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ows
were rst detected using direct Doppler measurements (Sheeley 1972) and also
can be inferred from the motion of small magnetic features (e.g. Brickhouse &
Labonte 1988).
Local helioseismology is a useful tool for studying
ows around sunspots (Du-
vall et al. 1996, Lindsey et al. 1996). Gizon, Duvall & Larsen (2000) used f-mode
time-distance helioseismology to study the moat
ows in the two Mm below the
surface. Comparisons between the near-surface
ows inferred from local helioseis-
mology with direct Doppler measurements have demonstrated the validity of the
near-surface inversions. Figure 15 shows the moat
ow at a depth of 1 Mm using
a time-distance inversion and all ridges from f through p 4(Gizon et al. 2009). At
this depth, the moat
ow has an amplitude of 250 m s1, which is consistent
with the motion of the small magnetic features. This out
ow is detected in the
top 4 Mm. Such measurements of the subsurface moat
ow have been conrmed
by ring-diagram analysis (Moradi et al. 2009, submitted).
The moat
ow is believed to be driven by a pressure gradient caused by the
blockage of heat transport by sunspots (e.g. Nye, Bruning & Labonte 1988). For a
more accurate description in terms of the surface cooling that causes the observed

ows, see Spruit (1997). Though much slower, the moat
ow appears physically
connected with the Evershed
ow in the penumbra. This is evident both from
the observations by, e.g., Cabrera Solana et al. (2006) and the realistic numerical
simulations of Heinemann et al. (2007) and Rempel et al. (2009).
6.3 Absorption of Solar Oscillations
The rst major discovery made by local helioseismology was that solar oscillations
(f and p modes) are absorbed by sunspots (Braun, Duvall & Labonte 1987).
This discovery was made using Fourier-Hankel analysis, which is based on the
decomposition of the waveeld around the sunspot into ingoing and outgoing
waves. Braun, Duvall & Labonte (1987, 1988) found that typical sunspots can28 Gizon, Birch & Spruit
absorb up to 50% of the incoming power.
Spruit & Bogdan (1992) proposed that mode conversion between the oscilla-
tions of the quiet Sun and the slow magneto-acoustic waves of the sunspot could
explain the observations of wave absorption by sunspots. Theoretical modeling
(Cally & Bogdan 1993, Crouch et al. 2005) and numerical simulations (Cally 2000;
Cally & Bogdan 1997; Cameron, Gizon & Duvall 2008) demonstrated that mode
conversion is indeed capable of removing a large fraction of the energy from the
helioseismic waves incident on a sunspot. The eciency of the mode conversion
is strongly dependent on the angle of the magnetic eld from the vertical, with a
maximum in the absorption occurring at an angle of about 30from the vertical
(Cally, Crouch & Braun 2003). Comparisons between observations and models
(e.g. Cameron, Gizon & Duvall 2008; Crouch et al. 2005) show that the expla-
nation of mode conversion is consistent with the observations of the reduction
of outgoing wave power. We note that plage regions are also known to 'absorb'
incoming waves.
6.4 Phase Shifts and Wave-Speed Perturbations
Fourier-Hankel analysis (Section 3.6) has demonstrated that outgoing waves from
sunspots have dierent phases than the corresponding ingoing waves (Braun 1995,
Braun et al. 1992). At xed radial order, the phase shifts increase roughly linearly
with increasing angular degree. Fan, Braun & Chou (1995) used a simple model,
in which the sunspot is treated as a local enhancement in the sound-speed, to
suggest that these phase shifts are indicative of a near-surface change in the
wave speed relative to quiet Sun. Crouch et al. (2005) showed that approximate
models of wave propagation in a model sunspot (embedded magnetic cylinders)
could simultaneously explain the absorption and phase shift measurements. The
success of these simple models suggests that the interaction of solar oscillations
with the sunspot magnetic elds may be the essential physics in understanding
both wave absorption (Section 6.3) and the phase shifts caused by sunspots.
Time-distance, holography, and ring-diagrams have all been used to infer changes
in the wave speed in sunspots (e.g. Basu, Antia & Bogart 2004; Couvidat, Birch
& Kosovichev 2006; Jensen et al. 2001; Kosovichev 1996; Kosovichev, Duvall &
Scherrer 2000, among a great many others). Interpretation of the helioseismic
measurements is a rapidly developing topic of current research. Figure 16 shows
a comparison between time-distance and ring-analysis inversions, forward mod-
els based on Fourier-Hankel analysis, and a realistic numerical simulation of a
sunspot. As was shown by Gizon et al. (2009) and Moradi et al. (2009, submit-
ted), there is not yet agreement among the dierent analysis methods, especially
between the time-distance and ring-diagram results. There are a number of pos-
sible explanations for this disagreement. Current inversions for the time-distance
and ring-diagram methods use sensitivity functions that do not explicitly includeLocal Helioseismology 29
the direct eects of the magnetic eld and both assume that wave-speed pertur-
bations are small. The time-distance sensitivity functions may not model the
reference power spectrum suciently accurately (convective background, mode
frequencies, relative mode amplitudes, line widths and asymmetries). Neither
method fully accounts for the details of the measurements procedure, especially
in the case of time-distance where the eects of the data analysis ltering in
Fourier space (e.g., phase-speed lters) are not fully accounted for. Except for
the time-distance inversion, all other methods are consistent with an increased
wave speed in the top 2 Mm and show wave-speed perturbations with amplitudes
less than about 2% at greater depths.
Direct simulation of wave propagation through sunspot models is useful to test
the validity of these models. Cameron, Gizon & Duvall (2008) used a three-
dimensional MHD code to compute the propagation of f modes through a model
sunspot. Here we show a computation for the propagation of a p 1wave packet
using the same simulation code. The simulated waveeld, solution to an initial
value problem, is compared to the observed cross-covariance in Figure 17 in
order to assess the validity of the underlying sunspot model. Like for the f
modes, the comparison with the observations is promising: the amplitude of the
transmitted waves is reduced and the waves travel faster in the sunspot than in
quiet Sun.
As mentioned in Section 4.2, sunspot magnetic elds strongly aect the nature

ows

were rst detected using direct Doppler measurements (Sheeley 1972) and also

can be inferred from the motion of small magnetic features (e.g. Brickhouse &

Labonte 1988).

Local helioseismology is a useful tool for studying

ows around sunspots (Du-

vall et al. 1996, Lindsey et al. 1996). Gizon, Duvall & Larsen (2000) used f-mode

time-distance helioseismology to study the moat

ows in the two Mm below the

surface. Comparisons between the near-surface

ows inferred from local helioseis-

mology with direct Doppler measurements have demonstrated the validity of the

near-surface inversions. Figure 15 shows the moat

ow at a depth of 1 Mm using

a time-distance inversion and all ridges from f through p 4(Gizon et al. 2009). At

this depth, the moat

ow has an amplitude of 250 m s1, which is consistent

with the motion of the small magnetic features. This out

ow is detected in the

top 4 Mm. Such measurements of the subsurface moat

ow have been conrmed

by ring-diagram analysis (Moradi et al. 2009, submitted).

The moat

ow is believed to be driven by a pressure gradient caused by the

blockage of heat transport by sunspots (e.g. Nye, Bruning & Labonte 1988). For a

more accurate description in terms of the surface cooling that causes the observed

ows, see Spruit (1997). Though much slower, the moat

ow appears physically

connected with the Evershed

ow in the penumbra. This is evident both from

the observations by, e.g., Cabrera Solana et al. (2006) and the realistic numerical

simulations of Heinemann et al. (2007) and Rempel et al. (2009).

6.3 Absorption of Solar Oscillations

The rst major discovery made by local helioseismology was that solar oscillations

(f and p modes) are absorbed by sunspots (Braun, Duvall & Labonte 1987).

This discovery was made using Fourier-Hankel analysis, which is based on the

decomposition of the waveeld around the sunspot into ingoing and outgoing

waves. Braun, Duvall & Labonte (1987, 1988) found that typical sunspots can28 Gizon, Birch & Spruit

absorb up to 50% of the incoming power.

Spruit & Bogdan (1992) proposed that mode conversion between the oscilla-

tions of the quiet Sun and the slow magneto-acoustic waves of the sunspot could

explain the observations of wave absorption by sunspots. Theoretical modeling

(Cally & Bogdan 1993, Crouch et al. 2005) and numerical simulations (Cally 2000;

Cally & Bogdan 1997; Cameron, Gizon & Duvall 2008) demonstrated that mode

conversion is indeed capable of removing a large fraction of the energy from the

helioseismic waves incident on a sunspot. The eciency of the mode conversion

is strongly dependent on the angle of the magnetic eld from the vertical, with a

maximum in the absorption occurring at an angle of about 30from the vertical

(Cally, Crouch & Braun 2003). Comparisons between observations and models

(e.g. Cameron, Gizon & Duvall 2008; Crouch et al. 2005) show that the expla-

nation of mode conversion is consistent with the observations of the reduction

of outgoing wave power. We note that plage regions are also known to 'absorb'

incoming waves.

6.4 Phase Shifts and Wave-Speed Perturbations

Fourier-Hankel analysis (Section 3.6) has demonstrated that outgoing waves from

sunspots have dierent phases than the corresponding ingoing waves (Braun 1995,

Braun et al. 1992). At xed radial order, the phase shifts increase roughly linearly

with increasing angular degree. Fan, Braun & Chou (1995) used a simple model,

in which the sunspot is treated as a local enhancement in the sound-speed, to

suggest that these phase shifts are indicative of a near-surface change in the

wave speed relative to quiet Sun. Crouch et al. (2005) showed that approximate

models of wave propagation in a model sunspot (embedded magnetic cylinders)

could simultaneously explain the absorption and phase shift measurements. The

success of these simple models suggests that the interaction of solar oscillations

with the sunspot magnetic elds may be the essential physics in understanding

both wave absorption (Section 6.3) and the phase shifts caused by sunspots.

Time-distance, holography, and ring-diagrams have all been used to infer changes

in the wave speed in sunspots (e.g. Basu, Antia & Bogart 2004; Couvidat, Birch

& Kosovichev 2006; Jensen et al. 2001; Kosovichev 1996; Kosovichev, Duvall &

Scherrer 2000, among a great many others). Interpretation of the helioseismic

measurements is a rapidly developing topic of current research. Figure 16 shows

a comparison between time-distance and ring-analysis inversions, forward mod-

els based on Fourier-Hankel analysis, and a realistic numerical simulation of a

sunspot. As was shown by Gizon et al. (2009) and Moradi et al. (2009, submit-

ted), there is not yet agreement among the dierent analysis methods, especially

between the time-distance and ring-diagram results. There are a number of pos-

sible explanations for this disagreement. Current inversions for the time-distance

and ring-diagram methods use sensitivity functions that do not explicitly includeLocal Helioseismology 29

the direct eects of the magnetic eld and both assume that wave-speed pertur-

bations are small. The time-distance sensitivity functions may not model the

reference power spectrum suciently accurately (convective background, mode

frequencies, relative mode amplitudes, line widths and asymmetries). Neither

method fully accounts for the details of the measurements procedure, especially

in the case of time-distance where the eects of the data analysis ltering in

Fourier space (e.g., phase-speed lters) are not fully accounted for. Except for

the time-distance inversion, all other methods are consistent with an increased

wave speed in the top 2 Mm and show wave-speed perturbations with amplitudes

less than about 2% at greater depths.

Direct simulation of wave propagation through sunspot models is useful to test

the validity of these models. Cameron, Gizon & Duvall (2008) used a three-

dimensional MHD code to compute the propagation of f modes through a model

sunspot. Here we show a computation for the propagation of a p 1wave packet

using the same simulation code. The simulated waveeld, solution to an initial

value problem, is compared to the observed cross-covariance in Figure 17 in

order to assess the validity of the underlying sunspot model. Like for the f

modes, the comparison with the observations is promising: the amplitude of the

transmitted waves is reduced and the waves travel faster in the sunspot than in

quiet Sun.

As mentioned in Section 4.2, sunspot magnetic elds strongly aect the nature