5.3 Supergranulation

5.3.1 Paradigms

Granulation, with a typical size of $1.5 Mm$ , is well understood as a convective phenomenon and can be studied with realistic numerical simulations (see, e.g., Stein and Nordlund, 2000 ). Supergranules, however, have remained puzzling since their detection by Hart (1954 , 1956 ). In a classic paper, Leighton et al. (1962 ) reported “large cells of horizontally moving material distributed roughly uniformly over the entire solar surface” with a characteristic scale of $30 Mm$ that are outlined by the chromospheric network; they suggested that the cells are a “surface manifestation of a supergranulation pattern of convective currents”. Unfortunately, there is no accepted theory that explains why solar convection should favor a $30 Mm$ scale. The solar convection zone is so highly turbulent and stratified that numerical modeling at supergranular scales has remained elusive (see, e.g., Rincon, 2004 , and references therein).

Simon and Leighton (1964 ) presumed that a convective instability due to the recombination of ionized Helium is at the origin of the distinct supergranular scale. Forty years later, this hypothesis has not been proved to be right or wrong. The depth of the supergranulation layer is not known either. We note that Parker (1973 ) believes that the effect of stratification on convection may imply that supergranules are a deep phenomenon, with depths in excess of their horizontal diameters. Antia and Chitre (1993 ) investigated the stability of linear convective modes in the solar convection zone and found that a few convective modes dominate the power spectrum, one of which was identified as supergranulation. Using 1D numerical caculations, Ploner et al. (2000 ) suggested that the scale of supergranulation may be due to the interaction and merging of individual granular plumes (see also Rast, 2003 ). A somewhat related model was proposed by Rieutord et al. (2000 , 2001 ) whereby supergranulation is the result of a non-linear large-scale instability of the granular flow, triggered by exploding granules. In the two previous models, supergranulation is not a proper scale of thermal convection. Regarding the influence of supergranular flows on magnetic fields, it is well established that a stationary cellular flow tends to expel the magnetic field from the regions of fluid motion and concentrate the magnetic flux into ropes at the cell boundaries (Parker, 1963 ; Galloway et al., 1977 ; Galloway and Weiss, 1981 ).

On the observational side, the original work of Leighton and coworkers has been refined. A variety of methods have been used to characterize the distribution of the cell sizes. A characteristic scale can be obtained from the spatial autocorrelation function (see, e.g., Hart, 1956 ; Simon and Leighton, 1964 ; Duvall Jr, 1980 ), the spatial Fourier spectrum (see, e.g., Hathaway, 1992 ; Beck, 1997 ), and segmentation or tessellation algorithms (Hagenaar et al., 1997 ). Although definitions vary, average cell sizes are in the range $15- 30 Mm$ . It is unclear whether there is a variation of cell sizes with latitude: Rimmele and Schroeter (1989 ) and Komm et al. (1993a ) report a possible decrease with latitude, Berrilli et al. (1999 ) an increase, and Beck (1997 ) no significant variation. The typical lifetime of the supergranular/chromospheric network is found to be in the range $1- 2 d$ (e.g Rogers, 1970 ; Worden and Simon, 1976 ; Duvall Jr, 1980 ; Wang and Zirin, 1989 ). The rms horizontal velocity of supergranular flows is known to be about $300 m s- 1$ (see, e.g., Hathaway et al., 2000 ). The vertical component of the flows, however, has been extremely difficult to measure (see, e.g., Giovanelli, 1980 ) or infer (November, 1989 ). Indeed, Miller et al. (1984 ) caution that Doppler velocity measurements at the cell boundaries may be polluted by the network field. Estimates of the rms vertical flow are provided by Chou et al. (1991 ) and Hathaway et al. (2002 ) who find speeds of about $30 m s-1$ (the topology of the vertical flows is largely unknown). Even more difficult to measure are the related temperature fluctuations. Observers have searched for the thermal signature of a convective process, i.e., rising hot material at the cell centers and sinking cool material at the cell boundaries. Unfortunately, answers vary too widely (see Lin and Kuhn, 1992 , and references therein).

Estimates of the pattern rotation rate of supergranulation can be obtained by correlation tracking techniques. Duvall Jr (1980 ) and Snodgrass and Ulrich (1990 ) used Doppler scans separated in time by $Dt = 24 hr$ . The results indicate that the equatorial rotation of the supergranulation pattern is faster than the spectroscopic rate (Snodgrass and Ulrich, 1990 ) by about 4% and faster than the magnetic features (Komm et al., 1993b ) by about 2%. Duvall Jr (1980 ) also considered same-day scans with $Dt = 6 hr$ and found a slightly smaller pattern rotation rate than for $Dt = 24 hr$ .

It was suggested by Foukal (1972 ) that the difference between the rotation of magnetic features and the spectroscopic rate may be due to magnetic structures being rooted in deeper, more rapidly rotating layers. Supergranules must also sense increase rotation in the shear layer below the surface. However, Beck (2000 ) remarks that the pattern rotation of supergranulation measured from correlation tracking with $Dt = 24 hr$ is significantly faster than the rotation of the solar plasma measured by helioseismology at any depth in the interior. It is especialy puzzling that the rotation of the magnetic network is significantly less than that of the supergranular pattern, since small magnetic elements are believed to be advected by supergranular flows. Hathaway (1982 ) suggested that a faster supergranular rate may be a direct consequence of the interaction of convection and rotation.

As explained in the following sections, local heliososeismology has become a powerful tool to study the structure and evolution of supergranular flows. In fact, helioseismological measurements have transformed our knowledge of the dynamics of supergranulation.

5.3.2 Horizontal flows and vertical structure

In order to resolve structures at supergranular scales, Duvall Jr et al. (1996 ) measured p-mode travel times at distances shorter than $10 Mm$ over an $8.5 hr$ time interval. Directional information was obtained by cross-correlating a point on the surface with surrounding quadrants centered on the four cardinal directions. In a first order approximation, the south-north and east-west travel time differences were converted into an apparent horizontal vector flow field without inversion. Duvall Jr et al. (1997 ) found that the line-of-sight projection of the inferred flow field is highly correlated with the mean MDI Dopplergram (correlation coefficient 0.74). This comparison was initialy used as a validation of the time-distance technique.

Three dimensional inversions of quiet-sun travel times have been presented by Duvall Jr et al. (1997 ), Kosovichev and Duvall Jr (1997 ), and Zhao and Kosovichev (2003a ). Tests of the ray-based inversion procedure of Zhao and Kosovichev (2003a ) show that the horizontal components of the velocity can be infered with some confidence in the upper $5 Mm$ (and perhaps down to $10 Mm$ ). The small vertical flows, on the other hand, cannot be inferred reliably near the surface due to significant “cross-talk” with the horizontal divergence signal.

Near-surface horizontal flows have been measured at a depth of $1 Mm$ with f-mode time-distance helioseismology (Duvall Jr and Gizon, 2000 ). In this case travel time differences are directly sensitive to horizontal flows because f modes propagate horizontaly. The f-mode time-distance technique gives results that are comparable to correlation tracking of granulation (De Rosa et al., 2000 ). Shown in Figure 22 is an inversion of f-mode travel times that employs 2D Born kernels (Gizon et al., 2000 ). The correlation coefficient between the estimated line-of-sight velocity and the surface Doppler image is about $0.7$ .

Figure 57:

Movie of the divergence signal (inward travel times minus outward times) with magnetic field signal overlaid. The magnetic field is displayed as green and red for the two polarities when the magnitude of the field is larger than $15 G$ . The gray scale is for the divergence signal with white shades for outflow and dark shades for inflow. The colorbar indicates the travel time difference in seconds. The line $Y = 0$ corresponds to the equator and $X = 0$ corresponds to Carrington longitude $o 180$ . The time-distance data is averaged over $8.5 hr$ starting at the time shown on top. The movie shows the time evolution of the supergranulation pattern over 6 days. From Duvall Jr and Gizon (2000

The holographic technique also measures flows at supergranular scales. As shown earlier in Figure 45

, supergranules are easy to identify as regions of outflows for a $3 Mm$ focus depth ( $24 hr$ time average).

A practical way to detect and display supergranulation is to plot the horizontal divergence of the flow field. Time-distance helioseismology applied to f modes is particularly well adapted to measuring the horizontal divergence of the velocity near the surface. Duvall Jr and Gizon (2000 ) measured the time it takes for solar f modes to propagate from any given point on the solar surface to a concentric annulus around that point. The difference in travel times between inward and outward propagating waves is a proxy for the local horizontal divergence. An example of the divergence signal (inward minus outward travel time) for one 8-hour interval analyzed is shown in Figure 57 with magnetic field information overlaid (MDI full-disk data). A white, or positive signal, corresponds to a horizontal outflow. From the size of the features present, their lifetime, and the presence of the magnetic field in the dark lanes regions, supergranulation is identified as the main contributor to the signal. This can also be seen by making a spatial power spectrum of the divergence signal: Power peaks near degree $l = 120$ . The movie associated with Figure (57 ) shows that the magnetic features stay in the regions of horizontal convergence over the time interval of the observations (one week).

Local heliososeismology opens prospects for mapping the structure of supergranular flows below the surface. A major goal is to answer the long-standing question of how deep supergranular flows persist below the surface.

Figure 58:

Flow field (arrows) and wave-speed perturbations (grey-scale background) inferred by Kosovichev and Duvall Jr (1997

) with time-distance helioseismology. Typically there are upflows in the hotter areas where the sound speed is higher, and downflows in the cooler areas. From Kosovichev and Duvall Jr (1997

Duvall Jr et al. (1997

) and Kosovichev and Duvall Jr (1997 ) presented three-dimensional inversions of quiet-sun travel times using ray-theoretical sensitivity kernels. Figure 58

shows a vertical cut through the velocity field; the horizontal sampling is $4.3 Mm$ and the vertical sampling is about $1 Mm$ . In the upper $5 Mm$ the flows are positively correlated with the direct surface observations. To assess the average structure of superganular flows with depth, Duvall Jr (1998

) used the inversion results from Duvall Jr et al. (1997 ) to measure the correlation of horizontal flows between the surface and any given depth. As seen in Figure 59

, the correlation drops to zero at a depth of $5 Mm$ and is negative in the range $5 -8 Mm$ , suggesting the existence of a “return flow”. Since the correlation seems to disappear below $8 Mm$ , Duvall Jr (1998

) concluded that the depth of supergranulation is $8 Mm$ . The same type of analysis was repeated by Zhao and Kosovichev (2003a

) with more recent (and presumably more reliable) inversions of travel times. They found that the correlation of the horizontal divergence falls off away from the surface, changes sign at a depth of about $6 Mm$ , and is negative at depths in the range $6 -14 Mm$ (see Figure 59

). In particular, Zhao and Kosovichev (2003a

) find a correlation coefficient of $- 0.5$ at a depth of $10 Mm$ . Zhao and Kosovichev (2003a

) concluded that supergranulation is about $15 Mm$ deep.

Figure 59:

Correlation coefficient between the horizontal divergence at each depth, derived from p-mode time distance helioseismology, and the divergence image of the uppermost layer. The dashed line is from Duvall Jr (1998 ) and the solid line is from Zhao and Kosovichev (2003a ).

We note that inversion results heavily rely on the assumed travel time sensitivity kernels. While most inversions use ray-based kernels, Jensen et al. (2000 ) and Birch and Kosovichev (2000 ) showed that finite-wavelength effects must be taken into account. It may be that wave-based kernels could yield depth inversion results that are qualitatively different from the ones presented above.

Figure 60:

Comparison of the observed horizontal divergence, derived from seismic holography (left-most panels), with those from a control computation which assumes that the Doppler signatures are concentrated at the solar surface (right-most panels). The middle panels show the weighted pupils used in the analysis. The companion Movie 61

shows the observed and control data as a function of focus depth. From Braun and Lindsey (2003

Figure 61:

Companion movie to Figure 60

, showing the observed and control data as a function of focus depth. From Braun and Lindsey (2003

Braun and Lindsey (2003

) used a $24 hr$ MDI data cube and seismic holography to make maps of convective flows at focus depths from $3 Mm$ to $37 Mm$ . The horizontal-divergence maps are shown in the left panels of Figure 60

. The behavior of the divergence maps with depth can be studied by making scatter plots of the divergence values observed at a given depth versus the near-surface values. The slope of a line fit to the scatter is observed to change sign at $10 Mm$ and to remain negative in the range $10- 37 Mm$ . Although it is tempting to interpret the reversal of the horizontal divergence signal at focus depths greater than $10 Mm$ , Braun and Lindsey (2003 ) suspect that this reversal could be due to the increasing contribution of oppositely directed surface flows as the pupil increases in size with focus depth. To test this hypothesis, they considered a control experiment in which the observed flows at $3 Mm$ depth are used to estimate the surface contribution to the seismic measurements at greater focus depths. The divergence maps of the control experiment, which are shown in the right panels of Figure 60

, ressemble the observed divergence maps (left panels). The good correspondence between the two suggests that the observations at depths are mostly due to leakage from the surface (the “showerglass effect”). Modelling efforts by Braun et al. (2004 ) have confirmed that the change of sign of the correlation below $10 Mm$ is a predominantly surface contamination of the velocity signal from neighboring supergranules. However, the fact that the control data do not reproduce exactly the observations may indicate that there is at least some sensitivity to velocities at depth.

It would appear that local helioseismology has not yet provided a definitive answer regarding the depth of supergranulation.

5.3.3 Rotation-induced vorticity

Since the typical lifetime of supergranules is significantly less than the solar rotation period, the influence of rotation on supergranular convection is expected to be small. Solar rotation effects in supergranules were illustrated by Hathaway (1982 ) in non-linear numerical simulations. The Coriolis force causes divergent and convergent horizontal flows to be associated with vertical components of vorticity of opposite signs (on average). In the northern hemisphere, cells rotate clockwise where the horizontal divergence is positive, while they rotate counterclockwise in the convergent flow towards the sinks. The sense of circulation is reversed in the southern hemisphere.

In general, it is not straightforward to predict the statistical properties of the vorticity in rotating turbulent convection: Vorticity production is due to the effect of the Coriolis force and to vortex stretching and tilting mechanisms. The importance of the Coriolis force is characterized by an inverse Rossby number, or Coriolis number, defined by

where $t c$ is a characteristic correlation time and $_O_ eq$ is the equatorial solar angular velocity. The choice $tc = 48 hr$ implies $Co -~ 1$ for supergranulation. At latitude $c$ , quasi-linear theory (Rüdiger et al., 1999 ) predicts that the vertical vorticity of the velocity (denoted below by ’ $curl$ ’) and the horizontal divergence of the horizontal velocity (denoted by ’ $div$ ’) are correlated according to

where the brackets denote an expectation value or longitudinal average. The latitudinal variations are given by the function

and are due to the projection of the local angular velocity vector onto the radial direction.

Figure 62:

Effect of the Coriolis force on supergranular flows. (a) Plot of the correlation coefficient, $C(c)$ , between the vertical vorticity ( $curl$ ) and the horizontal divergence ( $div$ ). (b) Horizontal averages of the vorticity, $<curl>+$ (solid) and $<curl>-$ (dashed), over regions with $div > 0$ and $div < 0$ , respectively. A vorticity of $1 Ms -1$ corresponds to an angular velocity of $2.5o day- 1$ or a typical circular velocity of $10 m s-1$ . From Gizon and Duvall Jr (2003

The effect of rotation on supergranules was first observed by Duvall Jr and Gizon (2000

) and Gizon and Duvall Jr (2003

)² . They computed the vertical vorticity and horizontal divergence from vector flow maps obtained with f-mode time-distance helioseismology ( $1 Mm$ deep, $8 hr$ time invervals). The vertical large-scale vorticity due to differential rotation and meridional circulation was removed to study the vorticity at supergranular scales. Panel (a) of Figure 62

shows the correlation coefficient between div and curl at latitude $c$ ,

where the angle brackets denote a spatial average over the area of a $10o$ longitudinal strip centered around $c$ . In the north, positive (negative) divergence is correlated with clockwise (anticlockwise) vorticity; the correlation changes sign in the south. Thus, away from the equator, the number of right-handed cyclones is not equal to the number of left handed cyclones. The sign and the latitudinal variation of $C(c)$ are both characteristic of the effect of the Coriolis force on the flows. Panel (b) of Figure 62

shows horizontal averages of the vertical vorticity $<curl>±$ versus $f (c)$ , where the averages $<.>+$ and $<.>-$ are restricted to the regions of positive and negative divergence, respectively. Note that there is a small difference between the total area covered by regions of positive and negative divergence (Duvall Jr and Gizon, 2000 ). A nearly perfect linear relationship between $<curl>±$ and $± f(c)$ is observed, with $- 1 <curl>± = ± 3f (c) Ms$ . This is again consistent with the interpretation as a Coriolis effect. Observations may be summarized by a measurement of the average correlation between the horizontal divergence and the vertical vorticity (Gizon and Duvall Jr, 2003

Since $_O_eq/tc = 2 × 10- 11 s-2$ for $tc = 48 hr$ , the naive prediction from Equation (91

) is one order of magnitude smaller than the measurements (Equation 94

). Recent numerical simulations of rotating convection by Egorov et al. (2004

) give values of $<divcurl>$ near the top of the convection zone that appear to be close to the observations. They show that a good agreement with the data is obtained for a reasonable Coriolis number (their definition of the Coriolis number is slightly different than ours; Egorov, 2004, private communication). Egorov et al. (2004 ) predict that the amplitude of the $curl$ - $div$ correlation should decrease fast with depth.

5.3.4 Pattern evolution

As mentioned above, the pattern rotation rate of supergranulation derived from correlation tracking of Doppler scans appears to be significantly faster than the spectroscopic rate (for a $24 hr$ time lag). Correlation tracking algorithms can also be applied to images of supergranular flows derived from local helioseismology, such as the horizontal divergence signal. Unlike raw Doppler images, the divergence signal has uniform sensitivity across the solar disk and is subject to few systematic errors.

Figure 63:

Anomalous pattern motion of supergranulation measured by tracking the horizontal divergence of the flow field observed with time-distance helioseismology. (a) Pattern rotation for time-lags $Dt = 6$ , $8$ , $16$ , and $22 hr$ . The cross shows the equatorial pattern rotation measured by Snodgrass and Ulrich (1990

) for $Dt = 24 hr$ from Mount Wilson Dopplergrams. (b) Meridional motion of the pattern as a function of time-lag $Dt$ . Notice that the meridional motion appears to be equatorward for $Dt > 20 hr$ . A P-angle correction was applied to the data. From Gizon and Duvall Jr (2003

Gizon and Duvall Jr (2003

) used a 60-day sequence of $90o × 90o$ divergence maps obtained with f-mode time-distance helioseismology ( $o 0.24$ spatial sampling). The original MDI Doppler velocity images were tracked at the Carrington rate to remove the main component of rotation. Small regions, apodized by a Gaussian surface with a full width at half maximum of $3.84o$ and separated by a time-lag $Dt$ , were spatially cross-correlated. For $Dt < 1.5 d$ the peak of maximum correlation is easy to identify, and the spatial displacement $Dx$ at correlation maximum gives an apparent pattern velocity $U LCT = Dx/Dt$ , where the subscript LCT stands for ’local correlation tracking’. Figure 63

shows that the apparent motion of the supergranulation pattern depends on $Dt$ . In particular, the pattern rotation rate increases rapidly with $Dt$ . The apparent meridional motion is poleward for $Dt < 18 hr$ with a very small amplitude ( $< 5 m s- 1$ ), while, suprisingly, it is slightly equatorward for greater time lags. The apparent rotation of supergranulation obtained by tracking the divergence maps confirms the original findings of Duvall Jr (1980 ) and Snodgrass and Ulrich (1990 ) obtained by tracking Dopplergrams.

Figure 64:

Movie showing the evolution of the spatial cross-correlation of the divergence signal at the equator as a function of $Dt$ from $0$ to $5.5 d$ . (Movie still for time lag $Dt = 24 hr$ .) The original MDI Doppler velocity images were tracked at the Carrington rate.

Figure 65:

Movie showing the evolution of the average map of the divergence signal in a $15o × 15o$ equatorial region as a function of tracking velocity in the range from $- 200 m s-1$ to $200 m s- 1$ with respect to the Carrington velocity. The movie still displays the temporal average ( $5 d$ ) tracked at a velocity $-1 125 m s$ above the Carrington velocity. This average is constructed from maps derived from time-distance helioseismology every $8 hr$ . Courtesy of J. Zhao.

Figure 66:

Ratio $sx/sy$ , as defined by Lisle et al. (2004

), at the equator as a function of tracking velocity $vt$ (offset with respect to Carrington velocity). This curve is an average over one month of data. The vertical dashed lines correspond to tracking offsets $- 8 m s-1$ and $123 m s- 1$ . Courtesy of J. Zhao.

Beck and Duvall Jr (2001 ) showed that the observed correlation function of the divergence signal switches sign at $Dt ~ 40 hr$ . The same analysis is repeated here with the 60-day sequence used by Gizon and Duvall Jr (2003

). Figure 64

shows a still at $Dt = 24 hr$ from a movie of the temporal evolution of the spatial correlation, with $Dt$ varying from zero to $5.5 d$ (equatorial region, $6 hr$ time steps, Carrington tracking rate). The spatial cross-correlation decays and oscillates as a function of time. For $Dt < 36 hr$ , the peak of maximum correlation is well defined and is surrounded by a ring of negative correlation due to adjacent supergranules. For $Dt > 48 hr$ a peak of negative correlation has appeared (east), which is surrounded by a ring of positive correlation. When the analysis is repeated away from the equator, one finds that the peak of negative correlation forms poleward of the east direction. For $Dt -~ 5.5 d$ the main correlation peak is again positive and is surrounded by a ring of negative correlation.

Earlier observations of solar convection assumed that supergranulation can be characterized by an autocorrelation function that exhibits a simple exponential decay in time (Harvey, 1985 ; Kuhn et al., 2000 ). It is now obvious that supergranulation does not follow such a simple model. The oscillation period of the correlation function, of the order of $6 d$ , suggests an underlying long-range order. As shown below, these puzzling observations are easier to describe in Fourier space (Section 5.3.5).

Another interesting method of analysis to study the evolution of the supergranulation pattern is described by Lisle et al. (2004 ). They used horizontal-divergence maps obtained from local correlation tracking of granules ( $1 min$ time lag). Lisle et al. (2004 ) constructed temporal averages ( $8 d$ ) of the divergence maps for different tracking velocities $vt$ . The spatial variations of a $o o 15 × 15$ section of the time-averaged divergence map were characterized by the rms values in the east-west ( $sx$ ) and south-north ( $sy$ ) directions. Lisle et al. (2004 ) found that at the equator the ratio $sx/sy$ is maximum at a tracking velocity $U LRT$ which is $110 m s-1$ above the Carrington velocity, i.e., faster than any rotation rate measured so far. They suggested that a ratio $sx/sy$ greater than 1 is an indication that supergranules are preferentially aligned in a north-south direction. Junwei Zhao has recently repeated their analysis using time-distance divergence maps. Figure 66 and the companion Movie ?? shows a $15o × 15o$ time-averaged divergence map for various tracking velocities. Figure 66 shows the ratio $sx/sy$ at the equator as a function of tracking velocity, averaged over all available data. The maximum occurs at a tracking rate $- 1 123 m s$ above the Carrington velocity. It is evident that there is another local maximum $8 m s-1$ below the Carrington velocity. This local maximum could only be hinted at from the plots of Lisle et al. (2004 ) due to lack of averaging. We will come back to this observation near the end of the next section.

5.3.5 Traveling-wave convection

The 3D power spectrum of the divergence signal was studied first by Gizon et al. (2003 ). The same analysis was extended by Gizon and Duvall Jr (2004 ) to cover the period from 1996 to 2002. They considered series of MDI full-disk Dopplergrams from the Dynamics campaigns (two to three months each year). Dopplergrams were tracked at the Carrington angular velocity to remove the main component of rotation. Every $12 hr$ , a $o o 120 × 120$ map of the horizontal divergence of the near-surface flows was obtained using f-mode time-distance helioseismology. At a given target latitude $c$ a longitudinal section of the data was extracted, $10o$ wide in latitude. Gizon and Duvall Jr (2004 ) rearranged the data in a frame of reference with angular velocity $_O_mag(c) = 14.43o- 1.77o sin2 c - 2.58o sin4c d -1$ (rotation rate of small magnetic features; Komm et al., 1993b ). This choice of reference is convenient, although arbitrary. The position vector is denoted by $x = (x,y)$ in the neighborhood of latitude $c$ , where coordinate $x$ is prograde and $y$ is northward, with a spatial sampling of $2.92 Mm$ in both coordinates.

Figure 67:

Power spectrum of the horizontal divergence signal for $c = 0$ (equator) in 1997. Divergence maps were tracked in a frame of reference with angular velocity $_O_mag(c = 0)$ (small magnetic features). Left panel: Cylindrical section at constant wavenumber $k = 115/Ro .$ , as a function of azimuth $y$ and frequency $w/2p$ . Middle panel: Fit to the data according to the model described in the text. The values of the parameters are measured at fixed $k$ by a two-dimensional fit to the power spectrum in ( $y$ , $w$ )-space. For this particular example, $j = 1.15$ and $ymax = - 2o$ (excess power in the prograde direction). Right panel: Ratio of the data to the fit; no bias is observed. From Gizon and Duvall Jr (2004

The divergence signal was decomposed into its harmonic components $exp(ik .x - iwt)$ through FFT, where $w$ is the angular frequency and $k = (kx,ky)$ is the horizontal wavevector. The power spectrum of the horizontal divergence signal, denoted by $Pdiv(k,w)$ , was computed for each target latitude $c$ with $5o$ steps. Gizon et al. (2003

) discovered that, in the range with $50 < kRo . < 200$ , the power spectrum of the solar divergence signal can be described accurately by the following parametric model:

where

and

The functions $A$ , $w0$ , $U$ , $g$ , and $B$ , described below, were determined from fits to the data. The fits took into account the $w$ -convolution of the solar power spectrum with the power in the observation window. The representation of $P$ in terms of $F$ is convenient as it implies $P (-k, - w) = P (k,w) div div$ , i.e., the divergence signal is real.

At fixed $k$ , the power spectrum of the horizontal divergence can be described by the sum of two Lorentz functions with independent amplitudes $A(k)$ and $A( - k)$ and with central frequencies $wr(k)$ and $-wr(- k)$ respectively. The function $A(k)$ depends on the direction of $k$ and can be parametrized as follows:

where the azimuth $y$ is the direction of the wavector such that $k = (k cosy,k siny)$ , e.g., $y = 0$ when $k$ points prograde and $y = p/2$ when $k$ points north). The $A2$ component includes instrumental astigmatism: a purely spatial distortion which can not easily be separated from a real solar signal. The $A1$ component, however, must be of solar origin. The power anisotropy may be defined by the ratio $j = 2A1/A0$ , while $ymax$ was called the azimuth of maximum power. The background noise $B$ , which is small, was assumed to be independent of frequency but affected by astigmatism. The half width $g$ at half maximum of the Lorentz profile corresponds to a characteristic $e$ -folding lifetime $t = 1/g$ . The frequency shift $k .U$ was interpreted to be a Doppler shift produced by a horizontal flow $U = (Ux, Uy)$ measured in the frame of reference (rotating at $_O_mag$ ), as is done in ring-diagram analysis (Section 4.2.2).

Figure 67 shows a cyclindrical cut in the equatorial power spectrum at a constant $k$ typical of the supergranulation, and a model fit to the data according to Equations (95 , 96 , 97 ). For each azimuth $y$ , the power has two peaks at frequencies $w+ = wr(k)$ and $w- = - wr(- k)$ . Observations show that the difference $w+ - w- = 2w0(k)$ is independent of azimuth (Gizon et al., 2003 ). No Galilean transformation can cause these peaks to coalesce, at zero frequency or otherwise. This implies that supergranulation undergoes oscillations. In the plane $y$ - $w$ , power is distributed along two parallel sinusoids at frequencies $± w0 + kUx cos y + kUy sin y$ . The presumption by Rast et al. (2004 ) that power is distributed along two intersecting sinusoids must be rejected: It is inconsistent with the observations. As one does in helioseismological ring analysis (Schou and Bogart, 1998 ), it is natural to interpret the frequency shift $k .U$ as a Doppler shift produced by an advective flow $U$ (some average flow in the supergranulation layer). The weak $k$ -dependence of the measured $U$ in the range $40 < kRo . < 180$ is consistent with this interpretation. Furthermore the functional form of the power is preserved for different tracking rates (i.e., different latitudes). As shown in Figure 68 , it is remarkable that the inferred rotation $Ux$ and meridional circulation $Uy$ are both similar to that of the small magnetic features (Komm et al., 1993b ,c ). This property is consistent with the view that magnetic fields are advected by supergranular flows. In addition, the torsional oscillations are seen with the same phase and amplitude as in global helioseismology. The temporal changes in the meridional circulations are also consistent with an inflows toward the mean latitude of activity (see Section 5.3.4).

Figure 68:

(a) Advective flow $Ux$ versus $c$ at $kRo . = 115$ . Each curve corresponds to a different year (from blue in 1996 to red in 2002). $Ux$ is measured with respect to the reference $R o. _O_mag cosc$ (small magnetic features). Only the north-south symmetric component is shown. (b) Residuals of $Ux$ after subtraction of a smooth fit (dashed line in panel a). The scale bar has units of m s-1. The black curve is an estimate of the mean latitude of activity. (c) Advective flow $U y$ versus $c$ at $kRo . = 115$ . Only the north-south antisymmetric component is shown. (d) Residual meridional circulation with respect to the dashed line in panel (c). The scale bar has units of m s-1.

Figure 69:

(a) Oscillation frequency $n0 = w0/(2p)$ versus $kRo .$ at latitudes $c = 0o$ (solid), $c = ± 25o$ (dotted), and $c = ±50o$ (dashed). For reference, the orange curve is $n = 1.8(kR /120)0.45m Hz o .$ . Also shown is the half width at half maximum (HWHM, $g/(2p)$ ) of the Lorentzian profiles for the same latitudes. The quality factor is $Q = w0/g -~ 2$ at $kRo . = 120$ . (b) Power spectrum corrected for rotation and meridional circulation and averaged over azimuth and latitude. The distribution of power as a function of frequency is affected only by the known temporal window function. From Gizon et al. (2003 ).

The dynamics of the supergranulation is best studied once the background flow $U$ has been removed. In a co-moving frame, each spatial component oscillates at a characteristic frequency $w0$ . There is a clear relationship between $w0$ and the wavenumber $k$ , well described by a power law (see panel (a) of Figure 69

This is a fundamental relationship as it is measured to be essentialy independent of $y$ , $c$ , and the phase of the solar cycle. The data are consistent with a spectrum of traveling waves with a dispersion relation $w = w0(k)$ . The azimuthally averaged power spectrum is shown in panel (b) of Figure 69

. In the range $60 < kRo . < 160$ , the peak of power away from zero temporal frequency is well resolved since the quality factor $Q = w0/g$ is larger than 1. Near $kRo . = 120$ we have $Q -~ 2$ . The variations of the oscillation frequency with latitude and time are less than 5% during the period 1996-2002 for $kR o. = 115$ . Since $w0$ and the dominant size of supergranules are observed to be essentially independent of latitude, the general dynamics determining the time scale and the spatial scale of supergranulation is not affected by the Coriolis force associated with the large scale vorticity (rotation). However, there is a pronounced anisotropy in the azimuthal distribution of wave power at fixed $k$ . The power anisotropy is measured to be $j -~ 1$ for $kRo . = 115$ . Furthermore, power is maximum in a direction equatorward of the prograde direction in both hemispheres: For $|c |< 25o$ observations show $ymax -~ -c$ , and for $|c|> 25o$ $y -~ - 25osignc max$ . The pattern therefore senses the effect of rotation. A snapshot of the divergence field would not reveal this as the sum of the powers measured in opposite directions is nearly isotropic. We recall that the effect of rotation on supergranulation was detected in the vorticity field.

The lifetime of supergranules is about $2.3 d$ at $kRo . = 115$ , i.e., a somewhat larger value than other estimates derived from the decay of the autocorrelation function. This is because the lifetime is not strictly given the decay of the correlation function at short time lags, but by the decay of the envelope of the correlation function, which oscillates. The lifetime decreases by about 20% at active latitudes (Gizon and Duvall Jr, 2004 ).

As mentioned earlier, estimates of supergranulation rotation obtained by tracking the Doppler pattern are systematically found to be higher than the rotation of the magnetic network (for large time lags, see panel (a) of Figure 63 ). This apparent superrotation of the pattern can now be understood as the result of the waves being predominantly prograde. The east-west motion of the pattern is effectively a power-weighted average of the true rotation and the non-advective phase speed $uw = w0/k ~ 65 m s-1$ for $kRo . = 120$ . Similarly, the excess of wave power toward the equator is reflected in the equatorward meridional motion of the pattern at large time lags (see panel (b) of Figure 63 ) even though the advective flow is poleward. Correlation tracking measurements must take into account the fact that the pattern propagates like a modulated traveling wave.

The measurement of the pattern rotation obtained by Lisle et al. (2004 ) is also a direct consequence of the power spectrum. It is straightforward to show that the ratio $sx/sy$ (see Section 5.3.4) can be expressed in terms of $P (k,w) div$ alone. Plugging in Equations (95 , 96 , 97 ), one finds that at the equator the variations of $sx/sy$ as a function of tracking velocity $vt$ should display two peaks at $vt -~ Ux ± uw$ , where $-1 uw -~ 65 m s$ is the dominant wave speed and $Ux$ is the advective velocity. The amplitudes of the peaks are approximately given by

using mean values $j - ~ 0.9$ and $Q -~ 2$ . The separation between the two peaks, $-1 2uw -~ 130 m s$ , is precisely the observed value (Figure 66

). The super-rotation of the pattern obtained by Lisle et al. (2004 ), $ULRT -~ Ux + uw$ , is understandably faster than measured by other methods, and faster than $Ux$ which is close to the magnetic feature rate. Figure 66

is not only a direct consequence of the observed power spectrum, but it is additional evidence for the existence of prograde and a retrograde components (in a frame rotating at $Ux$ ) as well as excess power in the prograde direction. Astigmatism ( $A2$ ) would also affect the ratio $sx/sy$ , but it was neglected in Equation (100

) for the sake of simplicity. It is worth noting that $sx/sy > 1$ is not a sufficient condition to imply that supergranules are aligned in the north-south direction, since $s /s x y$ depends on $P div$ only (phases are needed for pattern characterization).

It was a source of concern that the wavelike properties of supergranulation had not been seen previously in surface Doppler shifts. Schou (2003b ) showed that the same phenomenon can be observed directly using MDI Dopplergrams, thereby confirming the observations of Gizon and Duvall Jr (2003 ). In addition to confirming those results, Schou (2003b ) was able to extend the measurements f