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case is a plane-parallel version of Model S. The eigenfunctions are scaled by 1=2,
whereis the density (left-most panel). This scaling is used as we are interested
in the kinetic energy density of the modes, (U2+V2), which is a physically
relevant quantity. For the f mode the horizontal and vertical displacement eigen-
functions are equal. For the acoustic modes ( n > 0), the lower turning point,
zt, is the height at which the sound speed is equal to the horizontal phase speed
of the mode: c(zt) =!=k (neglecting the buoyancy frequency and the acoustic
cuto frequency, both of which are very small below a few Mm beneath the pho-
tosphere). Thus all the modes with a similar horizontal phase speed (a straightLocal Helioseismology 11
Figure 4: Density prole from Model S (left panel, green line) and mode eigen-
functionsUandVfor the radials order n= 0 { 4 (other panels) at the frequency
3:5 mHz. The lower turning points of the modes n= 1 { 4 are shown as thin hor-
izontal black lines. At xed frequency, the horizontal phase speed !=kincreases
with increasing radial order n, and therefore lower turning points increase with
increasingnas well. The functions UandVhave been scaled with 1=2as the
kinetic energy density is proportional to (U2+V2).
line through the origin in Figure 3 ) have a similar lower turning point and probe
essentially the same layers of the Sun.
3 LOCAL HELIOSEISMOLOGY
In this section we give an overview of the various methods of local helioseismology.
For an in-depth description of each method see e.g. Gizon & Birch (2005) and
references therein.
3.1 Ring Diagram Analysis
The rst operation in ring diagram analysis is to cover some fraction of the visible
solar disk with patches (overlapping or not) with circular areas with diameters in
the range 2{ 30. Each patch is tracked in longitude with a velocity close to the
solar surface rotation velocity to produce a time series of helioseismic observations
(Dopplergrams or intensity images). For each patch a three-dimensional local12 Gizon, Birch & Spruit
power spectrum of the solar oscillation, P, is computed according to Equations 1
and 2. The local power spectra re
ect the local physical conditions in the solar
interior, such as wave speed and horizontal
ows (Hill 1988). For example, a
constant horizontal
ow uwill introduce a Doppler shift of the power spectrum:
P(k;!) =P0(k;!ku); (4)
whereP0is the power spectrum in the absence of a
ow. This description is
highly simplied as
ows in the Sun do vary with horizontal position and depth.
A change in the structure of the solar interior produces a change in the dispersion
relation that does not depend on the direction of k, and thus a change in the
power spectrum that is also independent of the direction of k.
There are two ways to study local power spectra. The rst approach is to con-
sider cuts at constant frequency, !. In (kx;ky) space, wave power is concentrated
in rings, each corresponding to a dierent radial order n(Hill 1988). A ring dia-
gram is shown in Figure 5 . When there is no
ow, the radius of each ring is the
wavenumber kat which!n(k) =!and is sensitive to the local dispersion rela-
tion. Thus the ring radius is related to the local wave speed under the patch. As
shown above, a
ow will aect the local power spectrum. Linearizing Equation 4
for small ku, we nd that the change in the ring position is k=(k=vg)u,
wherevg=@k!nis the group speed. Hence the
ow amplitude and direction can
be estimated from the distortion and orientation of the distorted rings. Note that
the distortion of a ring depends on radial order in the case of a depth-varying

ow.
The second approach consists of considering cuts at constant wavenumber, k,
through the power spectra (Schou & Bogart 1998). The local power spectra are
then studied in ( ;!) space, where is the azimuth of the wave vector measured
from the prograde direction, ^ x. The modes appear as bands of power around the
resonant frequencies !n(k). According to Equation 4, a constant horizontal
ow
will Doppler shift the mode frequencies by !=ku=kuxcos +kuysin . As
a result, ucan be estimated from the frequency shifts at each kand radial order
n. A change in the wave speed will simply manifest itself as a change in the wave
frequencies that is independent of and can be disentangled from the eect of a
horizontal
ow.
Both approaches rely on tting a parametric model of the power spectrum to
the observed local power spectra. Several functional forms have been proposed
to t the observations. The most important tted parameters are the mode
frequencies, !(n;k), and the two
ow parameters, ux(n;k) anduy(n;k). There is
one set of parameters for each wavenumber and radial order. Details about the
tting procedures are given by, e.g., Basu, Antia & Tripathy (1999) for the rst
approach and, e.g., Haber et al. (2000) for the second approach.
The tted parameters are sensitive to the conditions in the solar interior, with aLocal Helioseismology 13
Figure 5: Slices through a model local power spectrum at constant fre-
quency!=2= 3:1 mHz ( left, ring diagram) and at constant wavenumber
k= 0:8 rad/Mm ( right) for the case of a depth-independent horizontal
ow u
with an amplitude of 1 km/s and in the direction 0that is thirty degrees north
of the prograde direction. In the left panel, the black arrow shows the direction of
the
ow. The dierent rings correspond to dierent radial orders; the outermost
ring is the f mode. The rings with large kare more strongly in
uenced by the

ow than those with small k(see text). In the right panel, the ridge frequencies
show a sinusoidal variation with and reach their maxima when = 0(shown
by the vertical black line). The frequency variation with is the same for all of
the ridges as as the
ow is independent of depth.
depth sensitivity that depends on the eigenfunction of the mode (e.g. Christensen-
Dalsgaard 2002). For example, the depth sensitivity to a horizontal
ow is ap-
proximately given by the kinetic energy density of the mode (e.g. Birch et al.

case is a plane-parallel version of Model S. The eigenfunctions are scaled by 1=2,

whereis the density (left-most panel). This scaling is used as we are interested

in the kinetic energy density of the modes, (U2+V2), which is a physically

relevant quantity. For the f mode the horizontal and vertical displacement eigen-

functions are equal. For the acoustic modes ( n > 0), the lower turning point,

zt, is the height at which the sound speed is equal to the horizontal phase speed

of the mode: c(zt) =!=k (neglecting the buoyancy frequency and the acoustic

cuto frequency, both of which are very small below a few Mm beneath the pho-

tosphere). Thus all the modes with a similar horizontal phase speed (a straightLocal Helioseismology 11

Figure 4: Density prole from Model S (left panel, green line) and mode eigen-

functionsUandVfor the radials order n= 0 { 4 (other panels) at the frequency

3:5 mHz. The lower turning points of the modes n= 1 { 4 are shown as thin hor-

izontal black lines. At xed frequency, the horizontal phase speed !=kincreases

with increasing radial order n, and therefore lower turning points increase with

increasingnas well. The functions UandVhave been scaled with 1=2as the

kinetic energy density is proportional to (U2+V2).

line through the origin in Figure 3 ) have a similar lower turning point and probe

essentially the same layers of the Sun.

3 LOCAL HELIOSEISMOLOGY

In this section we give an overview of the various methods of local helioseismology.

For an in-depth description of each method see e.g. Gizon & Birch (2005) and

references therein.

3.1 Ring Diagram Analysis

The rst operation in ring diagram analysis is to cover some fraction of the visible

solar disk with patches (overlapping or not) with circular areas with diameters in

the range 2{ 30. Each patch is tracked in longitude with a velocity close to the

solar surface rotation velocity to produce a time series of helioseismic observations

(Dopplergrams or intensity images). For each patch a three-dimensional local12 Gizon, Birch & Spruit

power spectrum of the solar oscillation, P, is computed according to Equations 1

and 2. The local power spectra re

ect the local physical conditions in the solar

interior, such as wave speed and horizontal

ows (Hill 1988). For example, a

constant horizontal

ow uwill introduce a Doppler shift of the power spectrum:

P(k;!) =P0(k;!ku); (4)

whereP0is the power spectrum in the absence of a

ow. This description is

highly simplied as

ows in the Sun do vary with horizontal position and depth.

A change in the structure of the solar interior produces a change in the dispersion

relation that does not depend on the direction of k, and thus a change in the

power spectrum that is also independent of the direction of k.

There are two ways to study local power spectra. The rst approach is to con-

sider cuts at constant frequency, !. In (kx;ky) space, wave power is concentrated

in rings, each corresponding to a dierent radial order n(Hill 1988). A ring dia-

gram is shown in Figure 5 . When there is no

ow, the radius of each ring is the

wavenumber kat which!n(k) =!and is sensitive to the local dispersion rela-

tion. Thus the ring radius is related to the local wave speed under the patch. As

shown above, a

ow will aect the local power spectrum. Linearizing Equation 4

for small ku, we nd that the change in the ring position is k=(k=vg)u,

wherevg=@k!nis the group speed. Hence the

ow amplitude and direction can

be estimated from the distortion and orientation of the distorted rings. Note that

the distortion of a ring depends on radial order in the case of a depth-varying

ow.

The second approach consists of considering cuts at constant wavenumber, k,

through the power spectra (Schou & Bogart 1998). The local power spectra are

then studied in ( ;!) space, where is the azimuth of the wave vector measured

from the prograde direction, ^ x. The modes appear as bands of power around the

resonant frequencies !n(k). According to Equation 4, a constant horizontal

ow

will Doppler shift the mode frequencies by !=ku=kuxcos +kuysin . As

a result, ucan be estimated from the frequency shifts at each kand radial order

n. A change in the wave speed will simply manifest itself as a change in the wave

frequencies that is independent of and can be disentangled from the eect of a

horizontal

ow.

Both approaches rely on tting a parametric model of the power spectrum to

the observed local power spectra. Several functional forms have been proposed

to t the observations. The most important tted parameters are the mode

frequencies, !(n;k), and the two

ow parameters, ux(n;k) anduy(n;k). There is

one set of parameters for each wavenumber and radial order. Details about the

tting procedures are given by, e.g., Basu, Antia & Tripathy (1999) for the rst

approach and, e.g., Haber et al. (2000) for the second approach.

The tted parameters are sensitive to the conditions in the solar interior, with aLocal Helioseismology 13

Figure 5: Slices through a model local power spectrum at constant fre-

quency!=2= 3:1 mHz ( left, ring diagram) and at constant wavenumber

k= 0:8 rad/Mm ( right) for the case of a depth-independent horizontal

ow u

with an amplitude of 1 km/s and in the direction 0that is thirty degrees north

of the prograde direction. In the left panel, the black arrow shows the direction of

the

ow. The dierent rings correspond to dierent radial orders; the outermost

ring is the f mode. The rings with large kare more strongly in

uenced by the

ow than those with small k(see text). In the right panel, the ridge frequencies

show a sinusoidal variation with and reach their maxima when = 0(shown

by the vertical black line). The frequency variation with is the same for all of

the ridges as as the

ow is independent of depth.

depth sensitivity that depends on the eigenfunction of the mode (e.g. Christensen-

Dalsgaard 2002). For example, the depth sensitivity to a horizontal

ow is ap-

proximately given by the kinetic energy density of the mode (e.g. Birch et al.