Figure 3: Global mean sea level contributions from Antarctica with a modified hydrofracturing scheme. Simulations follow two future greenhouse gas emissions scenarios, using our nominal model formulation of hydrofracturing used throughout the main text (solid lines), compared with an alternative formulation reducing meltwater influence on crevasse penetration in convergent (compressive) flow regimes (dashed lines). Reformulation of buttressing at grounding lines The hybrid ice sheet model used here heuristically blends vertically integrated shallow ice/shallow shelf approximations (SIA/SSA)20, with the seaward ice flux at grounding lines imposed as a boundary condition according to an analytical expression relating ice flux to ice thickness21. This expression includes a term θ representing buttressing by ice shelves, i.e., the amount of back stress caused by pinning points or lateral forces on the ice shelf further downstream. The buttressing factor θ is defined as the ratio of vertically averaged horizontal deviatoric stress normal to the grounding line, relative to its value if the ice shelf was freely floating with no back stress. The analysis for grounding-line flux and buttressing in ref.6 is limited to one-dimensional flowline geometry. In our standard model20, the expression is applied across individual one-grid-cell-wide segments separating pairs of grounded and floating grid cells, so that the orientation of each single- cell “grounding-line” segment is parallel to either the x or the y axis. Although this is consistent with the one-dimensional character of the formulation in ref.21, it neglects the actual orientation of the real, slightly wider-scale grounding line, and results in non-isotropic θ values for u and v staggered-grid velocities. Alternatively, a more rigorous, isotropic, treatment of 𝜃 can been implemented, by applying the expression in ref.21 to normal flow across a more realistic grounding-line orientation not constrained to one or the other grid axes, following equations 2 and 6 in ref.22 The alternative model treatments of 𝜃 are represented schematically by insets in Supplementary Figure 4a,b. We find that the new treatment of 𝜃 substantially improves the model’s performance23 in the idealized, relatively narrow fjord-like setting of the Marine Ice Sheet Model Intercomparison Project+ (MISMIP+)24, with regards to the transient pace of grounding line retreat and re-advance when compared with models using higher order or full-stokes treatments of englacial stresses. Our new results fall well within the envelope of the multi-model range in the MISMIP+ intercomparison24 (Supplementary Figure 4a,b). In contrast, at the continental scale the new, more rigorous treatment of 𝜃 has a very small effect on the pace of retreat (Supplementary Figure 4c,d), presumably because the dynamics of wide, major Antarctic outlets are adequately represented with the 1-dimensional formulation. The new treatment and further results are described in detail in ref.23. Supplementary Figure 4: Effect of reformulated buttressing. a, Time-evolving, mid-channel grounding line position in Experiment Ice1 of the MISMIP+ model intercomparison23, in an idealized, narrow fjord-like setting with reverse-sloped bedrock and channel width of 80 km (modified from Fig. 8b of ref.24). Blue and yellow colors show the response to oceanic basal melt rates applied at time 0, and red colors show the recovery after the basal melt rates are re-zeroed at year 100. Circles and squares show results of our standard model using the old θ method, with model resolution of 1 km and 10 km respectively. Shaded regions and solid lines show the envelope and mean of multiple other models in the MISMIP+ intercomparison (those using a similar Weertman-type basal sliding scheme). Our standard model retreats faster than other models in the intercomparison. b, Results with our model using the new, more rigorous θ method described above and 2-km model resolution. This substantially improves model performance relative to the other MISMIP+ models shown in a. Schematic diagrams representing the old versus new θ methods are shown at the bottom left of a and b, with the model grid represented by the thin black lines, arrows showing ice velocities across the grounding line, and the “actual” grounding line in the new method shown in grey. c and d, Continental-scale Antarctic simulations under RCP8.5 forcing, showing equivalent global mean sea level rise versus time corresponding to net Antarctic ice loss, without ice-cliff calving in c, and with ice-cliff calving in d. Unlike the idealized confined-fjord setting in a and b, these continental-scale Antarctic simulations show only small differences in net ice loss using the old vs. new θ methods. Without ice-cliff calving in c, the model using the new θ method (red curve) yields slightly faster ice loss after ~2300, but the differences are small and not important for the purposes of this paper. With ice-cliff calving in d, faster ice loss overwhelms any differences due to the θ method. The standard θ method (blue curves) is used in our main ensembles. Statistical emulation of model ensembles Here, we demonstrate the statistical robustness of the sea level estimates made with the ensembles presented in the main text. While the parameter sampling used in the ensembles is more dense than in our previous work2, many parameter values intermediate to the training set (Table 1) have not been tested, and the sea level projections are not fully probabilistic (i.e. intermediate values are implicitly ascribed zero-probability). To address this, we develop and sample from an Antarctic Ice Sheet model emulator, which is continuous across the prior range of the training data and may be used to generate a much larger ensemble of simulations. We also evaluate the importance of observational (modern and paleo) constraints for limiting emulated probabilistic projections of future sea level rise from Antarctica. Physically based and statistical emulation techniques have been used in several studies of sea level rise and climate change25,26 and specifically to calibrate complex models27,28. Our methodology has similarities to the recent methods of ref.29. We use Gaussian Process (GP) regression30 to construct a statistical emulator designed to mimic the behavior of the numerical ice-sheet model. GP regression is a non-parametric supervised machine learning technique which allows one to map model inputs (e.g., model parameters) to outputs (here, ice volume changes in global-mean sea level equivalent). In contrast to individual deterministic ice-sheet model simulations, GP regression is advantageous because the input parameter space and output prediction space are continuous, with emulation uncertainty inherently estimated for each output. For a set of untested inputs, the corresponding output and its uncertainty can be determined in a fraction of the time it takes to perform a single ice sheet model simulation. A full description and discussion of the emulator and its calibration are provided in a forthcoming manuscript31. The emulator is trained separately on two of the 196-member ensembles described in the main text: the Last Interglacial ensemble and the RCP8.5 scenario. We model the Antarctic ice-sheet contributions to global mean sea level (𝑓) as the sum of two terms, each with a mean-zero Gaussian process prior: 𝑓(𝜃1,𝜃2,𝑡) = 𝑓1(𝜃1,𝜃2) + 𝑓2(𝜃1,𝜃2,𝑡) (S1) The first term represents a parameter-specific intercept, the latter the temporal evolution of the contribution. The priors for each term are specified as: 𝑓 (𝜃 ,𝜃 )~𝒢𝒫(0,𝛼9𝐾 (𝜃 ,𝜃 ,𝜃@,𝜃@ ;l )) (S2) 𝑓 (𝜃 ,𝜃 ,𝑡)~𝒢𝒫(0,𝛼9𝐾 (𝜃 ,𝜃 ,𝜃@,𝜃@ ;l )𝐾 (𝑡,𝑡@ ;𝜏)) (S3) and where 𝜃7is normalized VMAX, 𝜃9 is normalized CREVLIQ, 𝛼E are amplitudes, lE are characteristic length scales in normalized parameter spaces, τ is the time scale and 𝐾 is a specified correlation function. Because the LIG training data is evaluated at a single time point, there is no temporal term and f2 is excluded in the LIG emulator construction. Each 𝐾E is defined to be a Matérn covariance function with a specified smoothness parameter, 𝛾 = 5/2, which governs how responsive the covariance function is to sharp changes in the training data30. Optimal hyperparameters (𝛼E , lE, and τ) of the GP model are found by maximizing the log- likelihood, given the training simulations (Supplementary Table 1). The optimized model can then be conditioned on the training data to predict LIG and RCP8.5 simulation results for parameter values intermediate to those run with the full ice sheet model.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
a9cea5dae349-13	𝑓(𝜃1,𝜃2,𝑡) = 𝑓1(𝜃1,𝜃2) + 𝑓2(𝜃1,𝜃2,𝑡) (S1) The first term represents a parameter-specific intercept, the latter the temporal evolution of the contribution. The priors for each term are specified as: 𝑓 (𝜃 ,𝜃 )~𝒢𝒫(0,𝛼9𝐾 (𝜃 ,𝜃 ,𝜃@,𝜃@ ;l )) (S2) 𝑓 (𝜃 ,𝜃 ,𝑡)~𝒢𝒫(0,𝛼9𝐾 (𝜃 ,𝜃 ,𝜃@,𝜃@ ;l )𝐾 (𝑡,𝑡@ ;𝜏)) (S3) and where 𝜃7is normalized VMAX, 𝜃9 is normalized CREVLIQ, 𝛼E are amplitudes, lE are characteristic length scales in normalized parameter spaces, τ is the time scale and 𝐾 is a specified correlation function. Because the LIG training data is evaluated at a single time point, there is no temporal term and f2 is excluded in the LIG emulator construction. Each 𝐾E is defined to be a Matérn covariance function with a specified smoothness parameter, 𝛾 = 5/2, which governs how responsive the covariance function is to sharp changes in the training data30. Optimal hyperparameters (𝛼E , lE, and τ) of the GP model are found by maximizing the log- likelihood, given the training simulations (Supplementary Table 1). The optimized model can then be conditioned on the training data to predict LIG and RCP8.5 simulation results for parameter values intermediate to those run with the full ice sheet model. Taking uniform priors over the input parameters that are consistent with those used by the numerical ice sheet model (i.e., CREVLIQ ~ 𝑈(0,195), VMAX ~ 𝑈(0,13) ) we then apply a Bayesian updating approach to estimate posterior probability distributions for these parameters, conditional upon observational constraints. To do this, we first take 20,000 Latin Hypercube samples from the prior distributions, then weight these based on two different constraints: a uniform LIG distribution, 𝑈(3.1 m, 6.1 m), and a uniform distribution of IMBIE32 trends, 𝑈(0.15 mm yrS7, 0.46 mm yrS7), over 1992-2017. As in the main text, the LIG constraint is based on the maximum Antarctic ice loss between 129 ka and 128 ka, equivalent to the ice loss at 128 ka. The results are posterior probabilities of CREVLIQ/VMAX pairs for each given constraint. These posteriors of CREVLIQ/VMAX are then used to estimate the posterior distributions of AIS sea-level contributions over time. The 5th, 50th, and 95th percentiles of these posterior distributions (in 2100 under RCP8.5) with no constraints, IMBIE constraints only, LIG constraints only, and combined IMBIE and LIG constraints are presented in Supplementary Table 2. The probability distribution over time from 20,000 samples of the combined (IMBIE and LIG) constrained emulator is shown in Supplementary Figure 5b. In Supplementary Figure 6 we show the emulated probability distributions in 2100, subject to each constraint and compared to a histogram of the training set. We note that the emulation results provided here are not directly comparable to the calibrated ensembles in the main text, because those ensembles add a third training constraint based on Pliocene sea level. Rather, these results are intended to complement the main paper by comparing projections that ignore the Pliocene constraints, and to demonstrate that statistically robust GP emulation compares favorably to the binary scoring approach used in Figure 1. Emulated distributions closely resemble that of the 196-member training ensemble, with some notable but minor differences that are ascribable to sampling limitations in the original ensemble (e.g., the conditioned training ensemble has 10 simulations at or below its 5th percentile, whereas the constrained ensemble has 1000). As with the training ensemble, the emulated probability distribution without constraints is positively skewed, with a long upper tail that stretches to 63 cm in the 95th percentile by 2100. We find that the prior distribution (Supplementary Figure 6) is qualitatively similar to the IMBIE- constrained distribution, and likewise the LIG-constrained distribution is similar to the IMBIE+LIG-constrained distribution. These results indicate that the IMBIE uniform distribution is not an adequately restrictive constraint on the emulator, although it does slightly reduce the upper bound of projections in 2100 by ~3 cm, shifting the distribution towards lower sea-level contributions. The IMBIE-constrained emulator is consistent with the conclusions of ref.33 that additional information from the satellite record is of limited utility (because simulated ice-mass losses by the end of the 21st century are only weakly correlated with loss trends at the beginning of the 21st century). In contrast, the uniform LIG constraint is more informative for calibrating emulated future projections of Antarctic sea-level contributions. Samples from parameter sets with CREVLIQ<45 and VMAX<4 fall outside the uniform LIG constraint, and the associated likelihoods are near or actually zero (not shown). Conversely, the VMAX/CREVLIQ parameter pairs above these values have greater (non-zero) likelihoods and the associated samples (which typically have higher RCP8.5 emulated sea-level contributions) are accordingly given more weight in the posterior. The resulting posterior distribution shifts towards the high end of the projections, with median projections in 2100 of 34 cm for the LIG-only constraint and 32 cm for the combined constraint distribution. Furthermore, the LIG-constrained distribution posterior has a narrower range than the prior starting in ~2060 and through 2100 (Supplementary Figure 5), demonstrating that future projections are less uncertain when the LIG constraint is applied. Importantly, we find the median of GP emulation results is within 1 cm of the projected GMSL contribution in 2100 when compared to the training ensemble (binary scoring) approach used in the main text (Supplementary Table 2, Extended Data Figure 4). The addition of a third training constraint (Pliocene sea level) in the main text slightly increases the central estimate of Antarctica’s GMSL contribution in 2100 from 32 cm (Supplementary Table 2) to 34 cm (Table 1), by further reducing the likelihood of both low and high VMAX/CREVLIQ parameter values. Supplementary Table 1: Optimized hyperparameters of the GP emulator found by maximizing the log- likelihoods, given the training ensembles Supplementary Table 2: The median and 5th / 95th percentiles of projected Antarctic ice-sheet contributions to GMSL in 2100 (m) Supplementary Figure 5: Emulated global mean sea level contributions from Antarctica. Fan charts of the range around the median (black line) in 10% increments from 20,000 RCP8.5 scenario emulator samples, from a the prior and b the posterior calibrated with combined LIG and IMBIE trend constraints using a Bayesian updating approach. Supplementary Figure 6: Probabilistic projections of global mean sea level contributions from Antarctica in 2100 under RCP8.5. Projections from 20,000 emulator samples (lines) weighted by different observational constraints. Shown are the prior distribution with no constraints (black), and distributions under the LIG uniform constraint (red), the IMBIE trend uniform constraint (cyan), and the combined LIG and IMBIE trend constraints (blue). Emulated distributions are shown using a kernel density estimation assumes a Silverman bandwidth divided by 2 (to prevent over-smoothing)34. The training ensemble from the main text is shown as a histogram (light blue) scaled for comparison to the emulated distributions.	https://sealeveldocs.readthedocs.io/en/latest/deconto21.html
d3958eec8aeb-0	Li et al. (2023) Title: Climate model differences contribute deep uncertainty in future Antarctic ice loss Corresponding author: Dawei Li Citation: Li, D., DeConto, R. M., & Pollard, D. (2023). Climate model differences contribute deep uncertainty in future Antarctic ice loss. Science Advances, 9(7), eadd7082. doi: 10.1126/sciadv.add7082 URL: https://www.science.org/doi/10.1126/sciadv.add7082 Abstract Future projections of ice sheets in response to different climate scenarios and their associated contributions to sea level changes are subject to deep uncertainty due to ice sheet instability processes, hampering a proper risk assessment of sea level rise and enaction of mitigation/adaptation strategies. For a systematic evaluation of the uncertainty due to climate model fields used as input to the ice sheet models, we drive a three-dimensional model of the Antarctic Ice Sheet (AIS) with the output from 36 climate models to simulate past and future changes in the AIS. Simulations show that a few climate models result in partial collapse of the West AIS under modeled preindustrial climates, and the spread in future changes in the AIS’s volume is comparable to the structural uncertainty originating from differing ice sheet models. These results highlight the need for improved representations of physical processes important for polar climate in climate models. Introduction Fluctuations of global mean sea level (GMSL) over the past few million years have been dominated by glacial-interglacial cycles. During the Last Glacial Maximum (21 ka ago), formation of the Laurentide Ice Sheet in North America and the Fennoscandian Ice Sheet in Northern Europe and, to a lesser extent, the expansion of Greenland and Antarctic ice sheets contributed to a ∼120-m drop in GMSL relative to today (1). GMSL during the interglacials was comparable to, although sometimes higher than, the present-day sea level. In contrast, the Last Interglacial (LIG; 129 to 116 ka ago) was not much warmer than preindustrial (−0.4° to 1.3°C) (2), but GMSL was 6 to 9 m higher (3, 4), of which ∼3.1 to 6.1 m may have been contributed by the Antarctic Ice Sheet (AIS) (2, 5). While the LIG is not a precise analog of future sea level, as Earth’s orbital parameters and polar insolation forcing likely played an important role (6), it still hints at a worrisome potential for future sea level rise (SLR) given the ∼1.2°C warming that has already occurred. At a rate of 3.58 mm year−1 over the period 2006–2015, the rise in GMSL is accelerating and is now dominated by melting of land ice, including glaciers and ice sheets (7). Projections of SLR over the 21st century and beyond have been made for various emission scenarios, but they are subject to substantial uncertainty, which becomes greater in scenarios with higher greenhouse gas emissions and hence more warming. Under the assessment by the Sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC-AR6), following the high greenhouse gas emission scenario of Shared Socioeconomic Pathway 5-8.5 (SSP5-8.5) (8), SLR is likely to reach 0.63 to 1.01 m by 2100, of which 0.03 to 0.34 m is expected to be contributed by the AIS (9). In contrast, another recent statistical analysis of multimodel ice sheet simulations indicates a smaller future contribution from Antarctica (−0.01 to 0.1 m likely range) under similar SSP5-8.5 forcing (10), highlighting ongoing uncertainty. As the largest source of uncertainty of SLR beyond 2100, ice loss from the AIS has evaded robust projections. Much of this uncertainty can be attributed to the diversity of numerical ice sheet models (ISMs), which differ not only in spatial resolution, equations of stress balance, numerical schemes, and initialization methods but also in their treatment of key physical processes including grounding line migration, calving, surface mass balance (SMB), and basal processes. The associated uncertainty in the AIS’s response to climate warming has been explored in a number of model intercomparisons, such as the Ice Sheet Model Intercomparison Project (ISMIP; the most recent phase being ISMIP6) (11–16). These projects have provided valuable insights by focusing on the difference in ISMs’ response to prescribed changes in climate boundary conditions. For instance, ISM initialization experiments show good agreement in the AIS’s response to changes in SMB, but a much greater spread in the response to ice shelf basal melt (13). Designed to assess how responses differ across the spectrum of ISMs under a nonexhaustive suite of modeled climates, ISMIP6 drove a variety of ISMs with climate fields from a subset of Coupled Model Intercomparison Project Phase 5 (CMIP5) models. As the succeeding generation CMIP6 model output became available to the climate research community, ISM intercomparison projects would benefit from using a more comprehensive set of climate models to take into account a wider and up-to-date range of intermodel uncertainty. It has been recognized that structural differences between climate models can produce divergent quasi-equilibrium states for the AIS in experiments where ISMs are forced by the output of climate models (17); however, there has been no comprehensive assessment of the uncertainty in projected future states of the AIS using the latest generation climate models. In existing ISM intercomparison projects, decentralized model development gives rise to ISMs across a wide spectrum, while often a small subset of available climate models is included to provide climate boundary conditions. Here, we take a complementary approach to evaluate the uncertainty in projected change of the AIS and its contribution to GMSL by driving a single three-dimensional ISM (18) with climate fields from 36 climate models in the CMIP6 archive. The ISM is fine-tuned so that it closely simulates the observed state of the AIS and rates of ice loss under present-day climate conditions (Experiment OBS_INV) (5). Assuming that the ISM is a “perfect” representation of the real AIS, the spread in ISM output reflects the uncertainty associated with past and future climate changes simulated by these CMIP6 models. Such “perfect model” framework has been widely used in climate research to evaluate model predictability, the performance of bias correction and statistical downscaling, etc. (19, 20) A series of ISM experiments under this perfect model framework, as documented in Table 1, were carried out to assess the effect of biases in modeled climate on the AIS’s equilibrium state and the uncertainty in past and future trajectories of the AIS due to divergent climate sensitivities displayed by CMIP6 models. We also provide additional ISM experiments to discuss an ice sheet instability mechanism, the effect of observed multidecadal warming of Antarctic subsurface ocean on the AIS, and an alternative modeling strategy with the ISM calibrated per climate model. A series of ISM experiments under this perfect model framework, as documented in Table 1, were carried out to assess the effect of biases in modeled climate on the AIS’s equilibrium state and the uncertainty in past and future trajectories of the AIS due to divergent climate sensitivities displayed by CMIP6 models. We also provide additional ISM experiments to discuss an ice sheet instability mechanism, the effect of observed multidecadal warming of Antarctic subsurface ocean on the AIS, and an alternative modeling strategy with the ISM calibrated per climate model. Table 1. List of experiments. BSC, basal sliding coefficient; OMF, ocean melt factor; BC, bias-corrected; MICI, marine ice cliff instability.*PD observation: Present-day (PD) climatological mean (averaged over 1981–2010), including monthly surface air temperature and precipitation from ERA5 and annual 400-m ocean temperature from World Ocean Atlas 2018 (WOA18).†PD: PD observed state of the AIS specified in Bedmap2.‡PI: Preindustrial climate. However, because not all CMIP6 models used in this study have a PI control run, climate fields averaged over the period 1850–1869 of historical runs are used here as approximately preindustrial.§1850–2100: Historical (1850–2014) and SSP5-8.5 (2015–2100) scenarios are combined to provide climate forcings for each year over the period 1850–2100. Results Near-equilibrium AIS under raw CMIP6 climates	https://sealeveldocs.readthedocs.io/en/latest/li23sciadv.html
d3958eec8aeb-1	A series of ISM experiments under this perfect model framework, as documented in Table 1, were carried out to assess the effect of biases in modeled climate on the AIS’s equilibrium state and the uncertainty in past and future trajectories of the AIS due to divergent climate sensitivities displayed by CMIP6 models. We also provide additional ISM experiments to discuss an ice sheet instability mechanism, the effect of observed multidecadal warming of Antarctic subsurface ocean on the AIS, and an alternative modeling strategy with the ISM calibrated per climate model. A series of ISM experiments under this perfect model framework, as documented in Table 1, were carried out to assess the effect of biases in modeled climate on the AIS’s equilibrium state and the uncertainty in past and future trajectories of the AIS due to divergent climate sensitivities displayed by CMIP6 models. We also provide additional ISM experiments to discuss an ice sheet instability mechanism, the effect of observed multidecadal warming of Antarctic subsurface ocean on the AIS, and an alternative modeling strategy with the ISM calibrated per climate model. Table 1. List of experiments. BSC, basal sliding coefficient; OMF, ocean melt factor; BC, bias-corrected; MICI, marine ice cliff instability.*PD observation: Present-day (PD) climatological mean (averaged over 1981–2010), including monthly surface air temperature and precipitation from ERA5 and annual 400-m ocean temperature from World Ocean Atlas 2018 (WOA18).†PD: PD observed state of the AIS specified in Bedmap2.‡PI: Preindustrial climate. However, because not all CMIP6 models used in this study have a PI control run, climate fields averaged over the period 1850–1869 of historical runs are used here as approximately preindustrial.§1850–2100: Historical (1850–2014) and SSP5-8.5 (2015–2100) scenarios are combined to provide climate forcings for each year over the period 1850–2100. Results Near-equilibrium AIS under raw CMIP6 climates In the experiment set CMIP6_RAW_PI_CTL (Table 1 and Methods), we investigate the effect of differing modeled preindustrial climates on the equilibrium state of the AIS. The AIS is initialized from the end state of an inverse simulation (21), which infers the characteristics of the AIS bed required to simulate a realistic AIS under the present-day observational climate (Experiment OBS_INV). Here, “present day” refers to the 3-year period 1981–2010, within the historical period per the CMIP6 protocol (1850–2014); “preindustrial” is defined as the 20-year period 1850–1869, under the assumption that, within the first 20 years, anthropogenic forcings had not changed the climate substantially from its pre-1850 state. A comparison of simulated present-day summer [December to February (DJF)] near-surface air temperature (T2m) and 400-m annual ocean potential temperature (θ400m) reveals substantial differences between CMIP6 models (Fig. 1). Deviations of modeled temperatures from observation display distinct spatial heterogeneities. For instance, ACCESS-ESM-1-5 has a warm bias in DJF T2m over the ice sheet but a cold bias over the ocean relative to ERA5 (Fig. 1A). Modeled θ400m can be too warm in one ocean sector but too cold in others (Fig. 1B). In addition to subsurface ocean temperatures, air temperature also strongly affects the stability of ice shelves in summer, when most surface melt occurs under the present-day climate. Rheological properties of the glacial ice are, in contrast, affected mainly by annual mean temperature, because seasonal variations in temperature only penetrate ∼1 m into the ice, a tiny fraction of the typical thickness of the ice sheet or ice shelves. We find substantial intermodel variation in simulated T2m with a warm bias over the ice shelves as much as 8°C in some models (Fig. 1A). Figure 1: Difference between modeled climate fields and observations. (A) Difference in January surface air temperature between 36 CMIP6 climate models and observations (ERA5). (B) Difference in annual mean 400-m ocean temperature between 36 CMIP6 climate models and observations [World Ocean Atlas 2018 (WOA18)]. Climate fields from CMIP6 models and observations are averaged over the period 1981–2010. The difference between simulated annual precipitation and that from ERA5 reanalysis dataset (22) generally shows patterns consistent with surface air temperature biases, with warmer models experiencing greater precipitation and vice versa (see the Supplementary Materials). The difference between modeled and observed subsurface ocean temperature at 400 m is less notable, but it is still substantial, as ice shelf basal melt rates are sensitive to ocean temperatures. Under the parameterization scheme used in the ISM, the basal melt rate has a quadratic dependence on θ400m (see Methods) so even modest intermodel differences can substantially change the basal mass balance of ice shelves with important consequences for the buttressed ice upstream. A myriad of quasi-equilibrium states of the AIS are reached in 10,000-year runs forced by 36 CMIP6 models’ preindustrial climates (Fig. 2A). In 17 simulations, near-complete collapse of the West AIS (WAIS) contributes >3 m of the GMSL rise (Fig. 3I). In addition, climate forcing from three models with a strong warm bias produces substantial retreat of the East AIS (EAIS), contributing >15 m of the GMSL rise. Climate models with a cold bias in subsurface temperature θ400m, in contrast, generally drive the ISM toward a quasi-equilibrium state with an expanded ice sheet and seaward advance of grounding lines onto continental shelves (Figs. 2A and 3). Figure 2: Simulated ice sheets under CMIP6 preindustrial climates. (A) Ice thickness by the end of the control runs forced by raw preindustrial climates from 36 CMIP6 models (Experiment CMIP6_RAW_PI_CTL). (B) Same as (A) but for simulations forced by bias-corrected CMIP6 preindustrial climates (Experiment CMIP6_BC_PI_CTL). Figure 3: Intermodel differences in CMIP6 climates and simulated AIS. Scatter plots show intermodel differences in modeled Antarctic climate and resulting states of the AIS forced by 36 CMIP6 climate models, represented by markers of different shapes and colors. (A to D) DJF near-surface (2-m) air temperature (T2m) (°C, vertical axes) averaged over ice shelves against Antarctic coastal ocean potential temperature at 400-m (°C, horizontal axes), (E to H) area of floating ice (10 × 106 km2, vertical axes) against area of grounded ice (106 km2, horizontal axes), and (I to L) contributions to GMSL change from the West AIS (WAIS) (m, vertical axes) against the East AIS (EAIS) (m, horizontal axes). (A) and (B) shows the raw (uncorrected) and bias-corrected preindustrial climates, respectively; (C) and (D) show the changes relative to the 1850–1869 period by year 2020 and 2100. Similarly, (E) and (F) and (I) and (J) show ISM results forced by the raw and bias-corrected climates, respectively; (G) and (H) and (K) and (L) show ice sheet changes from the initial preindustrial state at 2020 and 2100, respectively, forced by bias-corrected climates (Experiment CMIP6_BC_1850-2100). Gray squares show 16 to 84 percentile range of intermodel spread. These ISM control experiments highlight the room for improvement in CMIP6 models’ performance in the Antarctic region. The simulations also corroborate the established wisdom that the WAIS is especially sensitive to ocean temperatures: For example, the climate model NESM3 has a mean circum-Antarctic warm bias of 1.5°C in θ400m (Fig. 3A), but this is sufficient to drive a partial collapse of the WAIS in the ISM on long time scales (Fig. 3I).	https://sealeveldocs.readthedocs.io/en/latest/li23sciadv.html
d3958eec8aeb-2	Figure 3: Intermodel differences in CMIP6 climates and simulated AIS. Scatter plots show intermodel differences in modeled Antarctic climate and resulting states of the AIS forced by 36 CMIP6 climate models, represented by markers of different shapes and colors. (A to D) DJF near-surface (2-m) air temperature (T2m) (°C, vertical axes) averaged over ice shelves against Antarctic coastal ocean potential temperature at 400-m (°C, horizontal axes), (E to H) area of floating ice (10 × 106 km2, vertical axes) against area of grounded ice (106 km2, horizontal axes), and (I to L) contributions to GMSL change from the West AIS (WAIS) (m, vertical axes) against the East AIS (EAIS) (m, horizontal axes). (A) and (B) shows the raw (uncorrected) and bias-corrected preindustrial climates, respectively; (C) and (D) show the changes relative to the 1850–1869 period by year 2020 and 2100. Similarly, (E) and (F) and (I) and (J) show ISM results forced by the raw and bias-corrected climates, respectively; (G) and (H) and (K) and (L) show ice sheet changes from the initial preindustrial state at 2020 and 2100, respectively, forced by bias-corrected climates (Experiment CMIP6_BC_1850-2100). Gray squares show 16 to 84 percentile range of intermodel spread. These ISM control experiments highlight the room for improvement in CMIP6 models’ performance in the Antarctic region. The simulations also corroborate the established wisdom that the WAIS is especially sensitive to ocean temperatures: For example, the climate model NESM3 has a mean circum-Antarctic warm bias of 1.5°C in θ400m (Fig. 3A), but this is sufficient to drive a partial collapse of the WAIS in the ISM on long time scales (Fig. 3I). Despite that unrealistic AIS geometries were simulated by under many CMIP6-modeled climates, these experiments are not designed for evaluating CMIP6 climate models’ performance over Antarctica. Here, we have regarded reanalysis datasets as the “observational truth,” serving as a reference climate for calibrating the ISM model parameters, including the ocean melt rate coefficient and the basal sliding coefficients (BSCs). Because of the scarcity of observations available for Antarctica, reanalysis datasets may have substantial departures from the true climate state in some regions. These control experiments are run for 10,000 years, allowing the AIS to reach a quasi-equilibrium, but it is not clear how close the AIS was to such a state before the dawn of the Industrial Revolution, when anthropogenic climate forcing started to emerge. Furthermore, Earth’s orbital parameters drift substantially over 10,000 years, and the AIS is expected to respond accordingly. The availability of CMIP6 historical simulations, dating back to only 1850, makes the quasi-equilibrium assumption necessary for conducting an intercomparison of the AIS forced by different climate models, but the intrinsic uncertainty in the AIS’s natural variability cautions against judging these climate models based on their respective ISM simulations. Near-equilibrium AIS under bias-corrected CMIP6 climates The diverse polar climates simulated by CMIP6 models render the above approach unsuitable for assessing the uncertainty in the AIS’s future trajectory. An alternative strategy is to bias-correct CMIP6 climates against present-day observations. Spatially varying biases in CMIP6 monthly climate fields are calculated and subtracted from the raw model output (Methods). In this approach, we essentially remove CMIP6 models’ biases in present-day climates and focus on their changes from the reference period. However, because of CMIP6 models’ differing sensitivities to anthropogenic forcings, bias-corrected preindustrial climates for Antarctica still display significant intermodel variations, showing a large intermodel spread in preindustrial T2m and θ400m. Mean DJF T2m over ice shelves is up to 4.5 K lower than the present-day reference period. Simulated preindustrial annual mean θ400m averaged along the Antarctic coast is up to 1 K lower than present-day (Fig. 3B). Note that warming proceeds at a faster pace in the atmosphere than the subsurface ocean, underscoring complex processes at play in the Southern Ocean, where vigorous convection and upwelling around Antarctica may suppress the pace of warming (23). In the experiment set CMIP6_BC_PI_CTL, the ISM is initiated from the present-day AIS and runs for 15,000 years until it reaches a quasi-equilibrium but forced with CMIP6 bias-corrected preindustrial climates. Compared with the initial state, the modeled preindustrial AIS in quasi-equilibrium generally shows thinning of the EAIS, consistent with reduced snowfall in a colder preindustrial climate. Under most CMIP6 models, ice shelves around the AIS expand, which is also consistent with lower preindustrial ocean temperatures. Intermodel differences in ice volume of the EAIS and the WAIS are 0.6- and 1.2-m sea level equivalent (SLE), respectively (Fig. 3I). Projected changes in Antarctic climate and the AIS Under the SSP5-8.5 scenario, all CMIP6 models included in this study show substantial warming relative to preindustrial in both T2m and θ400m over this century (Figs. 3D and 4). DJF T2m averaged over all Antarctic ice shelf surfaces increases by 0.3 to 2.6 K in 2020 and by 1 to 10 K in 2100; θ400m averaged along the Antarctic coast increases by −0.1 to 0.5 K in 2020 and up to 1.6 K in 2100 (Fig. 3, C and D). The amplitude of warming in climate models reveals dependence on the state of simulated reference climate. For instance, CAMS-CSM1-0 and MIROC6 are among the models with the greatest warm bias in T2m (Figs. 1A and 3A), but they also show the least warming (<2 K) by 2100. One of the contributing factors might be that, in preindustrial climates, these models are mostly free of austral summer sea ice, reducing the strength of sea ice-albedo feedback in future warming scenarios. Figure 4: Simulated changes in ice thickness since 1850. (A) Changes in ice thickness since 1850 by year 2020 in simulations transiently forced by bias-corrected historical + SSP5-8.5 climates from 36 CMIP6 models (Experiment CMIP6_BC_1850–2100). (B) Same as (A) but for year 2100. Experiment set CMIP6_BC_1850-2100 are 250-year ISM runs under transient bias-corrected CMIP6 climates in combined historical (1850–2014) and SSP5-8.5 (2015–2100) scenarios, with the ice sheet initiated from the respective 15,000-year control simulation under the bias-corrected preindustrial climate described previously (Experiment CMIP6_BC_PI_CTL). Climate fields are bias-corrected and drive the ISM year by year, so that an evolution of the AIS is obtained for each CMIP6 model. In this approach, we essentially remove each CMIP6 model’s bias in simulated present-day climate and focus on the course of simulated climate change and associated impact on the AIS, especially on the uncertainty in the AIS’s future projections. Projected changes in the Antarctic climate from all CMIP6 models drive a reduction in both AIS volume and the extent of ice shelves (Figs. 3 and 5). The magnitude of ice loss, however, shows a large intermodel spread. CIESM shows the largest warming in atmospheric and oceanic temperatures and drives the most intense Antarctic ice loss. CESM2, CESM2-WACCM, and CNRM-CM6-1 are among the models with the largest warming in T2m by 2100 (Fig. 3C); they also drive some of the largest reductions in ice volume. Counterintuitively, the four variants of EC-Earth3 show greater oceanic warming, but they produce much less 21st century ice loss (Fig. 3, D and H). In the previous three models, ice surface melting and the loss of ice shelves overshadow sub-ice melting due to oceanic warming, which has been the focus of most recent studies on the sensitivity of the AIS, especially its marine-based WAIS portion (24). Climate models with the strongest atmospheric warming also produce the largest WAIS retreat, raising the GMSL by >0.25 m by 2100 (Fig. 3L). A contributing factor for this emerging correlation may be that the ISM used in this study resolves hydrofracturing and ice cliff failure processes, which make the ice shelves prone to collapse triggered by surface melting and thus increase the ISM’s sensitivity to atmospheric warming.	https://sealeveldocs.readthedocs.io/en/latest/li23sciadv.html
d3958eec8aeb-3	Experiment set CMIP6_BC_1850-2100 are 250-year ISM runs under transient bias-corrected CMIP6 climates in combined historical (1850–2014) and SSP5-8.5 (2015–2100) scenarios, with the ice sheet initiated from the respective 15,000-year control simulation under the bias-corrected preindustrial climate described previously (Experiment CMIP6_BC_PI_CTL). Climate fields are bias-corrected and drive the ISM year by year, so that an evolution of the AIS is obtained for each CMIP6 model. In this approach, we essentially remove each CMIP6 model’s bias in simulated present-day climate and focus on the course of simulated climate change and associated impact on the AIS, especially on the uncertainty in the AIS’s future projections. Projected changes in the Antarctic climate from all CMIP6 models drive a reduction in both AIS volume and the extent of ice shelves (Figs. 3 and 5). The magnitude of ice loss, however, shows a large intermodel spread. CIESM shows the largest warming in atmospheric and oceanic temperatures and drives the most intense Antarctic ice loss. CESM2, CESM2-WACCM, and CNRM-CM6-1 are among the models with the largest warming in T2m by 2100 (Fig. 3C); they also drive some of the largest reductions in ice volume. Counterintuitively, the four variants of EC-Earth3 show greater oceanic warming, but they produce much less 21st century ice loss (Fig. 3, D and H). In the previous three models, ice surface melting and the loss of ice shelves overshadow sub-ice melting due to oceanic warming, which has been the focus of most recent studies on the sensitivity of the AIS, especially its marine-based WAIS portion (24). Climate models with the strongest atmospheric warming also produce the largest WAIS retreat, raising the GMSL by >0.25 m by 2100 (Fig. 3L). A contributing factor for this emerging correlation may be that the ISM used in this study resolves hydrofracturing and ice cliff failure processes, which make the ice shelves prone to collapse triggered by surface melting and thus increase the ISM’s sensitivity to atmospheric warming. Figure 5: Simulated changes in the AIS’s area and sea level contribution. Top panels show changes in ice area relative to the preindustrial, where black, blue, and red lines represent all, grounded, and floating (shelf) ice, respectively. Bottom panels show changes in the AIS’s contribution to GMSL rise, where black, blue, and red lines are for the whole AIS, the EAIS, and the WAIS, respectively. Results from experiments with marine ice cliff instability (MICI) processes and forced with bias-corrected CMIP6 model climate (Experiment CMIP6_BC_1850-2100) are shown in the left column. Middle column shows results from experiments with MICI processes but forced with raw CMIP6 model climate, while the ISM is tuned separately for each CMIP6 model (Experiment CMIP6_RAW_1850-2100) (see Methods and figs. S13 to S16). Right column is for experiments forced with bias-corrected CMIP6 model climate, while MICI-related processes are turned off (Experiment CMIP6_BC_1850-2100_NO_MICI). In each panel, the full spread (0 to 100th percentile) in 36 simulations is shaded in light gray, and 16th to 84th percentile are in darker gray. The full spread and 16th to 84th percentile of respective variables for grounded ice/EAIS and floating ice/WAIS at 2100 are shown as blue and red boxplots, respectively, to the right of each panel. These 250-year AIS simulations using bias-corrected climates from 36 CMIP6 models reveal both accelerating retreat of the AIS and increasing uncertainty in its future trajectory. Relative to its preindustrial state, the multimodel median rate of ice loss increases by almost an order of magnitude from 2020 to 2100 (Fig. 3, G and H). EAIS and WAIS display contrasting changes over the early stage of warming before 2020: The WAIS loses mass and contributes to a SLR under all CMIP6 models’ bias-corrected climate trajectories, while, under most CMIP6 models (27 of 36), the EAIS gains mass and draws down GMSL (Fig. 3K). Between 1850 and 2020, the EAIS produces a small negative (~−0.01 m) multimodel median contribution to GMSL rise, while reduction in the WAIS is more consistent across models. As the 21st century warming proceeds, the EAIS is expected to reverse its trend later and begin to lose mass (Fig. 5C). By 2100, the multimodel median reduction in ice area increases to 6 × 105 km2, and the multimodel median sea level contribution of the AIS approaches 0.3 m (Fig. 5), with the highest modeled SLR exceeding 1 m. The full range of the AIS’s sea level contribution by 2100 greatly exceeds its multimodel median value as a result of the strong nonlinearity in the ice sheet’s response to temperature change. While the CMIP multimodel mean/median has been shown to produce an accurate representation of modern climate state, and multimodel median sea level projections remain more policy relevant than end-members, we should beware of the existence of low-probability, high-consequence scenarios in future SLR. Discussion Effect of MICI on projected ice loss The projected MMM rise in GMSL contributed by the AIS and associated uncertainty in these CMIP6-driven ISM simulations is noticeably greater than those assessed by ISMIP6 (15) and IPCC-AR6 (9). A possible factor might be the “marine ice cliff instability” (MICI) mechanism, which is accounted for in our ISM but has not been widely implemented in other ISMs. The ISM used in this study includes optional hydrofracturing and ice cliff failure mechanisms (25), which may give rise to MICI (5, 26, 27) under strong future warming scenarios but not in preindustrial and present-day climate conditions. MICI is a newly proposed mechanism, and there have been ongoing discussions concerning its validity. Self-sustaining ice loss triggered by MICI has been proposed to be necessary for explaining the Antarctic contribution to sea level high stands during the LIG and the Pliocene (5, 28) as well as the ice berg keel marks formed in deep water during the last deglaciation in the Amundsen Sea Embayment (29). On the other hand, some suggest that MICI is not well constrained and is not required to explain past sea level high stands (30), it may be mitigated by slow removal of ice shelves (31), and the progress of instability may be slowed by ice-mélange buttressing. Recent advances in modeling ice cliff failure reveal that MICI remains a feasible mechanism, but glacier models have shown a higher degree of complexity (32, 33) compared to the parameterization scheme originally implemented in our ISM. Although key parameters for hydrofracturing and cliff failure have been updated and constrained by sea level proxy data and observational records (5), considering their associated uncertainty, we also carried out alternative experiments without MICI processes (Exp. CMIP6_BC_1850-2100_NO_MICI). Without MICI, the ISM runs show smaller sea level contributions from the AIS by 2100, ranging from −0.05 to 0.2 m, with a median of 0.02 m, more in line with the findings of a recent study using statistical emulators of ISMs (10). In the absence of hydrofracturing and ice cliff failure, the warming in near-surface air temperature increases surface melt but does not trigger widespread collapse of ice shelves, and any tall ice cliffs that do emerge where ice shelves are lost remain intact in the model. Ignoring hydrofracturing and ice cliff failure processes puts our model in the lower range among ISMs in terms of its sensitivity to climate warming, so in these simulations without MICI, the resulting uncertainty in future sea level change reflects the combination of widely differing CMIP6 climate fields and a low-sensitivity ISM. However, even without MICI-related processes, the full range of climate-driven sea level uncertainty contributed by the AIS still amounts to 0.25 m by 2100, exceeding uncertainties from other major contributors, including sea water thermal expansion, mountain glaciers, and the Greenland Ice Sheet (7). Implications on observed ice sheet changes in recent decades	https://sealeveldocs.readthedocs.io/en/latest/li23sciadv.html
d3958eec8aeb-4	Although key parameters for hydrofracturing and cliff failure have been updated and constrained by sea level proxy data and observational records (5), considering their associated uncertainty, we also carried out alternative experiments without MICI processes (Exp. CMIP6_BC_1850-2100_NO_MICI). Without MICI, the ISM runs show smaller sea level contributions from the AIS by 2100, ranging from −0.05 to 0.2 m, with a median of 0.02 m, more in line with the findings of a recent study using statistical emulators of ISMs (10). In the absence of hydrofracturing and ice cliff failure, the warming in near-surface air temperature increases surface melt but does not trigger widespread collapse of ice shelves, and any tall ice cliffs that do emerge where ice shelves are lost remain intact in the model. Ignoring hydrofracturing and ice cliff failure processes puts our model in the lower range among ISMs in terms of its sensitivity to climate warming, so in these simulations without MICI, the resulting uncertainty in future sea level change reflects the combination of widely differing CMIP6 climate fields and a low-sensitivity ISM. However, even without MICI-related processes, the full range of climate-driven sea level uncertainty contributed by the AIS still amounts to 0.25 m by 2100, exceeding uncertainties from other major contributors, including sea water thermal expansion, mountain glaciers, and the Greenland Ice Sheet (7). Implications on observed ice sheet changes in recent decades Centennial and millennial trends in the AIS are dictated by long-term climate change, natural or anthropogenic, but internal variabilities of the climate system may still be important for multidecadal ice sheet changes, e.g., changes in polar ice sheets observed during the satellite era. Few of the ISM simulations driven by bias-corrected CMIP6 climates (Exp. CMIP6_BC_1850-2100) show an Antarctic contribution to GMSL over 1992–2017 consistent with that estimated by the Ice Sheet Mass Balance Intercomparison Exercise (IMBIE) team (34, 35). Forced by the multimodel mean outputs from 36 CMIP6 models, net contribution by the AIS during the IMBIE period 1992–2017 is minimal (Fig. 6A). Figure 6: Simulated rates of GMSL change. (A) Contributions to the mean rate of change in GMSL during the IMBIE period (1992–2017) by the AIS in simulations forced by bias-corrected CMIP6 model climates, where the likely ranges estimated by IMBIE are marked by horizontal blue bars. (B) Same as (A) but for the WAIS. (C) Same as (A) but for the EAIS. (D) Same as (A) but for the late-century period 2081–2100 under the SSP5-8.5 scenario. Results from each model are shown in gray bars, and the horizontal dashed lines represent the multimodel mean. Gray bars labeled as “CMIP6” represent an ice sheet simulation forced by the multimodel mean climate fields of 36 models, while hollow blue bars are for a similar simulation but with forcings during 1980–2019 replaced by observed fields from ERA5 and WOA18. In another simulation with the same climate forcing, but its 1980–2019 segment replaced by observational data (Exp. CMIP6_BC_MMM+OBS), Antarctica’s contribution to GMSL rise is more consistent with the IMBIE assessment, as a result of faster retreat of the WAIS and slower growth of the EAIS. A multidecadal warming trend since the 1970s in the circumpolar deep water (CDW) (fig. S12) (36), a relatively warm water mass circulating around Antarctica, may have enhanced basal melting of West Antarctic ice shelves. The ISM presents rates of ice loss comparable to IMBIE estimates when driven by the observed transient climate (fig. S12). Multimodel mean climate fields are essentially devoid of internal climate variabilities—provided that the number of models is large enough—due to cancellation of random phases from models. The observed multidecadal CDW warming trend, which may be partially caused by internal climate variability, cannot—and should not—be expected to be robustly reproduced in CMIP6 historical simulations, and its absence could be a factor for the generally small 1992–2017 trends from ISM simulations forced by CMIP6 models. ISM intercomparison projects A number of modeling studies concerning the uncertainty in future SLR contributed by the AIS have been carried out. The ISMIP6-Antarctica project (15) used ISMs from 13 modeling groups and six CMIP5 climate models. A smaller subset of CMIP6 models, all with an equilibrium climate sensitivity (ECS) near the upper end of climate models, has been used in similar ways to assess the future GMSL contributions by ice sheets under different emission scenarios (16). The work presented here complements the scope of existing ISM intercomparison projects. We have included 36 climate models from the CMIP6 ensemble, which encompass a wider range of ECS and more fully represent the contemporary understanding of the climate system and its future changes. CMIP6 models are known to have an overall higher ECS compared with CMIP5 models, primarily as a result of stronger positive cloud feedbacks from refined cloud schemes (37). Although only one ISM is used in this study, we have provided contrasting simulations with and without MICI processes, differing substantially in the sensitivity to atmospheric warming. Under the Representative Concentration Pathway (RCP) 8.5 scenario, a radiative forcing scenario similar to its CMIP6 successor SSP5-8.5, ISMIP6 simulations with 13 different ISMs give an Antarctic contribution to GMSL during the period 2015–2100 between −7.8 and 30 cm (15). In those simulations, WAIS retreat shows great variance among projections, up to 18-cm SLE, while the EAIS mass change varies between −6.1- and 8.3-cm SLE. These ISMIP6 projections present less ice loss and associated uncertainty compared with those in our simulations with the MICI mechanism, which is not considered in ISMIP6. Another contributing factor is the higher ECS of CMIP6 models used here, which generally warm more rapidly under SSP5-8.5 compared with CMIP5 models under RCP8.5. Effects of ISM calibration per climate model Results discussed so far are all from ISM runs in a “single-ISM” framework, where the ISM is calibrated on the basis of observational data, with its parameters fixed for all CMIP6 climate models. Nonetheless, calibrating an ISM’s parameters so that, under a prescribed climate, it could simulate that a target ice sheet state is a common practice in the ice sheet modeling community, in which ISM parameters may absorb part of the spread in climate boundary conditions. Ice sheet intercomparison projects, e.g., ISMIP6, were carried out in similar ways, in which ISMs from decentralized development were calibrated separately with their own targets. To assess the effect of ISM tuning on projected Antarctic ice loss, we carried out a series of experiments to tune key ISM parameters for the preindustrial climate simulated by each CMIP6 model (Methods). This essentially results in multiple ISMs, each tailored for the respective CMIP6 model. We then run future projections of the AIS with the raw climate output from CMIP6 models, rather than bias-corrected climate as we did previously. With this “multi-ISM” approach, the spread in simulated AIS forced by raw CMIP6 1850–2100 climate is smaller than that in single-ISM runs (Fig. 5). In comparison with ISMIP6 results, however, the dispersion of simulated Antarctic ice loss by 2100 in multi-ISM runs is still larger than that documented by ISMIP6-Antarctica. This may well be contributed to the more comprehensive set of climate models used in our study and CMIP6 models generally showing a higher climate sensitivity to elevated greenhouse gas levels, despite that the SSP5-8.5 scenario (8) used by CMIP6 models has slightly lower rates of greenhouse gas emissions than RCP8.5—its CMIP5 counterpart used by ISMIP6.	https://sealeveldocs.readthedocs.io/en/latest/li23sciadv.html
d3958eec8aeb-5	Effects of ISM calibration per climate model Results discussed so far are all from ISM runs in a “single-ISM” framework, where the ISM is calibrated on the basis of observational data, with its parameters fixed for all CMIP6 climate models. Nonetheless, calibrating an ISM’s parameters so that, under a prescribed climate, it could simulate that a target ice sheet state is a common practice in the ice sheet modeling community, in which ISM parameters may absorb part of the spread in climate boundary conditions. Ice sheet intercomparison projects, e.g., ISMIP6, were carried out in similar ways, in which ISMs from decentralized development were calibrated separately with their own targets. To assess the effect of ISM tuning on projected Antarctic ice loss, we carried out a series of experiments to tune key ISM parameters for the preindustrial climate simulated by each CMIP6 model (Methods). This essentially results in multiple ISMs, each tailored for the respective CMIP6 model. We then run future projections of the AIS with the raw climate output from CMIP6 models, rather than bias-corrected climate as we did previously. With this “multi-ISM” approach, the spread in simulated AIS forced by raw CMIP6 1850–2100 climate is smaller than that in single-ISM runs (Fig. 5). In comparison with ISMIP6 results, however, the dispersion of simulated Antarctic ice loss by 2100 in multi-ISM runs is still larger than that documented by ISMIP6-Antarctica. This may well be contributed to the more comprehensive set of climate models used in our study and CMIP6 models generally showing a higher climate sensitivity to elevated greenhouse gas levels, despite that the SSP5-8.5 scenario (8) used by CMIP6 models has slightly lower rates of greenhouse gas emissions than RCP8.5—its CMIP5 counterpart used by ISMIP6. Some CMIP6 models with a high climate sensitivity happen to display a warm bias in simulated present-day Antarctic climate. For instance, CESM2-WACCM drives one of the largest Antarctic ice loss by 2100 (∼1.05 m) in the single-ISM run, but the number reduces to only ∼0.25-m SLE in the multi-ISM run. Examining the tuned ocean melt factor (OMF) (fig. S13), we can see that, to compensate the warm bias in CESM-WACCM’s 400-m ocean temperature, the OMF has to be reduced to 0.72, much smaller than the OMF (5.0) used in single-ISM runs, which was calibrated on the basis of present-day observations. This, of course, greatly reduces the ISM’s sensitivity to oceanic warming. In the case of MIROC6, which has an exceptionally large warm bias in the surface air temperature, the temperature offset in the ISM’s positive-degree-day scheme (TPD) has to be increased to 4.64 K so that the modeled rate of surface meltwater production is around 100 Gt year−1. In other words, for the ISM tuned for MIROC6, ice and snow only melt at temperatures higher than 4.64 K. This is clearly unphysical but not an unexpected outcome of the tuning process. As warm bias in simulated modern polar climate is more prevalent than cold bias among CMIP6 models (Fig. 1A), tuning specifically for each CMIP6 model would generally reduce the ISM’s sensitivity to climatic warming and narrow the spread in projected ice loss. Then, we come to the question whether this multi-ISM approach, in comparison with the single-ISM way, is more appropriate in assessing the uncertainty in projected Antarctic ice loss associated with climate models. The multi-ISM way hides climate models’ biases under tailored ISM parameter settings but may resort to parameters that are unphysical or contradicting to observational evidence. Implications for Earth system model development The large spread in modeled polar climate in the current generation CMIP6 models would make it highly challenging to conduct intercomparisons of “Earth system models” with embedded, active ice sheets. It is not uncommon for climate models from different modeling centers to share components, and the same ISM or its close variants may be incorporated in several Earth system models. For instance, the Parallel Ice Sheet Model (PISM) is used in NASA GISS and MPI-ESM models, and the Grenoble ice sheet and land ice (GRISLI) model is used in CNRM-CM and IPSL-CM6 (11). Our ISM simulations forced by raw CMIP6 climates have demonstrated that, even with the same ISM, structural differences between atmosphere-ocean models can result in widely varying equilibrium states of the AIS. It has been recognized that simulated paleo–ice sheet volume, such as that during the mid-Pliocene, is highly dependent on climate model–based forcings (17). Results from our study highlight that biases in simulated polar climate from state-of-the-art climate models are large enough to drive the AIS to equilibrium states distinctly different from the present day, although the ISM simulates a realistic AIS with observational climate data. This poses a serious challenge to the practice of using paleo sea level to constrain the parameters of ice sheet processes, irrespective of the accuracy of ice volume and sea level reconstructions (38). Since the advent of numerical general circulation models in the 1960s, climate models have followed an evolutionary path of increasing complexity with ever more components added for explicit simulation (39). Spanning a hierarchy of models (40), climate modeling has now entered the Earth system model phase, where the most sophisticated models have added biogeochemical cycles and land ice sheets to the atmosphere-land-ocean system. Integrated ice sheet components embedded within Earth system models allow consistent simulations of crucial processes for polar climate change, e.g., the ice-albedo feedback, ice-elevation feedbacks associated with an evolving ice sheet topography, and the climate feedbacks associated with ice sheet meltwater (41–43). Results from our study, however, warn of substantial ongoing uncertainty among Earth system models with interactive ice sheets for the evaluation of future SLR. While progress has been made in ice sheet modeling, the uncertainty in future changes of the AIS and associated impacts on GMSL have not been reduced to a level needed for straightforward decision-making, and more work is required. Current greenhouse gas emissions put the climate on track of a >3°C warming by 2100, and the time window is shrinking for reducing carbon emissions to avoid rapid and unstoppable SLR (5). For more robust sea level projections, improved understanding of processes important for polar climate, including cloud radiative forcing and deep ocean circulations and mixing, is urgently needed. Methods Ice sheet model In this study, we use PSUICE3D (18), a numerical ISM with a hybrid approach to the dynamical equations governing ice sheet and ice shelf flow, which are described by the shallow ice and shallow shelf approximations, respectively, and are combined heuristically by an imposed mass flux condition across the grounding line (44). These hybrid ice dynamics capture the migration of grounding lines and essential mechanisms of ice sheet–ice shelf dynamics, e.g., the marine ice sheet instability (MISI) for an inward-deepening ice sheet bed, while allowing the model to be run on coarse grids (20 km in this study) so that a large ensemble of simulations can be carried out economically on centennial and millennial time scales and on continental spatial scales. Bedrock deformation under the weight of the ice sheet is represented by a local relaxation toward isostatic equilibrium and elastic lithospheric flexure. No explicit basal hydrology is implemented in the model other than allowing basal sliding where the basal temperature reaches the melt point. BSCs of the bed are obtained using an inverse method, in which the model is driven by present-day observational climate and the sliding coefficient at each grid point is tuned iteratively until the local ice thickness equilibrates toward the present-day observed value (21). Ice sheet SMB is calculated as snowfall minus surface melt, while sublimation at the ice surface is ignored. The fraction of precipitation falling as snow is determined by a parameterized formulation based on the corresponding monthly surface air temperature Ta (45). Ice surface melt is calculated from Ta using a standard positive-degree-day (PDD) scheme with a coefficient of 0.005 m per degree-day, but the temperature baseline for zero melt (parameter TPD) is set at −1°C in single-ISM runs so that, under present-day climate, the total surface melt rate of Antarctic ice shelves is within the observational range (46). Heavily parameterized in the current generation ISMs, ocean-induced ice shelf basal melt is recognized as a major source of uncertainty in the AIS’s response to climate change (11, 13, 15). Basal melt of Antarctic ice shelves is strongly influenced by the incursion of warm CDW, which occurs at ∼10-km spatial scales and daily to subdaily time scales (47) and cannot be faithfully simulated in a coarse resolution (∼100-km) ocean model typical of CMIP6 models. Recognizing these limitations, in this study, we use a simple parameterization scheme for basal melt rates, which assumes a quadratic dependence on the 400-m ocean temperature above the pressure melting point of ice (T_o − T_f)	https://sealeveldocs.readthedocs.io/en/latest/li23sciadv.html
e48670af6dee-0	Little et al. (2019) Title: The Relationship Between U.S. East Coast Sea Level and the Atlantic Meridional Overturning Circulation: A Review Key Points: The relationship between the AMOC and coastal sea level is important to flood risk projections and ocean circulation reconstructions The amplitude and pattern of sea level variability associated with AMOC variations is location, forcing, timescale, and model dependent Future research should address the complex spatiotemporal structure of AMOC and the role of near.coast ageostrophic processes Corresponding author: Little Citation: Little, C. M., Hu, A., Hughes, C. W., McCarthy, G. D., Piecuch, C. G., Ponte, R. M., & Thomas, M. D. (2019). The Relationship between U.S. East Coast sea level and the Atlantic Meridional Overturning Circulation: A review. Journal of Geophysical Research: Oceans, 124, 6435Ð6458. doi:10.1029/2019JC015152 URL: https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2019JC015152 Abstract Scientific and societal interest in the relationship between the Atlantic Meridional Overturning Circulation (AMOC) and U.S. East Coast sea level has intensified over the past decade, largely due to (1) projected, and potentially ongoing, enhancement of sea level rise associated with AMOC weakening and (2) the potential for observations of U.S. East Coast sea level to inform reconstructions of North Atlantic circulation and climate. These implications have inspired a wealth of model- and observation-based analyses. Here, we review this research, finding consistent support in numerical models for an antiphase relationship between AMOC strength and dynamic sea level. However, simulations exhibit substantial along-coast and intermodel differences in the amplitude of AMOC-associated dynamic sea level variability. Observational analyses focusing on shorter (generally less than decadal) timescales show robust relationships between some components of the North Atlantic large-scale circulation and coastal sea level variability, but the causal relationships between different observational metrics, AMOC, and sea level are often unclear. We highlight the importance of existing and future research seeking to understand relationships between AMOC and its component currents, the role of ageostrophic processes near the coast, and the interplay of local and remote forcing. Such research will help reconcile the results of different numerical simulations with each other and with observations, inform the physical origins of covariability, and reveal the sensitivity of scaling relationships to forcing, timescale, and model representation. This information will, in turn, provide a more complete characterization of uncertainty in relevant relationships, leading to more robust reconstructions and projections. Plain Language Summary Sea level along the U.S. East Coast is influenced by changes in the density and currents of the North Atlantic Ocean. Indeed, there are simple theoretical considerations that relate indices of basin-scale flow to coastal sea level. Such a relationship could be leveraged to predict future sea level changes and coastal flooding given an expected change in climate and ocean circulation. Alternatively, it could be used to reconstruct ocean circulation from sea level measurements. This paper reviews the nature of this relationship and whether, and when, it is evident in climate models and observations. Although the current generation of large-scale climate and ocean models generally show an antiphase relationship between basin-scale ocean current strength and coastal sea level, the spatial pattern of sea level change differs from theory and between models. Supported by existing and emerging research, the authors hypothesize that these deviations result from important physical processes occurring on the continental shelf and slope, and the complexities of the 3-dimensional ocean circulation. A quantitative assessment of the importance of these processes is critical for understanding past and future climate and sea level changes in this heavily populated and vulnerable region. Sea Level Variability Along the United States East Coast and Its Societal Importance The densely populated U.S. East Coast is especially vulnerable to the impacts of sea level change, with ~2.4 million people and ~1.4 million housing units between Maine and Florida less than 1 m above local mean high water (Strauss et al., 2012). Here, sea level rise is already having adverse environmental, societal, and economic consequences, including increases in the severity and frequency of coastal flooding (e.g., Ezer & Atkinson, 2014; Moftakhari et al., 2015; Ray & Foster, 2016; Sweet et al., 2018; Wdowinski et al., 2016). Regional rates of sea level rise, and their associated consequences, are projected to increase substantially over the coming century (Figure 1a; Brown et al., 2018; Dahl et al., 2017; Kopp et al., 2014; Little, Horton, Kopp, Oppenheimer, Yip, 2015; Ray & Foster, 2016; Vitousek et al., 2017). Figure 1: (a) Monthly mean tide gauge sea level (in millimeters relative to year 2000) at the Battery (New York City; blue line). Projections of relative sea level (RSL) change, relative to year 2000, for RCP 2.6 (blue) and RCP 8.5 emission scenarios (red; Kopp et al., 2014). Shading after the year 2000 indicates 17th to 83rd percentile range of RSL projections. (b) Annual mean RSL (in millimeters, with arbitrary offset) measured at 15 U.S. East Coast tide gauges (Holgate et al., 2013) with long and relatively complete records. (c) Linear trend in RSL along the U.S. East Coast from 1900Ð2017, in millimeters per year, from a Bayesian reconstruction (panel taken from Piecuch, Huybers, et al., 2018). Understanding the drivers of future change in relative sea level (RSL, i.e., that observed by tide gauges and relevant to coastal locations; see Gregory et al., 2019), and the ability of numerical models to represent such drivers, is critical. However, this is a complex task, given the many contributing processes that operate over different temporal and spatial scales, including, for example: freshwater input from land and the cryosphere, thermal expansion of sea water, glacial isostatic adjustment, and oceanic mass and volume redistribution (see Kopp et al., 2015; Milne et al., 2009; Stammer et al., 2013, for more thorough reviews of these processes). The relative contributions of these processes to U.S. East Coast RSL vary across space and through time. For example, vertical land motion (due primarily to glacial isostatic adjustment) accounts for the majority of the large-scale spatial variation in recent centennial trends and underlies the high rates of RSL rise in the Mid-Atlantic (Figure 1c; Karegar et al., 2017; Piecuch, Huybers, et al., 2018). However, ongoing climate-related processes - associated with net freshwater input, atmosphere-ocean momentum and buoyancy fluxes, and ocean mass and volume redistributionÑdominate the interannual to multidecadal, spatially variable, U.S. east coast RSL signals during the twentieth century (Figure 1b; Andres et al., 2013; Bingham & Hughes, 2009; Davis & Vinogradova, 2017; Ezer, 2013; Ezer et al., 2013; Frederikse et al., 2017; Goddard et al., 2015; Park & Sweet, 2015; Piecuch et al., 2016; Piecuch, Bittermann, et al., 2018; Piecuch & Ponte, 2015; Thompson & Mitchum, 2014; Woodworth et al., 2014; Yin & Goddard, 2013).	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-1	Understanding the drivers of future change in relative sea level (RSL, i.e., that observed by tide gauges and relevant to coastal locations; see Gregory et al., 2019), and the ability of numerical models to represent such drivers, is critical. However, this is a complex task, given the many contributing processes that operate over different temporal and spatial scales, including, for example: freshwater input from land and the cryosphere, thermal expansion of sea water, glacial isostatic adjustment, and oceanic mass and volume redistribution (see Kopp et al., 2015; Milne et al., 2009; Stammer et al., 2013, for more thorough reviews of these processes). The relative contributions of these processes to U.S. East Coast RSL vary across space and through time. For example, vertical land motion (due primarily to glacial isostatic adjustment) accounts for the majority of the large-scale spatial variation in recent centennial trends and underlies the high rates of RSL rise in the Mid-Atlantic (Figure 1c; Karegar et al., 2017; Piecuch, Huybers, et al., 2018). However, ongoing climate-related processes - associated with net freshwater input, atmosphere-ocean momentum and buoyancy fluxes, and ocean mass and volume redistributionÑdominate the interannual to multidecadal, spatially variable, U.S. east coast RSL signals during the twentieth century (Figure 1b; Andres et al., 2013; Bingham & Hughes, 2009; Davis & Vinogradova, 2017; Ezer, 2013; Ezer et al., 2013; Frederikse et al., 2017; Goddard et al., 2015; Park & Sweet, 2015; Piecuch et al., 2016; Piecuch, Bittermann, et al., 2018; Piecuch & Ponte, 2015; Thompson & Mitchum, 2014; Woodworth et al., 2014; Yin & Goddard, 2013). Of interest in this review paper is RSL variability related to changes in ocean circulation and density that may be causally coupled, or simply correlated, with the Atlantic Meridional Overturning Circulation (AMOC; see section 2). We thus focus on variability in “dynamic sea level” (DSL), that is, the height of the sea surface above the geoid, with the inverse barometer correction applied (Gregory et al., 2019). Secular DSL changes are evident in 21st century climate model simulations and are projected to be a principal driver of acceleration in 21st century sea level and its spatial variation along the east coast (Bilbao et al., 2015; Bouttes et al., 2014; Carson et al., 2016; Chen et al., 2018; Church et al., 2013; Kopp et al., 2014; Little, Horton, Kopp, Oppenheimer, Vecchi, et al., 2015; Little, Horton, Kopp, Oppenheimer, & Yip, 2015; Perrette et al., 2013; Slangen et al., 2014; Yin et al., 2009; Yin, 2012; Yin & Goddard, 2013). Various studies have shown these large-scale regional DSL anomalies to be correlated with a decline in AMOC strength (section 4). However, current-generation climate models also show a wide range in future projections of regional DSL rise. They may also exhibit systematic biases due to poorly resolved processes that influence near-coast DSL (section 6). An improved theoretical and observational basis for AMOC-DSL relationships would enable assessments of the reliability of individual model projections, and climate models more generally, allowing improved estimates of the magnitude, spatial pattern, and time of emergence of expected sea level rise. In addition, a robust “signature” of AMOC (or some other feature of the large-scale circulation) in coastal RSL could be leveraged to infer preinstrumental changes in AMOC and/or climate. Recent improvements to analysis of the tide gauge record, including approaches to cope with data gaps and account for vertical land motion and glacial isostatic adjustment (Kopp, 2013; Piecuch, Huybers, et al., 2017), have intensified the interest in exploiting this relationship to inform reconstructions of ocean variability (e.g., Butler et al., 2015; Kienert & Rahmstorf, 2012; McCarthy et al., 2015). Proxies that predate the tide gauge record offer the opportunity to extend these reconstructions over centennial to millennial timescales (e.g., Kemp et al., 2017, 2018). Here, motivated by these considerations, we review evidence for the covariation of AMOC and U.S. East Coast sea level. In section 2, we define AMOC and its relationship to the large-scale circulation of the North Atlantic Ocean. Section 3 presents a simple diagnostic scaling argument between AMOC strength and DSL. Section 4 surveys AMOC-DSL linkages in numerical simulations (where long-period relationships are able to be assessed) and includes a new analysis of the AMOC-DSL scaling coefficient in Coupled Model Intercomparison Project Phase 5 (CMIP5) simulations. Section 5 examines observational linkages between AMOC components and coastal sea level, clarifying the specific components of AMOC (e.g., Gulf Stream) invoked, the regional fingerprint of such linkages, and the timescales over which the relationship has been documented. In section 6, we suggest potential origins of along-coast variations, intersimulation differences in scaling relationships, and discrepancies between models and observations; section 7 highlights new research directions that can help assess these discrepancies more extensively and quantitatively. AMOC and the North Atlantic Ocean Circulation The U.S. East Coast borders the western boundary of the North Atlantic Ocean, which is characterized by a spatially and temporally complex system of surface and deep currents (Figure 2). Figure 2: Schematic of key AMOC.related components of the North Atlantic Ocean (modfied from Garcia-Ibanez et al., 2018). Abbreviations are as follows: NRG = Northern Recirculation Gyre; LC = Labrador Current; DWBC = Deep Western Boundary Current; IC = Irminger Current; EGIC = East Greenland.Irminger Current. Three source waters for North Atlantic Deep Water are noted: LSW = Labrador Sea Water; ISOW = Iceland-Scotland Overflow Water; DSOW = Denmark Straits Overßow Water. Box indicates the U.S. East Coast region. At U.S. East Coast latitudes, the large-scale ocean circulation is dominated by two opposing gyres. At subtropical latitudes, southward wind-driven transport in the interior of the gyre is closed by a western boundary current, composed of the Gulf Stream to the north and the Florida and Antilles currents further south. At subpolar latitudes, the North Atlantic Current (NAC) splits into various branches that flow northwards along the eastern side of the subpolar gyre (Rhein et al., 2011). These currents flow cyclonically around the subpolar gyre, contributing to the upper parts of the western boundary currents comprising the East and West Greenland Currents and the Labrador Current. Part of the NAC also flows into the Nordic Seas (e.g., Dickson & Brown, 1994; Sarafanov et al., 2012). Along these high-latitude branches, warm and salty surface waters originating from the tropical and subtropical Atlantic increase in density and transform into North Atlantic Deep Water through a variety of processes, including cooling, mixing, and convection (Marotzke & Scott, 1999; Spall & Pickart, 2001; Thomas et al., 2015). In addition to these large-scale flows, there are important currents along the U.S. East Coast continental shelf, shelf break, and slope: flowing northward over the continental shelf south of Cape Hatteras (the South Atlantic Bight) and southward along the shelf between Cape Hatteras and Nova Scotia (Figure 2). These currents are driven by a combination of local wind and buoyancy forcing as well as interactions with the larger-scale flow field (see section 6). In the South Atlantic Bight, interactions between the shelf current and the Gulf Stream are clearly important, but there is evidence of locally wind driven variability closer to the shore (Lee et al., 1991; Stegmann & Yoder, 1996; Yuan et al., 2017). To the north of Cape Hatteras, the Slope Current has its origins in the Labrador Current and the East Greenland Current (Chapman & Beardsley, 1989; Rossby et al., 2014). Its strength is therefore linked to the AMOC, through the strength of the Labrador Current, as well as through interactions with the Northern Recirculation Gyre (Andres et al., 2013; Zhang, 2008), the Deep Western Boundary Current (e.g. Zhang & Vallis, 2007), and the Gulf Stream (Ezer, 2015).	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-2	In addition to these large-scale flows, there are important currents along the U.S. East Coast continental shelf, shelf break, and slope: flowing northward over the continental shelf south of Cape Hatteras (the South Atlantic Bight) and southward along the shelf between Cape Hatteras and Nova Scotia (Figure 2). These currents are driven by a combination of local wind and buoyancy forcing as well as interactions with the larger-scale flow field (see section 6). In the South Atlantic Bight, interactions between the shelf current and the Gulf Stream are clearly important, but there is evidence of locally wind driven variability closer to the shore (Lee et al., 1991; Stegmann & Yoder, 1996; Yuan et al., 2017). To the north of Cape Hatteras, the Slope Current has its origins in the Labrador Current and the East Greenland Current (Chapman & Beardsley, 1989; Rossby et al., 2014). Its strength is therefore linked to the AMOC, through the strength of the Labrador Current, as well as through interactions with the Northern Recirculation Gyre (Andres et al., 2013; Zhang, 2008), the Deep Western Boundary Current (e.g. Zhang & Vallis, 2007), and the Gulf Stream (Ezer, 2015). In aggregate, these horizontal and vertical flows result in an “overturning” circulation that transports over 1 PW of heat poleward (Trenberth & Fasullo, 2017). In this paper, this AMOC is defined as the stream function of the zonally and cumulatively vertically integrated meridional velocity of the Atlantic Ocean north of 35°S (Buckley & Marshall, 2016; Zhang, 2010). In models and observations, the AMOC reveals upper and lower interhemispheric overturning cells of water that are sourced by high-latitude sites of deep water formation in the northern and southern hemispheres respectively (Figure 3). Figure 3: The AMOC, averaged over the 1959–2012 period, from a 1/12° resolution model simulation as described in Hughes et al. (2018). The flow is clockwise around positive values, and the stream function is calculated by integrating the southward velocity both zonally and upwards from the bottom. The black contour is at zero. The upper overturning cell reflects northward transport in the upper ocean currents, including those mentioned earlier in this section, compensated by southward flowing North Atlantic Deep Water at intermediate depths. In models, the maximum of the AMOC stream function is typically located around the latitude of the Gulf Stream separation and at approximately 1,000-m depth. Below this upper cell is a lower cell of Antarctic Bottom Water that originates from sources at high southern latitudes (Buckley & Marshall, 2016; Kuhlbrodt et al., 2007). See Buckley and Marshall (2016) and other reviews in this special issue, particularly Bower et al. (2019), for a more comprehensive description of AMOC structure and variability. A Simple Theoretical Basis for AMOC-DSL Covariability A diagnostic relationship between the AMOC and DSL can be derived from the zonal momentum equation: frac{rho}{r cos{phi}} frac{D}{Dt} left(u r cos{phi}right) - rho f v + rho f w cot{phi} = -frac{1}{r cos{phi}} frac{partial p}{partial lambda} + F_x, (1) where r is the Earth’s radius, u is the zonal velocity, v is the meridional velocity, w is the vertical velocity, f is the Coriolis frequency, phi is latitude, lambda is longitude, p is pressure, rho is density, F_x is the eastward viscous force per unit volume, and D/Dt is the material rate of change. For a derivation and discussion of the equations of motion see, for example, Vallis (2006, Chapter 2) and Gill (1982, Chapter 4). If we (1) zonally integrate over the basin and (2) neglect the advection of relative angular momentum (the first term), the term involving w (usually neglected in the Primitive Equations), and the viscous term (assuming we are below the surface Ekman layer, and that any bottom Ekman layer occupies only a small fraction of the zonal integralÑthis assumes that we are at depths where it is meaningful to consider the ocean to have sidewalls), this reduces to an integrated geostrophic balance: f T = p_E - p_W, (2) where T is the northward mass transport across the section (the zonal integral of rho v). Equation (2) relates the northward mass transport to the difference between pressure at the eastern end (p_E) and the western end (p_W) of the section. These pressures are bottom pressures, which become equivalent to DSL (with a scaling of approximately 1 cm/mbar of pressure) as the depth tends to zero at the coast. This zonally integrated geostrophic balance can be used to derive a simple scaling between the AMOC and DSL at the western boundary. First, we note that the eastern boundary pressure is very close to being a function of depth alone, independent of latitude, at least below a depth of around 100 m (Hughes et al., 2018; Hughes & de Cuevas, 2001). Subtracting off this reference function of depth in our definition of p (which now should be considered to be a pressure anomaly, referenced to the eastern boundary value), we find that p_E = 0. Then, integrating over depth from the surface (z = 0) to the depth of the maximum in the overturning streamfunction (z = -H), we find that the total northward mass transport above this depth is given by Q = int_{-H}^0{T dz} - frac{1}{f} int_{-H}^0{p_W dz} = -frac{H}{f}overline{p_W}, (3) where overline{p_W} is the western boundary pressure averaged over the depth range above the maximum overturning. The relationship to coastal sea level then follows from the assumption that the depth-averaged pressure in this zone is related to the boundary pressure near the surface, {p_W}_0, which is in turn related to inverse barometer-corrected boundary sea level h_W by rho_0 g h_W = {p_W}_0, where we use a reference density rho_0. Rewriting in terms of this near.surface western boundary pressure anomaly, we find Q = -frac{H_e}{f} {p_W}_0 = - frac{H_e}{f} rho_0 g h_W, (4) requiring the definition of an effective layer thickness H_e = int_{-H}^0{frac{p_W}{{p_W}_0}dz}, (5) which may be interpreted as the layer thickness used to multiply the near-surface boundary pressure anomaly (proportional to sea level), in order to get the correct depth-integrated pressure force on the sidewall. If the pressure anomaly (or equivalently the northward transport) is independent of depth above -H, H_e = H. If the zonally integrated flow (or pressure anomaly) is largest at the surface and decreases linearly to zero at the maximum of the overturning, H_e = 0.5 H. Rearranging (5), we find that the coastal sea level signal can be written as h_W = -frac{Q}{rho_0}frac{f}{g H_e}, (6) in which it is shown how the coastal sea level signal h_W is negatively related to the strength of the overturning Q/rho_0, and the size of the signal is larger if the effective layer thickness H_e is smaller. Figure 3 reveals a fairly uniform (or slowly decreasing with increasing depth) northward zonally integrated flow above about 1,000-m depth, balanced by a deeper return flow (with more complicated flows in the top few hundred meters, representing the wind-driven flows superimposed on the large-scale MOC). Assuming f = 10^{-4} s^{-1} (true at a latitude of about 43¡N), equation (6) predicts a sea level change of 1 cm/Sv of meridional transport (less for latitudes closer to the equator, and slightly more for more poleward latitudes). If, rather than constant transport per unit depth above 1,000 m (as in a simple two-layer model), we assume a linear rise from zero at 1,000 m to a maximum at the surface, then pressure at the surface is twice the depth average, leading to a scaling of -2 cm/Sv. Realistic scalings are likely to be between these limits, subject to the assumption of geostrophic balance in equation (2), and the approximation that the vertical profile of the flow remains constant (temporal variations in H_e are proportionally smaller than those in Q). The dependence on f means that this scaling should also lead to smaller sea level signals closer to the equator, again assuming that proportional variations in H_e are smaller than those in f. Evidence of an AMOC-DSL Relationship in Numerical Models	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-3	h_W = -frac{Q}{rho_0}frac{f}{g H_e}, (6) in which it is shown how the coastal sea level signal h_W is negatively related to the strength of the overturning Q/rho_0, and the size of the signal is larger if the effective layer thickness H_e is smaller. Figure 3 reveals a fairly uniform (or slowly decreasing with increasing depth) northward zonally integrated flow above about 1,000-m depth, balanced by a deeper return flow (with more complicated flows in the top few hundred meters, representing the wind-driven flows superimposed on the large-scale MOC). Assuming f = 10^{-4} s^{-1} (true at a latitude of about 43¡N), equation (6) predicts a sea level change of 1 cm/Sv of meridional transport (less for latitudes closer to the equator, and slightly more for more poleward latitudes). If, rather than constant transport per unit depth above 1,000 m (as in a simple two-layer model), we assume a linear rise from zero at 1,000 m to a maximum at the surface, then pressure at the surface is twice the depth average, leading to a scaling of -2 cm/Sv. Realistic scalings are likely to be between these limits, subject to the assumption of geostrophic balance in equation (2), and the approximation that the vertical profile of the flow remains constant (temporal variations in H_e are proportionally smaller than those in Q). The dependence on f means that this scaling should also lead to smaller sea level signals closer to the equator, again assuming that proportional variations in H_e are smaller than those in f. Evidence of an AMOC-DSL Relationship in Numerical Models Numerical simulations allow analysis of AMOC-DSL relationships that can be compared to the theoretical considerations of the previous section, while incorporating local and large-scale forcing, complex 3-D flows, and ageostrophic processes, to the extent permitted by their resolution. Most analysis of numerical simulations has focused on 21st century, centennial-timescale, AMOC-DSL relationships. In this section, we thus focus on longer timescales, although we contrast these results with selected studies that have examined covariability over shorter timescales, often with a focus on the historical record. The connection between U.S. East Coast sea level rise and the AMOC in coupled climate models was first established by Levermann et al. (2005) through “hosing” simulations (in which extreme freshwater forcing is applied to the subpolar North Atlantic). They found that, in a climate model with a relatively coarse (3.75° horizontal resolution) ocean, a weakened AMOC is associated with DSL rise in most of the Atlantic basin, with a scaling coefficient of up to −5 cm/Sv. Most subsequent numerical simulations that have assessed this relationship show a more complex spatial pattern of DSL change (e.g., Kienert & Rahmstorf, 2012; Landerer et al., 2007; Lorbacher et al., 2010; Yin et al., 2010), and a smaller (less negative) scaling coefficient (e.g., Bingham & Hughes, 2009; Little et al., 2017; Schleussner et al., 2011). However, the correlation between DSL rise over portions of the U.S. East Coast and a decline in AMOC (and, often, a rise in steric height in the western North Atlantic intergyre region) has been repeatedly noted, in simulations forced by future greenhouse gas emission scenarios, freshwater input into the subpolar North Atlantic, or both (e.g., Hu et al., 2009; Hu et al., 2011; Hu & Bates, 2018; Hu & Deser, 2013; Kienert & Rahmstorf, 2012; Krasting et al., 2016; Landerer et al., 2007; Lorbacher et al., 2010; Pardaens et al., 2011; Yin et al., 2010; Yin & Goddard, 2013). The AMOC weakens over the 21st century in most CMIP3 and CMIP5 simulations (Church et al., 2013), with a rate that varies widely across emissions scenarios and models (e.g., Figure 4a; Bakker et al., 2016; Cheng et al., 2013; Heuzé, 2017; Huber & Zanna, 2017; Schleussner et al., 2011; Weaver et al., 2012). Figure 4: (a) Change in maximum AMOC strength for a 28 Coupled Model Intercomparison Project Phase 5 model, RCP4.5-forced, ensemble, from 1976–2000 to 2076–2100, as calculated by Chen et al. (2018). (b) Ensemble mean dynamic sea level change (m) from 1976–2000 to 2076–2100. The amplitude and spatial pattern of DSL changes associated with 21st century AMOC weakening has been noted in several studies (Schleussner et al., 2011; Yin et al., 2009). However, such studies have generally considered the ensemble mean DSL change (Figure 4b), or a small subset of available models, and have often focused on the Northeast United States only, limiting analysis of intermodel or regional differences. An assessment of the robustness of the scaling of North Atlantic DSL to AMOC change across climate models is missing in the literature. To fill this gap, we perform a brief analysis using available datasets, including the results of Chen et al. (2018), who investigated the relationship between 21st century changes in DSL and the annual-mean maximum AMOC stream function below 500 m in a large (30-member) CMIP5 ensemble. Models included in this ensemble show an AMOC decline from 1976–2000 to 2076–2100 ranging from approximately zero to 8 Sv (Figure 4a). In Figure 5, we calculate the AMOC-DSL scaling coefficient for 25 CMIP5 models over this century-long period, at a 1° horizontal resolution. Figure 5: Map of the ratio of dynamic sea level change to AMOC change (m/Sv; 2076–2100 minus 1976–2000) for 25 RCP4.5-forced Coupled Model Intercomparison Project Phase 5 models with AMOC weakening larger than 2 Sv. There are broad similarities in the spatial pattern of scaling coefficients and that of the ensemble mean DSL change (Figure 4b), with only a few models showing dramatic differences from the subtropical high/subpolar and coastal low relationship (e.g., MRI-CGCM and FGOALS-g2). However, the amplitude of the scaling coefficient near the U.S. East Coast ranges widely, both north and south of Cape Hatteras, across the ensemble, along with substantial meridional gradients along these coastal regions within individual models. The diversity of model-specific scaling coefficients along the western boundary can also be shown with a regression of DSL change against AMOC change, that is, ∆DSL(x,y,m) = α(x,y) ∆AMOC(m) + ε(x,y,m) (7) where x and y are longitude and latitude, m is the model index, α is a local scaling coefficient, and ε is a residual. (Although the RCP 4.5 scenario is shown, spatial patterns of DSL change, and DSL change associated with AMOC change, do not exhibit strong RCP-dependence; Chen et al., 2018; Little, Horton, Kopp, Oppenheimer, & Yip, 2015; Yin, 2012; Yin et al., 2009). Local regression coefficients, shown in Figure 6b, indicate a meridional tripole in the North Atlantic; models with more AMOC weakening are associated with larger DSL rise in the subtropical gyre and larger DSL fall in most of the subpolar gyre and the tropics. This pattern bears some similarity to the dominant mode of sea surface height variability over the historical record (e.g., Hakkinen & Rhines, 2004; Yin & Goddard, 2013), the multimodel mean 21st century change observed in CMIP simulations (Figure 4b; Little, Horton, Kopp, Oppenheimer, & Yip, 2015; Yin, 2012; Yin et al., 2009, 2010), and a regression of DSL on AMOC strength in a model simulation of the historical period (Figure 6a). Coastal regression coefficients range from approximately −1.5 to 0 cm/Sv, with more negative values in U.S. East Coast regions north of Cape Hatteras.	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-4	There are broad similarities in the spatial pattern of scaling coefficients and that of the ensemble mean DSL change (Figure 4b), with only a few models showing dramatic differences from the subtropical high/subpolar and coastal low relationship (e.g., MRI-CGCM and FGOALS-g2). However, the amplitude of the scaling coefficient near the U.S. East Coast ranges widely, both north and south of Cape Hatteras, across the ensemble, along with substantial meridional gradients along these coastal regions within individual models. The diversity of model-specific scaling coefficients along the western boundary can also be shown with a regression of DSL change against AMOC change, that is, ∆DSL(x,y,m) = α(x,y) ∆AMOC(m) + ε(x,y,m) (7) where x and y are longitude and latitude, m is the model index, α is a local scaling coefficient, and ε is a residual. (Although the RCP 4.5 scenario is shown, spatial patterns of DSL change, and DSL change associated with AMOC change, do not exhibit strong RCP-dependence; Chen et al., 2018; Little, Horton, Kopp, Oppenheimer, & Yip, 2015; Yin, 2012; Yin et al., 2009). Local regression coefficients, shown in Figure 6b, indicate a meridional tripole in the North Atlantic; models with more AMOC weakening are associated with larger DSL rise in the subtropical gyre and larger DSL fall in most of the subpolar gyre and the tropics. This pattern bears some similarity to the dominant mode of sea surface height variability over the historical record (e.g., Hakkinen & Rhines, 2004; Yin & Goddard, 2013), the multimodel mean 21st century change observed in CMIP simulations (Figure 4b; Little, Horton, Kopp, Oppenheimer, & Yip, 2015; Yin, 2012; Yin et al., 2009, 2010), and a regression of DSL on AMOC strength in a model simulation of the historical period (Figure 6a). Coastal regression coefficients range from approximately −1.5 to 0 cm/Sv, with more negative values in U.S. East Coast regions north of Cape Hatteras. Figure 6: (a) From Woodworth et al. (2014). Regression coefficients of annual mean sea level and overturning transport (at the same latitude) for depths between 100 and 1,300 m using a 1° ocean model, for the period 1950–2009, without wind forcing. (b) Linear regression coefficient (α) of DSL change against the change in maximum AMOC strength for the models shown in Figure 5 (m/Sv). (c) Variance in DSL change explained by AMOC change (%). DSL = dynamic sea level; AMOC = Atlantic Meridional Overturning Circulation. However, regression coefficients in Figure 6b diverge from those obtained through a regression of annual mean DSL on AMOC over the 1950–2009 period (Figure 6a) and those predicted by equation 6, particularly along the western boundary, where the CMIP5-derived pattern does not show a universal anticorrelation of sea level and the AMOC. (We note that the single-valued AMOC index used in Figure 6b is different than the meridionally varying index used in Figure 6a. However, we would not expect this to affect the sign of regression coefficients, if AMOC transport changes are meridionally coherent). Perhaps more important than the spatial pattern in Figure 6b is the fact that only a small fraction of intermodel DSL variance is explained by differences in AMOC strength change (Figure 6c). In coastal regions, and in the subpolar gyre, factors unrelated to AMOC strength are principally responsible for the wide spread in 21st century projections of U.S. East Coast DSL rise (Yin et al., 2009, 2010; Kopp et al., 2014; Little, Horton, Kopp, Oppenheimer, & Yip, 2015; Minobe, 2017). It is possible that differences between Figures 6a and 6b originate in a timescale-dependent relationship. This was suggested by Yin and Goddard (2013, their Figure 3) based on (1) similarities between the DSL patterns of observed decadal trends and 21st century model trends; (2) similarities between observed and modeled Empirical Orthogonal Function (EOF) patterns (describing interannual variability); and (3) differences between DSL patterns associated with long-term trends and interannual variability. Similar conclusions were drawn by Lorbacher et al. (2010). Model-derived scaling coefficients for interannual AMOC-DSL relationships north of Cape Hatteras appear to be more consistent than those in Figure 5, and more consistent with the theoretical values in section 2. For example, Bingham and Hughes (2009) find a scaling of -1.7 cm Sv−1, Woodworth et al. (2014) find -1.5 cm Sv−1, and Little et al. (2017) obtain −1.8 cm/Sv. However, the wide spread in scaling coefficients across models under identical forcing (Figure 5) suggests that differences in model representation are critical over longer timescales. Although an analysis of these regional and inter-model differences is beyond the scope of this review, we highlight its importance and discuss possible explanations in sections 6 and 7. Evidence of an AMOC-DSL Relationship in Observations The first direct, continuous, basin-wide, observations of the AMOC began in 2004 with the RAPID project (Rapid Climate Change; Cunningham et al., 2007). This record is now complemented by two other basin-wide in situ programs: NOAC at 47°N (North Atlantic Changes; Mertens et al., 2014) and OSNAP, around 60°N (Overturning in the Sub-polar North Atlantic; Lozier et al., 2017). Although RAPID observations have revealed a wealth of information, they provide only a 13-year time series at 26°N at time of writing. This limited record hinders an observation-based assessment of AMOC-DSL relationships, especially over the decadal and longer timescales of primary interest here. Over shorter timescales, Ezer (2015) compared monthly RAPID observations to the Atlantic City-Bermuda tide gauge sea level difference, finding a correlation of 0.27. In the same analysis, Ezer noted substantial differences in correlations, and lag/lead relationships, between the sea level difference and the three individual components of AMOC observed by RAPID (Ekman, Florida Current, and Mid-Ocean transport). Piecuch et al. (2019) also note differing relationships between each of these AMOC components and New England coastal sea level, with only the Ekman component exhibiting strong coherence. In addition to the RAPID record, longer observations of elements of the North Atlantic circulation are available: for example, the Florida Current time series since 1982 (Meinen et al., 2010), the Oleander time series of Gulf Stream transport since 1992 (Rossby et al., 2014), and the position of the Gulf Stream Extension since 1955 (Joyce & Zhang, 2010). Studies based on models (e.g. Saba et al., 2016; Sanchez-Franks & Zhang, 2015) and observations (e.g., Kopp, 2013; McCarthy et al., 2015; Park & Sweet, 2015) have shown strong statistical relationships between US east coast sea level and these metrics at up to multidecadal timescales. We consider evidence in this section for relationships between DSL and these and other elements of the North Atlantic circulation, while emphasizing that changes in the latter do not necessarily imply changes in AMOC, as defined in section 2. We briefly discuss the nature of potential linkages with AMOC in section 6.3. Linkages Between DSL and AMOC Components The relationship between coastal DSL and the Gulf Stream has been assessed using theory, observations, and models. Studies have considered the roles of Gulf Stream transport, velocity, and position, both upstream and downstream of the detachment at Cape Hatteras, as well as the strength of the Florida Current. Early studies focused on the relationship between tide gauge observations and the Gulf Stream over seasonal timescales between Florida and Cape Hatteras. Two linkages between ocean circulation and DSL were considered: the cross-stream (shelf) sea level gradient, related to ocean circulation via geostrophy, and the downstream (along-coast) sea level gradient, related via the Bernoulli principle. Montgomery (1941) found little evidence for a relationship of the downstream sea level gradient to velocity (in the Gulf Stream).	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-5	In addition to the RAPID record, longer observations of elements of the North Atlantic circulation are available: for example, the Florida Current time series since 1982 (Meinen et al., 2010), the Oleander time series of Gulf Stream transport since 1992 (Rossby et al., 2014), and the position of the Gulf Stream Extension since 1955 (Joyce & Zhang, 2010). Studies based on models (e.g. Saba et al., 2016; Sanchez-Franks & Zhang, 2015) and observations (e.g., Kopp, 2013; McCarthy et al., 2015; Park & Sweet, 2015) have shown strong statistical relationships between US east coast sea level and these metrics at up to multidecadal timescales. We consider evidence in this section for relationships between DSL and these and other elements of the North Atlantic circulation, while emphasizing that changes in the latter do not necessarily imply changes in AMOC, as defined in section 2. We briefly discuss the nature of potential linkages with AMOC in section 6.3. Linkages Between DSL and AMOC Components The relationship between coastal DSL and the Gulf Stream has been assessed using theory, observations, and models. Studies have considered the roles of Gulf Stream transport, velocity, and position, both upstream and downstream of the detachment at Cape Hatteras, as well as the strength of the Florida Current. Early studies focused on the relationship between tide gauge observations and the Gulf Stream over seasonal timescales between Florida and Cape Hatteras. Two linkages between ocean circulation and DSL were considered: the cross-stream (shelf) sea level gradient, related to ocean circulation via geostrophy, and the downstream (along-coast) sea level gradient, related via the Bernoulli principle. Montgomery (1941) found little evidence for a relationship of the downstream sea level gradient to velocity (in the Gulf Stream). Attempts to relate the cross-stream gradient to Gulf Stream fluctuations were more successful. By examining tide gauges along the Florida coastline, and between Charleston and Bermuda, Montgomery (1938) concluded that fluctuations in Gulf Stream strength could be seen in cross-stream sea level measurements. The study of Iselin (1940) supported the utility of tide gauges south of Cape Hatteras (Key West to Charleston) for estimating the Gulf Stream strength. Both studies were based on comparison with shipboard hydrography and related a summer-to-fall increase in sea level to a drop in Gulf Stream transport. Hela (1951) revisited the two earlier studies to relate the annual cycle of sea level difference from Miami to Cat Cay, Bahamas to transport estimates of the Gulf Stream from ship drift (Fuglister, 1948), finding a high correlation (r = 0.95) between the zonal sea level gradient and meridional transport in the Gulf Stream. Blaha (1984) removed local effects of the inverse barometer, seasonal steric effects, river runoff, and local wind stress, to demonstrate that the residual sea level variability had a robust correlation with Gulf Stream transport on seasonal timescales. More recently, Park and Sweet (2015) found an interannual- to decadal-timescale relationship between Florida Current transport and tide gauge observations at three locations in Florida using empirical mode decomposition, with a scaling coefficient determined to be consistent with geostrophic balance. Similar techniques have been used to examine links between Gulf Stream transport variability and sea level in the Mid-Atlantic Bight. Ezer (2013) found a longer-period relationship between Mid-Atlantic Bight DSL and the sea surface gradient across the detached Gulf Stream. The offshore DSL gradient was found to be correlated with sea level at individual tide gauge locations over decadal timescales, as suggested by Yin and Goddard (2013). However, the robustness of these longer period relationships, found using statistical techniques including empirical mode decomposition, has been questioned (Chambers, 2015). Model-based support for observed Florida Current and Gulf Stream correlations is stronger on short timescales: for example, while idealized modeling studies show that an oscillatory transport of Gulf Stream is associated with coherent coastal sea level variations along the southeast U.S. coast (Ezer, 2016), Woodworth et al. (2017) do not see evidence of Florida Current transport variations in annual mean sea level, either averaged south of Cape Hatteras or in the difference of sea level averaged over the coastline north and south of Cape Hatteras. Coastal sea level has also been related to the position of the Gulf Stream on leaving the coast at Cape Hatteras, known as the Gulf Stream North Wall (GSNW; Fuglister, 1955). Indices of the GSNW based on sea surface temperature exist since 1966 (Taylor & Stephens, 1980) and based on temperature at 200 m since 1955 (Joyce & Zhang, 2010). The GSNW has been shown to exhibit quasi-decadal fluctuations that are similar to those in sea level data along the U.S. East Coast (McCarthy et al., 2019; Nigam et al., 2018). Kopp (2013) found a significant antiphase relationship between the GSNW index and DSL north of Cape Hatteras and a likely in-phase relationship between GSNW and DSL south of Cape Hatteras. McCarthy et al. (2015) noted the difference of sea level south and north of Cape Hatteras projected onto the surface velocity of the GSNW. Whether these sea level variations reflect AMOC strength changes relies upon an understanding of the interaction of different AMOC components: Early explanations associated an AMOC strengthening with a northward shift in the GSNW (e.g., Eden & Jung, 2001). However, recent literature indicates the inverse; AMOC strengthening drives a southward shift in the GSNW due to coupling between the Gulf Stream, Deep Western Boundary Current, and topography (Joyce & Zhang, 2010; Sanchez-Franks & Zhang, 2015; Yeager, 2015; Zhang & Vallis, 2007). AMOC variability may also be related to heat content and density variations in the subtropical and subpolar gyres (Williams et al., 2014). Such changes in gyre properties have been found to be correlated with U.S. East Coast sea level changes (Thompson & Mitchum, 2014). Frederikse et al. (2017) find that, after being adjusted for local atmospheric (wind and pressure) effects and smoothed on decadal timescales, sea level changes from tide gauges north of Cape Hatteras over 1965–2014 are correlated with upper-ocean steric height changes in the Labrador Sea and the deep midlatitude North Atlantic intergyre region. This is consistent with the strong relationship between U.S. coastal sea level and Labrador Sea level in the CMIP5 ensemble (Minobe et al., 2017). Other studies have considered property differences between gyres, in particular the meridional density gradient, as an indicator of AMOC strength (Butler et al., 2015; De Boer et al., 2010; Kienert & Rahmstorf, 2012; Rahmstorf, 1996; Rahmstorf et al., 2015; Sijp et al., 2012; Thorpe et al., 2001). The meridional density gradient can be related to the gyre-scale sea level gradient, which has been shown to be related to the strength of the AMOC over sufficiently long timescales (multidecadal and longer; Butler et al., 2015). This relationship was investigated by McCarthy et al. (2015), who used differences in DSL north and south of Cape Hatteras as an estimate of the meridional density gradient between the subtropical and subpolar gyres. The meridional gradient projected strongly onto the circulation in the intergyre region and changes in the subpolar heat content on interannual to decadal timescales. Output from a NEMO 0.25° simulation related the differences in DSL north and south of the modeled Gulf Stream separation to the meridional heat transport at 40°N, indicating a relationship to AMOC. Possible Sources of Regional, Intermodel, and Model-Observational Discrepancies The diagnostic geostrophic relationship between AMOC transport and U.S. East Coast sea level derived in section 3 implies a scaling coefficient of order −1 to −2 cm/Sv with little alongshore variation. Although some numerical simulations find coefficients within this range over portions of the U.S. East Coast, a uniform along-coast scaling of AMOC strength and DSL is not evident (section 4). These deviations from theory likely result from neglect of terms in the more complete zonal momentum balance (e.g., friction, nonlinearities, time dependence), or a breakdown in the assumption that U.S. East Coast sea level is related to the depth-averaged boundary pressure via a constant effective layer thickness (He in equation 6). Similarly, intermodel differences under identical forcing must originate in the relative magnitude of neglected dynamical terms, and their treatment in models. Observations of other components of the North Atlantic circulation offer general support for antiphase relationships between large-scale meridional transport and DSL along portions of the U.S. east coast but are constrained by their limited record length and indirect relationship with AMOC (section 5).	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-6	Possible Sources of Regional, Intermodel, and Model-Observational Discrepancies The diagnostic geostrophic relationship between AMOC transport and U.S. East Coast sea level derived in section 3 implies a scaling coefficient of order −1 to −2 cm/Sv with little alongshore variation. Although some numerical simulations find coefficients within this range over portions of the U.S. East Coast, a uniform along-coast scaling of AMOC strength and DSL is not evident (section 4). These deviations from theory likely result from neglect of terms in the more complete zonal momentum balance (e.g., friction, nonlinearities, time dependence), or a breakdown in the assumption that U.S. East Coast sea level is related to the depth-averaged boundary pressure via a constant effective layer thickness (He in equation 6). Similarly, intermodel differences under identical forcing must originate in the relative magnitude of neglected dynamical terms, and their treatment in models. Observations of other components of the North Atlantic circulation offer general support for antiphase relationships between large-scale meridional transport and DSL along portions of the U.S. east coast but are constrained by their limited record length and indirect relationship with AMOC (section 5). In this section, we highlight findings from three areas of research that can at least partially account for these regional, intermodel, and model-observational discrepancies via: (1) friction and bathymetry at the coast, (2) local forcing, and (3) temporal and spatial incoherence of AMOC and its components. In section 7, we suggest opportunities to better integrate these findings into the sea level literature. Friction and Topographic Influence on Coastal Sea Level Most analyses noted in sections 4 and 5 interpret AMOC-DSL relationships based on geostrophy. To understand offshore influences on coastal sea level, however, requires addressing ageostrophic flows and forcing on the slope and shelf, where water column thickness goes to zero and friction is important. Recently, Minobe et al. (2017) have addressed coastal DSL onshore of a western boundary current using a reduced gravity, vertical sidewall model. Such a framework bears similarity to that used in other studies of remotely forced coastal sea level variability in western boundary regions (e.g., Hong et al., 2000; Thompson & Mitchum, 2014). In this model, interior DSL gradients are moderated by friction within a coastal boundary layer. Their main result can be written as frac{h_W}{f} = (frac{h_W}{f})_0 + int_y^{y_0} frac{beta h_I}{f^2} dy’ where h_W and h_I are sea level as a function of latitude (y) at the western boundary and in the ocean interior respectively (h_I is taken near the boundary, but to the east of any western boundary current.) The integral is from the y value of interest to a reference point y_0 further north, where frac{h_W}{f} = (frac{h_W}{f})_0. The western boundary sea level is determined from a combination of the interior ocean sea level and the sea level from higher latitudes; coastal sea level anomalies are smaller than those in the interior and shifted toward the equator for reasons which become clearer if the vertical sidewall case is seen as a limiting case of a sloping continental shelf and slope. The southward shift and weakening of the “interior” sea level signal as it approaches the coast is reminiscent of the linear, barotropic case with a sloping sidewall as explored by Becker and Salmon (1997) following the ideas of Welander (1968). Instead of being controlled by contours of constant f, as in the flat-bottom case, the flow is controlled by contours of constant f/H, where H is the ocean depth. With varying bathymetry, the subpolar gyre intrudes between the coast and the extension of the subtropical gyre, resulting in a reversing pattern of currents along the continental slope, rather than a simple single-signed western boundary current. Similar behavior is found in highly nonlinear and baroclinic cases with a sloping sidewall (e.g., Jackson et al., 2006, their Figure 1). Sea level signals from the interior therefore appear further equatorward at the coast. Wise et al. (2018) assess the influence of continental shelf bathymetry, using linear dynamics and ocean bottom pressure as the central variable (equivalent to sea level in a single layer case). In this formalism, sea level is “advected” along contours of gH/f (such contours can be thought of as representing the stream function of a fictitious flow carrying the sea level signal toward the coast at a speed which becomes the long Rossby wave speed over a flat bottom). The “advection” is toward the west and then toward the equator along the slope, in competition with a “diffusion” by bottom friction (Figure 7). Figure 7: From Wise et al. (2018). Sea level contours (nondimensional; dashed negative) for a given idealized coastal bathymetry along the western boundary of an ocean basin, where x and y are the nondimensional across-shore and alongshore coordinates, respectively. Vertical dotted lines indicate the continental shelf break at x = S and continental slope floor at x = 1. Panels show sea level patterns for different Péclet numbers: (a) Pa = 0.1, (b) Pa = 0.1, (c) Pa = 10, and (d) Pa = 200. Panels (b)–(d) show only the coastal region. As the coast is approached, the geostrophic shoreward flow becomes balanced by an offshore flow in the bottom Ekman layer, as in Csanady (1978). Friction is required for alongshore sea level gradients to exist without a flow through the coast, as a purely geostrophic balance would imply. When shelf bathymetry is included, coastal sea level is still determined by the combination of a poleward reference value, and a weighted integral of interior sea level between that poleward latitude and the latitude of interest. However, the coastal sea level anomaly can be smaller than that predicted in the Minobe et al. (2017) configuration: the western pressure signal can all be on the continental slope, with shallower currents causing it to be cancelled out at the coast. Wise et al. (2018) find that coastal DSL depends crucially on the strength of the bottom friction and the shelf bathymetry. The major dependence is on a nondimensional number, the analogue Péclet number Pa = βHL/r, where H is the offshore layer thickness, L is the width of the topography, and r is a linear bottom friction coefficient (Figure 7). As friction weakens, the coastal signal shifts further south and becomes weaker compared to the interior sea level. Equation 8 is a limiting case for a vertical sidewall, in which the solution becomes independent of the strength or form of the friction. In this linear case, the vertical sidewall limit is found to produce the largest coastal signal, for a given upper layer thickness. The mechanism here can be considered to be a breakdown of the assumption that there exists a meaningful effective layer thickness He. Counterpropagating currents over topography mean the boundary pressure pW can change sign over the upper continental slope, so equation 5 shows that He can become larger than H. Thus, the coastal sea level can be smaller than that implied by the depth-averaged pressure divided by a meaningful effective layer depth. This reduction of the coastal signal can be interpreted as the result of the influence of coastal trapped waves, which carry the interior signal equatorward along the western boundary, as seen for periods of a few days in the model simulations of Ezer (2016). See Hughes et al. (2019) for more detail on the smoothing and “advective” effect of coastal trapped waves on boundary sea level. We should note, though, that although friction plays a crucial role in communicating sea level changes to the coast, it does so in a manner which does not affect the zonal momentum balance (equation 1), which remains geostrophic. Locally (Shelf-) Forced Sea Level Variability The presence of locally forced sea level variability along the shelf may interfere with the simple AMOC-DSL scaling. Similar to section 6.1, ageostrophic dynamics are relevant, although in this case they may also upset the zonal momentum balance. Local meteorological and terrestrial forcing mechanisms, namely, winds, barometric pressure, and river runoff, have long been shown to drive U.S. East Coast sea level variability. Part of this variability can be static in nature, as with the case of inverted barometer effects related to atmospheric pressure, which are found to contribute sizably to variability at many tide gauges (Piecuch & Ponte, 2015; Ponte, 2006). By definition, static signals are not directly related to circulation changes. As such, their separate treatment, and removal if possible, is useful when assessing the relation between tide gauge and AMOC variability.	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-7	Equation 8 is a limiting case for a vertical sidewall, in which the solution becomes independent of the strength or form of the friction. In this linear case, the vertical sidewall limit is found to produce the largest coastal signal, for a given upper layer thickness. The mechanism here can be considered to be a breakdown of the assumption that there exists a meaningful effective layer thickness He. Counterpropagating currents over topography mean the boundary pressure pW can change sign over the upper continental slope, so equation 5 shows that He can become larger than H. Thus, the coastal sea level can be smaller than that implied by the depth-averaged pressure divided by a meaningful effective layer depth. This reduction of the coastal signal can be interpreted as the result of the influence of coastal trapped waves, which carry the interior signal equatorward along the western boundary, as seen for periods of a few days in the model simulations of Ezer (2016). See Hughes et al. (2019) for more detail on the smoothing and “advective” effect of coastal trapped waves on boundary sea level. We should note, though, that although friction plays a crucial role in communicating sea level changes to the coast, it does so in a manner which does not affect the zonal momentum balance (equation 1), which remains geostrophic. Locally (Shelf-) Forced Sea Level Variability The presence of locally forced sea level variability along the shelf may interfere with the simple AMOC-DSL scaling. Similar to section 6.1, ageostrophic dynamics are relevant, although in this case they may also upset the zonal momentum balance. Local meteorological and terrestrial forcing mechanisms, namely, winds, barometric pressure, and river runoff, have long been shown to drive U.S. East Coast sea level variability. Part of this variability can be static in nature, as with the case of inverted barometer effects related to atmospheric pressure, which are found to contribute sizably to variability at many tide gauges (Piecuch & Ponte, 2015; Ponte, 2006). By definition, static signals are not directly related to circulation changes. As such, their separate treatment, and removal if possible, is useful when assessing the relation between tide gauge and AMOC variability. Effects of local winds have been extensively examined in the observational studies of Blaha (1984), Andres et al. (2013), Domingues et al. (2018), and others. Simple regression analyses suggest an important contribution of local winds, particularly the alongshore component, to observed tide gauge variability at interannual to decadal timescales. Setup from onshore winds can also contribute to static variability at the coast (e.g., Thompson, 1986), but separate estimation of these effects has not been examined in detail. Recent studies (Domingues et al., 2018; Li et al., 2014; Little et al., 2017; Piecuch et al., 2016; Woodworth et al., 2014) reinforce the importance of near-coastal winds and barotropic dynamics to explain US east coast tide gauge records over interannual to decadal timescales. Much less studied has been the effect of river runoff. Meade and Emery (1971) found that about 20–29% of variations in detrended annual mean sea level in U.S. East Coast tide gauges could be accounted for by changes in riverine input. Their results are consistent with the analysis by Piecuch, Bittermann, et al. (2018), who relate sea level signals to the buoyancy-driven geostrophic coastal currents associated with the runoff. Other studies focusing on different river systems and utilizing different data sets have concluded that riverine input is negligible. For example, Hong et al. (2000) found contributions from runoff to be unimportant relative to winds for tide gauges south of 38°N (see also Blaha, 1984). Calafat et al. (2018) did not find a relationship between river runoff and decadal modulations in the amplitude of the sea level annual cycle along the South Atlantic Bight. However, beyond the few studies noted here, most US east coast sea level studies have ignored riverine effects. Regardless of its origin, the presence of local forcing can lead to large sea level variations that mask the open ocean influence, and thus the emergence of AMOC-associated sea level variability relative to locally forced variability. For example, correlations of DSL and AMOC are weaker in simulations that include wind forcing, particularly close to the coast and along the Northeast U.S. shelf (Figure 8). Figure 8: From Woodworth et al. (2014). (a) Correlations of detrended values of annual mean sea level and overturning transport at the same latitude for depths between 100 and 1,300 m using the simulations shown in Figure 6a (without wind forcing). (b) As in Figure 8a, with winds. The fact that atmospheric variability has an almost white spectrum means that locally forced variability will tend to be the dominant influence at higher frequencies, with emergence of the open ocean influence at lower frequencies. Little et al. (2017) conclude, using a climate model ensemble, that coherence with AMOC emerges along the northeast U.S. coast at periods of around 20 years. This conclusion echoes Woodworth et al. (2014), who find that local winds dominate nearshore sea level variability on interannual timescales. We note, however, that local forcing may evolve over longer timescales and may be responsible for some of the model spread seen in Figure 5. For example, Woodworth et al. (2017) suggest that changes in the 20th century wind field may underlie long-period changes in coastal sea level. Furthermore, atmospheric forcing is spatially coherent over very large scales; changes in local forcing may be associated with large-scale patterns of change that also influence AMOC and/or remote regions of the ocean. Spatiotemporal Complexity of AMOC, Hydrography, and Current Changes The fact that AMOC is the residual of a spatially and temporally complex system of surface and deep currents (Figure 2; see other reviews in this volume) underscores the relevance of the previous two sections for interpretations of observations: any current used as a proxy for AMOC (e.g., the Florida Current) may be characterized by an ageostrophic momentum balance (e.g., due to inertial terms in western boundary currents, or frictional effects in coastal currents). In fact, it is likely that ageostrophic terms become more important at these smaller scales. An additional important consideration is that currents may be zonally or meridionally compensated, either over shorter timescales, or in the steady state. Observations and modeling studies reveal that changes in AMOC can arise from changes in any of its components, including the interior subtropical gyre (Duchez et al., 2014; Smeed et al., 2018; Zhao & Johns, 2014) and subpolar gyre (Kwon & Frankignoul, 2014; Yeager, 2015), western boundary currents (Beadling et al., 2018; Thomas et al., 2012), and the formation of deep water at high latitude (Medhaug et al., 2011). Additional changes and variability also arise through near-surface Ekman transports (Kanzow et al., 2007), their barotropic compensation (Jayne & Marotzke, 2001), and eddy transports (e.g., Thomas & Zhai, 2013). All of these exhibit varying degrees of zonal and meridional coherence, reflecting a multitude of forcings occurring over different timescales (Wunsch & Heimbach, 2013). For example, the Gulf Stream, by which we refer to the full western boundary current near southern Florida, has two branches: the Florida Current and the Antilles Current, which flows offshore of the Bahama Banks (Figure 2). While the Florida Current carries a larger mean transport (about 32 Sv compared with about 5 Sv in the Antilles Current), both exhibit comparable variability (Lee et al., 1996). Thus, the total western boundary current flow could be constant, but its effect on coastal DSL would vary depending upon the respective contributions of the Florida and Antilles currents. In addition, assessing trends in the volume transport of complex, evolving, western boundary currents is challenging. This difficulty underlies the debate surrounding Ezer et al.’s (2013) conclusion that a Gulf Stream decline was responsible for accelerated sea level rise in the mid-Atlantic Bight (Ezer, 2015; Rossby et al., 2014). The deviation in Gulf Stream transport calculations found across studies is perhaps not surprising, given longitudinally varying changes in the Gulf Stream velocity, width, and position (Dong et al., 2019), and the presence of Gulf Stream meanders, eddies, and recirculation gyres.	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-8	For example, the Gulf Stream, by which we refer to the full western boundary current near southern Florida, has two branches: the Florida Current and the Antilles Current, which flows offshore of the Bahama Banks (Figure 2). While the Florida Current carries a larger mean transport (about 32 Sv compared with about 5 Sv in the Antilles Current), both exhibit comparable variability (Lee et al., 1996). Thus, the total western boundary current flow could be constant, but its effect on coastal DSL would vary depending upon the respective contributions of the Florida and Antilles currents. In addition, assessing trends in the volume transport of complex, evolving, western boundary currents is challenging. This difficulty underlies the debate surrounding Ezer et al.’s (2013) conclusion that a Gulf Stream decline was responsible for accelerated sea level rise in the mid-Atlantic Bight (Ezer, 2015; Rossby et al., 2014). The deviation in Gulf Stream transport calculations found across studies is perhaps not surprising, given longitudinally varying changes in the Gulf Stream velocity, width, and position (Dong et al., 2019), and the presence of Gulf Stream meanders, eddies, and recirculation gyres. Understanding the timescales over which the AMOC indicators discussed in section 5 (e.g., the Gulf Stream, and gyre densities) and AMOC strength variations are coherent is critical to their use as proxies of AMOC. There is little evidence for seasonal and interannual variability of the Florida Current or the Gulf Stream (characteristic over the timescales of many studies cited in section 5) to be related to AMOC. Using evidence that Sverdrup balance holds on multiannual to decadal timescales in the interior subtropics (Gray & Riser, 2014; Thomas et al., 2014; Wunsch, 2011), it can be demonstrated that (subtropical) AMOC variability must be mirrored by changes in the western boundary current at these timescales (de Boer & Johnson, 2007; Thomas et al., 2012). The Gulf Stream can therefore be expected to concentrate decadal-period changes in both the wind-driven and the thermohaline circulations, both of which are predicted to weaken in the 21st century (Beadling et al., 2018; Lique & Thomas, 2018; Thomas et al., 2012). However, this finding only applies southward of approximately 35°N, since the ocean to the north is not in Sverdrup balance (Gray & Riser, 2014; Thomas et al., 2014). Furthermore, there is no satisfactory way of defining the boundary between a western boundary current and the ocean interior when the ocean is dominated by mesoscale eddies (Wunsch, 2008). Models and observations also reveal a strong gyre dependence of AMOC changes, with interannual variability dominating in the subtropical gyre and decadal variability in the subpolar gyre (e.g., Bingham et al., 2007; Wunsch, 2011; Wunsch & Heimbach, 2013; Zhang, 2010). Lozier et al. (2010) used a data-assimilating numerical model to further demonstrate that gyre-dependent AMOC changes might be important on up to multidecadal periods. Relatedly, there is evidence that property changes in the subpolar and subtropical gyres may not reflect changes in AMOC over certain timescales. Processes governing ocean density changes in this region on decadal timescales remain unclear (Williams et al., 2015; Buckley & Marshall, 2016; Menary et al., 2015; Piecuch, Ponte, et al., 2017; Robson et al., 2016); remote Rossby wave signals, local atmospheric forcing, changes in deep convection and water mass formation, mean flow advection, and gyre circulation “wobbles” all potentially play a role (Buckley & Marshall, 2016). Although data collected in the subpolar and subtropical gyres suggest southward propagation of deep hydrographic properties on advective (multiannual to decadal) timescales in the Labrador Current and Deep Western Boundary Current of the subtropical gyre (e.g., Molinari et al., 1998; Talley & McCartney, 1982; van Sebille et al., 2011), tracer studies have identified that the majority of water in the Labrador Current does not pass southwards into the subtropical gyre but instead cyclonically recirculates back around within the subpolar gyre (e.g., Bower et al., 2009; Rhein et al., 2002; Zou & Lozier, 2016). Of the deep subpolar water that is advected into the subtropical gyre, the intergyre pathway is not principally via the Deep Western Boundary Current but rather through the interior ocean (Bower et al., 2009; Lozier, 2010; Zhang, 2010), which is compensated by slow upper ocean advective pathways northwards out of the subtropical gyre that reach the greatest transport at depths of approximately 700 m (Burkholder & Lozier, 2011, 2014). Implications for Along-Coast Variations and Across-Model Differences Collectively, sections 6.1 to 6.3 indicate that U.S. East Coast continental shelf bathymetry, and the evolution of western boundary and coastal currents under local- and large-scale forcing, will influence the local coastal sea level expression associated with a given change in AMOC. The importance of these processes should be expected to vary regionally (e.g., north and south Cape Hatteras, but also within each region); future studies might probe the influence of these smaller scale along-coast variations on local sea level gradients (see section 7). Focusing on time-mean sea level on the shelf, Higginson et al. (2015) suggest that coarse resolution models may exhibit errors in the representation of coastal sea level due to inadequate horizontal resolution, the form of the coastal boundary condition, poor representation of processes in shallow water, and/or unresolved continental shelf atmospheric forcing. Sections 6.1 and 6.2 support the importance of the representation of these coastal processes, and imply that differences in the model resolution may underlie some of the spread shown in Figure 5. Over the global coastal ocean, Becker et al. (2016) find that climate models have a wide range of success in reproducing the spectral characteristics of observed tide gauge sea level variability. Little et al. (2017) specifically tested the ability of an initial condition ensemble of Community Earth System Model simulations to represent interannual U.S. East Coast DSL variability, finding that Community Earth System Model agrees well with observed tide gauge data along the Northeast U.S. coast, but poorly represents the time-mean and variability of DSL south of Cape Hatteras. The Minobe et al. (2017) framework (section 6.1) also exhibits disagreement with CMIP5 US east coast DSL changes south of ~35°N (see their Figure 10). This suggests that large-scale models might be particularly limited in the South Atlantic Bight. Here, in addition to complex shelf bathymetry, DSL variability may also be influenced by incoherence between the Gulf Stream and AMOC, the complex vertical and horizontal structure of western boundary currents, the potential effect of rapid western boundary current flow against the prevailing propagation of information in the direction of boundary waves, and the Antilles Current (section 6.3). Penduff et al. (2010) find that higher-resolution models (as fine as 0.25°) show improved representations of variability and time-mean Sea Surface Height (SSH), especially in the eddy rich regions, in comparison to altimetry. Coastal sea level variability also appears improved with finer resolution, and DSL change under strong external forcing appears to be moderated near the coastline in models of higher resolution (Liu et al., 2016). Other high-resolution simulations show substantial modification of the coastal sea level signal (e.g., the two MPI models in Figure 5). Such resolution effects deserve more investigation as simulations become available (see, e.g., Haarsma et al., 2016). In addition to the varied, resolution-dependent, representation of coastal processes and shelf bathymetry in models, which might be expected to disproportionately affect coastal DSL, the spatial variability in the “interior” DSL change in CMIP5 models implies that more complex changes in the 2-D overturning, or in the 3-D structure of the North Atlantic circulation, are relevant for determining patterns of DSL change. Bouttes et al. (2014) suggest that the underlying driver of differences in large-scale DSL change is related to locations of deep convection. Support for dependence on forcing is also evident in Kienert and Rahmstorf (2012), who find a substantially different DSL response to AMOC changes associated with different forcing (freshwater hosing, CO2 increases, Southern Ocean wind stress changes) within the same climate model. Perspective and Future Directions	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-9	Penduff et al. (2010) find that higher-resolution models (as fine as 0.25°) show improved representations of variability and time-mean Sea Surface Height (SSH), especially in the eddy rich regions, in comparison to altimetry. Coastal sea level variability also appears improved with finer resolution, and DSL change under strong external forcing appears to be moderated near the coastline in models of higher resolution (Liu et al., 2016). Other high-resolution simulations show substantial modification of the coastal sea level signal (e.g., the two MPI models in Figure 5). Such resolution effects deserve more investigation as simulations become available (see, e.g., Haarsma et al., 2016). In addition to the varied, resolution-dependent, representation of coastal processes and shelf bathymetry in models, which might be expected to disproportionately affect coastal DSL, the spatial variability in the “interior” DSL change in CMIP5 models implies that more complex changes in the 2-D overturning, or in the 3-D structure of the North Atlantic circulation, are relevant for determining patterns of DSL change. Bouttes et al. (2014) suggest that the underlying driver of differences in large-scale DSL change is related to locations of deep convection. Support for dependence on forcing is also evident in Kienert and Rahmstorf (2012), who find a substantially different DSL response to AMOC changes associated with different forcing (freshwater hosing, CO2 increases, Southern Ocean wind stress changes) within the same climate model. Perspective and Future Directions An antiphase relationship between large-scale North Atlantic meridional volume transport and U.S. East Coast DSL is broadly evident across a range of numerical simulations and observational analyses. This relationship can be interpreted using the simple geostrophic framework introduced in section 3. However, such a framework is insufficient to explain the widely differing along-coast AMOC-DSL scalings derived in models and observations, or variation across climate models. Furthermore, such an interpretation limits causal attribution: Geostrophy cannot provide information about the forces that drive sea level changes. In this review, we have noted some possible origins for regional, model, timescale, and forcing dependence (section 6). However, we are unable to assess the degree to which each is responsible for variations in local scaling coefficients. Explanations for these deviations are essential to improve confidence in reconstructions of North Atlantic variability derived from tide gauge observations or paleoproxies and projections of coastal sea level change from current-generation climate models. We thus encourage the sea level research community to pursue the following near-term goals: (1) an understanding of the relationship between AMOC and other North Atlantic currents; (2) an understanding of the vertical structure of the AMOC and its variation with respect to local bathymetry; (3) an assessment of the importance of ageostrophic processes to AMOC and related currents; and (4) an effort to connect these research results, including their region (latitude-), model, and timescale dependence, to their origins in heat, momentum, and buoyancy forcing. Such efforts should include new sea level studies, as well as the incorporation of existing and new findings from outside the sea level realm. A simple step toward the first and second goals involves broadening the features of the ocean circulation analyzed in models beyond a single AMOC metric (e.g., the basin-wide maximum overturning stream function). Modeled and observed DSL changes have often been compared to AMOC changes at a different latitude, which involves an implicit or explicit assumption that such changes are synchronous and meridionally coherent, which is not supported by the literature cited in section 6.3. Indeed, such a coarse characterization of AMOC may underlie some of the difference in scaling coefficients shown in Figure 5. As noted in section 4, another critical ambiguity of relevance, particularly important to the interpretation of observational analyses, is the coherence of AMOC and western boundary currents. Other important relationships include those between the GSNW and AMOC; Labrador Sea and subpolar gyre steric changes; and subpolar and subtropical gyre steric changes. Higher-resolution simulations can now represent the mean state and variability of coastal currents and indicate that climate-driven changes in these currents may differ from those in the large-scale (e.g., Saba et al., 2016). Although evidence in section 6.3 suggests that many components of AMOC, and subpolar and Nordic Seas buoyancy variability, may be coherent over multidecadal time frames (Pillar et al., 2016), there is evidence that interannual to decadal variability is not, particularly across the intergyre boundary. Modeling studies examining AMOC-DSL relationships can easily include metrics of some of these other AMOC components and indicators (possibly over different timescales), which would improve the scope of their results, and the ability to reconcile with observations. The direct observational record of AMOC variability is limited; in this review, we have focused on the longer observed record of AMOC components. However, the ever extending record of AMOC at 26°N is now complemented by the OSNAP array, providing some perspective on gyre dependence, the meridional coherence of AMOC, and the relationship with other AMOC components. These AMOC records are complemented by new observational campaigns over the U.S. East coast continental shelf and slope (e.g., Andres et al., 2018, Gawarkiewicz et al., 2018). In addition to these instrumental records, proxy records of both coastal sea level and AMOC are available that are able to resolve decadal-centennial fluctuations (Engelhart & Horton, 2012; Kemp et al., 2017, 2018; Rahmstorf et al., 2015; Thornalley et al., 2018). Complemented by model results, these proxy observations could provide valuable constraints on multidecadal to centennial AMOC-DSL covariability. With respect to the assessment of ageostrophic processes, we note that many modeling centers have begun to provide the output required to compute closed momentum budgets offline (Gregory et al., 2016; Wunsch & Heimbach, 2009; Yeager, 2015). Such budgets, both zonally integrated, and local, would clearly indicate the importance of ageostrophic processes, their time and latitude dependence, and (if possible) differences across a set of models. They could also include the effect of terms, including nonlinearity (Hughes et al., 2019), and, in higher-resolution models, eddy variability (Grégorio et al., 2015; Sérazin et al., 2015), that are not discussed in this review. High-resolution models also offer promise for better resolving the shelf and shelf processes, and they may constitute a means for testing the theories of coastal modulation of interior signals (section 6.1), under a wider range of conditions, forcing, and timescales. Such analyses also move beyond the purely diagnostic, degenerate, statement of force balance supplied by geostrophy, allowing an understanding of the local, regional, basin, and global scale forcing responsible for coastal sea level changes. The incomplete interpretation provided by geostrophy is evident in Goddard et al. (2015), who linked an “extreme” interannual sea level rise event in the northeast US with an abrupt 30% AMOC weakening. However, this event occurred coincident with an anomalously negative North Atlantic Oscillation (NAO) associated with atmospheric pressure and wind anomalies. Piecuch and Ponte (2015) and Piecuch et al. (2016) demonstrated that 50% of this event could be explained by the inverse barometer effect and the remainder could be partly explained by local winds. The 30% drop in the AMOC itself was observed in the Gulf Stream transport (Ezer, 2015) and was explained by wind forcing (Zhao & Johns, 2014). It is thus more appropriate to view the sea level anomaly as driven by all of the forcings (local and remote) associated with the extreme NAO anomaly. Over longer timescales, causality often remains unclear: for example, differences in observed sea level changes along the US east coast have been attributed to changes in Gulf Stream position and strength, AMOC strength, and steric changes. While these changes may be coupled, and serve as indicators of AMOC, they do not identify causal drivers. Even if causality can be established under certain forcing and timescales (e.g., interannual, driven by NAO), it does not imply that the same processes and AMOC components (and sea level signatures) are always relevant (e.g., on centennial timescales in the past or future). For example, Kenigson et al. (2018) find that the relationship between DSL and NAO is nonstationary, echoing the results of Andres et al. (2013). Looking farther into the future, 21st century changes in AMOC strength in climate models are principally forced by greenhouse gas-associated heat and buoyancy fluxes in the North Atlantic (Beadling et al., 2018; Bouttes et al., 2014; Slangen et al., 2015), rather than NAO-associated wind stress.	https://sealeveldocs.readthedocs.io/en/latest/little19.html
e48670af6dee-10	Such analyses also move beyond the purely diagnostic, degenerate, statement of force balance supplied by geostrophy, allowing an understanding of the local, regional, basin, and global scale forcing responsible for coastal sea level changes. The incomplete interpretation provided by geostrophy is evident in Goddard et al. (2015), who linked an “extreme” interannual sea level rise event in the northeast US with an abrupt 30% AMOC weakening. However, this event occurred coincident with an anomalously negative North Atlantic Oscillation (NAO) associated with atmospheric pressure and wind anomalies. Piecuch and Ponte (2015) and Piecuch et al. (2016) demonstrated that 50% of this event could be explained by the inverse barometer effect and the remainder could be partly explained by local winds. The 30% drop in the AMOC itself was observed in the Gulf Stream transport (Ezer, 2015) and was explained by wind forcing (Zhao & Johns, 2014). It is thus more appropriate to view the sea level anomaly as driven by all of the forcings (local and remote) associated with the extreme NAO anomaly. Over longer timescales, causality often remains unclear: for example, differences in observed sea level changes along the US east coast have been attributed to changes in Gulf Stream position and strength, AMOC strength, and steric changes. While these changes may be coupled, and serve as indicators of AMOC, they do not identify causal drivers. Even if causality can be established under certain forcing and timescales (e.g., interannual, driven by NAO), it does not imply that the same processes and AMOC components (and sea level signatures) are always relevant (e.g., on centennial timescales in the past or future). For example, Kenigson et al. (2018) find that the relationship between DSL and NAO is nonstationary, echoing the results of Andres et al. (2013). Looking farther into the future, 21st century changes in AMOC strength in climate models are principally forced by greenhouse gas-associated heat and buoyancy fluxes in the North Atlantic (Beadling et al., 2018; Bouttes et al., 2014; Slangen et al., 2015), rather than NAO-associated wind stress. Separation of local and remote wind-driven changes in circulation and sea level from remote buoyancy/deep water driven AMOC changes remains a key challenge. Such work will have to illuminate the timescales and climate forcing under which wind and buoyancy forcing are coupled. For example, Woodworth et al. (2014) indicate that wind forcing alone is largely responsible for decadal timescale sea level variability. However, since this study used a standalone ocean model, it is not clear what processes produce low-frequency wind variability. Furthermore, the large spatial scales of atmospheric forcing challenge efforts to isolate the AMOC-forced or remotely forced component of sea level change. Adjoint analyses or perturbation experiments (Heimbach et al., 2011; Pillar et al., 2016; Yeager & Danabasoglu, 2014) may help isolate the roles of wind and buoyancy forcing and elucidate the relevant pathways, state variables, and adjustment processes mediating connections between the open ocean and observed and projected US east coast sea level changes. To conclude, there are many productive areas of research that can help refine our understanding of the relationship between the large-scale climate, AMOC, and coastal sea level. Given their importance to future sea level changes on the U.S. East Coast, and reconstruction of preinstrumental ocean circulation and climate variability, we anticipate the research community will pursue them with vigor.	https://sealeveldocs.readthedocs.io/en/latest/little19.html
14d9a88b20c2-0	Gregory et al. (2019) Title: Concepts and Terminology for Sea Level: Mean, Variability and Change, Both Local and Global Key Points: Describes concepts and terminology related to sea level, its variability, and changes both locally and globally. Clarifies language and definitions for better communication in sea-level science Discusses the inconsistencies and ambiguities in sea-level terminology across different disciplines. Proposes new terminology and recommend replacing certain terms that are unclear or confusing. Keywords: sea level, concepts, terminology Corresponding author: Jonathan M. Gregory Citation: Gregory, J. M., Griffies, S. M., Hughes, C. W., Lowe, J. A., Church, J. A., Fukimori, I., et al. (2019). Concepts and Terminology for Sea Level: Mean, Variability and Change, Both Local and Global. Surveys in Geophysics, 40(6), 1251–1289. doi:10.1007/s10712-019-09525-z URL: https://link.springer.com/article/10.1007/s10712-019-09525-z Abstract Changes in sea level lead to some of the most severe impacts of anthropogenic climate change. Consequently, they are a subject of great interest in both scientific research and public policy. This paper defines concepts and terminology associated with sea level and sea-level changes in order to facilitate progress in sea-level science, in which communication is sometimes hindered by inconsistent and unclear language. We identify key terms and clarify their physical and mathematical meanings, make links between concepts and across disciplines, draw distinctions where there is ambiguity, and propose new terminology where it is lacking or where existing terminology is confusing. We include formulae and diagrams to support the definitions. Introduction and Motivation Changes in sea level lead to some of the most severe impacts of anthropogenic climate change (IPCC 2014). Consequently, they are a subject of great interest in both scientific research and public policy. Since changes in sea level are the result of diverse physical phenomena, there are many authors from a variety of disciplines working on questions of sea-level science. It is not surprising that sea-level terminology is inconsistent across disciplines (for example, “dynamic sea level” has different meanings in oceanography and geodynamics), as well as unclear or ambiguous even within a single discipline. (For instance, “eustatic” is ambiguous in the climate science literature.) We sometimes experience difficulty in finding correct and precise terms to use when writing about sea-level topics, or in understanding what others have written. Such communication problems hinder progress in research and may even confuse discussions about coastal planning and policy. This situation prompted us to revisit the meaning of key sea-level terms, and to recommend definitions along with their rationale. In so doing, we aim to clarify meanings, make links between concepts and across disciplines and draw distinctions where there is ambiguity. We propose new terminology where it is lacking and recommend replacing certain terms that we argue are unclear or confusing. Our goal is to facilitate communication and support progress within the broad realm of sea-level science and related engineering applications. In the next section, we outline the conventions and assumptions we use in our definitions and mathematical derivations. The following three sections (Sects. 3–5) contain the definitions, with a subsection for each major term defined, labelled with “N” and numbered consecutively throughout. In Sect. 3 we define five key surfaces: reference ellipsoid, sea surface, mean sea level, sea floor and geoid. We consider the variability and differences in these surfaces in Sect. 4, and quantities describing changes in sea level in Sect. 5. In Sect. 6, we show how relative sea-level change is related to other quantities in various ways. In Sect. 7 we describe how observational data are interpreted using the concepts we have defined. To facilitate sequential reading of this paper, the material of Sects. 3–7 is arranged to minimize forward references, though we were unable to avoid all. We give a list of deprecated terms with recommended replacements in Sect. 8, and a list in Sect. 9 of all terms defined, referring to the subsections where they are defined, thus providing an index that also includes our notation. The appendices contain further discussion of some aspects at greater length. The complexity of sea-level science is evident in the detail of the definitions and discussions in this paper. It may therefore be helpful to keep in mind that from the point of view of coastal planning and climate policy there are three quantities of particular interest. Extreme sea level along coasts (i.e., extreme coastal water level) or around offshore marine infrastructure (such as drilling platforms) is of great practical importance because of the enormous damage it can cause to human populations and their built environment and to ecosystems. The expected occurrence of extreme sea level under future climates is therefore relevant to decision-making on a range of time-horizons. The dominant factor in changes of future local extremes is relative sea-level change (RSLC), i.e., the change in sea level with respect to the land (Lowe et al. 2010; Church et al. 2013). Where there is relative sea-level rise, coastal defences have to be raised to afford a constant level of protection against extremes, and low-lying areas are threatened with permanent inundation. Although RSLC depends on many local and regional influences, the majority of coastlines are expected to experience RSLC within a few tens of per cent of global-mean sea-level rise (GMSLR) (Church et al. 2013). Projections of GMSLR are therefore of interest to global climate policy, with both adaptation and mitigation in mind. In general, greater GMSLR is projected for scenarios with higher rates of carbon dioxide emission and on longer timescales. Particular attention is paid, especially by risk-intolerant users, to the probability or possibility of future large changes in sea level (Hinkel et al. 2019). We recommend referring to these as high-end scenarios or projections of RSLC or GMSLR (rather than “extreme” scenarios), to avoid confusion with projections of extreme sea level. Conventions and Assumptions We here summarize the conventions and assumptions employed in the text and formulae of our definitions. First Appearance of Terms Within each definition subsection of Sects. 3–5, we use bold font for the first appearance of a term whose definition is the subject of another subsection. When we first define a term that does not have its own subsection, it appears with a slanted font. The reader can locate the definitions of terms marked in these ways by looking them up in Sect. 9. In the PDF of this paper, each bold term in Sects. 3–8 is a hyperlink to the relevant definition subsection. Time-Mean and Changes The sea surface varies on all the timescales of the Earth system, associated with sea-surface waves and tides, meteorological variability (from gustiness of winds to synoptic phenomena such as mid-latitude depressions and tropical cyclones), seasonal, interannual and longer-term internally generated climate variability (e.g., El Niño and the Interdecadal Pacific Oscillation), anthropogenic climate change and naturally forced changes (e.g., by volcanic eruptions, glacial cycles and tectonics). For various purposes of understanding and planning for sea-level variations it is helpful to draw a distinction between a time-mean state and fluctuations within that state. The definition of a “state” depends on the scientific interest or application. For sea level, the state might be defined by a time-mean long enough to remove tidal influence (about 19 years), or which characterizes a climatological state (conventionally 30 years), but it could be shorter, for example, if interannual variability were regarded as altering the state. Thus, the time-mean state cannot be absolutely defined, but the concept is necessary. In this paper, mean sea level refers to a time-mean state whose precise definition should be specified when the term is used, and which is understood to be long enough to eliminate the effect of meteorological variations at least. We use symbols with a tilde and time-dependence, e.g., 𝑋̃(𝑡) for time-varying quantities, and symbols without any distinguishing mark and no time-dependence, e.g., X for time-mean quantities that characterize the state of the system. On longer timescales, the state itself may change, for example, due to anthropogenic influence. We use the symbol Δ and the word “change” to refer to the difference between any two states; thus, Δ𝑋 is “change in X”, e.g., change in relative sea level between the time-mean of 1986–2005 and the time-mean of 2081–2100. Anthropogenic sea-level change comes mostly through climate change, but there are other influences too, such as impoundment of water on land in reservoirs. Local and Regional By a local quantity, we mean one which is a function of two-dimensional geographical location 𝐫, specified by latitude and longitude. For some applications, it is important to consider variations of local quantities over distance scales of kilometres or less. Other quantities do not have such pronounced local variation and are typically considered as properties of regions, with distance scales of tens to hundreds of kilometres. Global Mean Over the Ocean Surface Area	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-1	Thus, the time-mean state cannot be absolutely defined, but the concept is necessary. In this paper, mean sea level refers to a time-mean state whose precise definition should be specified when the term is used, and which is understood to be long enough to eliminate the effect of meteorological variations at least. We use symbols with a tilde and time-dependence, e.g., 𝑋̃(𝑡) for time-varying quantities, and symbols without any distinguishing mark and no time-dependence, e.g., X for time-mean quantities that characterize the state of the system. On longer timescales, the state itself may change, for example, due to anthropogenic influence. We use the symbol Δ and the word “change” to refer to the difference between any two states; thus, Δ𝑋 is “change in X”, e.g., change in relative sea level between the time-mean of 1986–2005 and the time-mean of 2081–2100. Anthropogenic sea-level change comes mostly through climate change, but there are other influences too, such as impoundment of water on land in reservoirs. Local and Regional By a local quantity, we mean one which is a function of two-dimensional geographical location 𝐫, specified by latitude and longitude. For some applications, it is important to consider variations of local quantities over distance scales of kilometres or less. Other quantities do not have such pronounced local variation and are typically considered as properties of regions, with distance scales of tens to hundreds of kilometres. Global Mean Over the Ocean Surface Area By global mean, we mean the area-weighted mean over the entire connected surface area of the ocean, i.e., excluding the land. The ocean includes marginal seas connected to the open ocean such as the Mediterranean Sea, Black Sea and Hudson Bay, but excludes inland seas such as the Caspian Sea, the African Great Lakes and the North American Great Lakes. It includes areas covered by sea ice and ice shelves, where special treatment is needed to define the level of the sea surface. We note that observational estimates of the global mean are often made from systems which lack complete coverage. For centennial timescales, we can assume the ocean surface area A is constant, with A = 3.625×10^{14} m^2 (Cogley 2012). It is altered substantially by global-mean sea-level changes of many metres, such as on glacial–interglacial timescales or possibly over future millennia due to ice-sheet changes, and on geological timescales due to plate tectonics. The formulae we give for some quantities describing global-mean changes are not exactly applicable under those circumstances. Sea-Water Density Many of the formulae in this paper involve sea-water density. Although sea-water density is a local quantity, we treat it in many contexts as a globally uniform constant with a representative value 𝜌∗ (e.g., 1028 kg m −3). For the density of freshwater added at the sea surface we use a constant 𝜌f=1000 kg m −3 for convenience, neglecting the variation of <1% in freshwater density due to temperature. Vertical Direction and Distance Before considering the vertical location of surfaces, or the local vertical distance between two surfaces, we need to specify the meaning of vertical. Geodesy is concerned with horizontal and vertical distances measured relative to the reference ellipsoid, which is a surface fixed with respect to the solid Earth. Geophysical fluid dynamics, including ocean circulation dynamics, is concerned with horizontal distances on surfaces of constant geopotential, and vertical distances measured perpendicular to such surfaces, especially the geoid. We discuss the two frames of reference (one relative to the reference ellipsoid and the other to the geoid) in the subsections describing those two surfaces. The distinction between the two frames is relevant only to the real world, because numerical ocean circulation models implicitly assume an idealized effective gravity field in which the geoid and the reference ellipsoid are identical (often spherical rather than ellipsoidal). In reality, the geoid has an irregular shape, whose vertical separation from the reference ellipsoid is ±100 m and varies over horizontal length scales of 100s km. Most of our formulae involve the local vertical coordinate of surfaces such as 𝑋(𝐫), for which we use the vertical distance above the reference ellipsoid (negative if below). We make this choice in order to give our formulae a well-defined interpretation. The choice of reference frame (with respect either to the reference ellipsoid or to the geoid) does not affect the geophysical definition of a surface, but the numerical value of its vertical coordinate at any given location is not the same in the two frames, because of the substantial difference between their reference surfaces. However, at a given location, there is negligible difference between the two frames regarding the local vertical direction. Hence, we can ignore the difference between the two definitions of “vertical” in evaluating a vertical gradient, the vertical distance 𝑌(𝐫)−𝑋(𝐫) between two surfaces, or the change in height Δ𝑋(𝐫) of a surface. Surfaces We here define five key surfaces used in sea-level studies. The reference ellipsoid and the associated terrestrial reference frame (depicted in Fig. 1) are geometrical constructions, chosen by convention. The other four surfaces (compared with the reference ellipsoid in Fig. 2) are geophysically defined and established with some uncertainty from observational data. These and other surfaces, such as datums defined by tides (e.g., mean lower-low water level), are located relative to the reference ellipsoid (Sect. 2.6), by their geodetic height as a function of geodetic location. Fig. 1: The reference ellipsoid, which is used to locate other surfaces in a terrestrial reference frame, whose origin is the centre of the Earth. The figure shows the construction which defines the geodetic coordinates of an arbitrary point 𝐱 in 3D space. The line between 𝐱 and 𝐫 is normal to the reference ellipsoid, on which 𝐫 lies. The equator is intersected at 𝐩 by the meridian through 𝐫, and at 𝐩0 by the prime meridian, which defines the zero of longitude N1 Reference ellipsoid: The surface of an ellipsoidal volume of revolution chosen to approximate the geoid. A reference ellipsoid is a conventional geometric construction used to specify locations in a terrestrial reference frame, i.e., relative to the solid Earth. Many reference ellipsoids have been defined by geodesists, and some are intended only for use over limited portions of the globe. A given specification of the reference ellipsoid is time-independent. For purposes relating to global sea level we make the following requirements of the reference ellipsoid. Its centre is the time-mean centre of mass of the Earth. Its semi-major axis lies in the equatorial plane and its semi-minor axis along the rotation (polar) axis of the Earth. Its axis of revolution is the rotation axis. It is fixed with respect to the solid Earth, and it rotates with the Earth. The International Earth Rotation and Reference Systems Service (www.iers.org) defines the International Terrestrial Reference Frame (ITRF). They recommend the GRS80 ellipsoid. For more precise geodetic purposes, the ITRF defines the coordinates and their rates of change of a set of stations on the Earth’s surface. The coordinates are time-dependent because of tectonic motions and true polar wander, i.e., the time-dependence of the Earth’s rotation axis with respect to the solid Earth. The latter phenomenon is neglected in the above specification of the ellipsoid. If the rotation axis is invariant, the last point in our specification above is not necessary because, being a volume of revolution, the reference ellipsoid is symmetrical with respect to rotation about the axis. To locate a point 𝐱 in 3D space in a reference frame based on the reference ellipsoid, we construct a straight line that passes through 𝐱 and is normal to the ellipsoid, which it intersects at 𝐫. The geodetic height of 𝐱 above the ellipsoid is the distance from 𝐫 to 𝐱 along this line, positive outwards. In our formulae, the vertical coordinate is the geodetic height, which is sometimes called ellipsoidal height. This is not the usual vertical coordinate for models of atmosphere and ocean circulation, which is instead defined relative to the geoid. The geodetic latitude, commonly referred to simply as latitude, is the angle between the equatorial plane and the normal to the ellipsoid. It is different from the geocentric latitude, which is the angle between the equatorial plane and the line from the centre of the Earth to 𝐱. Geodetic and geocentric latitudes are the same for the poles and the equator, but elsewhere geodetic latitude is larger (as can be appreciated from Fig. 1), by up to about 0.2˚.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-2	For purposes relating to global sea level we make the following requirements of the reference ellipsoid. Its centre is the time-mean centre of mass of the Earth. Its semi-major axis lies in the equatorial plane and its semi-minor axis along the rotation (polar) axis of the Earth. Its axis of revolution is the rotation axis. It is fixed with respect to the solid Earth, and it rotates with the Earth. The International Earth Rotation and Reference Systems Service (www.iers.org) defines the International Terrestrial Reference Frame (ITRF). They recommend the GRS80 ellipsoid. For more precise geodetic purposes, the ITRF defines the coordinates and their rates of change of a set of stations on the Earth’s surface. The coordinates are time-dependent because of tectonic motions and true polar wander, i.e., the time-dependence of the Earth’s rotation axis with respect to the solid Earth. The latter phenomenon is neglected in the above specification of the ellipsoid. If the rotation axis is invariant, the last point in our specification above is not necessary because, being a volume of revolution, the reference ellipsoid is symmetrical with respect to rotation about the axis. To locate a point 𝐱 in 3D space in a reference frame based on the reference ellipsoid, we construct a straight line that passes through 𝐱 and is normal to the ellipsoid, which it intersects at 𝐫. The geodetic height of 𝐱 above the ellipsoid is the distance from 𝐫 to 𝐱 along this line, positive outwards. In our formulae, the vertical coordinate is the geodetic height, which is sometimes called ellipsoidal height. This is not the usual vertical coordinate for models of atmosphere and ocean circulation, which is instead defined relative to the geoid. The geodetic latitude, commonly referred to simply as latitude, is the angle between the equatorial plane and the normal to the ellipsoid. It is different from the geocentric latitude, which is the angle between the equatorial plane and the line from the centre of the Earth to 𝐱. Geodetic and geocentric latitudes are the same for the poles and the equator, but elsewhere geodetic latitude is larger (as can be appreciated from Fig. 1), by up to about 0.2˚. To define the longitude of 𝐱 (“geodetic” and “geocentric” are the same for longitude), consider the meridian passing through 𝐫, which intersects the equator at point 𝐩, and the prime meridian (the Greenwich meridian), which intersects the equator at 𝐩0. Viewing the Earth from above, the longitude is the anticlockwise angle between the lines from the centre of the Earth to 𝐩0 and to 𝐩. Fig. 2: Relationship between surfaces relating to sea level. The normal to the reference ellipsoid defines the vertical in the terrestrial reference frame. The normal to the geoid is the vertical coordinate (z) for geophysical fluid dynamics, and anti-parallel to the local effective acceleration g due to gravity. The difference between these two definitions of the vertical direction is greatly exaggerated in this diagram; it is negligible in reality. The local vertical coordinates of mean sea level 𝜂, the geoid G and the sea floor 𝐹 are relative to the reference ellipsoid, while dynamic sea level 𝜁̃ is relative to the geoid. The local time-mean thickness of the ocean 𝐻 is the vertical distance between mean sea level and the sea floor. The deviation of atmospheric pressure 𝑝′𝑎 from its global mean causes the depression 𝐵 in sea level by the inverse barometer effect N2 Sea surface 𝜂̃: The time-varying upper boundary of the ocean. The sea-surface height is the geodetic height of the sea surface above the reference ellipsoid (a negative value if below). In ocean areas without floating ice (sea ice, ice shelves or icebergs), the liquid sea surface is the bottom boundary of the atmosphere. In such areas, the existence of a well-defined sea-surface height (SSH)𝜂̃(𝐫,𝑡), that can be represented by a continuous and single-valued mathematical expression, presupposes a space–time averaging, because the instantaneous surface is ill-defined in the presence of some short-timescale phenomena that produce foam and sea spray, such as breaking surface waves and conditions of intense wind. We assume such averaging when speaking about the sea surface. In ocean areas with floating ice, the liquid surface boundary is the bottom of the ice. For those areas, we define the SSH 𝜂̃ as the liquid-water equivalent sea surface𝜂̃LWE which the liquid would have if the ice were replaced by an equal mass of sea water of the density 𝜌s of the surface water in its vicinity. Following Archimedes’ principle, 𝜂̃LWE = 𝜂̃𝑠 + 𝑤𝑖𝑔𝜌s, (1) where 𝜂̃𝑠 is the geodetic height of the liquid sea-water surface (beneath the ice) and 𝑤𝑖 is the weight per unit area of floating ice. (The depression of 𝜂̃𝑠 relative to 𝜂̃ is called the “ inverse barometer effect of sea ice” by Griffies and Greatbatch 2012, and 𝜂̃LWE is their “effective sea level”.) Although the liquid-water equivalent sea-surface height is not directly measurable, it is a convenient construct for many practical purposes of sea-level studies, and dynamically justifiable because the hydrostatic pressure and gravity beneath the ice are largely unaffected by the replacement of ice with liquid water. The sea surface varies periodically with various frequencies due to tides . It varies also on all timescales due to sea-surface waves , atmospheric pressure, surface flux exchanges (with the atmosphere), river inflow, variability that is internally generated by ocean dynamics, motion of the sea floor and changes in the mass distribution within the ocean and solid Earth (discussed in many following entries). Note that ocean dynamic sea level and ocean dynamic topography are distinct concepts from sea-surface height and from each other. N3 Mean sea level (MSL) 𝜂: The time-mean of the sea surface . The period for the time-mean must be long enough to eliminate the effects of waves and other meteorologically induced fluctuations (as discussed in Sect. 2.2). Predicted tidal variations are subtracted if the period is not long enough to remove time-dependent tides , but permanent tides are included in MSL. For a precise definition of MSL, the period of the time-mean should be specified, and it could be described either with or without dependence on the time of year. MSL is located by its geodetic height 𝜂(𝐫) above the reference ellipsoid (a negative value if below). In ocean models which regard the geoid and the reference ellipsoid as coincident, 𝜂 is equally the orthometric height of MSL above the geoid. MSL is sometimes called “mean sea surface”. We recommend against using this term, in order to make a clear distinction from “sea-surface height”. N4 Sea floor 𝐹: The lower boundary of the ocean, its interface with the solid Earth. The sea floor is the part of the surface of the solid Earth (whether bedrock or consolidated sediment, and lying beneath any unconsolidated sediment, e.g., Webb et al. 2013) that is always or sometimes submerged under sea water. The level of the sea floor varies due to solid-Earth tides , accumulation of sediment (with eventual compaction) and vertical land movement on a range of timescales. We specify the instantaneous level of the sea floor by its geodetic height 𝐹̃(𝐫,𝑡) (negative over most of the ocean) relative to the reference ellipsoid . The local instantaneous thickness of the ocean (its vertical extent, the depth of the sea floor measured from a ship, a positive quantity sometimes called the depth of the water column) is given by 𝐻̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)−𝐹̃(𝐫,𝑡)≥0, (2) i.e., the vertical distance between the sea surface and the sea floor. The choice of reference surface for vertical coordinates does not affect the value of 𝐻̃, because it is the difference between two vertical coordinates; 𝐻̃ would be the same if SSH and the sea floor were located by heights relative to the geoid rather than the reference ellipsoid. The time-mean thickness of the ocean𝐻 𝐻(𝐫)=𝜂(𝐫)−𝐹(𝐫)≥0 (3) is likewise related to MSL 𝜂.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-3	MSL is located by its geodetic height 𝜂(𝐫) above the reference ellipsoid (a negative value if below). In ocean models which regard the geoid and the reference ellipsoid as coincident, 𝜂 is equally the orthometric height of MSL above the geoid. MSL is sometimes called “mean sea surface”. We recommend against using this term, in order to make a clear distinction from “sea-surface height”. N4 Sea floor 𝐹: The lower boundary of the ocean, its interface with the solid Earth. The sea floor is the part of the surface of the solid Earth (whether bedrock or consolidated sediment, and lying beneath any unconsolidated sediment, e.g., Webb et al. 2013) that is always or sometimes submerged under sea water. The level of the sea floor varies due to solid-Earth tides , accumulation of sediment (with eventual compaction) and vertical land movement on a range of timescales. We specify the instantaneous level of the sea floor by its geodetic height 𝐹̃(𝐫,𝑡) (negative over most of the ocean) relative to the reference ellipsoid . The local instantaneous thickness of the ocean (its vertical extent, the depth of the sea floor measured from a ship, a positive quantity sometimes called the depth of the water column) is given by 𝐻̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)−𝐹̃(𝐫,𝑡)≥0, (2) i.e., the vertical distance between the sea surface and the sea floor. The choice of reference surface for vertical coordinates does not affect the value of 𝐻̃, because it is the difference between two vertical coordinates; 𝐻̃ would be the same if SSH and the sea floor were located by heights relative to the geoid rather than the reference ellipsoid. The time-mean thickness of the ocean𝐻 𝐻(𝐫)=𝜂(𝐫)−𝐹(𝐫)≥0 (3) is likewise related to MSL 𝜂. The shape of the sea floor is sometimes called the bottom topography or the bathymetry, for example in describing it as “rough” or “smooth”. These two synonymous terms are also used as names for the quantities −𝐹, 𝐺−𝐹 and 𝜂−𝐹(=𝐻), i.e., the time-mean vertical distance of the sea floor beneath the reference ellipsoid, the geoid or MSL, respectively. In order to be precise, it should be stated which of these alternatives is intended, since G and the reference ellipsoid differ by ±100 m (Sect. 2.6), and G and MSL differ by ±1 m, following time-mean ocean dynamic sea level . N5 Geoid G: A surface on which the geopotentialΦ has a uniform value, chosen so that the volume enclosed between the geoid and the sea floor is equal to the time-mean volume of sea water in the ocean (including the liquid-water equivalent of floating ice). The geopotential is a field of potential energy per unit mass, accounting for the Newtonian gravitational acceleration due to the mass of the Earth plus the centrifugal acceleration arising from the Earth’s rotation. We define the sign of the geopotential such that work is required to move a sea-water parcel from a lower geopotential (deeper in the ocean) to a higher geopotential (shallower in the ocean). Note that this sign convention for the geopotential is opposite to that used in geodesy. The vertical gradient of the geopotential is equal to the local effective gravitational acceleration, g, 𝑔(𝐫,𝑡)=∂Φ/∂𝑧, usually referred to as “gravitational acceleration” in geophysical fluid dynamics. Hence, the effective gravitational acceleration is normal to the geoid, because the geoid is an equipotential surface, i.e., one on which the geopotential is constant. The Newtonian gravitational acceleration is time-dependent because the distribution of mass in the ocean (liquid and solid), on land (including the land-based cryosphere) and within the solid Earth is generally changing. The centrifugal acceleration is time-dependent because the Earth’s rotation rate and rotation axis are variable. For models of atmosphere and ocean circulation, the vertical unit vector is directed anti-parallel to the effective gravitational acceleration g (or equivalently it is parallel to the local gradient of the geopotential). The height above the geoid of some point 𝐱 is the distance z, measured along the local vertical unit vector, from the geoid to 𝐱. The coordinate z (Fig. 2) is also called the orthometric height of 𝐱. Strictly, the orthometric height is measured along a plumb line, which is always normal to equipotential surfaces, but this distance differs negligibly from that measured along the perpendicular to the reference ellipsoid . We define z such that 𝑧=0 is the geoid, 𝑧>0 is above the geoid, and 𝑧<0 is below. By horizontal we mean aligned with a surface of constant z. This is not strictly an equipotential surface, but the difference is locally negligible. It is, however, very different from a surface of constant geodetic height 𝑧′=𝑧+𝐺, where 𝐺(𝐫) is the geoid height above the reference ellipsoid (𝐺<0 where the geoid is below the ellipsoid). The sea surface would coincide with the geoid if the ocean were in a resting steady state in the rotating frame of the earth. Although defining the geoid in this way is conceptually attractive, it is not realistic or practically useful. (See Appendix 1.) The sea surface does not really coincide with the geoid because the ocean is not at rest. (See ocean dynamic sea level .) For example, mean sea level (MSL) north of the Antarctic Circumpolar Current (ACC) is at a higher geopotential than MSL south of the ACC; with respect to the geoid, MSL on the north side is roughly 2 m higher than on the south side. Referring to its definition, we choose the geoid as the equipotential surface (out of the infinite set of them) which satisfies ∫(𝐺−𝐹)d𝐴 = 𝑉 = ∫𝐻 d𝐴 = ∫(𝜂−𝐹) d𝐴, (4) using Eq. (3) for 𝐻, and where V is the volume of the global ocean and 𝐴=∫d𝐴 is its surface area. It follows from Eq. (4) that ∫𝜂 d𝐴 = ∫𝐺 d𝐴, (5) i.e., MSL and geoid height above the reference ellipsoid have equal global means. We define the geoid in terms of MSL 𝜂, rather than the sea-surface height 𝜂̃, in order to restrict changes in G and V to those occurring on the timescales of global-mean sea-level rise , rather than on shorter timescales related to meteorological, seasonal and interannual fluctuations. Our definition of the geoid treats it as a geophysical quantity which changes as the Earth system evolves. In some applications, the geoid is defined in a time-independent way as a particular geopotential surface within a particular model of the Earth’s gravity field. We define the geoid to include the permanent ocean tide. With this choice, time-mean ocean dynamic sea level 𝜁 is determined solely by ocean dynamics and density. With the zero-tide convention, which is common in gravity-field models, 𝜁 would include the permanent ocean tide, which is almost +0.1 m at the equator and −0.2 m at the poles. We define the geoid as 𝐺(𝐫)=𝐸(𝐫,Φ𝐺), with a choice of Φ𝐺 such that Eq. (4) is satisfied, where 𝐸(𝐫,Φ) is the geodetic height of the equipotential surface for geopotential Φ. The shapes of the equipotential surfaces, including the geoid, depend on the geographical distribution of mass over the Earth. According to Eq. (4), the global-mean geoid height must change by 1𝐴∫Δ𝐺 d𝐴 = 1𝐴∫Δ𝐹 d𝐴 + Δ𝑉 𝐴 (6) if there is global-mean vertical land movement Δ𝐹 affecting the sea floor, or a change Δ𝑉 in the volume of the global ocean, whether due to change in density or in mass. Consequently, Φ𝐺 must change such that	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-4	∫𝜂 d𝐴 = ∫𝐺 d𝐴, (5) i.e., MSL and geoid height above the reference ellipsoid have equal global means. We define the geoid in terms of MSL 𝜂, rather than the sea-surface height 𝜂̃, in order to restrict changes in G and V to those occurring on the timescales of global-mean sea-level rise , rather than on shorter timescales related to meteorological, seasonal and interannual fluctuations. Our definition of the geoid treats it as a geophysical quantity which changes as the Earth system evolves. In some applications, the geoid is defined in a time-independent way as a particular geopotential surface within a particular model of the Earth’s gravity field. We define the geoid to include the permanent ocean tide. With this choice, time-mean ocean dynamic sea level 𝜁 is determined solely by ocean dynamics and density. With the zero-tide convention, which is common in gravity-field models, 𝜁 would include the permanent ocean tide, which is almost +0.1 m at the equator and −0.2 m at the poles. We define the geoid as 𝐺(𝐫)=𝐸(𝐫,Φ𝐺), with a choice of Φ𝐺 such that Eq. (4) is satisfied, where 𝐸(𝐫,Φ) is the geodetic height of the equipotential surface for geopotential Φ. The shapes of the equipotential surfaces, including the geoid, depend on the geographical distribution of mass over the Earth. According to Eq. (4), the global-mean geoid height must change by 1𝐴∫Δ𝐺 d𝐴 = 1𝐴∫Δ𝐹 d𝐴 + Δ𝑉 𝐴 (6) if there is global-mean vertical land movement Δ𝐹 affecting the sea floor, or a change Δ𝑉 in the volume of the global ocean, whether due to change in density or in mass. Consequently, Φ𝐺 must change such that 1𝐴∫Δ𝐺 d𝐴 = ΔΦ 𝐺 𝐴∫∂𝐸(𝐫,Φ) ∂Φd𝐴 = ΔΦ 𝐺 𝑔, (7) if we approximate g as globally uniform. Variations and Differences in Surfaces In this section we define terms for time-dependent variations in surfaces (on timescales shorter than those of mean sea-level change) and differences between surfaces. N6 Tides: Periodic motions within the ocean, atmosphere and solid Earth due to the rotation of the Earth and its motion relative to the moon and sun. Ocean tides cause the sea surface to rise and fall. The astronomical tide is the dominant constituent of the ocean tides. It is caused by periodic spatial variations in local gravity. Tidal motion of the land surface and sea floor is due to elastic deformation of the solid Earth by gravitational tidal forces. The diurnal and annual cycles of insolation produce periodic variations in atmospheric pressure and winds (sea breezes), which cause the radiational tide in the atmosphere and ocean (e.g., Williams et al. 2018). The predicted tide is the sum of the astronomical and radiational constituents. Because the ocean and atmosphere are fluids, tidal forces within them cause tidal currents as well as displacements. Sea-surface height (SSH) can be greatly elevated during a storm by a storm surge , and the consequent extreme sea level is sometimes called a storm tide. The tidal height is the vertical distance of the SSH due to the predicted tide above a local benchmark or a surface which is fixed with respect to the terrestrial reference frame. Often, this surface is a tidal datum, defined by a extremum of the periodic tide, such as mean lower-low water level. The velocity of the ocean tidal currents depends on water depth. Therefore, relative sea-level change (RSLC) affects the tides. In most coastal locations, this interaction alters the tidal variations of the sea surface with respect to mean sea level by less than 10% of the RSLC (Pickering et al. 2017). In most locations, the constituent of the ocean tide with the largest amplitude is the lunar semi-diurnal tide. The orbit of the moon around the Earth modulates the semi-diurnal tide to produce a large amplitude (spring tide) at new and full moon, and a small amplitude (neap tide) at half-moon. There are many smaller periodic constituents associated with the sun and moon. The precession of the plane of the moon’s orbit causes tidal variations with an 18.6-year cycle (the nodal period), affecting extreme sea level on this timescale. There are longer tidal periods. The pole tide is caused by variations of the Earth’s rotation axis relative to the solid Earth, altering the centrifugal acceleration and local gravity. The two largest components of the pole tide have periods of 1 year and about 433 days. The latter is due to the Chandler wobble, which is not strictly periodic and arises from the mechanics of the Earth’s rotation alone (it is a free nutation), rather than being caused by the gravitation of other bodies in the solar system. The time-means of the tidal forces of the moon and sun are nonzero. Hence, in addition to the periodic constituents, the tides have a constant constituent called the permanent tide, which tends to make the Earth and sea surface more oblate. Our definitions of mean sea level and the geoid use the mean-tide convention, including the permanent tide. In gravity models, the zero-tide convention is more usual, in which the permanent ocean tide is subtracted, but the permanent elastic tidal deformation of the solid Earth is retained; an estimate of the latter too is subtracted in the tide-free convention used by GNSS measurements. N7 Inverse barometer (IB)𝐵: The time-dependent hydrostatic depression of the sea surface by atmospheric pressure variations, also called inverted barometer. The ocean is almost incompressible. (A uniform change of 1 hPa over the ocean causes a global-mean sea-level rise of roughly 0.16 mm.) Therefore changes in atmospheric pressure have a negligible effect on the total volume of the ocean. However, they do move sea water around, and the effect on the sea surface depends on the deviation of sea-level pressure 𝑝̃𝑎(𝐫) from its global (ocean) mean, given by 𝑝̃′𝑎(𝐫,𝑡) = 𝑝̃𝑎(𝐫,𝑡) − 1𝐴∫𝑝̃𝑎(𝐫,𝑡) d𝐴. (8) For timescales longer than a few days, we can assume the ocean to be in hydrostatic balance. Therefore, the depression of the sea-surface height (SSH) 𝜂̃ by IB is 𝐵̃=𝑝̃′𝑎/(𝑔𝜌s) where 𝑔(𝐫) is the acceleration due to gravity and 𝜌s(𝐫,𝜂̃) the surface sea-water density. That is, when 𝑝̃′𝑎>0 then sea level is depressed locally by 𝐵̃(𝐫), and it is raised when 𝑝̃′𝑎<0. The latter effect is an important contribution to storm surge . In a storm or cyclone, 𝑝̃𝑎 may fall by several 10 hPa, causing SSH to rise by several 100 mm. The global mean of 𝑔𝜌s is approximately 9.9×10−5 m Pa −1≡9.9 mm hPa −1, with spatial and temporal variations of about 1% around this value. Hence, for most purposes of sea-level studies we can neglect the spatial variations in g and 𝜌s, and replace them with constants; thus, 𝐵̃=𝑝̃′𝑎𝑔𝜌∗. (9) Hence, the global-mean IB correction is zero, ∫𝐵̃(𝐫,𝑡)d𝐴=0, (10) which follows by definition of 𝑝̃′𝑎.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-5	𝑝̃′𝑎(𝐫,𝑡) = 𝑝̃𝑎(𝐫,𝑡) − 1𝐴∫𝑝̃𝑎(𝐫,𝑡) d𝐴. (8) For timescales longer than a few days, we can assume the ocean to be in hydrostatic balance. Therefore, the depression of the sea-surface height (SSH) 𝜂̃ by IB is 𝐵̃=𝑝̃′𝑎/(𝑔𝜌s) where 𝑔(𝐫) is the acceleration due to gravity and 𝜌s(𝐫,𝜂̃) the surface sea-water density. That is, when 𝑝̃′𝑎>0 then sea level is depressed locally by 𝐵̃(𝐫), and it is raised when 𝑝̃′𝑎<0. The latter effect is an important contribution to storm surge . In a storm or cyclone, 𝑝̃𝑎 may fall by several 10 hPa, causing SSH to rise by several 100 mm. The global mean of 𝑔𝜌s is approximately 9.9×10−5 m Pa −1≡9.9 mm hPa −1, with spatial and temporal variations of about 1% around this value. Hence, for most purposes of sea-level studies we can neglect the spatial variations in g and 𝜌s, and replace them with constants; thus, 𝐵̃=𝑝̃′𝑎𝑔𝜌∗. (9) Hence, the global-mean IB correction is zero, ∫𝐵̃(𝐫,𝑡)d𝐴=0, (10) which follows by definition of 𝑝̃′𝑎. The inverse-barometer response of the sea surface compensates for the effect of 𝑝̃′𝑎 on hydrostatic pressure within the ocean, and the subsurface ocean does not feel the fluctuations in atmospheric pressure. Consequently, the ocean behaves dynamically as if the sea-surface height were 𝜂̃+𝐵̃, which is called IB-corrected sea-surface height. Its time-mean is 𝜂+𝐵, the IB-corrected mean sea level. In most climate models, atmospheric pressure variations are not communicated to the ocean. In these models 𝐵̃ must be subtracted from the simulated SSH to produce a quantity that varies with 𝑝̃𝑎 like the observed 𝜂̃ does. N8 Extreme sea level: The occurrence or the level of an exceptionally high or low local sea-surface height. Extremely high sea-surface height (SSH) is caused by meteorological conditions as a storm surge , by sea-surface waves due to various causes and by exceptionally high or low (although predictable) tidal height. When considering coastal impacts, extreme sea level may be called extreme coastal water level. For decadal timescales, the main influence on changes in the frequency distribution of extreme sea level is relative sea-level change (RSLC), whose effect outweighs that of changes in meteorological forcing (Lowe et al. 2010; Church et al. 2013; Vousdoukas et al. 2018). To avoid confusion, we recommend the phrase high-end sea-level change to describe projections of very large RSLC, instead of using the word “extreme” for such projections. N9 Storm surge: The elevation or depression of the sea surface with respect to the predicted tide during a storm. Storm surges are caused during tropical cyclones and deep mid-latitude depressions by low atmospheric pressure, by strong winds pushing water towards the shore (or away from the shore, causing a negative surge) and by sea-surface waves breaking at the coast. Wave effects are usually excluded or underestimated by tide-gauges. If the actual sea-surface height (SSH) at location 𝐫 and time t due to tide and surge combined (sometimes called the storm tide) is 𝜂̃(𝐫,𝑡), and the predicted SSH due to the tide alone is 𝜂̃tide(𝐫,𝑡), the storm-surge heightσ is σ(𝐫,𝑡)=𝜂̃(𝐫,𝑡)−𝜂̃tide(𝐫,𝑡), (11) also called the surge residual or non-tidal residual. The storm-surge height σ is the sum of three components: the inverse barometer (IB) effect of low atmospheric pressure, the wind setup caused by the wind-driven current, and the wave setup, which is the elevation of the sea surface due to breaking waves. All three effects are normally present, but intensified by storms. IB and wind setup tend to be more important on wide continental shelves, but wave setup can dominate in some cases (Pedreros et al. 2018), especially in areas of steep sea floor slope. The swash is the uprush and backwash of water over the solid surface (e.g., sand or pebbles) generated by each wave. During the uprush, the swash extends above the wave setup. Its maximum height above the predicted tide, called the wave runup, gives the highest water level of the storm surge. Particularly high SSH 𝜂̃=σ+𝜂̃tide occurs when the storm surge coincides with high tide. Without the meteorological forcing, storm-surge height σ would be zero, but since the tide level influences the propagation of the storm-forced signal, σ and 𝜂̃tide are not independent (Horsburgh and Wilson 2007). The skew-surge height is the elevation of the highest sea surface that occurs within a single tidal cycle above the predicted level of the high tide within that cycle. If the actual SSH is 𝜂̃(𝐫,𝑡) and the predicted SSH due to the tide alone is 𝜂̃tide(𝐫,𝑡), the skew-surge height is σ𝑘(𝐫,𝑡)= max 𝑡(𝜂̃(𝐫,𝑡))− max 𝑡(𝜂̃tide(𝐫,𝑡)), (12) where max 𝑡(𝑋(𝐫,𝑡)) means the maximum value of X that occurs at location 𝐫 during the interval of time t from one low tide to the next. For extreme-value analysis, the skew-surge height is preferable to the storm-surge height as a measure of the effect of the meteorological forcing alone in regions where skew-surge height is uncorrelated with tidal height (Williams et al. 2016). N10 Sea-surface waves: Waves on the surface of the ocean, usually surface gravity waves caused by winds. The amplitude of a wind wave depends on the strength of the wind, and the time and the distance of open ocean, called the fetch, over which the wind has blown. The sea surface typically exhibits a superposition of many waves of different amplitudes, velocities, frequencies and directions. A swell wave is a wind wave of low frequency which was generated far away. A tsunami or seismic sea wave is an extreme sea level event caused by an earthquake, volcano, landslide or other submarine disturbance that suddenly displaces a volume of water. The displacement propagates as a long-wavelength surface gravity wave, but is not a tidal phenomenon, despite it sometimes being called a “tidal wave”. The wave height is the vertical distance from the crest to the trough of a wave, respectively its highest and lowest points. The wave period is the interval of time between the passage of repeated features on the waveform such as crests, troughs or upward crossings of the mean level. The significant wave height is a statistic computed from wave measurements, defined as either the mean of the largest one-third of the wave heights, or four times the standard deviation of wave heights. (These statistics are approximately equal.) The significant wave period is the mean period of the largest one-third of the waves. N11 Ocean dynamic sea level 𝜁: The local height of the sea surface above the geoid G, with the inverse barometer correction 𝐵 applied. Instantaneous ocean dynamic sea level is defined by 𝜁̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)+𝐵̃(𝐫,𝑡)−𝐺(𝐫). (13) It is determined jointly by ocean density and circulation. The time-mean ocean dynamic sea level is 𝜁(𝐫)=𝜂(𝐫)+𝐵(𝐫)−𝐺(𝐫), (14) whose global mean 1𝐴∫𝜁(𝐫)d𝐴=0, (15) in view of Eqs. (5) and (10).	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-6	N10 Sea-surface waves: Waves on the surface of the ocean, usually surface gravity waves caused by winds. The amplitude of a wind wave depends on the strength of the wind, and the time and the distance of open ocean, called the fetch, over which the wind has blown. The sea surface typically exhibits a superposition of many waves of different amplitudes, velocities, frequencies and directions. A swell wave is a wind wave of low frequency which was generated far away. A tsunami or seismic sea wave is an extreme sea level event caused by an earthquake, volcano, landslide or other submarine disturbance that suddenly displaces a volume of water. The displacement propagates as a long-wavelength surface gravity wave, but is not a tidal phenomenon, despite it sometimes being called a “tidal wave”. The wave height is the vertical distance from the crest to the trough of a wave, respectively its highest and lowest points. The wave period is the interval of time between the passage of repeated features on the waveform such as crests, troughs or upward crossings of the mean level. The significant wave height is a statistic computed from wave measurements, defined as either the mean of the largest one-third of the wave heights, or four times the standard deviation of wave heights. (These statistics are approximately equal.) The significant wave period is the mean period of the largest one-third of the waves. N11 Ocean dynamic sea level 𝜁: The local height of the sea surface above the geoid G, with the inverse barometer correction 𝐵 applied. Instantaneous ocean dynamic sea level is defined by 𝜁̃(𝐫,𝑡)=𝜂̃(𝐫,𝑡)+𝐵̃(𝐫,𝑡)−𝐺(𝐫). (13) It is determined jointly by ocean density and circulation. The time-mean ocean dynamic sea level is 𝜁(𝐫)=𝜂(𝐫)+𝐵(𝐫)−𝐺(𝐫), (14) whose global mean 1𝐴∫𝜁(𝐫)d𝐴=0, (15) in view of Eqs. (5) and (10). In the Coupled Model Intercomparison Project (CMIP), 𝜁̃ is stored in the diagnostic named zos, which is defined to have zero mean (Equation H14 of Griffies et al. 2016). However, some models supply it with a nonzero time-dependent mean. If the global mean of the zos diagnostic is found to be nonzero, the global mean should be subtracted uniformly. N12 Ocean dynamic topography: An estimate of ocean dynamic sea level computed from the ocean density structure above a reference level where the velocity is either known or assumed to be zero. On any horizontal (see geoid for definition) level z within the ocean, the hydrostatic pressure is given by 𝑝̃(𝑧) = 𝑝̃𝑎 + 𝑔∫𝜁̃−𝐵̃𝑧𝜌̃(𝑧′)d𝑧′, (16) which is the sum of the atmospheric pressure 𝑝̃𝑎 on the sea surface and the weight per unit horizontal area of sea water between z and the sea surface. The coordinate of the sea surface in this case is not 𝜂̃ but 𝜂̃−𝐺=𝜁̃−𝐵̃ by Eq. (13), the height of the sea surface above the geoid, because we are using the orthometric vertical coordinate z, which is the natural choice for ocean dynamics. Equation (9) leads to the horizontal gradient of the atmospheric pressure ∇𝑝̃𝑎 = ∇𝑝̃′𝑎 = 𝑔𝜌∗∇𝐵̃. Consequently, the horizontal gradient of pressure within the ocean is given by ∇𝑝̃ = ∇(𝑝̃𝑎 + 𝑔∫𝜁̃−𝐵̃𝑧𝜌̃(𝑧′)d𝑧′) (17a) = 𝑔𝜌∗∇∇𝐵̃+(𝑔𝜌∗∇∇𝜁̃−𝑔𝜌∗∇∇𝐵̃)+𝑔∫𝜁̃−𝐵̃𝑧∇∇𝜌̃(𝑧′)d𝑧′ (17b) = 𝑔𝜌∗∇∇𝜁̃+𝑔∫𝜁̃−𝐵̃𝑧∇∇𝜌̃(𝑧′)d𝑧′. (17c) In the first step of this derivation, we used Eq. (9) for the inverse barometer correction 𝐵̃, approximated g and sea-surface 𝜌̃=𝜌∗ as constants, and applied Leibniz’s rule to differentiate the integral, which yields the two terms in parentheses in Eq. (17b), but no term for 𝜌̃ at z because ∇∇𝑧=0. From Eq. (17c) we obtain ∇𝜁̃(𝐫) = 1𝑔𝜌∗∇𝑝̃(𝐫,𝑧) − 1𝜌∗∫𝜁̃−𝐵̃𝑧∇𝜌̃(𝐫,𝑧′)d𝑧′, (18) which relates the horizontal gradient of ocean dynamic sea level 𝜁̃ to the horizontal hydrostatic pressure gradient at a reference level z and the horizontal density gradient above that level. The second term on the right-hand side is the horizontal gradient of the dynamic heightD 𝐷 = −1𝜌∗∫𝜁̃−𝐵̃𝑧𝜌̃𝑑𝑧′ (19) of the sea surface relative to z. In much of the ocean interior (below the boundary layer and away from coastal and other strong currents), and taking a time-mean sufficient to eliminate tidal currents, geostrophy is a reasonable approximation, meaning that there is a balance (no net acceleration) between the pressure gradient and Coriolis forces, and all other forces are negligible. Therefore, 𝐯̃≃𝐯̃g, with the geostrophic velocity 𝐯̃g defined by 𝑓𝐤×𝜌̃𝐯̃g=−∇∇𝑝̃, (20) where f is the Coriolis parameter and 𝐤 the vertical unit vector. If we can measure 𝐯̃ at some z and assume it is geostrophic, we arrive at ∇𝜁̃(𝐫)=−𝑓𝑔𝜌∗𝐤×𝜌̃𝐯̃(𝐫,𝑧)−1𝜌∗∫𝜁̃−𝐵̃𝑧∇𝜌̃(𝐫,𝑧′)d𝑧′, (21) from Eq. (18). Alternatively, if we do not know 𝐯̃ at any z, we assume there exists a level of no motion𝑧=−𝐿, which is a geopotential (horizontal) surface on which 𝐯̃=𝐯̃g=0, requiring the horizontal hydrostatic pressure gradient to vanish (∇∇𝑝̃=0) by Eq. (20). Therefore, ∇𝜁̃(𝐫)=−1𝜌∗∫𝜁̃ − 𝐵̃ − 𝐿∇∇𝜌̃(𝐫,𝑧)d𝑧, (22)	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-7	𝐷 = −1𝜌∗∫𝜁̃−𝐵̃𝑧𝜌̃𝑑𝑧′ (19) of the sea surface relative to z. In much of the ocean interior (below the boundary layer and away from coastal and other strong currents), and taking a time-mean sufficient to eliminate tidal currents, geostrophy is a reasonable approximation, meaning that there is a balance (no net acceleration) between the pressure gradient and Coriolis forces, and all other forces are negligible. Therefore, 𝐯̃≃𝐯̃g, with the geostrophic velocity 𝐯̃g defined by 𝑓𝐤×𝜌̃𝐯̃g=−∇∇𝑝̃, (20) where f is the Coriolis parameter and 𝐤 the vertical unit vector. If we can measure 𝐯̃ at some z and assume it is geostrophic, we arrive at ∇𝜁̃(𝐫)=−𝑓𝑔𝜌∗𝐤×𝜌̃𝐯̃(𝐫,𝑧)−1𝜌∗∫𝜁̃−𝐵̃𝑧∇𝜌̃(𝐫,𝑧′)d𝑧′, (21) from Eq. (18). Alternatively, if we do not know 𝐯̃ at any z, we assume there exists a level of no motion𝑧=−𝐿, which is a geopotential (horizontal) surface on which 𝐯̃=𝐯̃g=0, requiring the horizontal hydrostatic pressure gradient to vanish (∇∇𝑝̃=0) by Eq. (20). Therefore, ∇𝜁̃(𝐫)=−1𝜌∗∫𝜁̃ − 𝐵̃ − 𝐿∇∇𝜌̃(𝐫,𝑧)d𝑧, (22) using Eq. (18). There is no motion on 𝑧=−𝐿 so long as there is a compensation between undulations of dynamic sea level 𝜁̃ (on the left-hand side of Eq. 22), and variations of the density structure above 𝑧=−𝐿 (on the right-hand side). Such exact compensation does not generally occur in the ocean, and the level of no motion does not exist. However, it is a useful approximation in many situations. For example, an anomalous sea-surface high is associated with a depression of the pycnocline in the interior of subtropical gyres (e.g., Figure 3.3 of Tomczak and Godfrey 1994), thus leading to relatively weak flow beneath the pycnocline. In some regions, the approximation is not useful; in particular, sizeable 𝐯̃ occurs at all depths in the Southern Ocean. The ocean dynamic topography is the estimate of ocean dynamic sea level made using Eqs. (21) or (22). Since Eqs. (21) and (22) are unaffected by adding a constant to 𝜁̃, the method provides only the difference in 𝜁̃ between any two points (i.e., the gradient); it cannot give 𝜁̃ for individual points relative to the geoid. Furthermore, it is not applicable in regions where the reference level for motion is below the sea floor , nor for differences between points in basins which are separated by sills that are shallower than the reference level. Changes in Sea Level The relationships between quantities determining changes in sea level are summarized in Fig. 3. The phrases “sea-level change” (SLC) and “sea-level rise” (SLR) are often used in the literature. These make sense when referring to the phenomenon in general, but more specific terms such as relative sea-level change and global-mean sea-level rise should be preferred where relevant. Fig. 3: Relationships between quantities, defined in Sect. 5, that determine changes in sea level. The lengths of the arrows do not have any significance—they are only illustrative—and the dotted horizontal lines serve only to indicate alignment. All of the quantities are differences between two states, and all except h, ℎ𝜃 and ℎ𝑏 are functions of location 𝐫. Any closed circuit gives an equality, in which a term has a positive sign when traversed in the direction of its arrow, and a negative sign if in the opposite direction to its arrow. For example, Δ𝑅−Δ𝜂+Δ𝐹=0 (Eq. 24) is the circuit marked in red, Eq. (23) in orange, Eq. (38) in blue, and Eq. (54) in green N13 Ocean dynamic sea-level change Δ𝜁: The change in time-mean ocean dynamic sea level, i.e., the change in IB-corrected mean sea level relative to the geoid. For the difference between two time-mean states of the climate, Eq. (13) gives Δ𝜁(𝐫) = Δ𝜂(𝐫) + Δ𝐵(𝐫) − Δ𝐺(𝐫). (23) Since the time-mean ocean dynamic sea level 𝜁(𝐫) always has a zero global mean by Eq. (15), so does Δ𝜁 i.e., global-mean sea-level rise is excluded from ocean dynamic sea-level change. This property depends on Eq. (5) and thus requires a different choice of geopotential to define the geoid in the two states, if there is any change in global-mean sea level. N14 Geocentric sea-level change Δ𝜂: The change in local mean sea level with respect to the terrestrial reference frame. Geocentric sea-level change is the change in the height 𝜂(𝐫) of MSL relative to the reference ellipsoid . IB-corrected geocentric sea-level change is Δ𝜂 + Δ𝐵, i.e., the same with the inverse barometer correction added. Geocentric sea-level change must be distinguished from relative sea-level change . N15 Relative sea-level change (RSLC) Δ𝑅: The change in local mean sea level relative to the local solid surface, i.e., the sea floor . Relative sea-level change is also called “relative sea-level rise” (RSLR). (See Sect. 6 for an exposition of the relationship of RSLC to other quantities.) Both the MSL height 𝜂 and the sea floor height 𝐹 may change and thus alter RSL. Hence, RSLC is geodetically expressed as Δ𝑅(𝐫) = Δ𝜂(𝐫) − Δ𝐹(𝐫), (24) the difference between geocentric sea-level change Δ𝜂 and vertical land movement Δ𝐹 (VLM). IB-corrected relative sea-level change is Δ𝑅+Δ𝐵, i.e., RSLC with the inverse barometer correction. Relative sea-level change is the quantity registered by a tide-gauge, which measures sea level relative to the solid surface where it is attached. Since climate models do not include VLM, they do not distinguish between geocentric and relative sea-level change. In climate models where atmospheric pressure changes Δ𝑝𝑎 are not applied to the ocean, −Δ𝐵 must be added to include the effect of Δ𝑝𝑎 simulated by the atmosphere model. (Note that this adjustment should not be made to ocean dynamic sea-level change Δ𝜁, which by definition is IB-corrected; see Eq. 23.) The term “relative sea level” is not employed in an absolute sense, but only in conjunction with “change”, because 𝜂−𝐹 (the analogue of Eq. 24) is simply the depth of the sea floor below MSL, equal to the time-mean thickness of the ocean 𝐻 (Eq. 3). In view of Eq. (3), we may also write RSLC as Δ𝑅(𝐫)=Δ𝐻(𝐫), (25) i.e., the change in local ocean thickness, making it obvious that RSLC is not meaningful at locations which change from land to sea (transgression) or vice versa (regression), since 𝐻 is undefined on land.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-8	Both the MSL height 𝜂 and the sea floor height 𝐹 may change and thus alter RSL. Hence, RSLC is geodetically expressed as Δ𝑅(𝐫) = Δ𝜂(𝐫) − Δ𝐹(𝐫), (24) the difference between geocentric sea-level change Δ𝜂 and vertical land movement Δ𝐹 (VLM). IB-corrected relative sea-level change is Δ𝑅+Δ𝐵, i.e., RSLC with the inverse barometer correction. Relative sea-level change is the quantity registered by a tide-gauge, which measures sea level relative to the solid surface where it is attached. Since climate models do not include VLM, they do not distinguish between geocentric and relative sea-level change. In climate models where atmospheric pressure changes Δ𝑝𝑎 are not applied to the ocean, −Δ𝐵 must be added to include the effect of Δ𝑝𝑎 simulated by the atmosphere model. (Note that this adjustment should not be made to ocean dynamic sea-level change Δ𝜁, which by definition is IB-corrected; see Eq. 23.) The term “relative sea level” is not employed in an absolute sense, but only in conjunction with “change”, because 𝜂−𝐹 (the analogue of Eq. 24) is simply the depth of the sea floor below MSL, equal to the time-mean thickness of the ocean 𝐻 (Eq. 3). In view of Eq. (3), we may also write RSLC as Δ𝑅(𝐫)=Δ𝐻(𝐫), (25) i.e., the change in local ocean thickness, making it obvious that RSLC is not meaningful at locations which change from land to sea (transgression) or vice versa (regression), since 𝐻 is undefined on land. When considering sea-level change on geological timescales, in the absence of information about ocean dynamic sea level 𝜁̃ or the inverse barometer effect, we might approximate Δ𝜁̃≃0 and Δ𝐵≃0, in which case Δ𝜂≃Δ𝐺 from Eq. (23), and Δ𝑅≃Δ𝐺−Δ𝐹 from Eq. (24). This quantity is defined everywhere and thus gives an approximate meaning to RSLC in regions of transgression and regression. N16 Steric sea-level changeΔ𝑅𝜌: The part of relative sea-level change which is due to the change Δ𝜌 in ocean density, assuming the local mass of the ocean per unit area does not change. It is composed of thermosteric sea-level change Δ𝑅𝜃, which is the part due solely to the change Δ𝜃 in in-situ temperature, and halosteric sea-level changeΔ𝑅𝑆, which is the part due solely to the change Δ𝑆 in salinity. The time-mean local mass of the ocean per unit area is 𝑚 = ∫𝜂𝐹 𝜌 d𝑧 = 𝐻 ρ̄ with ρ̄ ≡ 1𝐻 ∫𝜂𝐹 𝜌 d𝑧, (26) where the first factor 𝐻 = 𝜂 − 𝐹 is the local time-mean thickness of the ocean (Eq. 3), and the second factor ρ̄ is the local vertical-mean time-mean density. If we change the density while keeping 𝑚 fixed, the thickness of the ocean changes, because 0 = Δ𝑚 = Δ𝐻\|𝑚ρ̄ + H Δρ̄∣𝑚 (27) (by making a linear approximation). Therefore, Δ𝐻\|𝑚 =−(𝐻/ρ̄)Δρ̄ with Δρ̄ = 1𝐻 ∫𝜂𝐹Δ𝜌d𝑧. (28) This would exactly define steric sea-level change in a situation where mass did not move horizontally. But in reality there are horizontal transports, making it impossible to separate density changes due to local changes in properties from those due to the movement of mass. For convenience, we approximate ρ̄ with the constant 𝜌∗. Since the RSLC is given by Δ𝑅 = Δ𝐻 (without inverse barometer correction, Eq. 25), steric sea-level change is Δ𝑅𝜌 = −1𝜌∗∫𝜂𝐹 Δ𝜌d𝑧 = Δ𝑅𝜃 + Δ𝑅𝑆 (29) with the density increment decomposed into thermal and haline components (by making a linear approximation) Δ𝜌 = ∂𝜌∂𝜃Δ𝜃 + ∂𝜌∂𝑆Δ𝑆, (30) and with the corresponding thermosteric and halosteric contributions Δ𝑅𝜃 = −1𝜌∗∫𝜂𝐹∂𝜌∂𝜃Δ𝜃d𝑧Δ𝑅𝑆 = −1𝜌∗∫𝜂𝐹∂𝜌∂𝑆Δ𝑆d𝑧. (31) Thermosteric sea-level change is often called thermal expansion, because ∂𝜌/∂𝜃<0, so increasing the temperature gives Δ𝑅𝜃>0. (See Appendix 2 regarding the dependence of density on salinity.) Relative sea-level change (without inverse barometer correction) is the sum of steric and manometric sea-level change (Eq. 35). N17 Global-mean thermosteric sea-level riseℎ𝜃: The part of global-mean sea-level rise (GMSLR) which is due to thermal expansion. This quantity is the global mean of local thermosteric sea-level change Δ𝑅𝜃 (due to temperature change, Eq. 31); thus, ℎ𝜃 = 1𝐴∫Δ𝑅𝜃d𝐴 = −1𝜌∗𝐴∫∫𝜂𝐹∂𝜌∂𝜃Δ𝜃(𝐫,𝑧)d𝑧d𝐴. (32) It is the change in global ocean volume due to change in temperature alone, divided by the ocean surface area. The CMIP variable zostoga is ℎ𝜃 calculated with respect to a fixed reference state. (Griffies et al. 2016 define the reference to be the initial state of the experiment for CMIP6.) Hence, differences in zostoga between two states give the global-mean thermosteric sea-level rise between those states. Although halosteric sea-level change Δ𝑅𝑆 (due to salinity change, Eq. 31) can be locally of the same order of magnitude as thermosteric, global-mean halosteric sea-level change is practically zero. In Appendix 2 we detail the physical arguments leading to this conclusion. Salinity change should be excluded when calculating ℎ𝜃, to avoid including a spurious global-mean halosteric sea-level change. (See Appendix 2 here as well as Appendix H9.5 of Griffies et al. 2016.) However, salinity change must of course be included when calculating Δ𝑅𝑆. It follows that global-mean steric sea-level change, which equals ℎ𝜃 because global-mean halosteric sea-level change is zero, cannot be calculated as the global mean of local steric sea-level change. This apparent contradiction is due to the inaccuracy of the approximations made following Eq. (28). N18 Manometric sea-level changeΔ𝑅𝑚: Definition A: The part of relative sea-level change (RSLC) which is not steric, or alternatively Definition B: The part of RSLC which is due to the change Δ𝑚(𝐫) in the time-mean local mass of the ocean per unit area, assuming the density does not change. In the following, we show that the two definitions are approximately the same.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-9	ℎ𝜃 = 1𝐴∫Δ𝑅𝜃d𝐴 = −1𝜌∗𝐴∫∫𝜂𝐹∂𝜌∂𝜃Δ𝜃(𝐫,𝑧)d𝑧d𝐴. (32) It is the change in global ocean volume due to change in temperature alone, divided by the ocean surface area. The CMIP variable zostoga is ℎ𝜃 calculated with respect to a fixed reference state. (Griffies et al. 2016 define the reference to be the initial state of the experiment for CMIP6.) Hence, differences in zostoga between two states give the global-mean thermosteric sea-level rise between those states. Although halosteric sea-level change Δ𝑅𝑆 (due to salinity change, Eq. 31) can be locally of the same order of magnitude as thermosteric, global-mean halosteric sea-level change is practically zero. In Appendix 2 we detail the physical arguments leading to this conclusion. Salinity change should be excluded when calculating ℎ𝜃, to avoid including a spurious global-mean halosteric sea-level change. (See Appendix 2 here as well as Appendix H9.5 of Griffies et al. 2016.) However, salinity change must of course be included when calculating Δ𝑅𝑆. It follows that global-mean steric sea-level change, which equals ℎ𝜃 because global-mean halosteric sea-level change is zero, cannot be calculated as the global mean of local steric sea-level change. This apparent contradiction is due to the inaccuracy of the approximations made following Eq. (28). N18 Manometric sea-level changeΔ𝑅𝑚: Definition A: The part of relative sea-level change (RSLC) which is not steric, or alternatively Definition B: The part of RSLC which is due to the change Δ𝑚(𝐫) in the time-mean local mass of the ocean per unit area, assuming the density does not change. In the following, we show that the two definitions are approximately the same. If we change the local mass 𝑚 per unit area while keeping density fixed, by Eq. (26) the thickness of the ocean changes by Δ𝐻\|𝜌=Δ𝑚/𝜌⎯⎯⎯, where 𝜌⎯⎯⎯ is the local vertical mean of 𝜌. (In reality, if the local mass per unit area changes, the density will probably change as well, since mass which is converging horizontally or through the sea surface is unlikely to have 𝜌=𝜌⎯⎯⎯ exactly.) Since RSLC Δ𝑅=Δ𝐻 (without inverse barometer correction, Eq. 25), if we approximate 𝜌⎯⎯⎯ with the constant 𝜌∗, we obtain Δ𝑅𝑚 ≃ Δ𝑚𝜌∗. (33) This is Definition B of “manometric sea-level change”. The local change in mass Δ𝑚 can be estimated from the gravity field, or from the bottom pressure𝑝𝑏, i.e., the hydrostatic pressure at the sea floor , according to Δ𝑚=(Δ𝑝𝑏−Δ𝑝𝑎)/𝑔. (See Appendix 3.) Because of its relationship to 𝑝𝑏, manometric sea-level change is sometimes referred to as the “bottom pressure term” in sea-level change. According to Definition B, the global mean of Δ𝑅𝑚 vanishes if the mass of the global ocean is constant, since 1/(𝐴𝜌∗)∫Δ𝑚d𝐴=0. However, Δ𝑅𝑚 may still be locally nonzero, due to rearrangement of the existing mass of the ocean. If the mass of the global ocean changes, the global mean of Δ𝑅𝑚 is nonzero and equals the barystatic sea-level rise (equality is approximate with Definition B of Δ𝑅𝑚, exact with Definition A), which is part of global-mean sea-level rise (GMSLR). Despite the correspondence between (local) manometric and (global) barystatic sea-level rise, we argue that these two concepts are sufficiently different to need distinct terms. (See the subsection for barystatic sea-level rise.) If mass and density are both allowed to change, Eq. (26) gives Δ𝑚 = Δ𝐻 ρ̄ + 𝐻 Δρ̄ => Δ𝐻 = (1/ρ̄ )(Δ𝑚 − ∫𝜂𝐹Δ𝜌d𝑧), (34) using the expression for Δρ̄ from Eq. (28). Again approximating ρ̄ as 𝜌∗ and substituting from Eqs. (25), (29) and (33), we obtain Δ𝑅 = Δ𝐻 ≃ Δ𝑅𝜌 + Δ𝑅𝑚, (35) i.e., RSLC (without inverse barometer correction) is the sum of steric sea-level change and manometric sea-level change, which are, respectively, the parts due to change in density and change in mass per unit area. Since 𝐻 is defined only in ocean areas, the formulae are not valid for locations which change from land to sea or vice versa. With Δ𝑅𝑚 defined by Eq. (33), Eq. (35) is only approximate, because of the replacement of 𝜌⎯⎯⎯ with 𝜌∗. We can make Eq. (35) exact if we retain the definition of Eq. (29) for steric sea-level change involving 𝜌∗ and adopt Definition A of “manometric sea-level change”, as Δ𝑅𝑚 ≡ Δ𝑅 − Δ𝑅𝜌, (36) i.e., Δ𝑅𝑚 is the part of RSLC that is not steric. We propose “manometric” as a new term because in the existing literature there is no unambiguous and generally used term for Δ𝑅𝑚. It may be described as the “mass effect on”, “mass contribution to”, “mass component of” or “mass term in” sea level or sea-level change, but these descriptions could equally well refer to GRD -induced sea-level change (the effects of a change in the geographical distribution of mass) or barystatic sea-level rise, so they can be confusing. (“Manometric” is an existing word, referring to the measurement of hydrostatic pressure using a column of liquid, a concept that is closely related to bottom pressure.) N19 Barystatic sea-level riseℎ𝑏: The part of global-mean sea-level rise (GMSLR) which is due to the addition to the ocean of water mass that formerly resided within the land area (as land water storage or land ice) or in the atmosphere (which contains a relatively tiny mass of water), or (if negative) the removal of mass from the ocean to be stored elsewhere. It is also called “barystatic sea-level change”. Land water storage, also called terrestrial water storage, is water on land that is stored as groundwater, soil moisture, water in reservoirs, lakes and rivers, seasonal snow and permafrost. Land ice means ice sheets, glaciers, permanent snow and firn. Barystatic sea-level rise includes contributions from changes in all of these. It does not include changes in the parts of ice shelves and glacier tongues whose weight is supported by the ocean rather than resting on land. (These floating parts constitute the majority of the mass of ice shelves and glacier tongues, but near the grounding line on the seaward side some part of the weight may be supported by the land-based ice.) Where land ice rests on a bed which is below mean sea level, it is already displacing sea water. Therefore, the land ice contribution to barystatic sea-level rise excludes the mass whose liquid-water equivalent volume equals the volume of sea water already displaced. The remainder, which is not currently displacing sea water, is often referred to as the ice mass or volume above flotation in glaciology. We define barystatic sea-level rise as ℎ𝑏 = Δ𝑀𝜌f𝐴, (37)	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-10	We propose “manometric” as a new term because in the existing literature there is no unambiguous and generally used term for Δ𝑅𝑚. It may be described as the “mass effect on”, “mass contribution to”, “mass component of” or “mass term in” sea level or sea-level change, but these descriptions could equally well refer to GRD -induced sea-level change (the effects of a change in the geographical distribution of mass) or barystatic sea-level rise, so they can be confusing. (“Manometric” is an existing word, referring to the measurement of hydrostatic pressure using a column of liquid, a concept that is closely related to bottom pressure.) N19 Barystatic sea-level riseℎ𝑏: The part of global-mean sea-level rise (GMSLR) which is due to the addition to the ocean of water mass that formerly resided within the land area (as land water storage or land ice) or in the atmosphere (which contains a relatively tiny mass of water), or (if negative) the removal of mass from the ocean to be stored elsewhere. It is also called “barystatic sea-level change”. Land water storage, also called terrestrial water storage, is water on land that is stored as groundwater, soil moisture, water in reservoirs, lakes and rivers, seasonal snow and permafrost. Land ice means ice sheets, glaciers, permanent snow and firn. Barystatic sea-level rise includes contributions from changes in all of these. It does not include changes in the parts of ice shelves and glacier tongues whose weight is supported by the ocean rather than resting on land. (These floating parts constitute the majority of the mass of ice shelves and glacier tongues, but near the grounding line on the seaward side some part of the weight may be supported by the land-based ice.) Where land ice rests on a bed which is below mean sea level, it is already displacing sea water. Therefore, the land ice contribution to barystatic sea-level rise excludes the mass whose liquid-water equivalent volume equals the volume of sea water already displaced. The remainder, which is not currently displacing sea water, is often referred to as the ice mass or volume above flotation in glaciology. We define barystatic sea-level rise as ℎ𝑏 = Δ𝑀𝜌f𝐴, (37) i.e., the change in mass Δ𝑀 of the global ocean from added freshwater, converted to a change in global ocean volume and divided by the ocean surface area A. Because global-mean halosteric change is negligible, the salinity of the existing sea water does not affect ℎ𝑏. Any contribution 𝛿𝑀 to barystatic sea-level rise can be expressed as its sea-level equivalent (SLE)𝛿𝑀/(𝜌f𝐴), using the same formula. The formula provides a convenient method of quantifying the changes in the mass of the ocean if A is constant. However, ℎ𝑏 and SLE may not accurately indicate the contribution of added mass to global-mean ocean thickness if there is a substantial change to A, as for example in the transition from glacial to interglacial. Calculating the global mean of manometric sea-level change Δ𝑅𝑚 from its Definition B (Eq. 33) gives 1/𝐴∫Δ𝑅𝑚d𝐴=Δ𝑀/(𝜌∗𝐴)≃ℎ𝑏, i.e., approximately equal to the barystatic sea-level change, but not exactly since 𝜌∗≠𝜌f. With Definition A, the global mean of Δ𝑅𝑚 exactly equals ℎ𝑏. (See global-mean sea-level rise .) Despite this relationship between manometric sea-level change and barystatic sea-level rise, we argue that we need distinct terms for them, rather than referring to the latter as the global mean of the former, for two reasons. First, barystatic sea-level rise is well defined by conservation of water mass on Earth and can be evaluated from the change in mass of other stores of water, e.g., ice sheets and glaciers, without considering the ocean. This has been the usual approach in observational studies of the budget of global-mean sea-level rise, and is the only possibility for diagnosing ℎ𝑏 from the majority of climate models whose ocean component is Boussinesq or has a linear free surface, and therefore does not conserve water mass. Secondly, the partitioning of RSLC into steric and manometric (Eq. 35) is somewhat arbitrary, because it depends on the choice of 𝜌∗ as a reference density. Neither reason for the distinction of Δ𝑅𝑚 and ℎ𝑏 applies to thermosteric sea-level change; its contribution to global-mean sea-level rise can only be conceived or evaluated as the global mean of the local Δ𝑅𝜃, whose definition by Eq. (31) is well defined. In recent literature, “eustatic” is often used as a synonym for “barystatic”, whereas in geological literature eustatic sea-level change means either global-mean sea-level rise or global-mean geocentric sea-level rise . Because of this confusion of meaning, we deprecate the term “eustatic”, following the last three assessment reports of the Intergovernmental Panel on Climate Change (Church et al. 2001; Meehl et al. 2007; Church et al. 2013). N20 Sterodynamic sea-level changeΔ𝑍: Relative sea-level change due to changes in ocean density and circulation, with inverse barometer (IB) correction. This term is the sum of ocean dynamic sea-level change (which includes the IB correction) and global-mean thermosteric sea-level rise , Δ𝑍(𝐫)=Δ𝜁(𝐫)+ℎ𝜃. (38) It can be diagnosed from ocean models (even those that do not conserve mass as per the commonly used Boussinesq models) as the sum of the changes in zos and zostoga. Sterodynamic sea-level change is the part of relative sea-level change that can be simulated with such models. (As discussed above for ocean dynamic sea-level change Δ𝜁, the change in zos calculated from CMIP data should have zero global mean.) “Sterodynamic” is a term which is newly introduced in this paper. We propose it because in the existing literature there is no clear, simple or generally used term for Δ𝑍. It is a concept that appears in the literature, where it is referred to by various cumbersome phrases, such as “the oceanographic part of sea-level change”, “steric plus dynamic sea-level change” or “sea-level change due to ocean density and circulation change”. N21 Vertical land movement (VLM)Δ𝐹: The change in the height of the sea floor or the land surface. VLM has several causes, including isostasy, elastic flexure of the lithosphere, earthquakes and volcanoes (due to tectonics). All of these involve a change in height of the existing solid surface. In contrast, landslides and sedimentation alter the solid surface and its height by transport of materials; some authors count them as VLM. Extraction of groundwater and hydrocarbons may cause subsidence (sinking of the solid surface) by compaction (the reduction in the liquid fraction in the sediment). These anthropogenic effects can be locally large, e.g., in Manila, and can exceed the natural effects by orders of magnitude. Where VLM occurs near the coast, it may cause emergence or submergence of land and thus alter the coastline. Isostasy or isostatic adjustment is the process of adjustment of the lithosphere (the crust and the rigid upper part of the mantle) towards a hydrostatic equilibrium in which it is regarded as floating in the asthenosphere (the underlying viscous mantle, which is of higher density than the lithosphere), with an equal pressure everywhere at some notional horizontal level beneath the lithosphere. On geological timescales, isostatic adjustment occurs in response to changes in the mass load of the lithosphere upon the mantle beneath (the asthenosphere and lower mantle), due to erosion, sedimentation or emplacement of igneous rocks. On climate timescales there are large changes in load due to the varying mass of ice on land during glacial–interglacial cycles. (See glacial isostatic adjustment .) Isostatic adjustment occurs over multi-millennial timescales determined by the viscous flow of the mantle beneath the lithosphere. An elastic response of the lithosphere, on annual timescales, occurs in response to changes in load. Although it is small compared with the eventual isostatic response, it is much more rapid, and hence responsible for significant VLM due to contemporary and recent historical changes in land ice, for instance in West Antarctica. N22 GRD: Changes in Earth Gravity, Earth Rotation (and hence centrifugal acceleration) and viscoelastic solid-Earth Deformation.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-11	N21 Vertical land movement (VLM)Δ𝐹: The change in the height of the sea floor or the land surface. VLM has several causes, including isostasy, elastic flexure of the lithosphere, earthquakes and volcanoes (due to tectonics). All of these involve a change in height of the existing solid surface. In contrast, landslides and sedimentation alter the solid surface and its height by transport of materials; some authors count them as VLM. Extraction of groundwater and hydrocarbons may cause subsidence (sinking of the solid surface) by compaction (the reduction in the liquid fraction in the sediment). These anthropogenic effects can be locally large, e.g., in Manila, and can exceed the natural effects by orders of magnitude. Where VLM occurs near the coast, it may cause emergence or submergence of land and thus alter the coastline. Isostasy or isostatic adjustment is the process of adjustment of the lithosphere (the crust and the rigid upper part of the mantle) towards a hydrostatic equilibrium in which it is regarded as floating in the asthenosphere (the underlying viscous mantle, which is of higher density than the lithosphere), with an equal pressure everywhere at some notional horizontal level beneath the lithosphere. On geological timescales, isostatic adjustment occurs in response to changes in the mass load of the lithosphere upon the mantle beneath (the asthenosphere and lower mantle), due to erosion, sedimentation or emplacement of igneous rocks. On climate timescales there are large changes in load due to the varying mass of ice on land during glacial–interglacial cycles. (See glacial isostatic adjustment .) Isostatic adjustment occurs over multi-millennial timescales determined by the viscous flow of the mantle beneath the lithosphere. An elastic response of the lithosphere, on annual timescales, occurs in response to changes in load. Although it is small compared with the eventual isostatic response, it is much more rapid, and hence responsible for significant VLM due to contemporary and recent historical changes in land ice, for instance in West Antarctica. N22 GRD: Changes in Earth Gravity, Earth Rotation (and hence centrifugal acceleration) and viscoelastic solid-Earth Deformation. These three effects are all caused by changes in the geographical distribution of ocean and solid mass over the Earth. They are often considered together because they occur simultaneously and may interact. Changes in gravitation and rotation alter the geopotential field and hence the geoid 𝐺(𝐫), while deformation of the solid Earth changes the sea floor topography 𝐹(𝐫) through vertical land movement . By altering G and 𝐹, GRD induces relative sea-level change (e.g., Tamisiea and Mitrovica 2011; Kopp et al. 2015), which redistributes but does not change the global ocean volume and thus causes no global-mean sea-level rise . GRD-induced relative sea-level changeΔ𝛤 is defined as Δ𝛤 = Δ𝐺′ − Δ𝐹′ (39) (derived in Sect. 6 as Eq. 51) where Δ𝐺′ and Δ𝐹′ are the deviations of the changes in the geoid and in the sea floor from their respective global (ocean) means. By construction, the global (ocean) means of 𝐺′ and 𝐹′ are each zero; hence, the global (ocean) mean of Δ𝛤 is zero. Whatever the cause, redistribution of the ocean mass itself has GRD effects, and thereby the ocean affects its own mass distribution and mean sea level (MSL). Thus, MSL, the geoid and the sea floor must all be related in a self-consistent solution, which in the context of glacial isostatic adjustment (GIA) is expressed by the sea-level equation (Farrell and Clark 1976). The ocean GRD effects are called self-attraction and loading (SAL), where “loading” means the weight on the solid Earth. SAL is caused by climatic change in ocean density and circulation (Gregory et al. 2013), which do not involve any change in the mass of the ocean. SAL is also a component of GRD which is caused by changes in land ice and in the solid Earth; thus, SAL contributes to the sea-level effects of GIA, contemporary GRD and mantle dynamic topography as well. We propose the new term “GRD” in the absence of any existing single term to describe this frequently discussed group of effects. GRD-induced relative sea-level change may be described as the “mass effect”, “mass contribution”, “mass component” or “mass term”, but these labels could equally well refer to manometric sea-level change if local, or barystatic sea-level rise if global, so they can be confusing. Moreover, “GRD” is helpful as a label for a concept which unifies SAL, GIA, contemporary GRD and mantle dynamic topography. N23 Glacial isostatic adjustment (GIA): GRD due to ongoing changes in the solid Earth caused by past changes in land ice. GIA is caused by the viscous adjustment of the mantle to changes in the load on the lithosphere that occurred when mass was transferred from land ice into the ocean, or the reverse. It is dominated by the ongoing effects of the deglaciation following the Last Glacial Maximum. Due to the reduction in the mass load on land, areas that were beneath former ice sheets are generally rising. This process is sometimes called post-glacial rebound, but that term is unsatisfactory because GIA involves remote vertical land movement as well, both upward and downward. Areas adjacent to the former ice sheets are subsiding as mantle material moves towards the areas of uplift, while land near to the coast is rising and the sea floor is generally subsiding as a result of the increase in the mass of the ocean. The ongoing widespread redistribution of mass also affects the geoid . Together, the changes in geoid and sea floor cause GIA-induced relative sea-level changeΔ𝛤GIA. Previous changes in land ice during the Holocene contribute to GIA as well, but GIA does not include the contributions from any ongoing change in land ice or ocean mass, whose effects we call contemporary GRD . N24 Contemporary GRD: GRD due to ongoing changes in the mass of water stored on land as ice sheets, glaciers and land water storage. Such transfers of mass cause instantaneous changes in the geoid , and vertical land movement (VLM) on annual timescales due to elastic deformation of the solid Earth, which causes further change to the geoid. Together, these effects produce relative sea-level change (RSLC). There are also slower responses, both VLM and geoid, due to viscous deformation of the asthenosphere. Note that contemporary GRD excludes GIA; the former arises from ongoing change in the mass of water on land, and the latter from past change. The elastic deformation and associated geoid contributions to contemporary GRD-induced relative sea-level change are separately proportional to the mass Δ𝑀 which has been added to the ocean. Hence, their sum is Δ𝛤𝑝(𝐫)=Δ𝑀𝛾𝑝(𝐫), (40) where 𝛾𝑝(𝐫) is a geographically dependent constant of proportionality, independent of Δ𝑀. Since the barystatic sea-level rise is Δ𝑀/𝜌f𝐴 (Eq. 37), the RSLC due to the combination of these three effects is Δ𝑅𝑝(𝐫) = Δ𝑀𝜙(𝐫) where 𝜙(𝐫) = 𝛾𝑝(𝐫) + 1/𝜌f𝐴. (41) The addition of freshwater to the ocean will induce sterodynamic sea-level change as well (e.g., Agarwal et al. 2015). The barystatic–GRD fingerprint 𝜙 is a constant geographical pattern, often called a sea-level fingerprint, or sometimes a static-equilibrium fingerprint to contrast it with the patterns of ocean dynamic sea-level change . “Fingerprint” without qualification can be easily confused with climate detection and attribution studies where the same word refers to the patterns caused by particular climate change forcing agents such as greenhouse gases. “Static-equilibrium” is not informative about the processes concerned. The part of contemporary GRD-induced RSLC due to viscous deformation and associated geoid change cannot be represented by a constant pattern because it depends on convolving the history of mass addition with the time-dependent solid-Earth response. N25 Mantle dynamic topography: GRD due to ongoing changes in the solid Earth caused by mantle convection and plate tectonics.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-12	The elastic deformation and associated geoid contributions to contemporary GRD-induced relative sea-level change are separately proportional to the mass Δ𝑀 which has been added to the ocean. Hence, their sum is Δ𝛤𝑝(𝐫)=Δ𝑀𝛾𝑝(𝐫), (40) where 𝛾𝑝(𝐫) is a geographically dependent constant of proportionality, independent of Δ𝑀. Since the barystatic sea-level rise is Δ𝑀/𝜌f𝐴 (Eq. 37), the RSLC due to the combination of these three effects is Δ𝑅𝑝(𝐫) = Δ𝑀𝜙(𝐫) where 𝜙(𝐫) = 𝛾𝑝(𝐫) + 1/𝜌f𝐴. (41) The addition of freshwater to the ocean will induce sterodynamic sea-level change as well (e.g., Agarwal et al. 2015). The barystatic–GRD fingerprint 𝜙 is a constant geographical pattern, often called a sea-level fingerprint, or sometimes a static-equilibrium fingerprint to contrast it with the patterns of ocean dynamic sea-level change . “Fingerprint” without qualification can be easily confused with climate detection and attribution studies where the same word refers to the patterns caused by particular climate change forcing agents such as greenhouse gases. “Static-equilibrium” is not informative about the processes concerned. The part of contemporary GRD-induced RSLC due to viscous deformation and associated geoid change cannot be represented by a constant pattern because it depends on convolving the history of mass addition with the time-dependent solid-Earth response. N25 Mantle dynamic topography: GRD due to ongoing changes in the solid Earth caused by mantle convection and plate tectonics. The dynamics of the interior of the Earth cause vertical land movement , such as the uplift of mid-ocean ridges by upwelling material and the formation of oceanic trenches due to subduction. At the same time, material with different density is redistributed within the Earth, altering the geoid . The consequent GRD-induced relative sea-level change can be very large on geological timescales, amounting to hundreds of metres. Mantle dynamic topography does not include glacial isostatic adjustment (although that is also due to ongoing changes in the solid Earth). Mantle dynamic topography is often called “dynamic topography” in the solid-Earth literature and also refers to changes in topography on land. We deprecate “dynamic topography” in a sea-level context because it could be confused with ocean dynamic topography . N26 Global-mean sea-level rise (GMSLR)h: The increase Δ𝑉 in the volume of the ocean divided by the ocean surface area A, also called “global-mean sea-level change” (GMSLC). Observational estimation of h is described in Sect. 7. By definition, GMSLR is ℎ = Δ𝑉/𝐴 = 1/𝐴Δ(∫(𝜂−𝐹)d𝐴) = 1/𝐴∫Δ𝑅(𝐫)d𝐴 = 1/𝐴∫Δ𝐻(𝐫)d𝐴, (42) which follows from Eqs. (4), (24) and (25). Hence, GMSLR is the global mean of relative sea-level change Δ𝑅 and equals the global mean of the change Δ𝐻 in the thickness (or “depth”) of the ocean. Note that GMSLR differs from global-mean geocentric sea-level rise because GMSLR is unaltered by a global-mean change (1/𝐴)∫Δ𝐹d𝐴 in the level of the sea floor 𝐹, provided the global ocean volume V does not change. The global ocean volume can change due to changes in ocean density or due to changes in ocean mass. Hence, GMSLR is the sum of global-mean thermosteric sea-level rise and barystatic sea-level rise , ℎ=ℎ𝜃+ℎ𝑏. (43) A satisfactory explanation of historical observed GMSLR in terms of thermosteric and barystatic contributions has been achieved in recent years thanks to improvements in both observations and models (Church et al. 2011; Gregory et al. 2013; Chambers et al. 2017). Equation (43) is implied by the Definition A of manometric sea-level change , as the part of relative sea-level change which is not steric (Eq. 36), whose global mean 1𝐴∫Δ𝑅𝑚d𝐴=1𝐴∫Δ𝑅d𝐴−1𝐴∫Δ𝑅𝜌d𝐴=ℎ−ℎ𝜃, (44) using Eq. (42) and recalling that global-mean steric sea-level change is purely thermosteric. By definition, the part of h which is not steric is the part which is due to addition of mass, so it must be the case that ℎ𝑏=1𝐴∫Δ𝑅𝑚d𝐴, (45) i.e., barystatic sea-level rise equals the global mean of manometric sea-level change by Definition A. The added mass is unlikely to have exactly the temperature of the existing water to which it is added, implying that changes will probably occur to temperature and hence to density. Because of the nonlinearity of the dependence of 𝜌 on 𝜃, there may be a nonzero contribution to ℎ𝜃 in consequence. Since this is a steric effect, by definition it is not part of ℎ𝑏. From Definition B (Eq. 33) we obtain an approximate expression for global-mean manometric sea-level change as (1/𝐴)∫Δ𝑚/𝜌∗d𝐴=Δ𝑀/(𝜌∗𝐴), where Δ𝑀 is the added mass. This is the same as the expression (Eq. 37) for ℎ𝑏 except that 𝜌f is replaced by 𝜌∗. This difference is the result of the approximation in Eq. (33) that 𝜌∗ ≃ ρ̄. Physically, it is because manometric sea-level change Δ𝑅𝑚 (a local quantity) is dominated by redistribution of existing sea water, for which 𝜌∗ is a good choice of representative density, whereas barystatic sea-level rise (a global quantity) is due only to addition or subtraction of freshwater of density 𝜌f, since the redistributive effect is zero in the global mean. On glacial–interglacial and geological timescales, the variation of ocean area cannot be neglected, so GMSLR is ill-defined. However, it is still meaningful to consider global-mean relative sea-level changeℎ𝑅, which is the change in global-mean ocean thickness ℎ𝑅 = 𝑉+Δ𝑉/(𝐴+Δ𝐴) − 𝑉/𝐴 = 𝑉/(𝐴+Δ𝐴) (Δ𝑉/𝑉−Δ𝐴/𝐴). (46) If A is constant, ℎ𝑅 = ℎ. An increase in A (Δ𝐴>0) gives a negative contribution to ℎ𝑅, counteracting the positive contribution from a concomitant increase in V. In geological literature, global-mean sea-level rise is sometimes called “eustatic sea-level change”. Following the last three assessment reports of the Intergovernmental Panel on Climate Change (Church et al. 2001; Meehl et al. 2007; Church et al. 2013), we deprecate “eustatic” because it has become a confusing term, which is also used to mean global-mean geocentric sea-level rise or barystatic sea-level rise . N27 Global-mean geocentric sea-level riseℎ𝐺: The global-mean change in mean sea level with respect to the terrestrial reference frame. This quantity is the global mean of Δ𝜂, the change in MSL relative to the reference ellipsoid . From Eqs. (4) and (5) we have ℎ𝐺 = 1/𝐴∫Δ𝜂d𝐴 = 1/𝐴∫Δ𝐺d𝐴 = 1/𝐴(Δ𝑉 + ∫Δ𝐹d𝐴) = ℎ + 1/𝐴 ∫Δ𝐹 d𝐴, (47)	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-13	On glacial–interglacial and geological timescales, the variation of ocean area cannot be neglected, so GMSLR is ill-defined. However, it is still meaningful to consider global-mean relative sea-level changeℎ𝑅, which is the change in global-mean ocean thickness ℎ𝑅 = 𝑉+Δ𝑉/(𝐴+Δ𝐴) − 𝑉/𝐴 = 𝑉/(𝐴+Δ𝐴) (Δ𝑉/𝑉−Δ𝐴/𝐴). (46) If A is constant, ℎ𝑅 = ℎ. An increase in A (Δ𝐴>0) gives a negative contribution to ℎ𝑅, counteracting the positive contribution from a concomitant increase in V. In geological literature, global-mean sea-level rise is sometimes called “eustatic sea-level change”. Following the last three assessment reports of the Intergovernmental Panel on Climate Change (Church et al. 2001; Meehl et al. 2007; Church et al. 2013), we deprecate “eustatic” because it has become a confusing term, which is also used to mean global-mean geocentric sea-level rise or barystatic sea-level rise . N27 Global-mean geocentric sea-level riseℎ𝐺: The global-mean change in mean sea level with respect to the terrestrial reference frame. This quantity is the global mean of Δ𝜂, the change in MSL relative to the reference ellipsoid . From Eqs. (4) and (5) we have ℎ𝐺 = 1/𝐴∫Δ𝜂d𝐴 = 1/𝐴∫Δ𝐺d𝐴 = 1/𝐴(Δ𝑉 + ∫Δ𝐹d𝐴) = ℎ + 1/𝐴 ∫Δ𝐹 d𝐴, (47) where h is global-mean sea-level rise (GMSLR) defined by Eq. (42). Thus, global-mean geocentric sea-level rise ℎ𝐺 differs from GMSLR because the latter is unaltered by a global-mean change (1/𝐴)∫Δ𝐹d𝐴 in the level of the sea floor 𝐹, provided the volume of the ocean does not change. On geological timescales, when the area of the ocean may change, the global-mean change in level of the sea floor is Δ((1/𝐴)∫𝐹d𝐴). Relationships Determining Relative Sea-Level Change Relative sea-level change (RSLC) is Δ𝑅 = Δ𝜂 − Δ𝐹 (Eq. 24). By applying the inverse barometer correction, we obtain IB-corrected RSLC Δ𝑅 + Δ𝐵 = Δ𝜂 + Δ𝐵 − Δ𝐹 (48a) = Δ[𝜂−𝐺+𝐵] + Δ[𝐺−𝐹] (48b) = Δ𝜁 + Δ[𝐺−𝐹], (48c) where in Eq. (48c) we used the definition of ocean dynamic sea-level change Δ𝜁 (Eq. 23) to rewrite the first term. From the definition of the geoid (Eq. 4) we obtain 1/𝐴 ∫ Δ(𝐺−𝐹)d𝐴 = Δ𝑉/𝐴 = ℎ, (49) the global-mean sea-level rise . Let us write Δ𝐺(𝐫)=Δ𝐺′(𝐫)+(1/𝐴)∫Δ𝐺d𝐴 and similarly for Δ𝐹, thus defining Δ𝐺′,Δ𝐹′ as the local deviations of Δ𝐺,Δ𝐹 from their respective global (ocean) means. Therefore, Δ(𝐺−𝐹) = Δ(𝐺′−𝐹′) + 1/𝐴 ∫Δ(𝐺−𝐹)d𝐴 = Δ(𝐺′−𝐹′)+ℎ. (50) This leads to our expression for GRD -induced relative sea-level change (Eq. 39) as Δ𝛤(𝐫)≡Δ[𝐺′(𝐫)−𝐹′(𝐫)]=Δ[𝐺(𝐫)−𝐹(𝐫)]−ℎ. (51) Substituting Eq. (51) in Eq. (48c) gives IB-corrected RSLC as Δ𝑅(𝐫)+Δ𝐵(𝐫)=Δ𝜁(𝐫)+ℎ+Δ𝛤(𝐫), (52) the sum of ocean dynamic sea-level change Δ𝜁, global-mean sea-level rise h and GRD-induced RSLC Δ𝛤. Using Eqs. (43) and (38) we obtain Δ𝜁(𝐫)+ℎ=Δ𝜁(𝐫)+ℎ𝜃+ℎ𝑏=Δ𝑍(𝐫)+ℎ𝑏, (53) Hence, IB-corrected RSLC is Δ𝑅(𝐫)+Δ𝐵(𝐫)=Δ𝑍(𝐫)+ℎ𝑏+Δ𝛤(𝐫), (54) the sum of sterodynamic sea-level change Δ𝑍(𝐫), barystatic sea-level rise ℎ𝑏 and GRD-induced RSLC. The contemporary GRD -induced RSLC due to a change 𝛿𝑀𝑖 in any of the stores of water on land (as land water storage or land ice, e.g., in a lake or an ice sheet) has both a barystatic and a GRD-induced effect on sea level, which are related and may interact (e.g., Gomez et al. 2012). Provided they are both proportional to 𝛿𝑀𝑖, we can rewrite Eq. (54) as Δ𝑅(𝐫) + Δ𝐵(𝐫) = Δ𝑍(𝐫) + ∑𝑖𝛿𝑀𝑖𝜙𝑖 + Δ𝛤GIA + Δ𝛤𝑍, (55) where 𝜙𝑖 is the barystatic–GRD fingerprint (Eq. 41) of store i of water, Δ𝛤GIA is GIA-induced RSLC, and Δ𝛤𝑍 is the GRD-induced RSLC of ocean mass redistribution (self-attraction and loading) associated with sterodynamic sea-level change. The last term is typically neglected. Equation (55) is the means by which MSL projections are derived from coupled atmosphere–ocean general circulation models (AOGCMs). These models do not simulate GRD-induced RSLC (because they have time-independent geoid and sea floor) and are not generally used to compute barystatic sea-level rise (because they do not include adequate representations of land ice or land water storage). RSLR projections are therefore obtained by combining sterodynamic sea-level change simulated by an AOGCM with separately calculated projections of barystatic sea-level rise and GRD-induced RSLC using climate change simulations from the AOGCM applied to models of glaciers, ice sheets and the solid Earth (Church et al. 2013; Kopp et al. 2014; Slangen et al. 2014).	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-14	the sum of sterodynamic sea-level change Δ𝑍(𝐫), barystatic sea-level rise ℎ𝑏 and GRD-induced RSLC. The contemporary GRD -induced RSLC due to a change 𝛿𝑀𝑖 in any of the stores of water on land (as land water storage or land ice, e.g., in a lake or an ice sheet) has both a barystatic and a GRD-induced effect on sea level, which are related and may interact (e.g., Gomez et al. 2012). Provided they are both proportional to 𝛿𝑀𝑖, we can rewrite Eq. (54) as Δ𝑅(𝐫) + Δ𝐵(𝐫) = Δ𝑍(𝐫) + ∑𝑖𝛿𝑀𝑖𝜙𝑖 + Δ𝛤GIA + Δ𝛤𝑍, (55) where 𝜙𝑖 is the barystatic–GRD fingerprint (Eq. 41) of store i of water, Δ𝛤GIA is GIA-induced RSLC, and Δ𝛤𝑍 is the GRD-induced RSLC of ocean mass redistribution (self-attraction and loading) associated with sterodynamic sea-level change. The last term is typically neglected. Equation (55) is the means by which MSL projections are derived from coupled atmosphere–ocean general circulation models (AOGCMs). These models do not simulate GRD-induced RSLC (because they have time-independent geoid and sea floor) and are not generally used to compute barystatic sea-level rise (because they do not include adequate representations of land ice or land water storage). RSLR projections are therefore obtained by combining sterodynamic sea-level change simulated by an AOGCM with separately calculated projections of barystatic sea-level rise and GRD-induced RSLC using climate change simulations from the AOGCM applied to models of glaciers, ice sheets and the solid Earth (Church et al. 2013; Kopp et al. 2014; Slangen et al. 2014). According to Eqs. (35) or (36), Δ𝑅=Δ𝑅𝜌+Δ𝑅𝑚, the sum of steric sea-level change Δ𝑅𝜌 and manometric sea-level change Δ𝑅𝑚, which are the parts due, respectively, to change in density and change in mass per unit area. In general, Δ𝑅𝑚≠0 even if ℎ𝑏=0, because ocean mass may be redistributed. In particular, because Δ𝑅𝑚=−1/𝜌s∫𝜂𝐹Δ𝜌d𝑧 is small on the continental shelves (where the ocean is shallow), but ocean dynamics will not permit a strong gradient in 𝜁 to develop across the shelf break, global-mean thermosteric sea-level rise demands a redistribution of ocean mass onto the shelves (Landerer et al. 2007; Yin et al. 2010), with consequent ocean GRD (Gregory et al. 2013). Observations of Sea-Level Change Estimates of global-mean sea-level rise (GMSLR) for the last century depend mainly on records from tide-gauges. These instruments register coastal relative sea-level change (RSLC) Δ𝑅=Δ𝜂−Δ𝐹 (Eq. 24), which is affected by local vertical land movement (VLM) Δ𝐹. VLM is large in some places, with strong geographical gradients. GMSLR is calculated as the global mean of RSLC, ℎ=(1/𝐴)∫Δ𝑅d𝐴 (Eq. 42). However, tide-gauges measure Δ𝑅 only at points on the coast and thus give a sparse, non-uniform and unrepresentative sampling of the global ocean area. The calculation therefore depends on physically based methods for extrapolation. Considering Eq. (52) in the form ℎ = Δ𝑅 + Δ𝐵 − Δ𝜁 − Δ𝛤 (56) we see that in principle h can be calculated from Δ𝑅 from any tide-gauge by applying the inverse barometer (IB) correction Δ𝐵, and subtracting local ocean dynamic sea-level change Δ𝜁 and local GRD -induced RSLC Δ𝛤. The global mean of each of these three adjustments is zero, so Eq. (42) is satisfied. In practice, using historical records, it is necessary to combine many tide-gauges in order to reduce the influence of unforced variability in 𝜁̃. The IB adjustment is fairly small and can be made accurately from atmospheric pressure records. Various methods are used to allow for the spatial pattern of Δ𝜁, for example by calculating the mean over sets of gauges presumed to be representative of large regions (e.g., Jevrejeva et al. 2008), or by using spatial patterns of 𝜂̃ variation observed by satellite altimetry during its shorter period of availability (e.g., Church and White 2011). Because glacial isostatic adjustment (GIA) is the only part of GRD (including VLM) for which a global field is available, most estimates of GMSLR exclude all tide-gauges where GIA is not the only significant contribution to VLM (those affected by earthquakes, anthropogenic subsidence, sediment compactions, etc.). At the tide-gauges which are retained, we adjust for GIA-induced RSLC Δ𝛤GIA(𝐫) (e.g., Figure 3a of Tamisiea and Mitrovica 2011), estimated by combining solid-Earth models, the sea-level equation and reconstructed histories of deglaciation. Alternatively, tide-gauge records may be corrected for VLM using vertical motion calculated from collocated GNSS (e.g., GPS) receivers. Effectively, this transforms RSLC to geocentric sea-level change Δ𝜂=Δ𝑅+Δ𝐹 (Eq. 24). Geoid adjustments must be applied to GNSS-corrected tide-gauge records just as for satellite altimetry, as described in the next paragraph. Geocentric sea-level change Δ𝜂(=Δ𝜁−Δ𝐵+Δ𝐺, Eq. 23) has been measured over most of the global ocean since the early 1990s by satellite radar altimetry, using instruments which are located in a terrestrial reference frame (equivalent to the reference ellipsoid), and measure their vertical distance from the sea surface. To study the contemporary causes of observed geocentric sea-level change we must subtract Δ𝐺GIA(𝐫), the effect of GIA on the geoid (e.g., Figure 3b of Tamisiea and Mitrovica 2011), from IB-corrected geocentric sea-level change, thus: Δ𝜂 + Δ𝐵 − Δ𝐺 GIA = Δ𝜁 + Δ𝐺NGIA, (57) where Δ𝐺NGIA = Δ𝐺 − Δ𝐺GIA is due to ongoing redistribution of water mass on the Earth’s surface. To convert global-mean geocentric sea-level rise ℎ𝐺=(1/𝐴)∫Δ𝜂d𝐴 to GMSLR h requires an adjustment for the global mean of Δ𝐹, according to Eq. (47). Although several processes can produce large local VLM, the only large global-mean effect is GIA. There is no contemporary GMSLR associated with GIA, so Eq. (49) gives (1/𝐴)∫Δ𝐹GIAd𝐴=(1/𝐴)∫Δ𝐺GIAd𝐴⇒(1/𝐴)∫Δ𝐺NGIAd𝐴=ℎ+(1/𝐴)∫Δ𝐹NGIAd𝐴. Hence, the global mean of Eq. (57) becomes ℎ𝐺 = ℎ + 1/𝐴∫Δ𝐹GIA d𝐴 + 1/𝐴∫Δ𝐹NGIA d𝐴, (58)	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-15	Geocentric sea-level change Δ𝜂(=Δ𝜁−Δ𝐵+Δ𝐺, Eq. 23) has been measured over most of the global ocean since the early 1990s by satellite radar altimetry, using instruments which are located in a terrestrial reference frame (equivalent to the reference ellipsoid), and measure their vertical distance from the sea surface. To study the contemporary causes of observed geocentric sea-level change we must subtract Δ𝐺GIA(𝐫), the effect of GIA on the geoid (e.g., Figure 3b of Tamisiea and Mitrovica 2011), from IB-corrected geocentric sea-level change, thus: Δ𝜂 + Δ𝐵 − Δ𝐺 GIA = Δ𝜁 + Δ𝐺NGIA, (57) where Δ𝐺NGIA = Δ𝐺 − Δ𝐺GIA is due to ongoing redistribution of water mass on the Earth’s surface. To convert global-mean geocentric sea-level rise ℎ𝐺=(1/𝐴)∫Δ𝜂d𝐴 to GMSLR h requires an adjustment for the global mean of Δ𝐹, according to Eq. (47). Although several processes can produce large local VLM, the only large global-mean effect is GIA. There is no contemporary GMSLR associated with GIA, so Eq. (49) gives (1/𝐴)∫Δ𝐹GIAd𝐴=(1/𝐴)∫Δ𝐺GIAd𝐴⇒(1/𝐴)∫Δ𝐺NGIAd𝐴=ℎ+(1/𝐴)∫Δ𝐹NGIAd𝐴. Hence, the global mean of Eq. (57) becomes ℎ𝐺 = ℎ + 1/𝐴∫Δ𝐹GIA d𝐴 + 1/𝐴∫Δ𝐹NGIA d𝐴, (58) recalling that the global means of Δ𝐵 and Δ𝜁 are zero. In response to the shift of mass from the land (as ice) into the ocean since the Last Glacial Maximum, and the consequent mantle adjustment, the sea floor is subsiding on average, giving a trend in (1/𝐴)∫Δ𝐹GIAd𝐴 of about −0.3 mm year −1 (Tamisiea and Mitrovica 2011). Thus, ℎ𝐺<ℎ due to GIA. Contemporary changes in land ice cause elastic deformation of the sea floor. This gives a negative (1/𝐴)∫Δ𝐹NGIAd𝐴 which reduces ℎ𝐺 by about 8% of the barystatic sea-level rise (Frederikse et al. 2017). Deprecated Terms and Recommended Replacements Deprecated term Recommended replacement Eustatic sea-level change Barystatic sea-level rise or barystatic sea-level change for global-mean sea-level rise due to change in the mass of the ocean, but not its density Global-mean sea-level rise for the global mean of relative sea-level change, due to the change in the volume of the ocean Global-mean geocentric sea-level rise for the global mean of change in mean sea level relative to the terrestrial reference frame, due to the combined effects of change in the volume of the ocean and change in the level of the sea floor Dynamic topography Ocean dynamic sea level for mean sea level above the geoid due to ocean dynamics Ocean dynamic topography for ocean dynamic sea level estimated from ocean density Sea-level change due to mantle dynamic topography for GRD-induced relative sea-level change due to solid-Earth dynamics Mean sea surface Mean sea level (MSL) Mean sea-level change or local sea-level change Relative sea-level change (RSLC) or relative sea-level rise (RSLR) for the change in mean sea level relative to the land Geocentric sea-level change for the change in mean sea level relative to the terrestrial reference frame Global sea-level change (GSLC) Gobal-mean sea-level rise (GMSLR) or global-mean sea- level change (GMSLC) for the global mean of relative sea- level change, due to the change in the volume of the ocean Global-mean geocentric sea-level rise for the global-mean change in mean sea level relative to the terrestrial reference frame Extreme sea level Extreme sea level for the occurrence of exceptionally high local sea surface due to short-term phenomena, or extreme coastal water level when considering coastal impacts. High-end sea-level change for projections or scenarios of very large RSLR or GMSLR Sea-level change due to thermal expansion Thermosteric sea-level change for contribution to relative sea-level change Global-mean thermosteric sea-level rise for contribution to global-mean sea-level change Sea-level change due to oceanographic processes or steric plus dynamic sea- level change Sterodynamic sea-level change for the change in relative sea-level due to change in ocean density and circulation Sea-level fingerprint or static-equilibrium fingerprint Barystatic–GRD fingerprint for the sum of the RSLC from GRD (elastic and geoid) and the barystatic sea-level rise due to the addition of a unit mass of water to the global ocean Post-glacial rebound (PGR) Glacial isostatic adjustment (GIA) Mass effect on, mass term in, mass component of, or mass contribution to sea level or to sea-level change Barystatic sea-level rise for the contribution to global- mean sea-level rise from the change of mass of the global ocean (associated with changes in mass of water and ice on land), as opposed to global-mean thermosteric sea-level rise Manometric sea-level change for the contribution to relative sea-level change due to change in the local mass of the ocean per unit area, as opposed to steric sea-level change GRD-induced relative sea-level change for the effects on relative sea level from geoid change and vertical land movement, as opposed to steric, ocean dynamic and barystatic sea-level change Bottom pressure term in sea-level change Manometric sea-level change List of Defined Terms and Notations This table gives the entry number or section (not the page number) in which each term is defined. In the PDF, each term is a hyperlink to the relevant text. The rows for which there is no entry number are included only to define notation. Above flotation N19 Altimetry Section 7 A Area of the global ocean Asthenosphere N21 Astronomical tide N6 𝑝𝑎 Atmospheric pressure at the sea surface ℎ𝑏 Barystatic sea-level rise N19 𝜙 Barystatic–GRD fingerprint N24 Bathymetry N4 Bottom pressure N18 Bottom topography N4 Chandler wobble N6 Compaction N21 Contemporary GRD N24 𝜌 Density of water Depth of the water column N4 D Dynamic height N12 g Effective gravitational acceleration N5 Ellipsoidal height N1 Equipotential surface N5 Eustatic sea-level change N19 Extreme coastal water level N8 Extreme sea level N8 Fetch N10 Free nutation N6 Geocentric latitude N1 Δ𝜂 Geocentric sea-level change N14 Geodetic height N1 Geodetic latitude N1 G Geoid N5 Geoid height N5 Φ Geopotential N5 Geostrophy N12 vg Geostrophic velocity N12 Δ𝛤GIA GIA-induced relative sea-level change N23 Glacial isostatic adjustment (GIA) N23 Global mean Section 2.4 ℎ𝐺 Global-mean geocentric sea-level rise N27 ℎ𝑅 Global-mean relative sea-level change N26 h Global-mean sea-level rise (GMSLR) N26 ℎ𝜃 Global-mean thermosteric sea-level rise N17 Gravitational acceleration N5 GRD N22 Δ𝛤 GRD-induced relative sea-level change N22 Greenwich meridian N1 Halosteric sea-level change Δ𝑅𝑆 N16 High-end N8 Horizontal N5 IB-corrected geocentric sea-level change N14 IB-corrected mean sea level N7	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-16	N19 Altimetry Section 7 A Area of the global ocean Asthenosphere N21 Astronomical tide N6 𝑝𝑎 Atmospheric pressure at the sea surface ℎ𝑏 Barystatic sea-level rise N19 𝜙 Barystatic–GRD fingerprint N24 Bathymetry N4 Bottom pressure N18 Bottom topography N4 Chandler wobble N6 Compaction N21 Contemporary GRD N24 𝜌 Density of water Depth of the water column N4 D Dynamic height N12 g Effective gravitational acceleration N5 Ellipsoidal height N1 Equipotential surface N5 Eustatic sea-level change N19 Extreme coastal water level N8 Extreme sea level N8 Fetch N10 Free nutation N6 Geocentric latitude N1 Δ𝜂 Geocentric sea-level change N14 Geodetic height N1 Geodetic latitude N1 G Geoid N5 Geoid height N5 Φ Geopotential N5 Geostrophy N12 vg Geostrophic velocity N12 Δ𝛤GIA GIA-induced relative sea-level change N23 Glacial isostatic adjustment (GIA) N23 Global mean Section 2.4 ℎ𝐺 Global-mean geocentric sea-level rise N27 ℎ𝑅 Global-mean relative sea-level change N26 h Global-mean sea-level rise (GMSLR) N26 ℎ𝜃 Global-mean thermosteric sea-level rise N17 Gravitational acceleration N5 GRD N22 Δ𝛤 GRD-induced relative sea-level change N22 Greenwich meridian N1 Halosteric sea-level change Δ𝑅𝑆 N16 High-end N8 Horizontal N5 IB-corrected geocentric sea-level change N14 IB-corrected mean sea level N7 IB-corrected relative sea-level change N15 IB-corrected sea-surface height N7 𝜃 In-situ temperature of ocean water B Inverse barometer (IB) N7 Inverted barometer N7 Isostasy N21 Isostatic adjustment N21 Land ice N19 Land water storage N19 Latitude N1 Level of no motion N12 Liquid-water equivalent sea surface N2 Lithosphere N21 Local Section 2.3 Longitude N1 Δ𝑅𝑚 Manometric sea-level change N18 Mantle dynamic topography N25 M Mass of the global ocean 𝜂 Mean sea level (MSL) N3 Mean-tide N6 Neap tide N6 Nodal period N6 Ocean Section 2.4 𝜁 Ocean dynamic sea level N11 Δ𝜁 Ocean dynamic sea-level change N13 Ocean dynamic topography N12 Ocean GRD N22 m Ocean mass per unit area Orthometric height N5 Permanent tide N6 Pole tide N6 Post-glacial rebound N23 Predicted tide N6 Prime meridian N1 Radiational tide N6 Reference ellipsoid N1 Regression N15 Δ𝑅 Relative sea-level change (RSLC) N15 𝜌∗ Representative density of ocean water S Salinity of ocean water F Sea floor N4 Sea surface N2 Sea-level equation N22 Sea-level equivalent (SLE) N19 Sea-level fingerprint N24 Sea-surface height (SSH) N2 Sea-surface waves N10 Seismic sea wave N10 Self-attraction and loading (SAL) N22 Significant wave height N10 Significant wave period N10 Skew-surge height N9 Spring tide N6 State Section 2.2 Static-equilibrium fingerprint N24 Δ𝑅𝜌 Steric sea-level change N16 Δ𝑍 Sterodynamic sea-level change N20 Storm surge N9 Storm tide N6 σ Storm-surge height N9 Subsidence N21 Surge residual N9 Swash N9 Swell wave N10 Terrestrial reference frame N1 Terrestrial water storage N19 Thermal expansion N16 Thermosteric sea-level change Δ𝑅𝜃 N16 H Thickness of the ocean N4 Tidal currents N6 Tidal datum N6 Tidal height N6 Tide-free N6 Tide-gauge N15 Tides N6 Transgression N15 True polar wander N1 Tsunami N10 Vertical N1 Δ𝐹 Vertical land movement (VLM) N21 V Volume of the global ocean Wave height N10 Wave period N10 Wave runup N9 Wave setup N9 Wind setup N9 Wind wave N10 Zero-tide N6 Appendix 1: No Resting Steady State Exists for a Realistic Ocean A state of rest requires zero acceleration parallel to the surface, which must therefore be an equipotential. This condition is satisfied by the geoid because it is an equipotential surface by definition. However, zero acceleration at the surface is not a sufficient condition for the ocean to remain at rest. If there are any horizontal density gradients within the ocean, there will be pressure gradients beneath the horizontal surface, producing forces that will set the ocean into motion. So another necessary condition for zero ocean circulation is the absence of density gradients along horizontal surfaces, i.e., density is a function of depth only. This configuration is quite unlike the real state of the ocean. Appendix 2: Why We Can Ignore Global Halosteric Sea-Level Change When freshwater enters the ocean, such as from melting continental ice sheets, it adds to the ocean mass and in turn increases global-mean sea level (barystatic sea-level rise). Ocean salinity also changes due to the dilution of sea water, thus suggesting a role for a global halosteric sea-level change (Munk 2003; Levitus et al. 2005). However, the net effect on global-mean sea level is almost entirely barystatic since the global halosteric effect is negligible (Lowe and Gregory 2006). We can understand why this is so by recognizing that freshwater entering the ocean sees its salinity increase while the ambient sea water is itself freshened. These compensating salinity changes (which are often ignored, as by Munk 2003 and Levitus et al. 2005) have corresponding compensating sea-level changes, thus bringing the global halosteric effect to near zero. We demonstrate this effect in the following subsections, by considering a two-bucket thought experiment where one bucket holds freshwater (bucket-1) and the other holds sea water (bucket-2). We ask how the total water volume changes upon homogenizing the water in the two buckets, while conserving the masses of freshwater and salt. As we will see, the total volume of homogenized water is very nearly equal to the sum of the volume initially in the two separate buckets (to within 0.1%). Conservation of Mass for Freshwater and Salt Let the two buckets contain water of mass 𝑀𝑛, volume 𝑉𝑛, salinity 𝑆𝑛, and density 𝜌𝑛, 𝑛=1,2, and assume they have equal Conservative Temperature and equal pressure. Now homogenize the water from the two buckets into a single larger bucket, and assume no change in pressure nor any heat of mixing so that Conservative Temperature also remains unchanged. The total mass of freshwater and salt is unchanged upon homogenizing, so that 𝑀 = 𝑀1 + 𝑀2; 𝑀𝑆 = 𝑀1𝑆1 + 𝑀2𝑆2, (59) where M is the total mass and S is the salinity of the homogenized water. Return the homogenized water to the original buckets, placing the same mass 𝑀1 back into the first bucket and mass 𝑀2 into the second bucket. Dependence of Density on Salinity	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-17	Conservation of Mass for Freshwater and Salt Let the two buckets contain water of mass 𝑀𝑛, volume 𝑉𝑛, salinity 𝑆𝑛, and density 𝜌𝑛, 𝑛=1,2, and assume they have equal Conservative Temperature and equal pressure. Now homogenize the water from the two buckets into a single larger bucket, and assume no change in pressure nor any heat of mixing so that Conservative Temperature also remains unchanged. The total mass of freshwater and salt is unchanged upon homogenizing, so that 𝑀 = 𝑀1 + 𝑀2; 𝑀𝑆 = 𝑀1𝑆1 + 𝑀2𝑆2, (59) where M is the total mass and S is the salinity of the homogenized water. Return the homogenized water to the original buckets, placing the same mass 𝑀1 back into the first bucket and mass 𝑀2 into the second bucket. Dependence of Density on Salinity The dimensionless coefficients 𝛼≡𝜌−1∂𝜌/∂𝜃 and 𝛽≡𝜌−1∂𝜌/∂𝑆, which are used to compute steric sea-level change (Eq. 31), measure the relative change in the in-situ density as a function of temperature and salinity. Because ∂𝜌/∂𝜃 is generally negative, the volume of a given parcel of sea water increases as its temperature rises; this phenomenon is called thermal expansion. By analogy, since ∂𝜌/∂𝑆>0, the corresponding effect for salinity is sometimes called “haline contraction”. This is a misleading analogy, because if the salinity has increased but the mass has not changed, some freshwater must have been replaced by salt, so the parcel is materially altered, unlike in the case of adding heat. If salt is added to the parcel but no mass is removed, the salinity, mass and volume of the parcel will all increase. The notion of “haline contraction” has led some previous authors to draw incorrect conclusions about the effect on sea level from adding freshwater to the ocean. We here aim to clarify this situation. The change in the volume of water due to homogenization depends on the value of 𝛽. For the surface ocean, representative values are 𝜌=1028 kg m −3 and 𝑆=0.035, while for freshwater 𝜌=𝜌f=1000 kg m −3 and 𝑆=0. Hence, a representative 𝛽≈(1/1000)(1028−1000)/(0.035−0)=28/35=0.8. This coefficient has a roughly 5% relative variation across the ocean, with most of that variation determined by temperature rather than salinity. (See Roquet et al. 2015, as well as Figure 1 in Griffies et al. 2014.) Since our concern is with salinity changes, we take 𝛽 to be constant in the following. Computing the Change in Total Volume The change in total volume upon homogenization is the sum of the changes in the two buckets, 𝛿𝑉=𝛿𝑉1+𝛿𝑉2. (60) Since the mass of water in the two buckets remains the same before and after homogenization, the volume in the two buckets is altered only due to changes in their respective densities 𝛿𝜌𝑛=𝛿(𝑀𝑛/𝑉𝑛)=−(𝜌𝑛/𝑉𝑛)𝛿𝑉𝑛⇒𝛿𝜌𝑛/𝜌𝑛=−𝛿𝑉𝑛/𝑉𝑛, (61) i.e., the relative density increases as the relative volume decreases. Because the buckets have the same temperature in our experiment, relative density changes occur only through salinity changes, according to 𝛿𝜌𝑛/𝜌𝑛=𝛽𝛿𝑆𝑛=−𝛿𝑉𝑛/𝑉𝑛, (62) in which case the change in total volume is 𝛿𝑉=𝛿𝑉1+𝛿𝑉2=−(𝑉1𝛽𝛿𝑆1+𝑉2𝛽𝛿𝑆2). (63) Mass conservation for salt means that 𝛿(𝑀𝑆)=𝛿(𝑀1𝑆1)+𝛿(𝑀2𝑆2)=0. (64) Furthermore, since mass in the two buckets is unchanged, salt conservation leads to 𝑀1𝛿𝑆1+𝑀2𝛿𝑆2=0, (65) so that salinity in one bucket rises while that in the other falls. Making use of this result in Eq. (63) leads to our desired expression for total volume change 𝛿𝑉=−𝛽𝛿𝑆1(𝑉1−𝑉2𝑀1/𝑀2)=−𝛽𝑉1𝛿𝑆1(1−𝜌1/𝜌2). (66) Connecting to Thickness Changes Equation (66) provides an expression for the change in total volume upon homogenizing two buckets of water with equal Conservative Temperatures, equal pressures, but with differing salinities. To connect to sea level, assume an equal cross-sectional area, A, for the buckets, so that the volume of water is given by 𝑉𝑛=𝐴ℎ𝑛, where ℎ𝑛 is the thickness of the water in the bucket. Equation (66) then says that upon homogenization, the thickness of water changes by 𝛿ℎ=−𝛽ℎ1𝛿𝑆1(1−𝜌1/𝜌2),(67) and that the total thickness of the homogenized water is given by ℎnew=ℎ1+ℎ2+𝛿ℎ=ℎ2+ℎ1[1−𝛽𝛿𝑆1(1−𝜌1/𝜌2)]. (68) As expected, we see that 𝛿ℎ=0 only when 𝛽=0 or 𝜌1=𝜌2. The first is never true, and the second is not true in the general case of differing temperatures (in which case there are thermosteric changes ignored in our discussion). An Ocean Example To explore the oceanic implications of Eq. (68), assume bucket-1 initially has freshwater with density 𝜌1=𝜌f, whereas bucket-2 initially has sea water with density 𝜌2=𝜌s=𝜌f+𝜌′. The salinity change for bucket-1 is 𝛿𝑆1=𝑆, since this bucket went from its original freshwater concentration to the homogenized sea water with salinity S. The halosteric-induced thickness change (Eq. 67) is thus given by 𝛿ℎ=−ℎ1𝛽𝑆(𝜌′/𝜌s)<0. (69) How large is this effect? For the case of an upper ocean with salt concentration 𝑆=0.035, sea-water density 𝜌s=1028 kg m −3⇒𝜌′=28 kg m −3 and 𝛽=0.8, we have 𝛿ℎ=−ℎ1×0.8×0.035×(28/1028)≈−ℎ1×7.6×10−4. (70) To within roughly 8 parts in 104, the change in thickness of the ocean column is nearly identical to the thickness of freshwater added to the ocean. For example, if we add one metre of freshwater into the upper ocean (ℎ1=1 m ), then the change in sea level is equal to one metre minus the tiny amount 0.76 mm . Hence, as emphasized by Lowe and Gregory (2006), we can generally ignore the contribution to global-mean sea level from global halosteric effects. Appendix 3: Bottom Pressure The hydrostatic pressure at the ocean sea floor is commonly referred to as the ocean bottom pressure 𝑝̃𝑏, usually calculated as 𝑝̃𝑏=𝑝̃𝑎+𝑔𝑚̃, (71)	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
14d9a88b20c2-18	An Ocean Example To explore the oceanic implications of Eq. (68), assume bucket-1 initially has freshwater with density 𝜌1=𝜌f, whereas bucket-2 initially has sea water with density 𝜌2=𝜌s=𝜌f+𝜌′. The salinity change for bucket-1 is 𝛿𝑆1=𝑆, since this bucket went from its original freshwater concentration to the homogenized sea water with salinity S. The halosteric-induced thickness change (Eq. 67) is thus given by 𝛿ℎ=−ℎ1𝛽𝑆(𝜌′/𝜌s)<0. (69) How large is this effect? For the case of an upper ocean with salt concentration 𝑆=0.035, sea-water density 𝜌s=1028 kg m −3⇒𝜌′=28 kg m −3 and 𝛽=0.8, we have 𝛿ℎ=−ℎ1×0.8×0.035×(28/1028)≈−ℎ1×7.6×10−4. (70) To within roughly 8 parts in 104, the change in thickness of the ocean column is nearly identical to the thickness of freshwater added to the ocean. For example, if we add one metre of freshwater into the upper ocean (ℎ1=1 m ), then the change in sea level is equal to one metre minus the tiny amount 0.76 mm . Hence, as emphasized by Lowe and Gregory (2006), we can generally ignore the contribution to global-mean sea level from global halosteric effects. Appendix 3: Bottom Pressure The hydrostatic pressure at the ocean sea floor is commonly referred to as the ocean bottom pressure 𝑝̃𝑏, usually calculated as 𝑝̃𝑏=𝑝̃𝑎+𝑔𝑚̃, (71) where the first term is the atmospheric pressure at the liquid-water equivalent sea surface and the second term is the weight of the mass per unit area 𝑚̃(𝐫) of sea water, given by Eq. (26). This formula makes two approximations. First, hydrostatic pressure does not exactly equal the weight per unit area of the fluid above it because of the curvature of the Earth (Ambaum 2008). Second, g should appear within the vertical integral of density to obtain m (Eq. 26), because g depends on z. However, these approximations are entirely adequate for sea-level studies. The difference in bottom pressure between two states Δ𝑝𝑏=Δ𝑝𝑎+𝑔Δ𝑚, (72) that is, the sum of the change in local atmospheric pressure and the change in the weight of the local ocean. Using Eqs. (26) and (25) to replace Δ𝑚, Δ𝑝𝑏=Δ𝑝𝑎+𝑔𝜌∗Δ𝐻+𝑔∫𝜂𝐹Δ𝜌(𝑧)d𝑧, (73) where 𝐻=𝜂−𝐹 is the time-mean thickness of the ocean (Eq. 3). This form separates the change in pressure due to sea water into one term (the second) due to the change in the local thickness of the ocean, and another (the third) which is proportional to the local vertical-mean change in sea-water density.	https://sealeveldocs.readthedocs.io/en/latest/gregory19.html
1a5e23fee427-0	Horton et al. (2018) Title: Mapping Sea-Level Change in Time, Space, and Probability Keywords: sea level, climate change, Holocene, Last Interglacial, Mid-Pliocene Warm Period, sea-level rise projections Corresponding author: Horton Citation: Horton, B. P., Kopp, R. E., Garner, A. J., Hay, C. C., Khan, N. S., Roy, K., & Shaw, T. A. (2018). Mapping Sea-Level Change in Time, Space, and Probability. Annual Review of Environment and Resources, 43(1), 481–521. doi:10.1146/annurev-environ-102017-025826 URL: https://www.annualreviews.org/doi/10.1146/annurev-environ-102017-025826 Abstract Future sea-level rise generates hazards for coastal populations, economies, infrastructure, and ecosystems around the world. The projection of future sea-level rise relies on an accurate understanding of the mechanisms driving its complex spatio-temporal evolution, which must be founded on an understanding of its history. We review the current methodologies and data sources used to reconstruct the history of sea-level change over geological (Pliocene, Last Interglacial, and Holocene) and instrumental (tide-gauge and satellite altimetry) eras, and the tools used to project the future spatial and temporal evolution of sea level. We summarize the understanding of the future evolution of sea level over the near (through 2050), medium (2100), and long (post-2100) terms. Using case studies from Singapore and New Jersey, we illustrate the ways in which current methodologies and data sources can constrain future projections, and how accurate projections can motivate the development of new sea-level research questions across relevant timescales. Introduction As recorded instrumentally and reconstructed from geological proxies, sea levels have risen and fallen throughout Earth’s history, on timescales ranging from minutes to millions of years. Sea-level projections depend on establishing a robust relationship between sea level and climate forcing, but the vast majority of instrumental records contain less than 60 years of data, which are from the late twentieth and early twenty-first centuries (1-3). This brief instrumental period captures only a single climate mode of rising temperatures and sea level within a baseline state that is climatically mild by geological standards. Complementing the instrumental records, geological proxies provide valuable archives of the sea-level response to past climate variability, including periods of more extreme global mean surface temperature (e.g., 4-7). Ultimately, information from the geological record can help assess the relationship between sea level and climate change, providing a firmer basis for projecting the future (8), but current ties between past changes and future projections are often vague and heuristic. Greater interconnections between the two sub-disciplines are key to major progress. The linked problems of characterizing past sea-level changes and projecting future sea-level rise face two fundamental challenges. First, regional and local sea-level changes vary substantially from the global mean (9). Understanding regional variability is critical to both interpreting records of past changes and generating local projections for effective coastal risk management (e.g., 9, 10). Second, uncertainty is pervasive in both records of past changes and in the physical and statistical modeling approaches used to project future changes (e.g., 11), and it requires careful quantification and statistical analysis (12). Quantification of uncertainty becomes particularly important for decision analysis related to future projections (e.g., 13). Here, we review the mechanisms that drive spatial variability, as well as their contributions to the uncertainty in mapping sea level on different timescales. We describe methodologies and data sources for piecing together lines of evidence related to past and future sea level to map changes in space, time, and probability. We review the sources and statistical methods applied to proxy [Pliocene, Last Interglacial (LIG) and Holocene] and instrumental (tide gauges and satellites) data, and the statistical and physical modeling approaches used to project future sea-level changes. Finally, we highlight two case study regions - Singapore and New Jersey - to illustrate the way in which proxy and instrumental data can improve future projections, and conversely how future projections can guide the development of new sea-level research questions to further constrain projections. Mechanisms for global, regional, and local relative sea-level changes Relative sea level (RSL) is defined as the difference in elevation between the sea surface and the land. Global mean sea level (GMSL) is defined as the areal mean of either RSL or sea-surface height over the global ocean. It is often approximated by taking various forms of weighted means of individual RSL records, sometimes with corrections for specific local processes. Over the twentieth century, GMSL trends (14) were dominated by increases in ocean mass due to melting of land-based glaciers (e.g., 15) and ice sheets (e.g., 16), and by thermal expansion of warming ocean water. Changes in land water storage due to dam construction and groundwater withdrawal also made a small contribution (e.g., 17). Over a variety of timescales, RSL differs from GMSL, because of key driving processes such as atmosphere/ocean dynamics, the static-equilibrium effects of ocean/cryosphere/hydrosphere mass redistribution on the height of the geoid and the EarthÕs surface, glacio-isostatic adjustment (GIA), sediment compaction, tectonics, and mantle dynamic topography (MDT). The driving processes are spatially variable and cause RSL change to vary in rate and magnitude among regions (Figure 1). Figure 1: Mapping uncertainty of sea-level drivers on different timescales based on available estimates. The length of colored bars along the x-axis represents the characteristic timescale over which a process may occur, rather than the total time duration over which the process has been active. The color scale represents the range in magnitude of relative sea-level change driven by a process over an event or observed/predicted timescale. It does not imply a specific relationship of the change in amplitude with timescale, given the nonlinear nature of many of these processes. The color scheme for glacial eustasy is also scaled to encompass predicted changes in global mean sea level of decimeters in the next several decades to meters over the next several centuries. (b) The uncertainty of instrumental and proxy recorders of sea level. The x (age) axis represents the time span over which the proxy may be used (given the temporal range of the dating method used to determine its age), rather than the proxy’s temporal uncertainty. To estimate the contribution of a given process, the vertical and temporal resolution of a chosen instrument or proxy cannot exceed the magnitude and rate of sea-level change driven by that process. Atmosphere/ocean dynamics are the dominant driver of spatial heterogeneity in RSL on annual and multidecadal timescales (18-21), as well as a significant driver on longer timescales during periods with limited land-ice changes, such as the Common Era (22-25). The highest rates of RSL rise over the past two decades (greater than 15 mm/year) have occurred in the western tropical PaciÞc (18, 26), although the pattern appears to have reversed since 2011 (27). Observations and numerical model simulations (18, 28) conÞrm that the intensification of trade winds, which occurs when the PaciÞc Decadal Oscillation (PDO) exhibits a negative trend, accounts for the amplitude and spatial pattern of RSL rise in the western tropical PaciÞc. In the western North Atlantic Ocean, changes in the strength and/or position of the Gulf Stream impact RSL trends differently north and south of North Carolina, where the Gulf Stream separates from the US Atlantic coast and turns toward northern Europe (19, 22, 23, 29). In fact, there is a .30-cm difference in sea-surface height between New Jersey and North Carolina (29). Climate models project that by the late twenty-Þrst century, associated with a decline in the Atlantic Meridional Overturning Circulation (AMOC), ocean dynamic sea-level rise of up to 0.2 to 0.3 m could occur along the western boundary of the North Atlantic (30). However, coastal ocean dynamic variability in the western North Atlantic has been largely driven over the past few decades by local winds, with limited evidence for coupling to AMOC strength (21, 31).	https://sealeveldocs.readthedocs.io/en/latest/horton18.html
1a5e23fee427-1	Atmosphere/ocean dynamics are the dominant driver of spatial heterogeneity in RSL on annual and multidecadal timescales (18-21), as well as a significant driver on longer timescales during periods with limited land-ice changes, such as the Common Era (22-25). The highest rates of RSL rise over the past two decades (greater than 15 mm/year) have occurred in the western tropical PaciÞc (18, 26), although the pattern appears to have reversed since 2011 (27). Observations and numerical model simulations (18, 28) conÞrm that the intensification of trade winds, which occurs when the PaciÞc Decadal Oscillation (PDO) exhibits a negative trend, accounts for the amplitude and spatial pattern of RSL rise in the western tropical PaciÞc. In the western North Atlantic Ocean, changes in the strength and/or position of the Gulf Stream impact RSL trends differently north and south of North Carolina, where the Gulf Stream separates from the US Atlantic coast and turns toward northern Europe (19, 22, 23, 29). In fact, there is a .30-cm difference in sea-surface height between New Jersey and North Carolina (29). Climate models project that by the late twenty-Þrst century, associated with a decline in the Atlantic Meridional Overturning Circulation (AMOC), ocean dynamic sea-level rise of up to 0.2 to 0.3 m could occur along the western boundary of the North Atlantic (30). However, coastal ocean dynamic variability in the western North Atlantic has been largely driven over the past few decades by local winds, with limited evidence for coupling to AMOC strength (21, 31). Gravitational, rotational, and elastic deformational effects - also called static-equilibrium effects - reshape sea level nearly instantaneously in response to the redistribution of mass be.tween the cryosphere, the ocean, and the terrestrial hydrosphere (32-35). These effects are linked to the change in self-gravitation of the ice sheets and liquid water, the response of the Earth’s rotational vector to the redistribution of mass at the Earth’s surface, and the elastic response of the solid Earth surface to changing surface loads (Figure 2b,c). Unique RSL change geometries, sometimes called “fingerprints,” can be associated with the melting of different ice sheets and glaciers, and this response scales linearly with the magnitude of a marginal ice-mass change (32, 34). The dominant self-gravitation signal will result in a RSL fall near a shrinking land ice mass, which will be compensated by a RSL rise in the far Þeld that will be greater than the GMSL signal expected from the water mass inßux. The exact spatial pattern of RSL change depends on the geometry of the melting undergone by the ice reservoir. Recent studies (36, 37) have examined how mass loss centered in different portions of an ice sheet or glacial region will affect RSL differently. For example, New Jersey experiences a RSL fall in response to mass loss in southern.most Greenland, even though it experiences a modest (approximately 50% of the global mean) RSL rise in response to uniform melting across Greenland (Figure 2b). Figure 2: (a) Dynamic sea-level contribution to sea surface height (millimeter/year) from 2006-2100 under the RCP8.5 experiment of the Community Earth System Model, as archived by Coupled Model Intercomparison Project Phase 5 (30). Elastic fingerprints of projections of (b) Greenland ice-sheet mass loss and (c) West Antarctic ice-sheet mass loss, presented as ratios of RSL change to GMSL change (140). (d) Contribution of glacio-isostatic adjustment (GIA) to present-day relative sea level (RSL) change (millimeter/year), calculated using the ICE-5G ice loading history (52), combined with a maximum-likelihood solid-Earth model identified through a Kalman smoother tide-gauge analysis (119). ModiÞed from Kopp et al. (24). Pliocene: epoch in the geologic timescale that extends from 5.3 million to 1.8 million years ago, during which the Earth experienced a transition from relatively warm climates to the prevailing cooler climates of the Pleistocene; includes the Mid-Pliocene Warm Period (.3.2 to 3.0 million years ago), which is the most recent period in geologic time with temperatures comparable to those projected for the twenty-first century. Last Interglacial: the interglacial stage prior to the current Holocene interglacial (an interglacial is a geological interval of warmer global average temperature, characterized by the absence of large ice sheets in North America and Europe); the Last Interglacial extends from approximately 129,000 to 116,000 years ago, corresponds to Marine Isotope Stage5e and is also known as the Eemian. Holocene: current geological epoch, beginning approximately 11,650 years ago, after the last glacial period; the start of the Holocene is formally defined by chemical (delta^{18}O) shifts in an ice core from northern Greenland that reflect climate warming. Relative sea level (RSL): height of the sea surface at a specific location, measured with respect to the height of the surface of the solid Earth. Dynamic sea level: sea-surface height variations produced by oceanic and atmospheric circulation and by temperature and salinity distributions. Global mean sea level (GMSL): areal average height of relative sea level (or, in some uses, sea-surface height) over the Earth’s oceans combined; influenced primarily by the volume of seawater and the size and shape of the ocean basins; in the geological literature, GMSL is classically referred to as “eustatic sea level”. Static-equilibrium effects: gravitational, elastic, and rotational effects of mass redistribution at the Earth surface, which lead to changes in both sea-surface height and the height of the solid Earth; these combined effects give rise to what are known as “sea-level fingerprints,” or the geographic pattern of sea-level change following the rapid melting of ice sheets and glaciers. Over longer timescales, GIA arises as the viscoelastic mantle responds to the transfer of mass between land-based ice sheets and the global ocean during a glacial cycle. GIA induces deformation of the solid Earth, as well as changes in the EarthÕs gravitational Þeld and its rotational state (38-43). After a change in surface load, the elastic component of the deformation is recovered nearly instantaneously, but the viscoelastic properties of the underlying mantle determine the characteristics of the recovery over longer timescales. In general, this recovery takes place over thousands of years, although in localized regions underlain by low-viscosity mantle, it can take place over decades to centuries (e.g., 44). On a global scale, at least in the Quaternary period, the system does not reach isostatic equilibrium, because it is interrupted by the initiation of another glacial cycle. The deformation that is observed today is the overprint of a series of glacial cycles that extend from the Pliocene through the Pleistocene glacial cycles and into the Holocene (7, 45, 46). GIA models simulate the evolution of the solid Earth as a function of the rheological structure and ice-sheet history (42, 47, 48). During the glacial phase of a glaciation-deglaciation cycle, the depression of land beneath ice sheets causes a migration of mantle material away from ice-load centers. This migration results in the formation of a forebulge in regions adjacent to ice sheets (e.g., the mid-Atlantic coast of the United States). Following the ice-sheet retreat, mantle material flows toward the former load centers. These centers experience postglacial rebound, while the forebulge retreats and collapses. In regions located beneath the centers of Last Glacial Maximum ice sheets (e.g., Northern Canada and Fennoscandia), postglacial uplift has resulted in RSL records characterized by a continuous fall; rates of present-day uplift greater than 10 mm/year occur in these near-Þeld locations (42). In the former forebulges, land is subsiding at a rate that varies with distance from the former ice centers. Along the US Atlantic coast, rates of present-day subsidence reach a maximum amplitude of close to 2 mm/year (49). In regions distal from the former ice sheet, the GIA signal is much smaller (50). These regions are characterized by present-day GIA-induced rates of RSL change that are near constant or show a slight fall (<0.3 mm/year), due to hydro-isostatic loading (continental levering) and to a global fall in the ocean surface linked to the hydro-and glacio-isostatic loading of the bottom of the Earth’s ocean basins (equatorial ocean syphoning) (Figure 2d, 51).	https://sealeveldocs.readthedocs.io/en/latest/horton18.html
1a5e23fee427-2	GIA models simulate the evolution of the solid Earth as a function of the rheological structure and ice-sheet history (42, 47, 48). During the glacial phase of a glaciation-deglaciation cycle, the depression of land beneath ice sheets causes a migration of mantle material away from ice-load centers. This migration results in the formation of a forebulge in regions adjacent to ice sheets (e.g., the mid-Atlantic coast of the United States). Following the ice-sheet retreat, mantle material flows toward the former load centers. These centers experience postglacial rebound, while the forebulge retreats and collapses. In regions located beneath the centers of Last Glacial Maximum ice sheets (e.g., Northern Canada and Fennoscandia), postglacial uplift has resulted in RSL records characterized by a continuous fall; rates of present-day uplift greater than 10 mm/year occur in these near-Þeld locations (42). In the former forebulges, land is subsiding at a rate that varies with distance from the former ice centers. Along the US Atlantic coast, rates of present-day subsidence reach a maximum amplitude of close to 2 mm/year (49). In regions distal from the former ice sheet, the GIA signal is much smaller (50). These regions are characterized by present-day GIA-induced rates of RSL change that are near constant or show a slight fall (<0.3 mm/year), due to hydro-isostatic loading (continental levering) and to a global fall in the ocean surface linked to the hydro-and glacio-isostatic loading of the bottom of the Earth’s ocean basins (equatorial ocean syphoning) (Figure 2d, 51). Global models of the GIA process have contributed to our knowledge of solid Earth geodynamics through the constraints they provide on the effective viscosity of the mantle. They also provide a means to constrain the evolution of land-based ice sheets and ocean bathymetry over a glacial cycle (e.g., 42, 52), which can in turn be used to provide boundary conditions for tests of global climate models under paleoclimate conditions (e.g., 53). Global GIA models have traditionally relied on a simple Maxwell representation of the EarthÕs rheology and on spherically symmetric models of the EarthÕs mantle viscosity. Research is currently underway to include complex rheologies and lateral heterogeneity in mantle viscosity in GIA models, features that may be of substantial importance in regions with a complex geological structure, such as West Antarctica (e.g., 44). Locally, RSL can also change in response to sediment compaction driven by natural processes and by anthropogenic groundwater and hydrocarbon withdrawal (e.g., 54, 55). Many coastlines are located on plains composed of unconsolidated or loosely consolidated sediments, which com.pact under their own weight as the pressure of overlying sediments leads to a reduction in pore space (54, 56). Sediment compaction can occur over a range of depths and timescales. In the Mississippi Delta, Late Holocene subsidence due to shallow compaction has been estimated to be as high as 5 mm/year (57). Anthropogenic groundwater or hydrocarbon withdrawal can accelerate sediment compaction. For example, from 1958-2006 CE, subsidence in the Mississippi Delta was 7.6 ± 0.2 mm/year, with a peak rate of 9.8 ± 0.3 mm/year prior to 1991 that corresponded to the period of maximum oil extraction (58). Many other deltaic regions, including the Ganges, Chao Phraya, and Pasig Deltas, experience high rates of subsidence, linked to a combination of natural and anthropogenic sediment compaction, and they exhibit some of the highest rates of present-day RSL rise (59). Deeper, often poorly understood, processes also contribute to coastal subsidence, including thermal subsidence and fault motion (60). On some coastlines, deformation caused by tectonics can be an important driver of RSL change. Indeed, reconstructions of RSL can be used to estimate the presence and rate of vertical land motion caused by coastal tectonics at regional spatial scales (e.g., 61). Coastlines may have near-instantaneous or gradual rates of uplift or subsidence due to coseismic movement associated with earthquakes or longer-term post-or interseismic deformation. Geodetic measurements of the 2011 Tohoku-oki and the 2004 Indian Ocean megathrust earthquakes revealed several meters of near-instantaneous coseismic vertical land motion (e.g., 62), which were followed by postseismic recovery that quickly exceeded the amount of coseismic change (63, 64). Based on a collection of bedrock thermochronometry measurements (65), gradual rates of vertical land motion from tectonics vary from <0.01 mm/year to 10 mm/year. Stable, cratonic regions (e.g., central and western Australia, central North America, and eastern Scandinavia) exhibit negligible vertical land motion rates of <0.01 mm/year. Rates are higher (0.01-0.1 mm/year) along passive margins (e.g., southeastern Australia, Brazil, and the US Atlantic coast) and the highest vertical land motion rates (1- 10 mm/year) are found in several tectonically active mountainous areas (e.g., the Coastal Mountains of British Columbia, Papua New Guinea, and the Himalayas). Most of these rate estimates are integrated over several millions to tens of millions of years [and may include influence from other low-frequency signals such as MDT or karstification (66)], and therefore have insufficient resolution to reveal temporal variations on shorter timescales (65). A common approach to calculate long-term tectonic vertical land motion uses LIG shore.lines, often inappropriately assuming that, once tectonically corrected, the elevation of GMSL at the LIG was .5 m above present. However, when calculating long-term tectonics from LIG shorelines, uncertainty in LIG GMSL and departures from GMSL due to GIA in response to glacial-interglacial cycles and excess polar ice-sheet melt relative to present-day values must be considered (67, 68). Instrumental observations (e.g., global positioning system, interferometric synthetic-aperture radar) of vertical land motion can provide the resolution to decipher temporal variations in rates, although the observation period is too short to capture the full period over which these processes operate. MDT refers to the surface undulations induced by mantle ßow (69, 70). One of the most important consequences of MDT studies is their influence on estimates of long-term GMSL and RSL change (69). Several geophysical approaches have been developed to model global and regional MDT. Husson & Conrad (71) proposed that the dynamic effect of longer-term (~10^8 years) change in tectonic velocities on GMSL could be up to ~80 m from a model based on boundary layer theory. Conrad & Husson (72) used a forward model of mantle flow based on the present-day mantle structure and plate motions to estimate that the current rate of GMSL rise induced by MDT is <1 m/million years. An increasing body of evidence suggests that MDT can contribute to regional RSL change at rates of >1 m/100 kyr (e.g., 73, 74). The uncertainties due to MDT in RSL reconstructions become increasingly large further back in time (73, 75). Observationally, MDT is essentially indistinguishable from long-term tectonic change. Glacial isostatic adjustment (GIA): response of the solid Earth to mass redistribution during a glacial cycle; isostasy refers to a concept whereby deformation takes place in an attempt to return the Earth to a state of equilibrium; GIA refers to isostatic deformation related to ice and water loading during a glacial cycle Sediment compaction: reduction in the volume of sediments caused by a decrease in pore space, which has the effect of lowering the height of the solid Earth surface; can occur naturally or due to the anthropogenic extraction of fluids (such as water and fossil fuels) from the pore space. Mantle dynamic topography (MDT): differences in the height of the surface of the solid Earth caused by density-driven flow within the Earth’s mantle. Past and current observations of sea-level change Reconstructions of Relative Sea Levels from Proxy Data Geological reconstructions of RSL are derived from sea-level proxies, the formation of which was controlled by the past position of sea level (76). Sea-level proxies, which have a systematic and qunatifiable relationship with contemporary tides (77), include sedimentary, geomorphic, archeological, and fixed biological indicators, as well as coral reefs, coral microatolls, salt-marsh flora, and salt-marsh fauna (Figure 1b). The relationship of a proxy to sea level is deÞned by its “indicative meaning,” which describes the central tendency (reference water level) and vertical range (indicative range) of its relationship with tidal level(s).	https://sealeveldocs.readthedocs.io/en/latest/horton18.html

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