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lioseismology in the top 20 Mm (increasing angular velocity with depth), below
that depth the measurements showed instrument dependent systematic errors.
8.2 Meridional Flow
Doppler measurements (e.g. Hathaway 1996) reveal a surface meridional
ow
with an amplitude of about 20 m s1from the equator to the poles. It has also
been measured by tracking magnetic elements (Komm, Howard & Harvey 1993a),
with essentially the same result. This surface meridional
ow implies a subsurface
return (i.e., equatorward)
ow.
In
ux-transport dynamo models (e.g. Charbonneau 2005, Dikpati & Char-
bonneau 1999), the meridional
ow is responsible for the redistribution of
ux
from the active latitudes to the poles (at the surface) and in some models also
from the poles to equator (at the base of the convection zone). Hathaway et al.
(2003) argued that the equatorward drift of sunspots during the course of the
solar cycle (the butter
y diagram) implies the existence of a 1 :2 m s1return
ow
at the bottom of the convection zone. However Sch ussler & Schmitt (2004) argue
that the butter
y diagram could be reproduced by a traditional model of dynamo
waves without transport of magnetic
ux by a
ow.
In addition to its role in some dynamo theories, the meridional
ow is also32 Gizon, Birch & Spruit
thought to play an important role in transporting angular momentum and thus
in maintaining the dierential rotation (for a recent review see Miesch 2005).
The meridional
ow produces a second-order shift in the frequencies of the
global modes (unlike, for example, the dierential rotation which produces a
rst-order shift) and is therefore very dicult to measure using traditional global
helioseismology. The meridional
ow does however produce a rst-order change
in the eigenfunctions of the modes of the Sun, and thus produces rst-order eects
in local helioseismic measurements.
Giles et al. (1997) used time-distance helioseismology to obtain the rst detec-
tion of the subsurface meridional
ow. Imposing a mass conservation constraint,
inversions of the time-distance measurements suggested a meridional
ow that
lls the convection zone, is equatorward below about 0 :8R, and has a strength
of about 2 m s1at the base of the convection zone (Giles 2000). These deep
results, however, are not direct measurements.
Figure 20 shows measurements of the meridional
ow close to the surface,
using ring-diagram analysis and (a variant of) time-distance helioseismology. The
maximum amplitude of the meridional
ow is about 15{20 m/s. The eleven years
of data show that the meridional
ow varies signicantly (up to 50% of its mean
value). The solar-cycle dependence of the meridional
ow will be discussed in the
next section.
As discussed by Braun & Birch (2008), one of the fundamental diculties in
measuring the deep meridional
ow is that the noise level, due to the stochastic
nature of solar oscillations, is very large compared to the weak signal expected
from a
ow of a few m s1(for comparison, the sound speed at the base of the
convection zone is roughly 230 km s1).
Numerical simulations of convection in rotating shells (Miesch, Brun & Toomre
2006) have roughly reproduced the overall amplitude of the meridional
ow, al-
though in these simulations the meridional
ow is highly variable and has a
multiple-cell structure. These simulations rely on the presence of a small lati-
tudinal entropy gradient to establish solar-like dierential rotation as suggested
by Rempel (2005). This gradient is imposed at the base of the convection zone
model as an adjustable part of the boundary conditions. The dierential rotation
in these models is thus what in geophysics is called a 'thermal wind', much like
the global atmospheric circulation on earth is due to the pole-equator temper-
ature dierence. In the Sun the cause for such latitudinal entropy variation is
not quite clear, however. Independent of this unsolved question, the fact that
the simulations so far appear unable to reproduce the solar dierential rotation
without an imposed entropy gradient is signicant. It implies that models of the
Sun's dierential rotation based on -eect or anisotropic turbulence formalisms
so far are not substantiated by numerical simulations.Local Helioseismology 33
8.3 Solar-Cycle Variations
The time varying component of rotation shows bands of slower and faster rota-
tion (10 m s1) that migrate in latitude with the phase of the solar cycle. This
pattern, called torsional oscillations, was rst seen in direct Doppler measure-
ments (Howard & Labonte 1980). The torsional oscillations have two branches.
At latitudes less than about 45, the bands of increased and decreased rotation
move towards the equator at the same rate as the active latitudes, with the ac-
tive latitudes located on the poleward side of the fast band. At high latitudes,
the bands move towards the poles. Global helioseismology has shown that the
torsional oscillations have their maximum amplitude close to the surface, but
extend throughout much of the convection zone (e.g. Vorontsov et al. 2002). The
torsional oscillations tend to be roughly uniform along contours of constant ro-
tation rate (e.g. Howe et al. 2005). As reviewed by Gizon & Birch (2005), local
helioseismology has been used to conrm many of these results for the shallow
component of the torsional oscillations. Figure 21 shows the time residuals of
the zonal
ows near the surface (time-distance helioseismology).
In addition, local helioseismology has shown that there are

lioseismology in the top 20 Mm (increasing angular velocity with depth), below

that depth the measurements showed instrument dependent systematic errors.

8.2 Meridional Flow

Doppler measurements (e.g. Hathaway 1996) reveal a surface meridional

ow

with an amplitude of about 20 m s1from the equator to the poles. It has also

been measured by tracking magnetic elements (Komm, Howard & Harvey 1993a),

with essentially the same result. This surface meridional

ow implies a subsurface

return (i.e., equatorward)

ow.

In

ux-transport dynamo models (e.g. Charbonneau 2005, Dikpati & Char-

bonneau 1999), the meridional

ow is responsible for the redistribution of

ux

from the active latitudes to the poles (at the surface) and in some models also

from the poles to equator (at the base of the convection zone). Hathaway et al.

(2003) argued that the equatorward drift of sunspots during the course of the

solar cycle (the butter

y diagram) implies the existence of a 1 :2 m s1return

ow

at the bottom of the convection zone. However Sch ussler & Schmitt (2004) argue

that the butter

y diagram could be reproduced by a traditional model of dynamo

waves without transport of magnetic

ux by a

ow.

In addition to its role in some dynamo theories, the meridional

ow is also32 Gizon, Birch & Spruit

thought to play an important role in transporting angular momentum and thus

in maintaining the dierential rotation (for a recent review see Miesch 2005).

The meridional

ow produces a second-order shift in the frequencies of the

global modes (unlike, for example, the dierential rotation which produces a

rst-order shift) and is therefore very dicult to measure using traditional global

helioseismology. The meridional

ow does however produce a rst-order change

in the eigenfunctions of the modes of the Sun, and thus produces rst-order eects

in local helioseismic measurements.

Giles et al. (1997) used time-distance helioseismology to obtain the rst detec-

tion of the subsurface meridional

ow. Imposing a mass conservation constraint,

inversions of the time-distance measurements suggested a meridional

ow that

lls the convection zone, is equatorward below about 0 :8R, and has a strength

of about 2 m s1at the base of the convection zone (Giles 2000). These deep

results, however, are not direct measurements.

Figure 20 shows measurements of the meridional

ow close to the surface,

using ring-diagram analysis and (a variant of) time-distance helioseismology. The

maximum amplitude of the meridional

ow is about 15{20 m/s. The eleven years

of data show that the meridional

ow varies signicantly (up to 50% of its mean

value). The solar-cycle dependence of the meridional

ow will be discussed in the

next section.

As discussed by Braun & Birch (2008), one of the fundamental diculties in

measuring the deep meridional

ow is that the noise level, due to the stochastic

nature of solar oscillations, is very large compared to the weak signal expected

from a

ow of a few m s1(for comparison, the sound speed at the base of the

convection zone is roughly 230 km s1).

Numerical simulations of convection in rotating shells (Miesch, Brun & Toomre

2006) have roughly reproduced the overall amplitude of the meridional

ow, al-

though in these simulations the meridional

ow is highly variable and has a

multiple-cell structure. These simulations rely on the presence of a small lati-

tudinal entropy gradient to establish solar-like dierential rotation as suggested

by Rempel (2005). This gradient is imposed at the base of the convection zone

model as an adjustable part of the boundary conditions. The dierential rotation

in these models is thus what in geophysics is called a 'thermal wind', much like

the global atmospheric circulation on earth is due to the pole-equator temper-

ature dierence. In the Sun the cause for such latitudinal entropy variation is

not quite clear, however. Independent of this unsolved question, the fact that

the simulations so far appear unable to reproduce the solar dierential rotation

without an imposed entropy gradient is signicant. It implies that models of the

Sun's dierential rotation based on -eect or anisotropic turbulence formalisms

so far are not substantiated by numerical simulations.Local Helioseismology 33

8.3 Solar-Cycle Variations

The time varying component of rotation shows bands of slower and faster rota-

tion (10 m s1) that migrate in latitude with the phase of the solar cycle. This

pattern, called torsional oscillations, was rst seen in direct Doppler measure-

ments (Howard & Labonte 1980). The torsional oscillations have two branches.

At latitudes less than about 45, the bands of increased and decreased rotation

move towards the equator at the same rate as the active latitudes, with the ac-

tive latitudes located on the poleward side of the fast band. At high latitudes,

the bands move towards the poles. Global helioseismology has shown that the

torsional oscillations have their maximum amplitude close to the surface, but

extend throughout much of the convection zone (e.g. Vorontsov et al. 2002). The

torsional oscillations tend to be roughly uniform along contours of constant ro-

tation rate (e.g. Howe et al. 2005). As reviewed by Gizon & Birch (2005), local

helioseismology has been used to conrm many of these results for the shallow

component of the torsional oscillations. Figure 21 shows the time residuals of

the zonal

ows near the surface (time-distance helioseismology).

In addition, local helioseismology has shown that there are