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surface. Because they propagate horizontally, f modes are well suited to mea-
sure horizontal
ows and their horizontal divergence. The
ows from f-mode
time-distance helioseismology compare well with
ows estimated from local cor-
relation tracking of mesogranulation (De Rosa, Duvall & Toomre 2000). Recently,
Woodard (2009) has shown that direct modeling can be used to detect convection
on scales of space and time that are smaller than those of supergranulation.
Figure 12 shows the most recent inversion of travel-times (Jackiewicz, Gi-
zon & Birch 2008) using modes f through p 4. This inversion incorporates a full
treatment of nite-wavelength eects (rst-order Born approximation), modeling
of the details of the measurement procedure, and a treatment of the statistical
properties of noise. The vector
ow eld, averaged over T= 3 days, is domi-
nated by long-lived supergranules. As seen in the gure, the divergent
ows are
co-spatial with up
ows with about 30 m s1rms velocity (with maximum values
of50 m s1). Near the surface, the vertical velocity can be measured in super-
granules with a noise level of about 10 m s1for 24 hr averages and a horizontal
resolution of about 10 Mm. Estimates of the vertical velocity in supergranules
from direct Doppler measurements can only be obtained near disk center and are
in the range 10 { 30 m s1(Hathaway et al. 2002, and references therein), which
is consistent with the results of local helioseismology.
Because noise reduction requires time averaging, the nite lifetime of super-
granulation implies a strict limitation on the maximum depth at which we can
probe the
ow eld before it evolves substantially. Using the f and p 1{p4modes,
Woodard (2007) found that random noise dominates below about 4 Mm. Prob-
ing supergranules at greater depth involves statistical analysis over large elds of
view and many supergranulation lifetimes in order to reduce the noise: this allows
the study of the average properties of the
ows at depth. Inversions of convec-
tive
ows tens of Mm below the surface are challenging as they require excellent
models of the sensitivity of travel times to subsurface
ows (see Braun & Lindsey
2003, for a discussion) and claims of the detection of a supergranulation return

ow are apparently inconsistent (e.g. Duvall 1998, Zhao & Kosovichev 2003).
The pattern of divergent
ows in the surface layers is outlined by a network of
small magnetic features (see Braun & Lindsey 2003, Duvall & Gizon 2000, and
Supplemental Movie 7 ). This can be understood as the magnetic eld is swept24 Gizon, Birch & Spruit
by the
ows and concentrates at the boundaries of supergranules (e.g. Galloway,
Proctor & Weiss 1977). The connections between the magnetic network and the
propagation behavior of acoustic waves in the solar chromosphere can be studied
by cross-correlating the observations of solar oscillations at multiple heights in
the solar atmosphere (Finsterle et al. 2004b). Jeeries et al. (2006) showed that
inclined magnetic eld lines at the boundaries of supergranules provide `portals'
through which low-frequency ( <5 mHz) slow MAG waves can propagate up into
the solar chromosphere (see Figure 13 ). This is because the cut-o frequency is
lowered in the magnetic network relative to the quiet Sun by a factor cos , where
is the inclination of the magnetic eld to the vertical. These low-frequency
upward traveling waves have been suggested to act as a source of heating of the
quiet-Sun chromosphere { as an alternative to Joule heating due to magnetic eld
reconnection or mechanical heating due to high-frequency waves.
5.2 Rotation-Induced Vorticity
Rotation is expected to have a small eect on the dynamics of the supergranula-
tion through the Coriolis force. As solar convection is highly turbulent, this eect
can only be studied in a statistical sense using several months of data. For ex-
ample, in the northern hemisphere, divergent
ows are expected to have a slight
positive correlation with clockwise vertical vorticity. Duvall & Gizon (2000) and
Gizon & Duvall (2003) used time-distance helioseismology to make the rst mea-
surement of this small eect. After removing the average rotation and meridional
circulation from the inferred
ows, Gizon & Duvall (2003) studied the relationship
between the horizontal divergence of the
ows, denoted by `div', and the vertical
component of vorticity, denoted by `curl'. Figure 14ashows the latitudinal de-
pendence ofhcurli+andhcurli, respectively dened as the averages of the curl
over regions of positive and negative div. In the northern hemisphere, diverg-
ing
ows preferentially rotate clockwise, whereas converging
ows preferentially
rotate counter-clockwise. This pattern is reversed in the southern hemisphere.
This situation is an expected consequence of the Coriolis force. Furthermore, the
latitudinal dependence of hcurli+andhcurliare observed to be nearly exactly
proportional to the radial component of the solar angular velocity, sin( )
(),
whereis latitude and
is the solar angular velocity.
Figure 14bshows that the average of the product of div by curl is given by
hdiv curli' 31010sin()
()=
eqs2; (13)
where
eqis the equatorial angular velocity. Simple dimensional analysis of
the equations of motion predicts that hdiv curli Co()=2where Co() =
2
() sin () is the local Coriolis number and is the characteristic correla-
tion time of the turbulence. For example, = 2 days implies Co( )=2Local Helioseismology 25
31011sin()
()=
eqs2. It is not clear if this dierence of a factor of
ten indicates missing physics or simply re
ects the uncertainty in such estimates.

surface. Because they propagate horizontally, f modes are well suited to mea-

sure horizontal

ows and their horizontal divergence. The

ows from f-mode

time-distance helioseismology compare well with

ows estimated from local cor-

relation tracking of mesogranulation (De Rosa, Duvall & Toomre 2000). Recently,

Woodard (2009) has shown that direct modeling can be used to detect convection

on scales of space and time that are smaller than those of supergranulation.

Figure 12 shows the most recent inversion of travel-times (Jackiewicz, Gi-

zon & Birch 2008) using modes f through p 4. This inversion incorporates a full

treatment of nite-wavelength eects (rst-order Born approximation), modeling

of the details of the measurement procedure, and a treatment of the statistical

properties of noise. The vector

ow eld, averaged over T= 3 days, is domi-

nated by long-lived supergranules. As seen in the gure, the divergent

ows are

co-spatial with up

ows with about 30 m s1rms velocity (with maximum values

of50 m s1). Near the surface, the vertical velocity can be measured in super-

granules with a noise level of about 10 m s1for 24 hr averages and a horizontal

resolution of about 10 Mm. Estimates of the vertical velocity in supergranules

from direct Doppler measurements can only be obtained near disk center and are

in the range 10 { 30 m s1(Hathaway et al. 2002, and references therein), which

is consistent with the results of local helioseismology.

Because noise reduction requires time averaging, the nite lifetime of super-

granulation implies a strict limitation on the maximum depth at which we can

probe the

ow eld before it evolves substantially. Using the f and p 1{p4modes,

Woodard (2007) found that random noise dominates below about 4 Mm. Prob-

ing supergranules at greater depth involves statistical analysis over large elds of

view and many supergranulation lifetimes in order to reduce the noise: this allows

the study of the average properties of the

ows at depth. Inversions of convec-

tive

ows tens of Mm below the surface are challenging as they require excellent

models of the sensitivity of travel times to subsurface

ows (see Braun & Lindsey

2003, for a discussion) and claims of the detection of a supergranulation return

ow are apparently inconsistent (e.g. Duvall 1998, Zhao & Kosovichev 2003).

The pattern of divergent

ows in the surface layers is outlined by a network of

small magnetic features (see Braun & Lindsey 2003, Duvall & Gizon 2000, and

Supplemental Movie 7 ). This can be understood as the magnetic eld is swept24 Gizon, Birch & Spruit

by the

ows and concentrates at the boundaries of supergranules (e.g. Galloway,

Proctor & Weiss 1977). The connections between the magnetic network and the

propagation behavior of acoustic waves in the solar chromosphere can be studied

by cross-correlating the observations of solar oscillations at multiple heights in

the solar atmosphere (Finsterle et al. 2004b). Jeeries et al. (2006) showed that

inclined magnetic eld lines at the boundaries of supergranules provide `portals'

through which low-frequency ( <5 mHz) slow MAG waves can propagate up into

the solar chromosphere (see Figure 13 ). This is because the cut-o frequency is

lowered in the magnetic network relative to the quiet Sun by a factor cos , where

is the inclination of the magnetic eld to the vertical. These low-frequency

upward traveling waves have been suggested to act as a source of heating of the

quiet-Sun chromosphere { as an alternative to Joule heating due to magnetic eld

reconnection or mechanical heating due to high-frequency waves.

5.2 Rotation-Induced Vorticity

Rotation is expected to have a small eect on the dynamics of the supergranula-

tion through the Coriolis force. As solar convection is highly turbulent, this eect

can only be studied in a statistical sense using several months of data. For ex-

ample, in the northern hemisphere, divergent

ows are expected to have a slight

positive correlation with clockwise vertical vorticity. Duvall & Gizon (2000) and

Gizon & Duvall (2003) used time-distance helioseismology to make the rst mea-

surement of this small eect. After removing the average rotation and meridional

circulation from the inferred

ows, Gizon & Duvall (2003) studied the relationship

between the horizontal divergence of the

ows, denoted by `div', and the vertical

component of vorticity, denoted by `curl'. Figure 14ashows the latitudinal de-

pendence ofhcurli+andhcurli, respectively dened as the averages of the curl

over regions of positive and negative div. In the northern hemisphere, diverg-

ing

ows preferentially rotate clockwise, whereas converging

ows preferentially

rotate counter-clockwise. This pattern is reversed in the southern hemisphere.

This situation is an expected consequence of the Coriolis force. Furthermore, the

latitudinal dependence of hcurli+andhcurliare observed to be nearly exactly

proportional to the radial component of the solar angular velocity, sin( )

(),

whereis latitude and

is the solar angular velocity.

Figure 14bshows that the average of the product of div by curl is given by

hdiv curli' 31010sin()

()=

eqs2; (13)

where

eqis the equatorial angular velocity. Simple dimensional analysis of

the equations of motion predicts that hdiv curli Co()=2where Co() =

2

() sin () is the local Coriolis number and is the characteristic correla-

tion time of the turbulence. For example, = 2 days implies Co( )=2Local Helioseismology 25

31011sin()

()=

eqs2. It is not clear if this dierence of a factor of

ten indicates missing physics or simply re

ects the uncertainty in such estimates.