alx-ai/arxiv_papers · Datasets at Hugging Face

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Movie 1 shows a time series of full-disk Dopplergrams. In high-resolution mode,
the resolution is better by a factor of three but the eld of view is reduced.
Because of limited telemetry, the full-disk Dopplergrams have only been trans-
mitted at full cadence for about two to three months each year since 1996, while
the high-resolution Dopplergrams are reserved for targeted campaigns of obser-
vation. The rest of the time, the Dopplergrams are spatially ltered onboard
and converted into lower-resolution 256 256 images, in order to save telemetry
('medium-degree' data).
GONG and MDI are complemented by other data sets, e.g. from the Taiwan
Oscillation Network (TON, Chou et al. 1995), from campaigns of observations at
the South Pole with the Magneto-Optical lter at Two Heights (MOTH, Finsterle
et al. 2004a), and from the Hinode satellite (e.g. Mitra-Kraev, Kosovichev & Sekii
2008).
In many applications of local helioseismology, the standard procedure consists
of choosing a relatively small region of the Sun and following (or `tracking') it
in a frame that is co-rotating with the Sun. This gives a time series of Doppler
images that are centered on the region of interest, like a magnetic active region.
In this process each individual image is mapped onto a common spatial grid.
For local studies it is often convenient to neglect the curvature of the solar
surface and work in plane-parallel geometry. With this simplication, it is nat-
ural to study the oscillations in three-dimensional Fourier space. The oscillation
signal, denoted by (r;t), where r= (x;y) is the horizontal position vector and8 Gizon, Birch & Spruit
Figure 2: SOHO/MDI observations on 22 January 2008 at 17:36:00 remapped
using Postel's azimuthal equidistant projection with a map scale of 0 :12 deg/pixel
or 1:46 Mm/pixel. The sunspot in Active Region NOAA 9787 is at the center of
projection. ( a) 512512 pixel subeld of the continuum intensity, normalized to
unity at disk center (plus sign). The box around the sunspot has size 147 Mm
73 Mm. ( b) Line-of-sight component of the magnetic eld in kG (truncated gray
scale). ( c) Line-of-sight Doppler velocity in km s1. Supergranulation is visible
toward the edges of the frame. ( d) Doppler velocity in the sunspot box. The
black contours give the outer edges of the umbra and penumbra of the sunspot.
The center-to-disk component of the Evershed out
ow is visible in the penumbra.
(e) Doppler velocity as a function of time at the two locations denoted by the
crosses in panel d. The ve minute period of the solar oscillations is evident. The
oscillations have reduced amplitudes in the sunspot.
tis time, is decomposed into harmonic components
(k;!) =Z
Ad2rZT
0dt(r;t)eikr+i!t; (1)
whereAis the area of study, Tis the total observation time, the vector k=
(kx;ky) is the horizontal wavevector, and !is the angular frequency. The hori-Local Helioseismology 9
zontal wavenumber is k=kkk. By convention, the xcoordinate is positive in the
direction of rotation (prograde) and the ycoordinate points north. The power
spectrum of solar oscillations is dened as
P(k;!) =j(k;!)j2: (2)
We note that the spatial Fourier transform should be replaced by a spherical
harmonic transform when curvature eects cannot be ignored, as in global helio-
seismology.
An example power spectrum of solar oscillations is shown in Figure 3 . Power is
distributed along well-dened discrete ridges in wavenumber-frequency space and
peaks around 3 mHz. The rst ridge at low frequencies shows the \fundamental"
(f) modes. These are surface gravity waves with exponential eigenfunctions and
a dispersion relation !2=gk, whereg= 274 m s2is the acceleration of gravity
at the solar surface; they are similar to waves at the surface of a deep ocean.
All other ridges correspond to pressure (p) modes, i.e. acoustic waves modied
by gravity. The existence of discrete ridges, !=!n(k) withn >0, re
ects the
fact that p modes are trapped in the vertical direction. At xed wavenumber,
the peaks of power are labeled p 0, p1, p2, etc. with increasing frequency. A
mode pnis such that the number of radial nodes of the mode displacement is
n(the radial order). By convention, the f modes are labeled with n= 0. All
ridges have reduced power above 5 :3 mHz, which is the cut-o frequency above
which waves are not re
ected back into the Sun but escape into the atmosphere.
The frequency width of a ridge is inversely proportional to the mode lifetime. A
recent description of the mode parameters, including mode lifetimes, is provided
by Korzennik, Rabello-Soares & Schou (2004).
2.2 Modes
In order to better understand the diagnostic capability of each mode, it is useful
to consider simple solar models. For our purpose, a simple solar model is a
reference standard solar model, which only depends on height (or radius), such
as Model S (Christensen-Dalsgaard et al. 1996). In plane-parallel models that
are isotropic and translation invariant in the horizontal directions, the normal
modes of the oscillations of the model vary horizontally as exp( ikr). For the
case of p and f modes, it is convenient to introduce the mode eigenfunctions
Un(z;k) andVn(z;k) such that the complex displacement eigenfunction of the
mode characterized by radial order nand horizontal wavevector kis
n(r;z;k) =h
^ zUn(z;k) +i^kVn(z;k)i
eikr; (3)
wherezis height, ^ zis the unit vector pointing upwards, and ^kis the horizontal
unit vector pointing in the direction of k. Zero height corresponds to the photo-10 Gizon, Birch & Spruit
Figure 3: Cut at ky= 0 through an average power spectrum of MDI/SOHO
high-resolution Doppler velocity data as a function of frequency and kxR. The
horizontal dashed line shows the acoustic cuto frequency. In order to reduce ran-
dom noise, an average was carried out over eight individual power spectra, each of
durationT= 4 hr and covering an apodized region of area A(500 Mm)2. Since
ky= 0, only waves traveling in the east or west directions are showed. The power
below1:5 mHz is due to solar convection, granulation and supergranulation.
sphere (zis depth). The representation of the displacement eigenfunctions in
terms of only the two functions UandVis possible as neither the f nor the p
modes have horizontal motions that are perpendicular to k. As a result of the
assumed isotropy, the functions UandVdo not depend on the direction of k
and the mode frequencies !n(k) only depend on wavenumber. The time evolution
of the mode ( n;k) is given by exp[ i!n(k)t]. In models that include attenua-
tion, the frequencies are complex, while they are real for the case of adiabatic
oscillations and standard boundary conditions.
Figure 4 shows the horizontal and vertical eigenfunctions corresponding to
the rst ve radial orders at a frequency of 3 :5 mHz. The solar model for this

Movie 1 shows a time series of full-disk Dopplergrams. In high-resolution mode,

the resolution is better by a factor of three but the eld of view is reduced.

Because of limited telemetry, the full-disk Dopplergrams have only been trans-

mitted at full cadence for about two to three months each year since 1996, while

the high-resolution Dopplergrams are reserved for targeted campaigns of obser-

vation. The rest of the time, the Dopplergrams are spatially ltered onboard

and converted into lower-resolution 256 256 images, in order to save telemetry

('medium-degree' data).

GONG and MDI are complemented by other data sets, e.g. from the Taiwan

Oscillation Network (TON, Chou et al. 1995), from campaigns of observations at

the South Pole with the Magneto-Optical lter at Two Heights (MOTH, Finsterle

et al. 2004a), and from the Hinode satellite (e.g. Mitra-Kraev, Kosovichev & Sekii

2008).

In many applications of local helioseismology, the standard procedure consists

of choosing a relatively small region of the Sun and following (or `tracking') it

in a frame that is co-rotating with the Sun. This gives a time series of Doppler

images that are centered on the region of interest, like a magnetic active region.

In this process each individual image is mapped onto a common spatial grid.

For local studies it is often convenient to neglect the curvature of the solar

surface and work in plane-parallel geometry. With this simplication, it is nat-

ural to study the oscillations in three-dimensional Fourier space. The oscillation

signal, denoted by (r;t), where r= (x;y) is the horizontal position vector and8 Gizon, Birch & Spruit

Figure 2: SOHO/MDI observations on 22 January 2008 at 17:36:00 remapped

using Postel's azimuthal equidistant projection with a map scale of 0 :12 deg/pixel

or 1:46 Mm/pixel. The sunspot in Active Region NOAA 9787 is at the center of

projection. ( a) 512512 pixel subeld of the continuum intensity, normalized to

unity at disk center (plus sign). The box around the sunspot has size 147 Mm

73 Mm. ( b) Line-of-sight component of the magnetic eld in kG (truncated gray

scale). ( c) Line-of-sight Doppler velocity in km s1. Supergranulation is visible

toward the edges of the frame. ( d) Doppler velocity in the sunspot box. The

black contours give the outer edges of the umbra and penumbra of the sunspot.

The center-to-disk component of the Evershed out

ow is visible in the penumbra.

(e) Doppler velocity as a function of time at the two locations denoted by the

crosses in panel d. The ve minute period of the solar oscillations is evident. The

oscillations have reduced amplitudes in the sunspot.

tis time, is decomposed into harmonic components

(k;!) =Z

Ad2rZT

0dt(r;t)eikr+i!t; (1)

whereAis the area of study, Tis the total observation time, the vector k=

(kx;ky) is the horizontal wavevector, and !is the angular frequency. The hori-Local Helioseismology 9

zontal wavenumber is k=kkk. By convention, the xcoordinate is positive in the

direction of rotation (prograde) and the ycoordinate points north. The power

spectrum of solar oscillations is dened as

P(k;!) =j(k;!)j2: (2)

We note that the spatial Fourier transform should be replaced by a spherical

harmonic transform when curvature eects cannot be ignored, as in global helio-

seismology.

An example power spectrum of solar oscillations is shown in Figure 3 . Power is

distributed along well-dened discrete ridges in wavenumber-frequency space and

peaks around 3 mHz. The rst ridge at low frequencies shows the \fundamental"

(f) modes. These are surface gravity waves with exponential eigenfunctions and

a dispersion relation !2=gk, whereg= 274 m s2is the acceleration of gravity

at the solar surface; they are similar to waves at the surface of a deep ocean.

All other ridges correspond to pressure (p) modes, i.e. acoustic waves modied

by gravity. The existence of discrete ridges, !=!n(k) withn >0, re

ects the

fact that p modes are trapped in the vertical direction. At xed wavenumber,

the peaks of power are labeled p 0, p1, p2, etc. with increasing frequency. A

mode pnis such that the number of radial nodes of the mode displacement is

n(the radial order). By convention, the f modes are labeled with n= 0. All

ridges have reduced power above 5 :3 mHz, which is the cut-o frequency above

which waves are not re

ected back into the Sun but escape into the atmosphere.

The frequency width of a ridge is inversely proportional to the mode lifetime. A

recent description of the mode parameters, including mode lifetimes, is provided

by Korzennik, Rabello-Soares & Schou (2004).

2.2 Modes

In order to better understand the diagnostic capability of each mode, it is useful

to consider simple solar models. For our purpose, a simple solar model is a

reference standard solar model, which only depends on height (or radius), such

as Model S (Christensen-Dalsgaard et al. 1996). In plane-parallel models that

are isotropic and translation invariant in the horizontal directions, the normal

modes of the oscillations of the model vary horizontally as exp( ikr). For the

case of p and f modes, it is convenient to introduce the mode eigenfunctions

Un(z;k) andVn(z;k) such that the complex displacement eigenfunction of the

mode characterized by radial order nand horizontal wavevector kis

n(r;z;k) =h

^ zUn(z;k) +i^kVn(z;k)i

eikr; (3)

wherezis height, ^ zis the unit vector pointing upwards, and ^kis the horizontal

unit vector pointing in the direction of k. Zero height corresponds to the photo-10 Gizon, Birch & Spruit

Figure 3: Cut at ky= 0 through an average power spectrum of MDI/SOHO

high-resolution Doppler velocity data as a function of frequency and kxR. The

horizontal dashed line shows the acoustic cuto frequency. In order to reduce ran-

dom noise, an average was carried out over eight individual power spectra, each of

durationT= 4 hr and covering an apodized region of area A(500 Mm)2. Since

ky= 0, only waves traveling in the east or west directions are showed. The power

below1:5 mHz is due to solar convection, granulation and supergranulation.

sphere (zis depth). The representation of the displacement eigenfunctions in

terms of only the two functions UandVis possible as neither the f nor the p

modes have horizontal motions that are perpendicular to k. As a result of the

assumed isotropy, the functions UandVdo not depend on the direction of k

and the mode frequencies !n(k) only depend on wavenumber. The time evolution

of the mode ( n;k) is given by exp[ i!n(k)t]. In models that include attenua-

tion, the frequencies are complex, while they are real for the case of adiabatic

oscillations and standard boundary conditions.

Figure 4 shows the horizontal and vertical eigenfunctions corresponding to

the rst ve radial orders at a frequency of 3 :5 mHz. The solar model for this