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horizontal and vertical components of the
ows was implemented by Jackiewicz,
Gizon & Birch (2008), in which the Born kernels and the noise covariance matrix
are both consistent with the denition of the observed travel times. We note
that the vertical component of velocity has been indirectly estimated from ring-
diagram inversion by requiring mass conservation (e.g. Komm et al. 2004).Local Helioseismology 21
4.2 Strong Perturbation Regime
The solar atmosphere is permeated by concentrations of magnetic eld with
strengthB > 1 kG. This magnetic eld profoundly aects the solar atmosphere
as well as the solar oscillations in the upper layers (Bogdan & Cally 1995, and
references therein). The eects on the waves are not small and it is formally
not justied to employ a single-scattering approximation to describe them. For
example, the rst Born approximation is not expected to capture the interaction
of f and p modes with the near-surface layers of sunspots (e.g. Gizon, Hanasoge
& Birch 2006).
The Lorentz force is an additional restoring force that permits the existence of
new oscillation modes. In the case of a spatially uniform model with no gravity,
it is possible to identify three types of magneto-hydrodynamics (MHD) waves:
the fast, slow, and Alfv en waves. In gravitationally stratied magnetized atmo-
spheres, this classication can only be applied locally. The waves couple near the
layer where the sound speed and the Alfv en speed, a=B=p4, are equal (e.g.
Schunker & Cally 2006). In a typical sunspot, the a=csurface is only a few
hundred km below the quiet-Sun photosphere.
The ray approximation can be extended in order to study wave propagation in
MHD problems (e.g. Schunker & Cally 2006). The magnetic eld aects travel
times, mode frequencies, and amplitudes. These eects depend sensitively on
the geometry, in particular the angle between the incident wave vector and the
magnetic eld vector at the a=clayer. Figure 10 shows two example ray
calculations for the case of an incoming acoustic wave approaching the solar
surface from below (the magnetic eld, wave vector, and gravity are in the same
plane). In these two calculations all of the parameters are the same except that
the angle of the magnetic eld has been changed. In Figure 10a, the wavevector
of the acoustic wave is nearly aligned with the magnetic eld at the a=clayer.
In this case, most of the energy is transmitted to the acoustic (slow) mode of the
a > c region. This acoustic mode escapes along the magnetic eld lines (ramp
eect). Some energy is, however, converted to the fast (magnetic) mode which is
then refracted by the increase of the Alfv en speed with height. In Figure 10b,
the incident wavevector makes a large angle with the magnetic eld. At the
a=clevel, the acoustic wave converts mostly into a magnetic (fast) mode that
is refracted back into the Sun and then mostly becomes a downward propagating
magnetic (slow) mode, while a small fraction of energy continues as a fast mode.
The slow mode is not seen again at the surface, and thus removes energy from
the surface waveeld. Supplemental Movies 3 { 6 illustrate generalized ray
theory for various values of the attack angle between the wave vector and the
magnetic eld. We note that in three dimensions, where the magnetic eld, wave
vector, and gravity are not all co-planar, strong coupling to the A
v en wave also
occurs (Cally & Goossens 2008).22 Gizon, Birch & Spruit
Numerical simulations are an important tool to study waves in magnetized
regions and sunspots. Two dierent approaches are employed. The rst ap-
proach is numerical simulations of wave propagation through prescribed back-
ground models (e.g. Cally & Bogdan 1997; Cameron, Gizon & Daiallah 2007;
Hanasoge 2008; Khomenko & Collados 2006; Parchevsky & Kosovichev 2009).
This approach permits the study of wave propagation without the complications
of solving for convection and it gives the freedom to choose various background
models. Typically, these codes solve the equations of motion for small-amplitude
waves. The second approach is realistic numerical simulations of magnetoconvec-
tion (e.g. Rempel et al. 2009). Such simulations include self-excited waves and
aim at simulating realistic solar active regions. This approach is very promising,
but computer intensive.
Figure 11 shows an example calculation of the propagation of a p 1wave packet
through a simple sunspot model using the three-dimensional code of Cameron,
Gizon & Daiallah (2007). This example shows that the transmitted wave packet
is phase-shifted by the sunspot (increased eective wave speed) and has a reduced
amplitude compared to the quiet-Sun value. Also seen is the partial conversion
of incoming p modes into downgoing slow modes. This process is responsible for
absorption of acoustic energy by sunspots (Cally & Bogdan 1997, Crouch & Cally
2005, Spruit & Bogdan 1992, and Section 6). An f mode wave packet is aected
in a similar fashion (Cally & Bogdan 1997; Cameron, Gizon & Duvall 2008).
The problem of inferring the subsurface structure and dynamics of solar active
regions is a dicult one. In principle, standard linear inversions cannot be used
because surface magnetic perturbations are not small with respect to a quiet Sun
reference model. No non-linear inversion has been implemented yet. Instead
there have been attempts to construct simple parametric models of magnetic re-
gions, which have a helioseismic signature that is consistent with the observations.
This approach does not require that perturbations be small. Forward models of
sunspots have been proposed by, e.g., Crouch et al. (2005) and Cameron, Gizon
& Duvall (2008); they will be discussed in more detail in Section 6.4.
Now that we have some understanding of the methods of local helioseismology
and their diagnostic potential, we turn to a description of the main observational
results: near-surface convection (Section 5), sunspots (Section 6), extended
ows
around active regions (Section 7), global scales (Section 8), farside imaging (Sec-
tion 9), and
are-excited waves (Section 10).
5 NEAR-SURFACE CONVECTION
5.1 Supergranulation and Magnetic Network
Solar supergranules are horizontal out
ows with a typical size of 30 Mm, outlined
by the chromospheric network (e.g. Leighton, Noyes & Simon 1962). They haveLocal Helioseismology 23
horizontal velocities of order 200 m s1and lifetimes of one to two days.
Duvall et al. (1997) found that p-mode travel times contain information at
supergranular length scales ( Figure 7 ). As a demonstration of this, the line-
of-sight component of velocity, estimated from the travel times assuming that
vertical motions are negligible, was found to be highly correlated with the time
averaged Dopplergram, thus conrming that local helioseismology is capable of
probing convective
ows at supergranulation length scales. Duvall & Gizon (2000)
extended the analysis to f modes to infer horizontal
ows within 2 Mm below the

horizontal and vertical components of the

ows was implemented by Jackiewicz,

Gizon & Birch (2008), in which the Born kernels and the noise covariance matrix

are both consistent with the denition of the observed travel times. We note

that the vertical component of velocity has been indirectly estimated from ring-

diagram inversion by requiring mass conservation (e.g. Komm et al. 2004).Local Helioseismology 21

4.2 Strong Perturbation Regime

The solar atmosphere is permeated by concentrations of magnetic eld with

strengthB > 1 kG. This magnetic eld profoundly aects the solar atmosphere

as well as the solar oscillations in the upper layers (Bogdan & Cally 1995, and

references therein). The eects on the waves are not small and it is formally

not justied to employ a single-scattering approximation to describe them. For

example, the rst Born approximation is not expected to capture the interaction

of f and p modes with the near-surface layers of sunspots (e.g. Gizon, Hanasoge

& Birch 2006).

The Lorentz force is an additional restoring force that permits the existence of

new oscillation modes. In the case of a spatially uniform model with no gravity,

it is possible to identify three types of magneto-hydrodynamics (MHD) waves:

the fast, slow, and Alfv en waves. In gravitationally stratied magnetized atmo-

spheres, this classication can only be applied locally. The waves couple near the

layer where the sound speed and the Alfv en speed, a=B=p4, are equal (e.g.

Schunker & Cally 2006). In a typical sunspot, the a=csurface is only a few

hundred km below the quiet-Sun photosphere.

The ray approximation can be extended in order to study wave propagation in

MHD problems (e.g. Schunker & Cally 2006). The magnetic eld aects travel

times, mode frequencies, and amplitudes. These eects depend sensitively on

the geometry, in particular the angle between the incident wave vector and the

magnetic eld vector at the a=clayer. Figure 10 shows two example ray

calculations for the case of an incoming acoustic wave approaching the solar

surface from below (the magnetic eld, wave vector, and gravity are in the same

plane). In these two calculations all of the parameters are the same except that

the angle of the magnetic eld has been changed. In Figure 10a, the wavevector

of the acoustic wave is nearly aligned with the magnetic eld at the a=clayer.

In this case, most of the energy is transmitted to the acoustic (slow) mode of the

a > c region. This acoustic mode escapes along the magnetic eld lines (ramp

eect). Some energy is, however, converted to the fast (magnetic) mode which is

then refracted by the increase of the Alfv en speed with height. In Figure 10b,

the incident wavevector makes a large angle with the magnetic eld. At the

a=clevel, the acoustic wave converts mostly into a magnetic (fast) mode that

is refracted back into the Sun and then mostly becomes a downward propagating

magnetic (slow) mode, while a small fraction of energy continues as a fast mode.

The slow mode is not seen again at the surface, and thus removes energy from

the surface waveeld. Supplemental Movies 3 { 6 illustrate generalized ray

theory for various values of the attack angle between the wave vector and the

magnetic eld. We note that in three dimensions, where the magnetic eld, wave

vector, and gravity are not all co-planar, strong coupling to the A

v en wave also

occurs (Cally & Goossens 2008).22 Gizon, Birch & Spruit

Numerical simulations are an important tool to study waves in magnetized

regions and sunspots. Two dierent approaches are employed. The rst ap-

proach is numerical simulations of wave propagation through prescribed back-

ground models (e.g. Cally & Bogdan 1997; Cameron, Gizon & Daiallah 2007;

Hanasoge 2008; Khomenko & Collados 2006; Parchevsky & Kosovichev 2009).

This approach permits the study of wave propagation without the complications

of solving for convection and it gives the freedom to choose various background

models. Typically, these codes solve the equations of motion for small-amplitude

waves. The second approach is realistic numerical simulations of magnetoconvec-

tion (e.g. Rempel et al. 2009). Such simulations include self-excited waves and

aim at simulating realistic solar active regions. This approach is very promising,

but computer intensive.

Figure 11 shows an example calculation of the propagation of a p 1wave packet

through a simple sunspot model using the three-dimensional code of Cameron,

Gizon & Daiallah (2007). This example shows that the transmitted wave packet

is phase-shifted by the sunspot (increased eective wave speed) and has a reduced

amplitude compared to the quiet-Sun value. Also seen is the partial conversion

of incoming p modes into downgoing slow modes. This process is responsible for

absorption of acoustic energy by sunspots (Cally & Bogdan 1997, Crouch & Cally

2005, Spruit & Bogdan 1992, and Section 6). An f mode wave packet is aected

in a similar fashion (Cally & Bogdan 1997; Cameron, Gizon & Duvall 2008).

The problem of inferring the subsurface structure and dynamics of solar active

regions is a dicult one. In principle, standard linear inversions cannot be used

because surface magnetic perturbations are not small with respect to a quiet Sun

reference model. No non-linear inversion has been implemented yet. Instead

there have been attempts to construct simple parametric models of magnetic re-

gions, which have a helioseismic signature that is consistent with the observations.

This approach does not require that perturbations be small. Forward models of

sunspots have been proposed by, e.g., Crouch et al. (2005) and Cameron, Gizon

& Duvall (2008); they will be discussed in more detail in Section 6.4.

Now that we have some understanding of the methods of local helioseismology

and their diagnostic potential, we turn to a description of the main observational

results: near-surface convection (Section 5), sunspots (Section 6), extended

ows

around active regions (Section 7), global scales (Section 8), farside imaging (Sec-

tion 9), and

are-excited waves (Section 10).

5 NEAR-SURFACE CONVECTION

5.1 Supergranulation and Magnetic Network

Solar supergranules are horizontal out

ows with a typical size of 30 Mm, outlined

by the chromospheric network (e.g. Leighton, Noyes & Simon 1962). They haveLocal Helioseismology 23

horizontal velocities of order 200 m s1and lifetimes of one to two days.

Duvall et al. (1997) found that p-mode travel times contain information at

supergranular length scales ( Figure 7 ). As a demonstration of this, the line-

of-sight component of velocity, estimated from the travel times assuming that

vertical motions are negligible, was found to be highly correlated with the time

averaged Dopplergram, thus conrming that local helioseismology is capable of

probing convective

ows at supergranulation length scales. Duvall & Gizon (2000)

extended the analysis to f modes to infer horizontal

ows within 2 Mm below the