The Sun’s Supergranulation
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Abstract
The Sun’s supergranulation refers to a physical pattern covering the surface of the quiet Sun with a typical horizontal scale of approximately 30,000 km and a lifetime of around 1.8 d. Its most noticeable observable signature is as a fluctuating velocity field of 360 m s^{t1} rms whose components are mostly horizontal. Supergranulation was discovered more than fifty years ago, however explaining why and how it originates still represents one of the main challenges of modern solar physics.
A lot of work has been devoted to the subject over the years, but observational constraints, conceptual difficulties and numerical limitations have all concurred to prevent a detailed understanding of the supergranulation phenomenon so far. With the advent of 21st century supercomputing resources and the availability of unprecedented highresolution observations of the Sun, a stage at which key progress can be made has now been reached. A unifying strategy between observations and modelling is more than ever required for this to be possible.
The primary aim of this review is therefore to provide readers with a detailed interdisciplinary description of past and current research on the problem, from the most elaborate observational strategies to recent theoretical and numerical modelling efforts that have all taken up the challenge of uncovering the origins of supergranulation. Throughout the text, we attempt to pick up the most robust findings so far, but we also outline the difficulties, limitations and open questions that the community has been confronted with over the years.
In the light of the current understanding of the multiscale dynamics of the quiet photosphere, we finally suggest a tentative picture of supergranulation as a dynamical feature of turbulent magnetohydrodynamic convection in an extended spatial domain, with the aim of stimulating future research and discussions.
Keywords
Solar Surface Dissipation Scale Turbulent Convection Oscillatory Convection Solar Convection Zone1 Introduction
The story of supergranulation really started in Oxford when Avril B. Hart reported in 1953 the existence of a “noisy” fluctuating velocity field on top of the mean rotation speed of the solar equator that she was measuring (Hart, 1954). Actually, it is most probable that this “noise”was already detected as early as 1915 by Plaskett (1916). Two years after her first detailed report, Hart (1956) confirmed the discovery and was able to give the first estimate^{1} of 26 Mm for the typical horizontal length scale of these “velocity fluctuations” (sic). Supergranulation was further recognised as a characteristic feature of the whole surface of the quiet Sun (the regions of weak magnetic fields, which represent the most important part of the solar surface) after the seminal work of Leighton et al. (1962), who published the first Doppler images of the Sun (and also the first detection of the five minutes oscillations). This work was soon after completed by another important paper by Simon and Leighton (1964) who showed, amongst other results, the intimate relation between supergranulation and the magnetic network of the quiet Sun.
It is remarkable that all the fundamentals of supergranulation have basically been uncovered over that 1954 – 1964 decade. Since then, progresses have been much less spectacular, especially on the theoretical side. This is certainly why supergranulation is still a fascinating subject. We are still wondering where it comes from, what its exact relation with magnetic fields is, if it is a universal feature of solar type stars, or of stellar surface convection, if it plays a role in the solar dynamo(s), etc. All these questions are pending fifty years after the discovery. There are many reasons why the solar physics community has not yet managed to answer them, several of which are not actually specific to the supergranulation problem. Most of these reasons are described in detail in this review, but it is worth pointing out a few important issues here as an introduction.
For a long time only a limited set of observables and restricted timerecords of the evolution of the supergranulation pattern have been available, may it be on short (24 consecutive hours) or long timescales (a solar cycle). This has somewhat hindered the study of the detailed spatial structure of supergranulation and the identification of the physical factors that affect it (buoyancy, stratification, magnetic fields, rotation). From the theoretical and numerical perspectives, on the other hand, the strongly nonlinear physical nature of magnetised thermal convection in the outer layers of the Sun makes it extremely difficult to come up with a simple, unique, verifiable physical model of the process (a similar problem arises in many subfields of solar physics, if not of astrophysics). Overall, these observational uncertainties and theoretical or numerical limitations have somewhat negatively interfered to prevent a rigorous solution to the supergranulation problem. The currently fairly obfuscated state of affairs may nevertheless get clarified in a near future, as the solar physics community is now armed with both highresolution solar observatories such as Hinode^{2} and large supercomputers that allow for increasingly realistic numerical simulations of the complex solar surface flows. However, it is clear that stronger connections between theory, numerics, and observations need to be established for the problem to be resolved.
In this review, we would like to introduce readers to this subject by first describing the full range of past and current research activities pertaining to the solar supergranulation, from the breadth of historical and modern observational results to the most elaborate numerical models of supergranulation convection. We particularly wish to provide a useful guide to the abundant literature related to that theme, to point out the important findings in the field, but also to stress the limitations and difficulties that have been encountered over the years in order to help overcome them. To this end, we attempt to discuss the already existing or possible connections between various pieces of research and try to identify some important questions whose answer may be crucial to understand how and why supergranulation originates.
The review is divided into eight parts, including this introduction. The next two sections offer some introductory material on the physics of deep and surface convection in the Sun (Section 2) and a brief recap on smallscale flows, namely granulation and mesogranulation (Section 3.1). Section 4 is dedicated to a presentation of observational facts that have been collected on supergranulation. We then carry on with the discussion of existing theoretical models to explain the origin of supergranulation in Section 5. In Section 6, we discuss related numerical experiments. Our current knowledge on supergranulation is summarised and commented in Section 7 for the convenience of hurried readers. In the light of the present understanding of multiscale dynamics of the quiet photosphere, we finally suggest a tentative picture of supergranulation as a dynamical feature of turbulent magnetohydrodynamic (MHD) convection in an extended spatial domain, with the aim of stimulating future research and discussions. We notably propose several numerical and observational diagnostics that could help make important progress on the problem in the near future (Section 8).
We tried to make the paper readable by all astrophysicists, assuming only little background in that field and trying to avoid as much as possible the solar physicists jargon, or to explain it when necessary.
2 General Considerations
2.1 Solar convection
To better appreciate the following discussions, we find it useful to start this section with a brief, nonexhaustive recap of the general properties of solar convection. Readers interested in various specific aspects of solar convection such as deep convection or the fine structure of granulation will find much more exhaustive accounts on these matters in the reviews by Miesch (2005) and Nordlund et al. (2009).
The outer layers of the Sun, 30% in radius, 2% in mass, are strongly stratified: the density ratio between the visible surface and the 0.3R_{⊙}. depth is close to 10^{6} (see Stix, 2004). This region is unstable with respect to thermal convection and is hence referred to as the solar convection zone (SCZ). The solar luminosity is mostly transported by fluid motions driven by thermal buoyancy in the SCZ: physically, the strongly nonadiabatic cooling of the surface layers, like the top cold plate of a convection experiment, imposes a strong negative entropy gradient below the surface, where the heat flux coming from the solar interior cannot be evacuated by microscopic heat diffusion alone. This entropy gradient gets smaller at larger depth as a consequence of the efficiency of convective mixing, but remains negative down to ∼ 0.3R_{⊙}, where it changes sign and becomes stable against convection.
2.2 Scales in turbulent convection

The injection range is the typical range of scales at which kinetic energy is injected into turbulent motions. In turbulent convection, injection of kinetic energy is due to the work of the buoyancy force. The scale most representative of the injection range under these conditions is called the Bolgiano scale L_{ B } (Bolgiano, 1959; L’vov, 1991; Chillá et al., 1993; Rincon, 2006). It can be shown, based on purely dimensional arguments and scaling considerations for heat transport in turbulent convection, to be almost always of the same order (up to some order one prefactor) as the local typical scale height (Rincon, 2007). In an incompressible thermal convection experiment, this corresponds to the distance between the hot and cold plates, but in the strongly stratified SCZ, a more sensible estimate is the local pressure scale height. Close to the surface, the Bolgiano scale is therefore comparable to or slightly larger than the granulation scale L_{ G } ∼ 1 Mm (Section 3.1 below). As one goes to deeper layers, the pressure scale height gets larger and larger as a consequence of the strong stratification, and so does the Bolgiano scale. So, in the SCZ, the injection scale basically increases with depth and ranges from 1 Mm close to the surface to 100 Mm close to the bottom of the SCZ.

A rough estimate for the viscous dissipation scale l_{ v } can be obtained from the Kolmogorov phenomenology of turbulence (Frisch, 1995) via the expression l_{ v } ∼ Re ^{3/4}L, where L stands for the injection scale and Re = LV/v, V being the typical velocity at the injection scale^{3}. In the SCZ, where v ∼ 10^{−3} m^{2} s^{−1} (Rieutord, 2008), we find l_{ v } ∼ 10^{−3} m at the surface, assuming L ∼ L_{G} ∼ 1 Mm and V ∼ 1 km s^{−1} (the typical granulation scale and velocity). At the bottom of the SCZ, where the injection scale is much larger, one can estimate similarly that l_{ v } ∼ 0.1 m. Hence, l_{ v } is everywhere extremely small and not available to observations.

In MHD, the relative value of the magnetic dissipation scale l_{ n } with respect to the viscous cutoff scale l_{ v } depends on the ordering of dissipative processes (see, e.g., Schekochihin et al., 2007) in the fluid. When the magnetic diffusivity η is much larger than the kinematic viscosity v, as is the case in the Sun, one may use l_{ n }/l_{ v } ∼ Pm^{3/4} (Moffatt, 1961), where Pm = v/η is called the magnetic Prandtl number^{4} and we have assumed a Kolmogorov scaling for the velocity field. The magnetic diffusivity in the subsurface layers of the Sun is η ∼ 10^{2} m^{2} s^{−1} (Spruit, 1974; Rieutord, 2008), so Pm ∼ 10^{−5}. Consequently, l_{ n } ∼ 100 m close to the surface (see also Pietarila Graham et al., 2009). This is also very small in comparison to the resolution of current observations, but is much larger than l_{ v }. Close to the bottom of the SCZ, Pm ∼ 10^{−1} – 10^{−2}, so l_{ n } ∼ 1 m.

The thermal dissipation scale l_{ K } is very important in the solar context, as it is the largest of all dissipation scales in the problem. In the SCZ, the thermal diffusivity K is everywhere much larger than the kinematic viscosity v, so the thermal Prandtl number Pr = v/K is very small. Under these conditions, we may estimate l_{ k } from the expression^{5} l_{ k } / l_{ v } ∼ Pr^{3/4}, once again assuming a Kolmogorov scaling for the velocity field. Thermal diffusion in the Sun is insured by photons, and thus depends strongly on the opacity of the fluid. In the deep SCZ, Pr ∼ 10^{−4} – 10^{−6}, so l_{ k } ∼ 500 m. In the very surface layers, l_{ k } is comparable to the scale of granulation L_{G} ∼ 1 Mm, at which heat advection and radiation are comparable (see Section 3.1 below). This is not small anymore in terms of solar observations, but remains nevertheless smaller than the typical scale of supergranulation L_{SG}.
2.3 The supergranulation puzzle
From the previous discussion, the scales that can be constructed from standard arguments on turbulence and convection all appear to be much smaller than the supergranulation scale in the surface layers. Besides, as one goes deeper into the stratified SCZ, the injection scale increases smoothly and monotonically with depth, so the supergranulation scale does not show up as a special scale in deep layers either in this simple scenario. Overall, we may therefore conclude that understanding supergranulation from simple arguments based on available theories of turbulent convection is not possible. In this respect, it is worth pointing out that supergranulation lies at the largescale edge of the injection range of turbulence at the solar surface, and might therefore not be directly correlated with hydrodynamic turbulent processes.
Uncovering the origin of supergranulation probably requires invoking physical processes that are not present in the most simple descriptions of turbulence. These processes may be specific to the solar context (e.g., surface radiation, chemical composition, ionisation states) or may have more general dynamical origin (instabilities related to the interaction with rotation, magnetic fields, shear, nonlocal scale interactions in a turbulent flow). As we shall see in Sections 5, 6, and 8, a variety of qualitative physical scenarios based on one or several such processes has been proposed in the past but, as yet, they have not provided a fully comprehensive, predictive, and verifiable theoretical model for supergranulation.
If we succeed one day in explaining the origin of the solar supergranulation, we may very well gain some new insight into turbulent convection or discover completely new physical effects at the same time. Attempting to solve the supergranulation problem therefore represents a very exciting challenge not only from the astrophysics point of view, but also from a fundamental physics perspective.
3 Flows at Scales Smaller than the Supergranulation
Before we head into the specific topic of supergranulation, a brief overview of smallerscale flows at the Sun’s surface, the wellknown granulation and the more controversial mesogranulation, is required to set the stage completely.
3.1 Granulation
The solar granulation is an intensity pattern with a contrast around 15%, which displays cellular convective motions with length scales ranging from ∼0.5 Mm to 2 Mm. From recent observations with the Solar Optical Telescope on board the Hinode satellite (Ichimoto et al., 2004; Suematsu et al., 2008), Rieutord et al. (2010) showed that power spectra of both intensity fluctuations and vertical velocities have a maximum amplitude at a scale around 1.7 Mm, with a weak dependence on photospheric height for V_{ z } Previous work had shown that the typical lifetime of granules is 10 min and that the associated velocities range from 0.5 to 1.5 km s^{−1} (Title et al., 1989). Hundreds of observations of the solar granulation have been done, and several reviews are dedicated to this subject. We notably refer the reader to Spruit et al. (1990) and Nordlund et al. (2009).
The granulation pattern is certainly the best understood feature of solar convection. Most notably, it is well reproduced by numerical simulations (Section 6). A remarkable property is that the advection of heat by the velocity field and the radiation of heat proceed on comparable timescales in a granule, so the corresponding Péclet number is order unity. As already discussed in Section 2.2, this means that the scale of granulation is comparable to the thermal dissipation scale. Physically, the granulation pattern corresponds to a thermal boundary layer formed in the strongly nonadiabatic surface region of the SCZ where the solar plasma becomes optically thin (see Figure 1).
3.2 Mesogranulation
Mesogranulation refers to flows at scales between the granulation and the supergranulation scales. It was first reported by November et al. (1981), who identified a pattern of prominent vertical motions at scales of the order of 8 Mm by timeaveraging Doppler images. This was thought as a significant finding because it seemed to provide the missing piece in the theory of multiscale convection at the solar surface (see Section 5) formulated by Simon and Leighton (1964): granulation was associated with the ionisation of Hydrogen, while mesogranulation and supergranulation were associated to the first and second ionisation of Helium, respectively (viz. November et al., 1981). Subsequent groundbased observations (November and Simon, 1988) and spacebased observations using the SOUP instrument on SpaceLab 2 (Title et al., 1989) gave some extra weight to this result.
However, the very existence of mesogranulation as a specific distinguishable convection scale at the solar surface remains a source of debate in the solar community. Wang (1989), Chou et al. (1991), and Straus et al. (1992) did not find any local maximum in the power spectrum (the scalebyscale distribution of energy) of solar convection that would correspond to mesogranulation. Ginet and Simon (1992) and Chou et al. (1992) came to an opposite conclusion. Power spectra computed from MDI observations by Hathaway et al. (2000) only revealed two peaks at granulation and supergranulation scales.
November (1994) wrote that the term “mesogranulation” was misleading and instead suggested to interpret this feature as “the vertical component of the supergranular convection”. Then, Straus and Bonaccini (1997) argued that mesogranulation was a mere powerful extension of granulation at large scales and Roudier et al. (1999b) and Rieutord et al. (2000) suggested that mesogranulation was likely an artefact produced by the correlation tracking algorithm. This view was disputed by Shine et al. (2000) because they found mesogranules in the range of 4 – 7 Mm, with a lifetime of 3 to 6 h (they used local correlation tracking on a 45 h MDI record of wide field images). Soon after, Lawrence et al. (2001), using a new technique based on wavelets applied to MDI images, found what they called a mesogranulation peak, but at 4 Mm, somewhat shorter compared to previous values.
The very recent work of Matloch et al. (2009) and Rieutord et al. (2010) may finally bring this debate to a conclusion. Indeed, Matloch et al. (2009) devised a simple model of granulation which mimics very well the fusion and splitting of granules. A conclusion of this work is that the statistical properties and behaviour of mesogranulation structures are consistent with the results of spatial and temporal averaging of random data. This conclusion underlines the fact that previous detection of mesogranulation were very likely mislead by the weird consequences of averaging procedures. On the other hand, using Doppler measurements of vertical velocities from Hinode/SOT, Rieutord et al. (2010) did not find any spectral signature of a distinguishable scale in between granulation and supergranulation.
To conclude, it is very likely that mesogranulation is a ghost feature of surface convection generated by averaging procedures. In our opinion, the most recent observational results strongly argue against the existence of a genuine surface feature similar to granulation or supergranulation. To avoid any misunderstanding, we shall hereafter refer to the scales in the range of 4 to 12 Mm as the mesoscales. These length scales are between the smallest scale of supergranulation (12 Mm, see Section 4.2 below) and the largest scale of granulation (4 Mm).
4 Observational Results on Supergranulation
We shall now review the current observational knowledge on solar surface convection from granulation to supergranulation scales. We first introduce the main detection methods of solar surface flows (Section 4.1), which are central to the identification and characterisation of the supergranulation pattern and expose the main observational results.We then review the numerous observational findings on the scales of supergranulation (Section 4.2), the measurements of supergranulationscale intensity variations (Section 4.3), the inferred depth of the pattern (Section 4.4), and its interactions with rotation (Section 4.5) and magnetic fields (Section 4.6). Some words of conclusion follow.
4.1 Methods
Let us recall that supergranulation is first of all a feature of the surface velocity field at the surface of the quiet Sun. Three methods are currently able to detect the associated signal. These are Doppler imaging, granule tracking, and local helioseismology.
4.1.1 Dopplergrams
4.1.2 Tracking
Another way to infer the velocity fields of the solar plasma in the photosphere is to track various structures visible at the surface. The idea is that smallscale structures like granules (see Section 3.1 below) are simply advected by largescale flows. This technique is used in three different algorithms: the local correlation tracking (LCT), the coherent structure tracking (CST), and the balltracking (BT). The first one determines the motion of features on an image by maximising the correlation between small subimages (November and Simon, 1988). The second method identifies coherent structures in the image by a segmentation process and then measures their displacement (e.g., Roudier et al., 1999a; Rieutord et al., 2007; Tkaczuk et al., 2007). The third algorithm (BT) follows the displacement of floating balls over the intensity surface of images. The motion of the floating balls traces the mean motion of granules; this is presumably more effective computationally speaking than LCT and CST (Potts et al., 2004).
The principles and accuracy of granule tracking with LCT or CST were tested by Rieutord et al. (2001) using synthetic data extracted from numerical simulations. They showed that flows at scales larger than 2.5 Mm are well reproduced by the displacements of granules. At shorter scales, the random motion of granules (which are dynamical structures) generates a noise that blurs the signal. The 2.5 Mm lower limit was recently confirmed by Rieutord et al. (2010) with observations using the Hinode/SOT data.
4.1.3 Local helioseismology
This method uses the propagation of acoustic or surface gravity waves (fmodes) to determine the velocity of the medium over which they propagate. Basically, if the wave velocity is c and that of the fluid is V, a plane wave travelling downstream shows a velocity V + c whereas the one travelling upstream moves with a velocity V  c. The sum of the two measured velocities gives that of the fluid. However, the phase velocity of the waves is not directly measurable: the observable quantity is the local oscillation of the fluid which results from the superposition of many travelling waves. A proper filtering is thus needed to select the desired wave; this operation requires a true machinery. We refer the reader to the review of Gizon and Birch (2005) for a detailed introduction to this subject. Here, we just want to recall some basic information about the output of this technique: the spatial resolution at which velocity fields can be measured is around 5 Mm, and the time resolution for timedistance helioseismology is around 8 h. This is lower than what can be achieved with other methods but, in exchange, this technique is the only one that can probe the vertical profiles of the velocities and image the subphotospheric dynamics. Typically, vertical variations can be evaluated down to 10 – 15 Mm below the surface, but the accuracy of measurements deeper than 10 Mm is still debated. A comparison between the tracking and helioseismic reconstructions of largescale solar surface flows was done by Švanda et al. (2007), who found very good agreement between the two.
4.2 The scales of supergranulation
4.2.1 Length scales
The horizontal scale of the supergranulation velocity field was the very first physical characteristic of the pattern to be measured. Using correlation of the signal, Hart (1956) found a typical length of 26 Mm. Since this pioneering work, this value has oscillated around 30 Mm. To appreciate correctly the values that are given in the literature, one should have in mind that supergranulation is a fluctuating, disordered pattern, hence only its statistical properties make sense. We should also remember that each technique has its own biases and gives values according to these biases.
A first technique to determine the typical length scale of the supergranulation is to measure the position of the maximum of spectral power or the correlation length of the horizontal velocity fields. The autocorrelation of Dopplergrams was first used in the seminal work of Leighton et al. (1962) and Simon and Leighton (1964), who gave the value of 32 Mm for the supergranulation length scale. The following major step was realized with the data from the SOHO/MDI instrument. The major progress made with this instrument has been the tremendous increase of the size (and quality) of the data set leading to very good statistics. Using data collected in May – June 1996, i.e., at solar minimum, Hathaway et al. (2000, 2002) determined the power spectrum of the lineofsight velocity, finding a peak at 36 Mm (spherical harmonic l = 120). This peak extends from 20 Mm up to 63 Mm as given by the width at halfmaximum.
Using the granule tracking method, Rieutord et al. (2008) also determined the characteristics of the spectral peak of supergranulation. They found a similar length scale of 36 Mm and an extension between 20 Mm and 75 Mm (the epoch is March 2007, also at solar minimum). The data set in this case was much smaller (7.5 h and a field of view of 300 × 200 Mm^{2}), but still a hundred of supergranules were captured, giving good statistics. A similar measurement by Rieutord et al. (2010) using the small field of view of Hinode (76 × 76 Mm^{2}) encompassing only four supergranules, gave a peak at 30 Mm.
Other authors, like DeRosa et al. (2000) and DeRosa and Toomre (2004), used local correlation tracking to determine the horizontal flows from the Doppler signal of SOHO/MDI and identified supergranules with horizontal divergences. From these data, they derived a rather small “diameter” in the 12 – 20 Mm range. Using a similar technique, Meunier et al. (2007c) found a mean value for supergranule diameters around 30 Mm. As underlined in these papers, the size of supergranules very much depends on the smoothing procedure used in the data processing.
Another set of independent measurements was performed by Del Moro et al. (2004) using data from local helioseismology. The technique is based on the fact that local helioseismology gives (more easily than the velocity itself) the local horizontal divergence of the flows, as this quantity appears as a difference between wave travel times. Thus, using a similar data set as DeRosa and Toomre (2004), Del Moro et al. (2004) extracted the horizontal divergence from the local propagation of waves and could also determine the statistics of supergranule sizes. They found a mean diameter at 27 Mm with a peak in the distribution at 30 Mm. These latter results have been confirmed by Hirzberger et al. (2008) using an even larger set of data (collecting more than 10^{5} supergranules).
Alternatively, several authors used tesselation algorithms or thresholdbased identification techniques to capture individual supergranulation cells and subsequently study their geometrical properties and spatial arrangement. Such techniques have mostly been applied to maps of the chromospheric network (e.g., Hagenaar et al., 1997; Schrijver et al., 1997; Berrilli et al., 1998), whose relationship to supergranulation is further described in Section 4.6.1. Following this approach, Schrijver et al. (1997) notably found that the patterns of granulation and supergranulation are very similar when properly rescaled. Their results are “nearly compatible with an essentially random distribution of upflow centers”. Comparisons between the spatial arrangement of supergranulation cells and granulation cells were also performed by Berrilli et al. (2004), who found that the supergranules distribution is well represented by a “hard sphere random close packing model” and by Hirzberger et al. (2008), whose result differ markedly from those of Berrilli et al. (2004) and are compatible with a field of “nonoverlapping circles with variable diameters”.
To conclude this paragraph, we would like to stress an important difference between the various techniques used to characterize the scale and spatial distribution of supergranules. The first technique consists in determining the scale at which the kinetic energy spectral density or correlation length of horizontal motions is maximal, while the second technique relies on identifying coherent structures using tesselation algorithms and threshold conditions (such as the FWHM of autocorrelation functions) to study the size statistics of the resulting distribution. Unsurprisingly, the two methods provide slightly different values for the supergranulation “length scale”. As noticed by Leighton et al. (1962), thresholdbased detection gives an estimate of the size of supergranules, whereas the location of the kinetic energy spectrum is an indication of the average distance between supergranules (assumed as to be the energycontaining structures).
4.2.2 Time scales
After supergranulation was discovered, one of the first questions was that of the lifetime of the structures. Here too, we would have to distinguish the lifetime of the coherent structures and the spectral power in a given time scale. However, this latter quantity, being too difficult to derive, is not available. Thus, the time scales discussed below are based on coherent cellular structures.
Worden and Simon (1976) suggested a lifetime of 36 h for the lifetime of supergranulation and reported a detection of vertical velocity fields only at the edge of supergranulation cells, confirming earlier work by Frazier (1970). Later, Wang and Zirin (1989) showed that supergranulation lifetime estimates depended strongly on the choice of tracer or proxy. They obtained 20 h using Dopplergrams, two days using direct counting techniques of supergranulation cells and 10 h using the tracking of magnetic structures (see also Section 4.6). Here again, SOHO/MDI data have dramatically increased the statistics and thus quality of the determinations. The latest results of Hirzberger et al. (2008) lead to a lifetime around 1.6 ± 0.7 or 1.8 ± 0.9 d, depending on the technique used. These values are somewhat longer than the previous ones, but the length of the time series associated with the size of the sample enable a better representation of longliving supergranules.
4.2.3 Velocity scales
A typical velocity associated with supergranules can be derived from the ratio between the previously discussed typical length and time scales. Taking 30 Mm for the former and 1.7 d for the latter, we find 205 m s^{−1} as the typical horizontal velocity. This estimate is in reasonable agreement with more direct inferences of the supergranulation velocity field from observations: the original work of Hart (1954) inferred 170 m s^{−1}, Simon and Leighton (1964) mentioned 300 m s^{−1} and more recently Hathaway et al. (2002) evaluated this amplitude at ∼360 m s^{−1}.
The preceding values are obtained from Doppler shifts. They are quite imprecise because they always mix the horizontal and vertical components of the flow. Granule tracking does not suffer from such a problem, however we here face the remaining problem of the scale dependence of the velocity. The obtained values depend on the way data are filtered.
The horizontal velocity needs to be completed by the vertical velocity. This latter quantity is unfortunately much harder to extract, because the signal is noised by the 5 min oscillations and by the presence of magnetic field concentrations at supergranule boundaries, where up and downflows tend to be localised (see Section 4.6 below). November (1989, 1994) advocated that this vertical component was in fact the mesogranulation that he detected some years before on radial velocities at disc centre (November et al., 1981). The rms value of this quantity was then estimated to be 60 m s1. More recently this quantity was evaluated using the SOHO/MDI data by Hathaway et al. (2002). They derived an estimate of 30 m s^{−1}. This value is in line with the results of Rieutord et al. (2010) obtained from Hinode/SOT data using power spectra of lineofsight velocities.
4.3 Intensity variations in supergranules
Next, one may wonder if supergranulationscale motions are associated with any temperature or intensity fluctuations, as this information may give an important clue to understand the origin of supergranulation.
As shown by Worden (1975), the thermal signature of supergranulation, if any, must be very faint. The intensity contrast between the border and the centre of supergranules probably does not exceed a few percents, at least in the infrared. For comparison, rms intensity variations at the granulation scale have been shown to be up to 27% byWedemeyerBöhm and Rouppe van der Voort (2009) using the recent data of Hinode. Several studies (Beckers, 1968; Frazier, 1970; Foukal and Fowler, 1984; Lin and Kuhn, 1992) found an increase of intensity at the edge of supergranulation cells, corresponding to a negative correlation between the supergranulation horizontal divergence maps and intensity maps. These results, which tend to rule out a convective origin for supergranulation, are however subject to caution because supergranulation vertices are strongly correlated with magnetic bright points (see Section 4.6 below). To circumvent this difficulty, Rast (2003a) considered only areas with low magnetic fields and found a small decrease of intensity at the edge of supergranules. The problem was reconsidered in detail by Meunier et al. (2007b, 2008) using MDI intensity maps. There too, the influence of the magnetic network was carefully eliminated. In contrast to most previous studies, they report a very small but significant intensity decrease from the centre to the edge of supergranulation cells (in the range 0.8 – 2.8 K). Goldbaum et al. (2009) recently came to the same conclusion using a different methodology. In addition, Meunier et al. (2008) noticed that the radial temperature profile at the surface of a supergranule is very similar to that of a simulated granule. These observations are consistent with a driving of supergranulation by buoyancy.
4.4 Supergranulation depth
The aforementioned supergranulation properties were all inferred from observations at the surface level (optical depth τ = 1). But one may also learn something about the origin of supergranulation by trying to infer its vertical extent.
Combining Dopplergrams and correlation tracking inferences with the above considerations on the continuity equation, November (1994) made the noteworthy prediction that the supergranulation flow should disappear at depths larger than 2.4 Mm below the visible surface. Note that his suggestion that the mesogranulation signal detected in power spectra at a horizontal scale of 7 Mm corresponded to the vertical flow component of convective supergranulation cells was part of the same argument. More recently, Rieutord et al. (2010) did the same exercise with divergences and velocity fields derived from Hinode data and found a vertical velocity scale height of ∼1 Mm, indicating a very shallow structure.
The advent of local helioseismology in the late 1990s made it possible to probe the supergranulation flow at opticallythick levels. Duvall Jr et al. (1997), using preliminary MDI data, only detected flows at supergranulation scales in the first few Mm below the surface. Duvall Jr (1998) further estimated that the depth of supergranulation was 8 Mm. Zhao and Kosovichev (2003) reported evidence for converging flows at 10 Mm and estimated the supergranulation depth to be 15 Mm. Woodard (2007) reported a detection of the flow pattern down to 5 Mm corresponding to the deepest layers accessible with their data set. Using new Hinode data, Sekii et al. (2007) recently found that a supergranulation pattern, monitored for 12 h in a small field of 80 × 40 Mm^{2}, does not persist at depths larger than 5 Mm. The existence of a return flow at depths larger than 5 Mm has also been suggested but remains unclear (Duvall Jr, 1998; Zhao and Kosovichev, 2003). Note that imaging deep convection using helioseismic techniques is not an easy task. Braun and Lindsey (2003) and Lindsey and Braun (2004) provide a detailed description of the shortcomings and artefacts of helioseismic inversions in this context (see also Gizon and Birch, 2005).
To summarise, the determination of the vertical extent of the supergranulation below the surface is still in a preliminary phase. The few results mentioned above point to a shallow structure but they are affected by large uncertainties associated with both the intrinsic difficulty to perform such measurements and with their weak statistical significance. It is clear that a decisive step forward regarding this problem requires a careful study of the systematics and the processing of a very large amount of data to reduce the impact of the fluctuating nature of the flows.
4.5 Rotational properties of supergranules
The first reports on the rotational properties of supergranulation focused on the rotation rate of the supergranulation pattern (Duvall Jr, 1980; Snodgrass and Ulrich, 1990). Using Dopplergrams, they found, surprisingly, that supergranulation is rotating 4% faster than the plasma. This is now referred to as the superrotation of supergranules. In recent years, local helioseismology has proven extremely useful to study the rotational properties of supergranules. Their superrotation was confirmed by Duvall Jr and Gizon (2000) using the timedistance technique applied to fmodes. Beck and Schou (2000) estimated that the supergranulation rotation rate is larger than the solar rotation rate at any depth probed by helioseismology. Analysing time series of divergence maps inferred from timedistance helioseismology applied to MDI data, Gizon et al. (2003) found that the supergranulation pattern had wavelike properties with a typical period of 6 – 9 d, fairly longer than the lifetime of individual supergranules. They showed that the power spectrum of the supergranulation signal close to the equator presented a power excess in the prograde direction (with a slight equatorwards deviation in both hemispheres), thus explaining the anomalous superrotation rate of the pattern. The dispersion relation for the wave appears to be only weakly dependent on the latitude (Gizon and Duvall Jr, 2004). Schou (2003) confirmed these findings with direct Doppler shift measurements and found that wave motions were mostly aligned with the direction of propagation of the pattern. These results brought some extremely interesting new light on the supergranulation phenomenon and led to the conjecture that supergranulation could be a manifestation of oscillatory convection, a typical property of convection in the presence of rotation and/or magnetic fields (see Section 5).
However, Rast et al. (2004) and Lisle et al. (2004) questioned the interpretation of the observed power spectrum in terms of oscillations and suggested an alternative explanation in terms of two superimposed steady flow components identified as mesogranulation and supergranulation advected by giant cell circulations. According to Gizon and Birch (2005), this interpretation is not supported by observations. They argue that the finding of Lisle et al. (2004) that supergranules tend to align in the direction of the Sun’s rotation axis under the influence of giant cells can be explained naturally in terms of wave dynamics. Even more recently, Hathaway et al. (2006) argued that the supergranulation pattern superrotation inferred from Doppler shifts was due to projection effects on the lineofsight signal. Using correlation tracking of divergence maps derived from intensity maps (Meunier et al., 2007c) and comparing it with direct Doppler tracking, Meunier and Roudier (2007) confirmed the existence of projection effects with the latter method, but found that the supergranulation pattern inferred from divergence maps was still superrotating, albeit at smaller angular velocities than those inferred by Duvall Jr (1980) and Snodgrass and Ulrich (1990). For a detailed discussion on the identification of supergranulation rotational properties with helioseismology, we refer the reader to the review article by Gizon and Birch (2005) on local helioseismology.
For the sake of completeness on the topic of rotation, we finally mention the observations by Kuhn et al. (2000) of smallscale 100 m high “hills” at the solar surface, which they interpreted as Rossby waves. Recently, Williams et al. (2007) argued that these structures actually resulted from the vertical convective motions associated with supergranules.
4.6 Multiscale convection and magnetic fields
As shown in Section 2.2, the magnetic dissipation scale at the solar surface is l_{ n } ∼ 100 m or slightly less. Hence, convection at the solar surface is strongly coupled to the Sun’s magnetic dynamics at all observable scales, including that of supergranulation. The particular role played by magnetic fields in the supergranulation problem and the large amount of observational information available on this topic justify a dedicated subsection. In the following, we first look at the correlations between supergranulation and the magnetic network and then describe the properties of internetwork fields, whose dynamics can hardly be dissociated from the formation of the magnetic network. After a short detour to the observations of the interactions between supergranulation and active regions, we finally review several studies of the dependence of supergranulation on the global solar magnetic activity.
4.6.1 Supergranulation and the magnetic network
These results suggest that the formation of the magnetic network is in some way related to the supergranulation flow. It is however probably too simplistic and misleading to make a onetoone correspondence between the single scale of supergranulation and the network distribution of magnetic bright points. Several studies with the Swedish Solar Telescope at La Palma observatory indicate that strong correlations between flows at scales comparable to or smaller than mesoscales (i.e., significantly smaller than supergranulation) and intense magnetic elements exist (Domínguez Cerdeña, 2003; Domínguez Cerdeña et al., 2003). A recent study by Roudier et al. (2009), combining spectropolarimetric and photometric Hinode measurements, also demonstrated a very clear correlation between the motions at mesoscales and those of the magnetic network (see also de Wijn and Müller, 2009).
4.6.2 Internetwork fields
One of the major advances on solar magnetism in the last ten years has been the detection of quiet Sun magnetic fields at scales much smaller than that of granulation (e.g., Domínguez Cerdeña et al., 2003; Berger et al., 2004; Trujillo Bueno et al., 2004; Rouppe van der Voort et al., 2005; Lites et al., 2008). The ubiquity of these fields and their energetics suggest that the dynamics of internetwork fields could also be an important piece of the supergranulation puzzle (see also Section 4.6.5 below). It is therefore useful to recall their main properties before discussing the physics of supergranulation in the next sections. Note that the following summary is not meant to be exhaustive. For a dedicated review, we refer the reader to the recent work of de Wijn et al. (2009).
Internetwork fields refer to mixedpolarity fields that populate the interior of supergranules. Their strength is on average thought to be much weaker than that of network fields, but magnetic bright points are also observed in the internetwork, (e.g., Muller, 1983; Nisenson et al., 2003; de Wijn et al., 2005; Lites et al., 2008). Besides, network and internetwork fields are known to be in permanent interaction (e.g., Martin, 1988). In the light of nowadays highresolution observations, the historical dichotomy between network and internetwork fields appears to be rather blurred (this point will be further discussed in Section 8.1).
Internetwork magnetism was originally discovered by Livingston and Harvey (1971, 1975) and subsequently studied by many authors (e.g., Martin, 1988; Keller et al., 1994; Lin, 1995) at resolutions not exceeding 1” (730 km). Observations with the solar telescope at La Palma observatory revealed the existence of such fields at scales comparable and even smaller than the granulation scale (Domínguez Cerdeña et al., 2003; Roudier and Muller, 2004; Rouppe van der Voort et al., 2005). Recent studies based on Hinode observations (Orozco Suárez et al., 2007; Lites et al., 2008) reported magnetic field variations at scales comparable to or smaller than 100 km.
The strength of internetwork fields, their distribution at granulation and subgranulation scales and their preferred orientation are still a matter of debate. Almost every possible value in the 5 – 500 G range can be found in literature for the typical field strengths within the internetwork (Martin, 1988; Keller et al., 1994; Lin, 1995; Domínguez Cerdeña et al., 2003; Trujillo Bueno et al., 2004; Lites et al., 2008). This wide dispersion is explained by several factors. The most important one is certainly that Zeeman spectropolarimetry, one of the most frequently used tools to study solar magnetism, is affected by cancellation effects when the magnetic field reverses sign at scales smaller than the instrument resolution (Trujillo Bueno et al., 2004; de Wijn et al., 2009). Hence, very smallscale fields still partially escape detection via this method. Recent Zeeman spectropolarimetry estimates of the average field strength based on Hinode observations (Lites et al., 2008) are 11 G for longitudinal fields and 60 G for transverse fields (horizontal fields at disc centre), but wide excursions from these average values are detected and the observed signatures may also be compatible with stronger, less spacefilling magnetic fields. On the side of Hanle spectropolarimetry, Trujillo Bueno et al. (2004) report an average field strength of 130 G, with stronger fields in the intergranular lanes and much weaker fields in the bright centres of granules.
The previously mentioned Zeeman estimates seem to indicate that internetwork fields have a tendency to be horizontal (Orozco Suárez et al., 2007; Bommier et al., 2007; Lites et al., 2008), sometimes even bridging over granules, but other studies have come to the opposite conclusion that internetwork fields are mostly isotropic (Martínez González et al., 2008; Asensio Ramos, 2009; Bommier et al., 2009). Using Zeeman and Hanle diagnostics in a complementary way, López Ariste et al. (2010) very recently came to the conclusion that internetwork fields are mostly isotropic and highly disordered, with a typical magnetic energy containing scale of 10 km.
4.6.3 The magnetic power spectrum of the quiet photosphere
The scalebyscale distribution of magnetic energy and the power spectrum of magnetic fields in the quiet photosphere are other important quantities to look at, as they may give us some clues on the type of MHD physics at work in the subgranulation to supergranulation range. Based on various types of analysis (structure statistics, wavelets, etc.), several authors have notably argued that solar magnetic fields, from the global solar scales to the smallest scales available to observations, may have a fractal or multifractal structure (Lawrence et al., 1995; Komm, 1995; NesmeRibes et al., 1996; Meunier, 1999; Janßen et al., 2003; Stenflo and Holzreuter, 2002, 2003a,b; Abramenko, 2005).
Explicit studies of the power spectrum of the quiet Sun are currently limited to the range 1 – 100 Mm and to the lineofsight component of the magnetic field. Most spectra available in literature have been obtained from either groundbased observations or SOHO/MDI magnetograms. We have been unable to find any study of the magnetic power spectrum of the quiet Sun covering scales well below 1 Mm, at which internetwork fields can now be detected with Hinode.
At scales below 10 Mm, the magnetic power spectrum of the quiet photosphere has been found to be rather flat and decreasing with decreasing scales. Scalings in that mesoscale interval range from k^{−1} to k^{−1.4} (Lee et al., 1997; Abramenko et al., 2001; Harvey et al., 2007; McAteer et al., 2009; Longcope and Parnell, 2009). At scales larger than 10 Mm, a slightly positive flat slope ∼ k^{−1.3} – k^{0} has been reported by several authors (Lee et al., 1997; Abramenko et al., 2001; Longcope and Parnell, 2009).
4.6.4 Supergranulation and flows in active regions
Proceeding along the description of the interactions between supergranulation and magnetic fields, one may also consider the properties of surface flows at scales comparable to supergranulation within active regions and in the vicinity of sunspots. The reason for this is twofold. First, we may wonder how the supergranulation pattern evolves locally during the formation or decay of an active region. Second, the properties of flows around sunspots may give us some hints of the effect of strong magnetic flux concentrations on the flow dynamics in the quiet photosphere.
As far as the first point is concerned, the information is fairly scarce at the moment. Rieutord et al. (2010) recently reported the disappearance of the supergranulation spectral peak in the kinetic energy power spectrum of solar convection during the emergence of two magnetic pores. While the pores (of a size comparable to that of a granule) are emerging, the supergranulation flow becomes very weak just like if the surrounding magnetic flux associated with the pores had a significant impact on the flow. A related observation by Hindman et al. (2009) shows that the fairly regular tiling of the surface of the quiet Sun associated with supergranulation is somewhat disorganised and washed away within magnetic active regions.
Several authors studied the structure of the moat flow using Doppler signal (Sheeley Jr and Bhatnagar, 1971; Sheeley Jr, 1972), by tracking surface features, such as granules (Muller and Mena, 1987; Shine et al., 1987) or smallscale magnetic elements (Sheeley Jr, 1972; Harvey and Harvey, 1973; Hagenaar and Shine, 2005), or using helioseismology (Gizon et al., 2000). One of the conclusions of these studies is that the outflow has properties similar to those of supergranulation (see notably Brickhouse and Labonte, 1988), albeit with a larger velocity ∼ 1 km s^{−1}. It is however unclear whether or not this flow has anything to do with the regular supergranulation, as the outflow is centred on a strong field region in that case whereas it is the supergranulation inflow vertices that coincide with magnetic flux concentrations in the quiet Sun. As far as supergranulation is concerned, nevertheless, the lesson to be learned from helioseismology of sunspot regions is that magnetoconvection in strong fields has the naturally ability to produce a variety of coherent outflows and inflows at various horizontal and vertical scales in the vicinity of regions of strong magnetic flux. This phenomenology may be worth exploring further in the somewhat scaleddown system consisting of the supergranulation flow and local flux concentrations associated with the magnetic network in the quiet Sun (see Section 8.2 in this review).
4.6.5 Supergranulation variations over the solar cycle
In view of the association between supergranulation and the magnetic network, it is finally natural to wonder if and how the size of supergranules varies with solar activity.
Singh and Bappu (1981), studying spectroheliograms spanning a period of seven solar maxima, found a decrease of the typical size of the chromospheric network between the maxima and the minima of the cycle. Their results are in line with those of Kariyappa and Sivaraman (1994), Berrilli et al. (1999) and Raju and Singh (2002), but appear to be at odds with those of Wang (1988) and Münzer et al. (1989), who both reported an increase of network cell sizes in regions of stronger magnetic activity, and with those of Meunier (2003), who found from MDI magnetograms spanning the first half of Cycle 23 an increase of the size of magnetic elements at supergranulationlike scales with solar activity (note that Berrilli et al., 1999 also used data obtained at the beginning of Cycle 23 close to the activity minimum). These somehow contradicting results show that magnetic tracers must be used with care for this kind of measurements. The results are indeed sensitive to the thresholds used to identify the various field components (e.g., network or internetwork). Disentangling all these effects is not an easy task.
Recent studies have thus attempted to use proxies independent of magnetic tracers of supergranulation to measure its size, notably velocity features like positive divergences. DeRosa and Toomre (2004), using two data sets obtained at periods of different levels of magnetic activity, found smaller supergranulation cell sizes in the period of high activity. A similar conclusion was reached by Meunier et al. (2008). Meunier et al. (2007a) found a decrease of the typical cell sizes with increasing field strength within supergranules, but noted that larger supergranulation cells were associated with stronger network fields at their boundaries. Hence, it seems that a negative or a positive correlation can be obtained, depending on whether the level of magnetic activity is defined with respect to internetwork or network fields. Meunier et al. (2007a) also reported the absence of large supergranulation cells for supergranules with large internetwork magnetic field strengths, indicating that internetwork fields do have a dynamical influence on supergranules. We refer the reader to Meunier et al. (2007a) for a more exhaustive discussion of the previous results and of the possible shortcomings and biases of these various studies.
Finally, on the helioseismic side, the dispersion relation for the supergranulation oscillations found by Gizon et al. (2003) appears to be only weakly dependent on the phase of the solar cycle (Gizon and Duvall Jr, 2004). However, the same authors reported a decrease in the lifetime and power anisotropy of the pattern from solar minimum to solar maximum.
4.7 Conclusions
Since the launch of SOHO and observations with the MDI instrument, the interest in supergranulation has been renewed. Most of its main observational properties, like its size, lifetime and the strength of the associated flows are now well determined. However, other aspects of supergranulation dynamics, like the vertical dependence of the flow, the vertical component of the velocity at the edge of supergranules and the connections between supergranulation and magnetic fields are still only very partially constrained by observations. They all require further investigations.
As far as velocity measurements are concerned, we may anticipate progress in the near future on the question of the depth of supergranulation thanks to local helioseismology applied to higherresolution observations. However, characterising vertical flows at the supergranulation scale is a more complex task since such flows are faint and very localised in space. Improving the diagnostics of the latitudinal dependence of the supergranulation pattern may also prove useful, in particular to help understand if subsurface shear plays a significant role in shaping the supergranulation flow.
Another point worth studying in more detail is the distribution of kinetic energy at scales between supergranulation and the Sun’s radius. This has already been attempted using supergranules as passive tracers of largerscale velocity patterns (Švanda et al., 2006, 2008). Even more accurate studies of this kind could become feasible soon by using tracking techniques applied to images obtained with widefield cameras imaging the full solar disc with subgranulation resolution.
The case of the interactions with magnetic fields deserves a lot of further attention on the observational side in our view. It is now well established that flows at scales larger than granulation advect internetwork fields and tend to concentrate magnetic elements into the network, along the boundaries of supergranules. This process is essentially kinematic, in the sense that the magnetic field only has a very weak feedback on the flow. But what we observe ultimately is probably a nonlinear statistically steady magnetised state, in which the magnetic field provides significant feedback on the flow. Hence, it would be useful to have more quantitative observational results on the relations between the properties of supergranules and the surrounding magnetic fields (internal and boundary flux, filling factors, strength, size) to characterise this feedback more precisely (we refer the reader to Section 8 for an exhaustive discussion on supergranulation and MHD). Most importantly, a precise determination of the magnetic energy spectrum of the quiet Sun over a very wide range of scales would be extremely precious to understand the nature of MHD interactions between supergranulation, network and internetwork fields. Finally, it would also be interesting to have more documented observational examples of the interactions of supergranulation with magnetic regions of various strengths (active regions, polar regions) to gain some insight into the dynamical processes at work in the problem. This latter point is important from the perspective of the global solar dynamo problem, as it would help better constrain the transport of magnetic field by turbulent diffusion at the surface of the Sun.
5 Theoretical Models of Supergranulation
We now turn to the description of existing theoretical models of supergranulation. These models are basically of two types: those that postulate that supergranulation has a convective origin (i.e., it is driven by thermal buoyancy), and those that do not. In order to set the stage for upcoming discussions, we start with a brief description of the rotating MHD RayleighBénard problem (Section 5.1), which provides the simplest mathematical description of rotating magnetoconvection in a fluid. We then review various thermal convection models of supergranulation (Section 5.2) and discuss other possible physical mechanisms involving collective “turbulent” dynamics of smallerscale convection (Section 5.3). A few concluding remarks follow.
Before we start, it is perhaps useful to mention that most of these models are unfortunately only very qualitative, in the sense that they either rely on extremely simplified theoretical frameworks (like linear or weakly nonlinear theory in two dimensions, or simple energetic arguments) or on simple dynamical toy models designed after phenomenological considerations. The looseness of theoretical models, combined with the incompleteness of observational constraints and shortcomings of numerical simulations, has made it difficult to either validate or invalidate any theoretical argument so far. What numerical simulations tell us and how the theoretical models described below fit with numerical results and observations will be discussed in detail in Section 6.
5.1 The rotating MHD RayleighBénard convection problem
5.1.1 Formulation
The simplest formulation of the problem of thermal convection of a fluid is called the Rayleigh. Bénard problem. It describes convection of a liquid enclosed between two differentially heated horizontal plates, each held at a fixed temperature. The mathematical model is derived under the Boussinesq approximation, which amounts to assuming that the flow is highly subsonic and that density perturbations δρ to a uniform and constant background density ρ_{o} are negligible everywhere except in the buoyancy term \(\delta \rho \vec g\), where \(\vec g =  g\vec e_z \) stands for the gravity (Chandrasekhar, 1961). The equilibrium background state is a linear temperature profile with temperature decreasing from the bottom to the top of the layer. This case is in many respects different and simpler than the strongly stratified SCZ case, which treatment requires using more general compressible fluid and energy equations than those given below (Nordlund, 1982), but is sufficient to discuss many of the important physical (Section 5.2) and numerical (Section 6.1) issues pertaining to supergranulation.
5.1.2 Linear instability and the solar regime
In the simplest nonrotating hydrodynamic case (Ta = Q = 0, no induction), when the Rayleigh number is less than a critical value Ra_{crit} that depends on the particular choice of boundary conditions, diffusive processes dominate over buoyancy: the hydrostatic solution is stable, i.e., any velocity or temperature perturbations decays. For Ra > Ra_{crit}, convection sets in as a linear instability and perturbations grow exponentially in the form of convection rolls or hexagons with a horizontal spatial periodicity comparable to the convective layer depth .. in most cases. The effects of magnetic fields and rotation on the linear stability analysis are discussed in the next paragraphs.
It should be noted that Ra, Q, and Ta are all extremely large numbers in the Sun, if they are computed from microscopic transport coefficients (Section 2.2). So, in principle, there is no reason why solar convection should be close to the instability threshold. However, theoretical studies of largescale convection (such as supergranulation) commonly assume that viscous, thermal, and magnetic diffusion at such scales are determined by turbulent transport, not microscopic transport. This leads to much larger transport coefficients (which can be estimated, for instance, using the typical scale and velocity of the granulation pattern) and much smaller “effective” Ra, Q, and Ta, so the “largescale” system is generally considered not too far away from criticality. Making this (strong) meanfield assumption serves to legitimate using the standard toolkits of linear and weakly nonlinear analysis to understand the largescale behaviour of solar convection.
5.2 Convective origin of supergranulation
Following its discovery in the 1950s and further studies in the 1960s, supergranulation was quickly considered to have a convective origin, very much like the solar granulation. Since then, many theoretical models relying on the basic phenomenology of thermal convection sketched in Section 5.1.2 have been devised to explain the apparently discretescales regime of the dynamics of the solar surface (namely the scales of granulation and supergranulation, but also that of mesogranulation, discussed in Section 3.2).
5.2.1 Multiple mode convection
The simplest model for the emergence of a set of special scales is that of multiple steady linear or weakly nonlinearly interacting modes of thermal convection forced at different depths. The first theoretical argument of this kind is due to Simon and Leighton (1964), who suggested that supergranulationscale motions corresponded to simple convection cells driven at the depth of He^{++} recombination and just advecting granulationscale convection. Schwarzschild (1975) invoked an opacity break, He^{+} and H^{+} recombinations as the drivers of supergranulationscale convection. Simon and Weiss (1968) and Vickers (1971), on the other hand, suggested that deep convection in the Sun had a multilayered structure composed of deep, giant cell circulations extending from the bottom of the convection zone to 40 Mm deep, topped by a shallower circulation pattern corresponding to supergranulation. In this second theory, recombination is not a necessary ingredient. Bogart et al. (1980) attempted to match a linear combination of convective eigenmodes to the solar convective flux but did not find that supergranulation came out as a preferred scale of convection in this quasilinear framework.
Antia et al. (1981) argued that turbulent viscosity and diffusivity should be taken into account in linear calculations, as they alter the growth and scales of the most unstable modes of convection. In their linear calculation with microscopic viscosity and thermal diffusivity coefficients replaced by their turbulent counterparts, granulation, and supergranulation show up as the two most unstable harmonics of convection. Calibrating the amplitudes of a linear superposition of convective modes to match mixinglength estimates of the solar convective flux in the spirit of Bogart et al. (1980), Antia and Chitre (1993) further argued that they could reproduce the main characteristics of the power spectrum of solar surface convection.
Gierasch (1985) devised a onedimensional energy model for the upper solar convection zone from which he argued that turbulent dissipation takes place and deposits thermal energy at preferred depths, thereby intensifying convection at granulation and supergranulation scales. On this subject, we also mention the work of Wolff (1995), who calculated that the damping of rmodes in the Sun should preferentially deposit heat 50 Mm below the surface as a result of the ionisation profile in the upper solar convection zone. This process might in turn result in convective intensification at similar horizontal scales.
5.2.2 Effects of temperature boundary conditions
An interesting theoretical suggestion on the problem of supergranulation was made by Van der Borght (1974), who considered the case of steady finiteamplitude thermal convection cells in the presence of fixed heat flux boundary conditions imposed at the top and bottom of the layer. He showed that the convection pattern in this framework has much smaller temperature fluctuations than in the standard RayleighBénard model with fixed temperature boundary conditions. This makes this case quite interesting for the supergranulation problem, considering that intensity fluctuations at supergranulation scales are rather elusive (see Section 4.3).
Even more interestingly, fixed heat flux boundary conditions naturally favour marginally stable convection cells with infinite horizontal extent compared to the layer depth, or convection cells with a very large but finite horizontal extent when a weak modulation of the heat flux is allowed for (Sparrow et al., 1964; Hurle et al., 1967; Van der Borght, 1974; Busse and Riahi, 1978; Chapman and Proctor, 1980; Depassier and Spiegel, 1981). This case is therefore very different from the standard RayleighBénard case with fixed temperature boundary conditions, which gives rise to cells with aspect ratio of order unity. In this framework, there is no need to invoke deep convection to produce supergranulationscale convection.
This idea was carried on with the addition of a uniform vertical magnetic field threading the convective layer. Contrary to the hydrodynamic case described above, where zerowavenumber solutions are preferred linearly (albeit with a zero growthrate), convection cells with a long but finite horizontal extent dominate in the magnetised case, provided that the magnetic field exceeds some threshold amplitude. The horizontal scale of the convection pattern in the model is subsequently directly dependent on the magnetic field strength. Murphy (1977) was the first to suggest that this model might be relevant to supergranulation. The linear problem in the Boussinesq approximation was solved by Edwards (1990). Rincon and Rieutord (2003) further solved the fully compressible linear problem numerically and revisited it in the context of supergranulation. Using typical solar values as an input for their model parameters (density scale height, turbulent viscosity etc.) they showed that the magnetic field strength (measured in the nondimensional equations by the Chandrasekhar number Q) required for compressible magnetoconvection with fixed heat flux to produce supergranulationscale convection was of the order 100 G.
5.2.3 Oscillatory convection and the role of dissipative processes
The discovery by Gizon et al. (2003) that supergranulation has wavelike properties (Section 4.5) opened some new perspectives for theoretical speculation. In particular, it offered an opportunity to revive the interest for several important theoretical findings pertaining to the issue of oscillatory convection, which we now attempt to describe.
The existence of timedependent oscillatory modes of thermal convection has been known for a long time (Chandrasekhar, 1961 provides an exhaustive presentation of linear theory on this topic). In many cases, such a behaviour requires the presence of a restoring force acting on the convective motions driven by buoyancy. It can be provided by Coriolis effects (rotation) or magnetic field tension for instance. The existence of oscillatory solutions is also known to depend very strongly on how various dissipative processes (viscous friction, thermal diffusion, and ohmic diffusion) compete in the flow. This is usually measured or parametrised in terms of the thermal Prandtl number Pr = v/k, where v is the kinematic viscosity and k is the thermal diffusivity, the magnetic Prandtl number Pm = v/η, where η is the magnetic diffusivity, and the “third” Prandtl number^{7} ζ = η/k = Pr/Pm. In the Sun, Pr ∼ 10^{−4} – 10^{−10}, Pm ∼ 10^{−2} – 10^{−5} (see Section 2.2), and ζ ≪ 1 at the photosphere.
5.2.4 Oscillatory convection, rotation, and shear
As mentioned in Section 4.5, the supergranulation flow seems to be weakly influenced by the global solar rotation. In the presence of a vertical rotation vector, overstable oscillatory convection is preferred to steady convection provided that Pr is small (Chandrasekhar, 1961). In more physical terms, an oscillation is only possible if inertial motions are not significantly damped viscously on the thermalization timescale of rising and sinking convective blobs.
Busse (2004, 2007) suggested on the basis of a local Cartesian analysis that the drift of supergranulation could be a signature of weakly nonlinear thermal convection rotating about an inclined axis and found a phase velocity consistent with the data of Gizon et al. (2003), assuming an eddy viscosity prescription consistent with solar estimates (based on the typical sizes and velocity of granulation). Earlier work on the linear stability of a rotating spherical Boussinesq fluid layer heated by internal heat sources showed that the most rapidly growing perturbations are oscillatory and form a prograde drifting pattern of convection cells at low Prandtl number in high Taylor number regimes corresponding physically to large rotation (Zhang and Busse, 1987).
A directly related issue is that of the influence of differential rotation on supergranulation. Green and Kosovichev (2006) considered the possible role of the solar subsurface shear layer (Schou et al., 1998) by looking at the effect of a vertical shear flow on the onset of convection in a strongly stratified Cartesian layer using linear theory. They found that convective modes in the nonsheared problem become travelling when a weak shear is added. Some previous work found that this behaviour is possible either at low Pr (Kropp and Busse, 1991) or if some form of symmetry breaking is present in the equations (Matthews and Cox, 1997). Since linear shear alone cannot do the job, it is likely that density stratification plays an important role in obtaining the result. Green and Kosovichev (2006) also report that the derived phase speeds for their travelling pattern are significantly smaller than those inferred from observations by Gizon et al. (2003).
Note that the relative orientations and amplitudes of rotation, shear, and gravity are fundamental parameters in the sheared rotating convection problem. It should therefore be kept in mind that the results (e.g., the pattern phase velocity and wavelength) of local Cartesian theoretical models of supergranulation incorporating solarlike rotation effects are expected to depend on latitude, as the orientation of the rotation vector changes from horizontal at the equator to vertical at the pole, and the subsurface rotational velocity gradient varies with latitude in the Sun. We recall that there is as yet no conclusive observational evidence for a latitudinal dependence of the scales of supergranulation (see Section 4.5), so it is unclear if local models of sheared rotating convection can help solve the problem quantitatively. Global spherical models do not necessarily suffer from this problem, as they predict global modes with a welldefined phase velocity.
5.2.5 Oscillatory convection and magnetic fields
Magnetoconvection in a uniform vertical magnetic field is also known to preferentially take on the form of timeoscillations at onset provided that ζ ≪ 1 (e.g., Chandrasekhar, 1961; Proctor and Weiss, 1982), a process referred to as magnetic overstability. Oscillatory magnetoconvection is also known to occur for nonvertical magnetic fields (e.g., Matthews et al., 1992; Hurlburt et al., 1996; Thompson, 2005, and references therein). Physically, field lines can only be bent significantly by convective motions and act as a spring if they do not slip too much through the moving fluid, which requires, in this context, that the magnetic diffusivity of the fluid be small enough in comparison to its thermal diffusivity. Since ζ ≪ 1 in the quiet photosphere, oscillatory magnetoconvection represents a possible option to explain the wavy behaviour of supergranulation. On this topic, Green and Kosovichev (2007) recently built on the work of Green and Kosovichev (2006) and considered the linear theory of sheared magnetoconvection in a uniform horizontal (toroidal) field shaped by the subsurface shear layer. They report that the phase speed of the travelling waves increases in comparison to the hydrodynamic case studied by Green and Kosovichev (2006) and argue that the actual phase speed measured by Gizon et al. (2003) can be obtained for a uniform horizontal field of 300 G.
5.2.6 Other effects
Finally, it is known theoretically and experimentally that even in the absence of any effect such as magnetic couplings, rotation or shear, the value of the thermal Prandtl number can significantly affect the scales and time evolution of convection, both in the linear and nonlinear regimes. Its value notably controls the threshold of secondary oscillatory instabilities of convection rolls (Busse, 1972). At very low Prandtl numbers, Thual (1992) showed that a very rich dynamical behaviour resulting from the interactions between the primary convection mode and the secondary oscillatory instability takes place close to the convection threshold. This includes travelling and standing wave convection.
Most theoretical studies of supergranulation to date have been either ideal (no dissipation) or for Pr ∼ Pm ∼ 1. For this reason, some important physical effects relevant to supergranulationscale convection may well have been overlooked until now.
5.2.7 Shortcomings of simple convection models
The previous models are interesting in many respects but it should be kept in mind that they all have very important shortcomings. First, they rely on linear or weakly nonlinear calculations, which is hard to justify considering that the actual Reynolds number in the solar photosphere is over 10^{10} and that power spectra of solar surface flows show that the dynamics is spread over many scales. A classical meanfield argument is that smallscale turbulence gives rise to effective turbulent transport coefficients, justifying that the largescale dynamics be computed from linear or weakly nonlinear theory. Even if it is physically appealing, this argument still lacks firm theoretical foundations. Assuming that turbulent diffusion can be parametrised by using the same formal expression as microscopic diffusion is a strong assumption, and so is the neglect of direct nonlocal, nonlinear energy transfers between disparate scales. Dedicated numerical simulations of this problem are therefore more than ever required to justify or to discard using this kind of assumptions.
Models with poor thermally conducting boundaries have the interesting feature of producing fairly shallow convection cells, with a large horizontal extent in comparison to their vertical extent. If it is confirmed that supergranulation is indeed a shallow flow (Section 4.4), investigating the theoretical basis of this assumption may prove useful to understand the origins of supergranulation. Of course, the main problem is that it remains to be demonstrated that fixed heat flux boundaries represent a good approximation of the effect of granulationscale convection on largerscale motions beneath the solar surface.
Finally, magnetoconvection models all assume the presence of a uniform field (either horizontal or vertical) threading the convective layer, which is certainly an oversimplified zerothorder prescription for the magnetic field geometry in the quiet Sun. Numerical modelling probably provides the only way to incorporate more complex magnetic field geometries, timeevolution and dynamical feedback in supergranulation models.
5.3 Largescale instabilities and collective interactions
Besides thermal convection scenarios, a few other theoretical arguments have been put forward to explain the origin of the solar supergranulation. These ideas are all based on the possible collective effects of smallscale structures such as granules, which might lead to a largescale instability injecting energy into the supergranulation range of scales.
5.3.1 Rip currents and largescale instabilities
The first work along this line of thought was published by Cloutman (1979). He proposed to explain the origin of supergranulation using the physical picture of rip currents on the beaches of oceans: the repeated breaking of waves on beaches induces currents (rip currents) flowing parallel to the coast line. On the Sun, he identified breakers with the rising flows of granules breaking into the stably stratified upper photosphere.
The rip current model provides an illustration of the suggestion of Rieutord et al. (2000) that the collective interaction of solar granules may give rise to a largescale instability driving supergranulation flows. The idea finds its root in theoretical work on energy localisation processes in nonlinear lattices (Dauxois and Peyrard, 1993) and largescale instabilities (“negative eddy viscosity instabilities”) of periodic flows, such as the Kolmogorov flow (Meshalkin and Sinai, 1961; Sivashinsky and Yakhot, 1985) or the decorated hexagonal flow (Gama et al., 1994). Asymptotic theory on simple prescribed vortical flows can be performed under the assumption of scale separation (Dubrulle and Frisch, 1991) between the basic periodic flow and the largescale instability mode. In such theories, the sign and amplitudes of the turbulent viscosities is found to be a function of the Reynolds number. For instance, an asymptotic theory based on a large aspect ratio expansion was developed by Newell et al. (1990) for thermal convection. In this problem, largescale instabilities take on the form of a slow, longwavelength modulation of convection roll patterns. Their evolution is governed by a phase diffusion equation with tensorial viscosity. In the case of negative effective parallel diffusion (with respect to the rolls orientation), the Eckhaus instability sets in, while the zigzag instability is preferred in the case of negative effective perpendicular diffusion.
5.3.2 Plume interactions
Another way of explaining the origin of supergranulation assumes that the pattern results from the collective interaction of plumes. The word plume usually refers to buoyantly driven rising or sinking flows. Plumes can be either laminar or turbulent, however the turbulent ones have by far received most of the attention because of their numerous applications (see Turner, 1986).
The first numerical simulations of compressible convection at high enough Reynolds numbers (e.g., Stein and Nordlund, 1989a; Rast and Toomre, 1993) quite clearly showed the importance of vigorous sinking plumes. These results prompted Rieutord and Zahn (1995) to study in some details the fate of these downdrafts. Unlike the downflows computed in early simulations, solar plumes are turbulent structures, which entrain the surrounding fluid (see Figure 1). As Rieutord and Zahn (1995) pointed it out, the mutual entrainment and merging of these plumes naturally leads to an increase of the horizontal scale as one proceeds deeper.
In this context, toy models have been elaborated to investigate the properties of “nbody” dynamical advectioninteractions between plumes. For instance, Rast (2003b) developed a model in which a twodimensional flow described by a collection of individual divergent horizontal flows (“fountains”) mimicking granules is evolved under a simple set of rules governing the merging of individual elements into larger fountains and their repulsion^{8}. For some parameters typical of the solar granulation (individual velocities and radius of the fountains notably), he argued that the clustering scales of the flow after a long evolution of the system resembled that of mesogranulation and supergranulation. A similar model incorporating simplified magnetic field dynamics was designed by Crouch et al. (2007). They observed some magnetic field organisation and polarity enhancement at scales similar to that of supergranulation in the course of the evolution of the model. The main caveat of these toy models is of course that they do not rely on the exact dynamical physical equations.
The previous concepts and models are appealing but theoretical and numerical support has been lacking so far to weigh their relevance to the supergranulation problem. In particular, analytical developments for fluid problems have been restricted to low Reynolds numbers and very simple analytical flow models. It is unclear if and how some quantitative progress can be made along these lines. Future highresolution numerical simulations may offer some insight into the nonlinear dynamics at work at supergranulation scales but, as will be shown in Section 6, they have not yet given us any indication that largescale instabilities or nonlinear granulation dynamics can generate a supergranulation flow. In particular, all hydrodynamic simulations so far tend to show that plume merging is a selfsimilar process that does not naturally produce a flow at welldefined scale that could be identified with supergranulation.
5.4 Conclusions
In the previous paragraphs, we have reviewed the current state of affairs on the theoretical understanding of supergranulation. As cautioned already in the introduction, a breadth of simple models and ideas has been suggested over the years, but the theoretical landscape is extremely fuzzy. A shared property of all models is the looseness of the approximations on which they rely (e.g., linear theory with turbulent viscosity parametrisation, or handwaving physical arguments on the nature of dynamical interactions between granules and their potential largescale instabilities). Consequently, completely distinct theoretical arguments can easily be tuned to produce results that are all broadly consistent with observations. This degeneracy makes it impossible to discriminate between the various scenarios.
This issue is of course not specific to the supergranulation problem. The theoretical approach to the global solar dynamo, for instance, is more or less affected by the same syndrome. Escaping this difficult situation probably requires a significant improvement of the numerical modelling of supergranulationscale convection. The status of this specific field is reviewed in detail in the next section.
6 Numerical Modelling
Our understanding of turbulent convection in general and turbulent convection at the solar surface in particular has improved significantly with the advent of largescale computing resources. As far as supergranulation is concerned, even though numerical modelling has been and is still confronted with several major computing limitations, it is now on the verge of making very interesting progress towards discriminating between various physical scenarios and ideas such as those presented in Section 5. In this section, we review the evolution of numerical simulations of solar surface convection over the last thirty years and how they have helped us make progress on the specific problem of supergranulation. We notably discuss the advantages and limitations of the various types of numerical models used to study solar surface convection, in order to identify what numerical simulations have really told us (or not told us) on the problem of supergranulation and to provide the reader with a (hopefully) clear understanding of the current important modelling issues.
The section starts with two introductory paragraphs on the numerical simulation of turbulent thermal convection in the RayleighBénard framework (Section 6.1) and in the solar context (Section 6.2). We then recall the main results obtained from granulationscale simulations (6.3), before describing in detail the history and current status of “largescale” hydrodynamic and MHD simulations (6.4) aiming at understanding the dynamics at scales comparable to that of supergranulation.
6.1 Numerical simulations of turbulent RayleighBénard convection
Even if the hydrodynamic nonrotating RayleighBénard problem (presented in Section 5.1) is much simpler than the solar convection problem, its numerical modelling in highly nonlinear regimes still represents a major challenge of nowadays fluid dynamics. It is therefore worth recalling the main issues and numerical requirements for this problem before discussing the solar case.
6.1.1 RayleighBénard and NavierStokes
In the nonrotating hydrodynamic case (Ta = Q = 0, no induction), the Boussinesq equations (2) are very similar to the forced Navier.Stokes equations, except that the forcing term is not an external body force but is determined selfconsistently from the timeevolution of the temperature fluctuations. Based on both experimental and numerical evidence at order one Prandtl number, several authors (Rincon, 2006; Lohse and Xia, 2010) have argued that the basic phenomenology of Rayleigh.Bénard turbulence at scales below the typical Bolgiano injection scale (see Section 2.2) should be similar to that of NavierStokes turbulence in the inertialrange (i.e., Kolmogorov turbulence plus possible intermittency corrections). Hence, at a very good first approximation, the numerical issues and requirements to simulate the Rayleigh.Béenard problem at very high Rayleigh numbers are the same as those pertaining to the simulation of forced NavierStokes turbulence at high Reynolds numbers.
6.1.2 Stateoftheart modelling
The performances of turbulent convection simulations are often measured by the ratio of the imposed Rayleigh number to its critical value Ra/Ra_{crit} as the larger this quantity is, the more turbulent is the flow (the larger the Reynolds number). The highest values attained so far in simulations of the RayleighBénard problem are approximately Ra/Ra_{crit} ∼ 10^{11} (Verzicco and Camussi, 2003; Amati et al., 2005), but most “routine” simulations of the problem are in the much softer range Ra/Ra_{crit} ∼ 10^{5} – 10^{7}. Achieving highly turbulent regimes first requires using highorder numerical methods such as spectral methods (Canuto et al., 2006), which provide very good numerical accuracy and constitute an ideal tool to resolve all the dynamics from the injection to the dissipation range of incompressible turbulence. The second price to pay is to use very high spatial resolutions to discretize the problem (currently, high resolutions mean ∼512^{3} to ∼1000^{3}) on a nonuniform grid (to resolve boundary layers). This in turn imposes correspondingly small time steps, so the integration times of very high Rayleigh number convection are limited to a few turbulent turnover times, possibly not enough to resolve some of the longtime, largescale, meanfield dynamics.

there is a very marked evolution of the pattern from moderate to very large Ra, showing that an asymptotic large Ra behaviour has not yet been attained in spite of the very high numerical resolution used,

one notices the emergence of two halfcell circulations in the highest Ra regime, revealing that some largescale coherent dynamics naturally emerges from the smallscale turbulent disorder as one goes to very strongly nonlinear regimes. How this kind of dynamics changes as a larger aspect ratio is allowed for is not understood at the moment. This point will be discussed in more detail in Section 6.4.3 in the context of supergranulationscale simulations.
Hence, we conclude this paragraph by emphasising that even the most advanced numerical simulations to date of the simplest turbulent convection problem at hand do not yet allow to study fully comprehensively the high Rayleigh number asymptotic behaviour of convection and the largescale dynamical evolution of the system.
6.2 Numerical simulations of solar convection
After this presentation of the stateoftheart numerical modelling of RayleighBénard convection, we now come back to the more specific problem of the numerical simulation of solar convection. The first direct numerical simulations of astrophysical convection date back from the 1970s. In the solar context, an early noteworthy study is the one of Graham (1975) who revealed an asymmetry of up and downflows in twodimensional simulations of compressible convection in a stratified medium (i.e., not in the Boussinesq approximation), as observed for granulation. The Rayleigh number was just ten times supercritical in this simulation. A similar numerical model was devised by Chan et al. (1982). Massaguer and Zahn (1980) used a truncated numerical model of convection to study this physical effect specific to compressible convection, which is now referred to as buoyancy braking. These early contributions, however, did not provide a fully realistic framework to understand the complexity of solar surface convection. The first attempts to incorporate some specificities of surface convection in stellar environments, like radiative transfer, are due to Nordlund (1982). Since then, astrophysical convection simulations have split into two families that define the main trends in the field nowadays.
A short remark is in order regarding the following “classification”. Any numerical simulation of solar convection corresponds to a specific physical setup tailored for the purpose of studying specific physical processes. In general, a given setup resembles one of the two families of models described below but any mix between the two is obviously possible in practice.
6.2.1 Idealised simulations
The first family of numerical models is in the spirit of the original experiments by Graham (1975) and can be referred to as “idealised” simulations. These simulations rely on simple models of stratified atmospheres such as polytropes and implement the standard incompressible or compressible fluid dynamics equations, including viscosity, thermal, and magnetic diffusivities in a bounded domain with idealised boundary conditions. For this purpose, they often make use of numerical spectral methods (see Section 6.1 above), which are extremely wellsuited for the numerical simulation of incompressible homogeneous turbulence (Vincent and Meneguzzi, 1991; Ishihara et al., 2009) but face some important problems when it comes to the simulation of stratified compressible flows. For instance, they cannot capture shocks easily and one is confronted with the problem of projecting the inhomogeneous stratified direction on a spectral basis. For this reason, the vertical direction is often treated using highorder finite differences or compact finite differences with spectrallike precision (e.g., Rincon et al., 2005). A spectral decomposition onto Chebyshev polynomials can nevertheless be used in this context for largely subsonic flows or if the equations are solved in the anelastic approximation (Clune et al., 1999).
6.2.2 Realistic simulations
The second family of models, which started to flourish in the solar and stellar communities after the pioneering contribution of Nordlund (1982), is now commonly referred to as “realistic” numerical simulations. These simulations attempt to take into account simultaneously the flow dynamics and other important physical processes of particular importance in the solar context, most notably radiative transfer, solarlike density stratifications, and realistic equations of state including Helium and Hydrogen ionizations. Unlike idealised simulations, they usually ignore the physical plasma viscosity and rely on numerical viscosity to avoid numerical blowup. These features, coupled to the use of handmade boundary conditions, makes this kind of simulations more versatile and allows to simulate compressible stratified flows at the solar surface in a more straightforward way. Indeed, as a result of including the physics of radiative transfer, the output of these simulations can be directly compared with solar observations.
There is of course a price to pay for this versatility. The first is enhanced grid dissipation (e.g., numerical viscosity): at equal resolution, the turbulent dynamics is usually much more vigorous in the idealised setups. This might not be crucial in some specific cases but, as has been already briefly mentioned in Section 6.1, probing the largescale dynamics may require achieving extremely nonlinear regimes. The second is that “realistic simulations” could be in the wrong regime, simply because they do not take into account rigorously the disparity of time and length scales of dissipative processes. This remark particularly applies to the simulation of solar MHD problems. In recent years, direct numerical simulations have shown that the dependence of the statistical properties of various turbulent MHD flows on dissipative processes can be particularly important. This has been shown for instance in the context of the fluctuation dynamo (Schekochihin et al., 2007), which might be responsible for the generation of internetwork fields (Vögler and Schüssler, 2007; Pietarila Graham et al., 2009), and for the problem of angular momentum transport mediated by magnetorotational turbulence in accretion discs (Lesur and Longaretti, 2007; Fromang et al., 2007). This point will be discussed further in Section 8.2 in the context of solar convection.
6.2.3 Current limitations
Finally, it is perhaps worth recalling that the finite capacities of computers make it completely impossible for any type of simulation, even today, to approach flow regimes characteristic of the solar surface and to span all the range of time and spatial scales involved in this highly nonlinear problem. It should therefore be constantly kept in mind when studying the results of numerical simulations of solar convection (and more generally of astrophysical turbulence) that neither of these types of models is perfect and that we are not actually “simulating the Sun” but a fairly quiet toy model of it. In particular, all simulations of solar convection to date are much less nonlinear than the RayleighBénard simulations presented in Section 6.1, which as we have seen are not themselves asymptotic in several respects either.
6.3 Simulations at granulation scales
6.3.1 Stratified convection and flows
The first realistic threedimensional simulation of solar surface convection by Nordlund (1982) mentioned earlier was followed by an improved version at higher resolution (Stein and Nordlund, 1989b). These studies were primarily devoted to studying and understanding the thermal structure and observational properties of granulation. Amongst other observations, Stein and Nordlund (1989b) noticed that convective plumes in a stratified atmosphere merged into larger plumes at larger depth, producing increasingly large convective patterns deeper and deeper. Their results also demonstrated the influence of stratification on the horizontal extent of granules and on the typical plasma velocity within granules. All these results were later confirmed by Stein and Nordlund (1998) thanks to much higher resolution simulations.
On the front of idealised simulations, Chan et al. (1982) and Hurlburt et al. (1984), using 2D numerical simulations of stratified convection at Rayleigh numbers up to 1000 times supercritical and Chan and Sofia (1989, 1996) and Cattaneo et al. (1991), using 3D idealised simulations in strongly stratified atmospheres, confirmed and refined the results of Graham (1975) and Massaguer and Zahn (1980). All the results of the early idealised simulations (some of them being fairly strongly stratified) are qualitatively in line with those of Stein and Nordlund (1989b) as far as the deep, largescale dynamics is concerned. This suggests that a detailed modelling of physical processes such as radiative transfer is not required to understand the turbulent dynamics and scaleinteractions in solar convection^{9}. On the other hand, the surface features of granulationscale convection are much better understood with realistic simulations, which are precisely tailored to this specific purpose.
Readers interested in the particular problem of granulationscale convection will find much more detailed information in the recent review by Nordlund et al. (2009). The important point to remember from this paragraph, as far as the topic of this review is concerned, is that both types of simulations predict strong asymmetries between up and downflows and the formation of largerscale convection at greater depth as a result of the imposed density stratification. Such results provide a numerical confirmation of the qualitative arguments put forward in the discussion of Section 2.2 on the scales of solar convection.
6.3.2 Main successes and caveats
Granulation is currently the best understood observational feature of solar convection, thanks mostly to the increasingly accurate simulations described above. Realistic simulations have in particular been extremely good at capturing the surface physics of radiative transfer and the thermodynamics (Stein and Nordlund, 1998), which is crucial to understand the strongly thermally diffusive nature of granules (as shown in Section 2.2, the thermal dissipation scale is similar to the granulation scale in the photosphere). The mild Péclet number regime typical of the solar granules is in this respect very helpful to numericists, as it allows for a proper numerical resolution of heat transfer processes at the granular scale.
However, the simulated granules drastically differ from the solar ones with respect to at least one important parameter, namely the Reynolds number. Because of the limited number of available grid points, the Reynolds number of a simulated granule is presently a few hundreds, while real granules have Reynolds numbers larger than 1010. This does not seem to affect the thermal physics too much: the remarkable fit of simulated and observed absorption lines (Asplund et al., 2009) demonstrates that this physics is indeed correctly captured. But research on fluid turbulence and RayleighBénard convection, as shown in Section 6.1, tells us that mild Reynolds number regimes are definitely not asymptotic with respect to the dynamics, most importantly the largescale parts of it.
Overall, it therefore remains to be demonstrated whether or not largescale simulations (reviewed in the next paragraph) based on the current generation of numerical models of granulation incorporate enough nonlinearity to make it possible to probe the actual dynamics at supergranulation scales.
6.4 Largescale simulations
6.4.1 Introduction
All simulations of the 1980s and 1990s could only address either granulationscale dynamical issues or global convection dynamics (giant cells and larger). Only in the last ten years did it become possible to start probing the dynamics at scales larger than individual granules. It is interesting in retrospect to recall (as a short anecdotal digression) the following optimistic citation, extracted from a paper by Nordlund (1985): “There is a need for numerical simulations at the scale of supergranulation [...] This is probably feasible with present day computers and numerical methods.”
Various objectives motivate largescale simulations of stellar convection. They serve to understand the global deep dynamics of spherical stellar envelopes, such as differential rotation, meridional circulation, giant cells, angular momentum transport (see Miesch (2005) for an exhaustive review), to characterise the distribution and generation of globalscale stellar magnetic fields in the presence of turbulent convection, but are also developed to study the dynamics at intermediate scales, such as sunspot scales (Heinemann et al., 2007; Rempel et al., 2009) or supergranules.
In our view, one of the most important limitations that numericists face today when it comes to simulating the Sun’s supergranulation is the following. Two different geometrical approaches are possible: local Cartesian simulations (taking a small patch of the solar surface) and global simulations in a spherical shell. In the local approach, the box size of the largest simulations to date (i.e., the largest scale of the simulation) is roughly comparable to the scale of supergranulation. Furthermore, such a configuration can only be achieved if the resolution of the turbulent processes at vigorous convective scales comparable to or smaller than the scale of granulation is sacrificed. Note also that the dynamics at supergranulation scales is tightly constrained by (periodic) lateral boundary conditions in this kind of setup. In the global spherical approach, in contrast, the smallest scales of the most recent simulations are comparable to the scale of supergranulation, which means that the “supergranulation” dynamics is strongly dissipative, in sharp contrast with the solar case (as mentioned in Section 4.2, the turbulent spectrum of solar surface convection reveals that supergranulation is located at the largescale edge of the injection range of turbulence, not in the dissipation range). In this second approach, the vigorous dynamics at granulation scales can simply not be included at the moment.
To summarise, the specific limitations of each type of simulations do not yet allow us to investigate the nonlinear dynamics and transfers of energy taking place at supergranulation scales fully consistently. These simulations nevertheless already provide us with useful informations on the largescale dynamics of convection and magnetic dynamics in the quiet Sun.
6.4.2 Global spherical simulations
New simulations of solarlike convective shells by Miesch et al. (2008) at higher spherical harmonics resolution (L_{max} = 682) have recently revealed the presence of intense cyclonic downdrafts at scales comparable to those of giant cells, a very likely reminiscence of the interactions between largescale convection and rotation described in Section 4.5. Looking at the spectrum of these simulations though, it is clear that supergranulation scales are still located in the dissipative range even at such high resolutions. This raises the concern that global simulations may not provide the most straightforward route to understand supergranulationscale dynamics.
6.4.3 Local hydrodynamic Cartesian simulations: mesoscale dynamics
The problem can also be approached by the other end, i.e., by devising models in which supergranulation scales correspond to the largest scales of the numerical domain and are consequently much larger than the actual resolution of the simulations. The general philosophy of these models is to attempt to solve for the entire nonlinear dynamics from subgranulation scales (typically 10 to 100 km with present day computers) to supergranulation scales. Computing limitations then impose that we sacrifice the global physics at scales much larger than supergranulation. Since the subgranulation to supergranulation range is expected to be fairly insensitive to curvature effects, a reasonable assumption is to perform such simulations in Cartesian geometry. As mentioned earlier, this kind of setup is also currently subject to resolution issues: if a fair amount of the available numerical resolution is devoted to the description of turbulent dynamics in the granulation range, then the dynamics at supergranulation scales is necessarily confined to the largest scales of the numerical domain and necessarily feels the artificial lateral boundary conditions (usually taken periodic).
Proceeding along these lines, Cattaneo et al. (2001) attempted to study the dynamics up to “mesoscales” (to be defined below). They performed threedimensional idealised turbulent convection simulations in the Boussinesq approximation with an aspect ratio (the ratio between the largest horizontal and vertical scales in the numerical domain) up to 20 for a Rayleigh number 5 × 10^{5} (roughly 1000 times supercritical). Note that their simulations are actually dynamo ones (see Section 6.4.6 below) but for the purpose of the discussion, we only discuss the hydrodynamic aspects of their results in this paragraph. Cattaneo et al. (2001) did not find any trace of a supergranulationlike pattern in their simulations but reported the existence of a slowly evolving granuleadvecting velocity field at a scale five times larger than the scale of granulation^{10}, corresponding to a “mesogranulation”. The typical correlation time at this mesoscale is much longer than the typical turbulent turnover time at granulation scales and the energy at this scale is also much larger than that contained in the superficial granulationscale motions. The authors suggested that the process might result from dynamical interactions at smaller scales, in the spirit of the theoretical concepts presented in Section 5.3. They also pointed out that the physical process responsible for the formation of these scales does not require that density stratification be taken into account (since it is not included in the Boussinesq approximation). The formation of mesoscale structures in large aspect ratio simulations of turbulent Boussinesq convection was subsequently confirmed by several studies (Hartlep et al., 2003; Parodi et al., 2004; von Hardenberg et al., 2008).
Motivated by these various results, Rincon et al. (2005) extended threedimensional local hydrodynamic simulations to a very wide aspect ratio ∼42, using a fully compressible polytropic setup with modest density stratification, Rayleigh numbers comparable to those of Cattaneo et al. (2001) and walltype boundary conditions at the top and bottom of the numerical domain. They did not find any trace of a “supergranulation bump” in the largescale end of the velocity power spectrum either, even though their setup allowed for such scales, but confirmed the existence of longlived and very powerful mesoscale flows with a horizontal scale also five times larger than that of granules in idealised simulations. The temperature pattern associated with these flows is clearly visible on the left side of Figure 12. They further showed that the horizontal scale of this flow increases slowly throughout the simulation, on timescales comparable or larger than the vertical thermal diffusion timescale. This slow evolution raises the issue of the thermal relaxation of all largescale simulations to date.
Another result of the study by Rincon et al. (2005) is that, from the strict scalebyscale energetics point of view, these flows are effectively driven by thermal buoyancy. During the early linear regime, basic linear stability tells us (rightly) that the growth of the convective eigenmode takes place at scales comparable to the vertical scale of the system. But, once in the nonlinear regime, the injection of energy continuously shifts to larger horizontal scales than in the linear regime. This result therefore suggests that the mesoscale flow is not directly driven by nonlinear interactions amongst smaller scales, but that nonlinearity plays a central role in the process of scale selection, possibly by controlling the strength of turbulent transport processes acting on the largescale dynamics.
Mesoscale circulations very likely correspond to the thermal winds observed in all laboratory experiments on convection (e.g., Krishnamurti and Howard, 1981; Sano et al., 1989; Niemela et al., 2001; Xi et al., 2004, and references therein). The phenomenology of this process has been shown to be very subtle, as several authors argue that the flow results from a nonlinear clustering process (Xi et al., 2004; Parodi et al., 2004) of distinct buoyant plumes. Whether or not this kind of circulations exist in the Sun and what would be their typical scale in the solar context remains an open question.
6.4.4 Stateoftheart local hydrodynamic Cartesian simulations
The most recent numerical efforts to date, as far as local hydrodynamic Cartesian simulations are concerned, are those by Ustyugov (2008) and Stein et al. (2009a). The latter ran a realistic simulation in a 96 Mm wide and 20 Mm deep threedimensional numerical box (see also Benson et al., 2006 and Georgobiani et al., 2007 for detailed reports on simulations of half this size) and found a monotonic smooth increase of the size of convective structures with depth, in agreement with the results presented in Section 6.3, and no or very little power enhancement at supergranulation scales in the surface power spectrum. They subsequently argued, similarly to Spruit et al. (1990), that there was no reason why a particular scale should popup in the continuum of scales present in the simulation (see Nordlund et al., 2009 and Georgobiani et al., 2007 for representations of the power spectra of the simulations). Ustyugov (2008) performed a similar experiment in a 60 Mm wide and 20 Mm deep threedimensional box, using a subgrid scale model to emulate the unresolved smallscale dynamics. He reported similar results, namely a gradual monotonic increase of the convection scale with depth.
It is worth pointing out that the ionisation states of Helium and Hydrogen are part of the model of Stein et al. (2009a), which allowed them to test for the first time the first theoretical explanation for the origin of the supergranulation (Simon and Leighton, 1964) presented in Section 5.2. Considering the gradual largescale decrease of energy in the power spectrum of their simulations, one may conclude that the existence of recombination layers of ionised elements does not have any noticeable impact on the surface flows. This is probably the most important conclusion relative to the supergranulation puzzle that can be drawn from these simulations.
6.4.5 Local hydrodynamic Cartesian simulations with rotation
Only a few local simulations have addressed the issue of the interactions between supergranulation and rotation. Hathaway (1982) made an early attempt at simulating this problem in the Boussinesq approximation, using a numerical box elongated in the horizontal direction (to the best of our knowledge, this is the first local numerical simulation of thermal convection at large aspect ratio, A = 10). He made the interesting observation that mean flows are generated in the presence of a tilted rotation axis and generate a subsurface shear layer (differential rotation). Since then, this kind of effects has been studied in a lot of details with local simulations at much higher numerical resolution (e.g., Brummell et al., 1996, 1998; Käpylä et al., 2004; Brandenburg, 2007). The focus of these papers is not specifically on the supergranulation problem but Brandenburg (2007) suggested that the travellingwave properties of supergranulation could be due to the radial subsurface shear (see also the paper by Green and Kosovichev, 2006 mentioned in Section 5). In a more dedicated study of this kind, Egorov et al. (2004) reported a good agreement between the divergencevorticity correlations obtained from simulations of rotating convection and those inferred from observations of the supergranulation flow field (discussed in Section 4.5).
6.4.6 Local MHD Cartesian simulations
Several local simulations have been devoted to the study of MHD convection at scales larger than granulation and notably to the process of network formation. As they are the most relevant for the problem of supergranulationscale MHD, we restrict attention to threedimensional simulations performed over the last ten years. An important distinction is in order here between two types of simulations. The first kind includes magnetoconvection simulations in an imposed mean magnetic field or with a magnetic flux introduced “by hand” at the beginning of the run. The second kind are turbulent dynamo simulations, in which the magnetic field is spontaneously generated by the turbulent convection flow starting from an infinitesimal seed field. These two types of simulations may produce qualitatively different results, as the dynamical feedback and induction terms in the equations behave in a different way for these various configurations.
One of the first of these “largescale” MHD numerical experiments was carried out by Tao et al. (1998). They performed idealised simulations of strongly stratified magnetoconvection in strong imposed magnetic fields (large Q) for various aspect ratios up to A = 8. In the simulations with largest aspect ratio (equivalently largest horizontal extent in their setup), they observed that magnetic fields tended to separate from the convective motions (flux separation). Strongfield magnetoconvection simulations are not directly relevant to the formation of the quiet Sun network though, but their phenomenology presents some interesting similarities with umbral dot or dark nuclei formation in sunspots and plage dynamics. Weiss et al. (2002) extended this work to much weaker field regimes and found that magnetic flux tended to organise into a network at scales larger than granulation, much like in the quiet Sun. The scale at which this “network” forms in their simulations seems to correspond to that of the mesoscale circulations observed in all idealised simulations (Section 6.4.3).
Largescale simulations of Boussinesq MHD convection in more turbulent regimes were performed by Cattaneo (1999), Emonet and Cattaneo (2001) and Cattaneo et al. (2003). In the absence of a mean field threading the layer (the dynamo setup mentioned at the beginning of the paragraph), they found that smallscale disordered magnetic fields generated by turbulent dynamo action organise into largerscale “mesoscale” magnetic structures. In the opposite limit of a strong mean field, they found that the dominant scales of turbulent convection are tightly constrained and reduced by magnetic tension. On this topic, we also mention the work of Bushby and Houghton (2005), Bushby et al. (2008) and Stein and Nordlund (2006), who investigated the formation process of magnetic ribbons and pointlike flux concentrations using mesoscale simulations (accommodating for just a few granules). The first group followed the idealised approach of simulations of threedimensional compressible magnetoconvection in weak field regimes and the second group a realistic approach, starting their simulation with a uniform horizontal magnetic field. Another noteworthy effort towards an improved modelling of MHD convection at scales larger than granulation is by Vögler and Schüssler (2007), who studied the generation and distribution of magnetic fields by the fluctuation dynamo process using realistic numerical simulations at moderate aspect ratio.
Another very recent attempt is by Stein et al. (2009b), who performed a set of MHD simulations in 20 Mm deep and 48 Mm wide box in which a magnetic field is introduced initially at the bottom of the numerical domain in the form of a uniform horizontal flux. Even though the time extent of their simulations is just comparable with the supergranulation timescale, their results show that the sweeping of magnetic field elements at the boundaries of supergranularlike structures leads to the formation of a magnetic network, very much like in the simulations of Ustyugov (2009).
6.5 Conclusions
The most recent largescale threedimensional models represent numerical tours de force and are increasingly successful at reproducing observational features of solar surface magnetoconvection in the granulation to supergranulation range. One of their main achievements has been the test of standard helioseismic diagnostic tools with numerical data sets: the results compare reasonably well with those extracted from real data (e.g., Georgobiani et al., 2007; Couvidat and Birch, 2009). Another important point on the topic of “virtual observations” is that largescale simulations have helped validate granuletracking techniques to reconstruct velocity fields at the solar surface (Rieutord et al., 2001).
On the specific problem of the origin of supergranulation, an important result is that the supergranulation scale does not seem to emerge as a particular scale in purely hydrodynamic simulations incorporating the ionisation of Hydrogen and Helium, which tends to disprove the “classical” Simon and Leighton (1964) supergranulation theory. Finally, it is encouraging for the future that the most recent MHD simulations to date (Ustyugov, 2009; Stein et al., 2009b) start to be large enough for meaningful numerical studies of the process of network formation and supergranulationscale MHD to be possible.
It is fair to say, however, that even the most advanced and impressive numerical efforts to date can only be considered as preliminary with respect to the supergranulation puzzle. As shown in Section 4.2, recent observations indicate that the supergranulation scale is an undeniable feature of the horizontal velocity power spectrum of solar surface convection, whatever method is used to compute the spectrum. This observation has not been reproduced by any numerical simulation so far. The superrotation rate of supergranules is another open question that no simulation can address at the moment. Finally, the interactions between supergranulation, the magnetic network, and internetwork fields are still poorly understood.
7 Summary of Current Knowledge and Issues
This long tour of the main observational, theoretical and numerical results on the problem of supergranulation being completed, we are now in a position to provide hurried (as well as less hurried) readers with a synthetic presentation of what has been learned so far on the supergranulation phenomenon and what are the current issues. The presentation of a more personal outlook and suggestions for future research is deferred to Section 8.
7.1 Observations

The length scale of the supergranulation flow, as given by the kinetic energy power spectrum of the horizontal component of solar surface flows, is in the range of 20 – 70 Mm with a preferred scale of 36 Mm (Section 4.2.1). These results come from both Dopplergrams (Hathaway et al., 2000) and from granule tracking in widefield high resolution image series (Rieutord et al., 2008). The size of the field must be sufficiently large to secure the statistical convergence of the results.

The typical size of supergranules, defined as coherent diverging flow cells at the solar surface, is in the range 10 – 30 Mm (Hirzberger et al., 2008). The derived average size is sensitive to the method used to identify supergranules.

The kinetic energy excess associated with supergranulation in the power spectrum of solar surface flows lies on the largescale side of the injection range of photospheric turbulence located at the granulation scale (Section 4.2.1 and Figure 4).

The most recent estimate of the lifetime of supergranules, based on the largest sample of supergranules collected so far, is 1.6 ± 0.7 d (Hirzberger et al., 2008). The dispersion in the measurements of supergranules lifetimes is also fairly large (Section 4.2.2).

Rms horizontal velocities at supergranulation scale are of the order of 350 m s^{−1}, while rms vertical velocities are around 30 m s^{−1} (Hathaway et al., 2002). As velocities depend on the scales considered, the relation between amplitude and scale, namely the power spectrum, provides the most suitable observable to estimate the amplitude of the supergranulation velocity field (Section 4.2.3).

Local helioseismology indicates that supergranules are shallow structures (Section 4.4), possibly not deeper than 5 Mm (Sekii et al., 2007). The mean vertical profile of the supergranulation flow is not very well constrained at the moment. More precise determinations are definitely called for.
Note that the foregoing determinations are not independent of each other, because velocity scales can be derived from the combination of length and time scales. Namely, 30 Mm divided by 1.7 d gives 205 m s^{−1}, which is in reasonable agreement with direct measurements of supergranulationscale velocities. In our view, the computation of the power spectra of solar surface flows provides one of the most robust methods to make progress on the determination of these various quantities in the future. Most notably, an accurate determination of the vertical velocity spectrum of vertical velocities in the supergranulation range is still lacking.

Horizontal intensity fluctuations at supergranulation scales are very faint (Section 4.3). The latest studies indicate that supergranules are slightly warmer at their centre. The temperature drop is less than 3 K (Meunier et al., 2007b; Goldbaum et al., 2009).

Supergranulation is affected by the global solar rotation (Section 4.5). Locally, supergranules are anticyclonic structures (Gizon and Duvall Jr, 2003), their mean vertical vorticity is negative in the northern hemisphere and positive in the southern one. The supergranulation pattern has been observed to propagate anisotropically in the prograde direction (Gizon et al., 2003).

Supergranulation has dynamical interactions with the magnetic fields of the quiet Sun. Most notably, supergranules are strongly correlated with the magnetic network (Section 4.6). Correlations between the size of supergranules and the strength of network and internetwork fields have been evidenced recently (Meunier et al., 2007a). The solarcycle dependence of the pattern remains uncertain though, as various papers have been giving contradicting results.
The bounds on intensity variations seem to be well established now. The proper rotation of supergranules, as measured by their local mean vertical vorticity, is also well constrained by local helioseismology. On the other hand, we believe that more work is required to quantitatively constrain the interaction of supergranules with magnetic fields. A determination of the magnetic energy spectrum of the quiet Sun over a wide range of scales would be extremely useful to put constraints on the physical processes at the origin of the network and internetwork fields and on their interactions with supergranulation (see Section 8 below).
7.2 Theory

Thermal convection scenarios, in which buoyancy is the main driver of the supergranulation flow (Section 5.2). Various effects (magnetic fields, shear, rotation, effective boundary conditions) have been explored within the framework of linear and weakly nonlinear theory to explain the size of of the supergranulation pattern, its weak thermal signature, its oscillations and propagation.

Collective interaction scenarios, whereby supergranulation emerges as a largescale coherent pattern triggered by nonlinear interactions of vigorous smallerscale structures like granulation (see Rieutord et al., 2000 and Section 5.3). These scenarios have mostly been explored quantitatively through “toy model” simulations (Rast, 2003b; Crouch et al., 2007) that do not incorporate the full complexity of dynamical MHD equations. In our view, direct numerical simulations provide the most promising way of making progress on this side in the future.
Finally, it is possible but certainly not obvious that supergranulation can be explained quantitatively by a simple mathematical theoretical model. In any case, one of the most urgent tasks to overcome some of the previously mentioned shortcomings is to figure out if the basic assumptions and arguments on which current theoretical models rely (linear theory, effective boundary conditions, convection in uniform magnetic fields, etc.) are justified, and to test them quantitatively with the help of largescale numerical simulations.
7.3 Numerical simulations
The complexity and nonlinearity of the physical environment of supergranulation is extraordinary: vigorous turbulent smallscale flows in a strongly stratified atmosphere, ionisation physics, rotation, shear, and tortuous magnetic fields geometries at all observable scales may all have something to do with the supergranulation phenomenon. As argued several times in this review, numerical simulations have a unique potential for approaching this complexity. They have now become an unavoidable tool to uncover the real nature of supergranulation and to test the various qualitative theoretical pictures described in Section 5.

Global spherical simulations (DeRosa et al., 2002) exhibit a supergranulationlike pattern, but the scale of this pattern is dangerously close to the grid scale of the simulations (Section 6.4.2).

In local largescale idealised simulations (Cattaneo et al., 2001; Rincon et al., 2005), two patterns can be singled out of the continuum of turbulent scales: a granulation pattern forming in the upper thermal boundary layer, and a largerscale, extremely energetic mesoscale pattern, which extends through the whole convective layer (Figure 12). Whether or not this pattern has anything to do with supergranulation or with the hypothetical solar mesogranulation is not understood (see Section 6.4.3 for an indepth discussion).

Local largescale realistic simulations of hydrodynamic convection (Stein et al., 2009a) do not exhibit any significant energy excess at supergranulation scales in spite of the presence of Hydrogen and Helium ionizations in the model (Section 6.4.4). This result therefore tends to disprove the “classical” Simon and Leighton (1964) supergranulation theory.

Local largescale realistic simulations of MHD convection reveal the formation of a magnetic network at scales ranging from mesoscales to supergranulation scales (Section 6.4.6). What sets the scale of this network and the emergence of supergranulation as a special scale in these simulations has not been investigated yet, but a recent study (Ustyugov, 2009) suggests that strong magnetic flux concentrations play a significant role in the scaleselection process.
Numericists will have to address several important issues in the forthcoming years. One of the main problems is that all dedicated simulations to date are still fairly dissipative (much more than the RayleighBénard simulations described in Section 6.1, for instance). Local largescale simulations, for instance, barely accommodate 10 grid points within a granule. This kind of resolution is not sufficient to capture all the dynamics of solar surface flows, as the viscous and magnetic dissipation scales are both much smaller than 100 km (Section 2.2) at the solar surface. As mentioned in Sections 6.2 and 6.5, resolving dissipation scales properly has recently turned out to be essential to make progress on several turbulent MHD problems, such as magnetic field generation (dynamo action) by nonhelical turbulent velocity fields. A related point is that uncovering the full dynamical physics of large scales and avoiding spurious finitebox effects requires both very large numerical domains and large integration times of the simulations, which is not ensured in today’s experiments. This point is easily illustrated by the supergranulationscale dichotomy between global and local simulations discussed in Section 6.4.
Overall, the current computing limitations are such that numerical simulations are still far away from the parameter regime typical of the Sun. Hence, one cannot exclude that all simulations to date miss some critical multiscale dynamical phenomena, either purely hydrodynamic or MHD. Largescale simulations are also currently too expensive for any decent scan of the parameter space of the problem to be possible. However, it is fair to say that the perspective of petaflop computations holds the promise of significant numerical breakthroughs in a tenyears future.
8 Discussion and Outlook
Arriving at the end of this review, we cannot escape the conclusion that the solution to the supergranulation puzzle is not as yet conspicuous. However, the combination of observational and numerical results indicates that a likely key to solve the problem is to understand the largescale interactions between magnetic fields and velocity fields in the quiet Sun. In order to stimulate future work and discussions, we would like to conclude this work by dwelling on the suggestion that supergranulation is a feature of statistically steady saturated turbulent MHD convection in an extended domain.
We start with a few contextual comments that motivate this suggestion (Section 8.1) and then explore the main features and critical points of such a scenario (Section 8.2). We finally propose a set of numerical and observational studies whose results would significantly help make progress on the understanding of largescale MHD turbulence in the quiet Sun (Section 8.3).
8.1 Preliminary comments
8.1.1 The largescale tail of the kinetic power spectrum
The energetic signature of supergranulation lies in the 10 – 100 Mm range of the horizontal kinetic energy spectrum of solar surface convection, i.e., in the largescale tail of the spectrum extending beyond the injection range (see Figure 4 and Figure 22 in et al., Nordlund 2009for instance). Both observationally and numerically derived kinetic energy spectra show us that there is some kinetic energy in that range, even though it lies beyond the typical injection scale12. But, in all hydrodynamic simulations of supergranulationscale convection to date, the kinetic energy of the flow in that range has been observed to be a monotonically decreasing, quasiselfsimilar function of the horizontal scale. In the solar photosphere, estimates for the associated typical horizontal velocities range from 1 – 2 km ^{s1} at granulation scale to 50 – 100 m s^{−1} at 100 Mm.
These results suggest two conclusions. First, horizontal flows are naturally generated at all scales in the 10 – 100 Mm range by an essentially hydrodynamic process. Second, supergranulation scales do not appear to be singled out by that process, to the best of today’s knowledge. Hence, even though we cannot yet rule out that a purely hydrodynamic process would make supergranulation a special scale, we should consider alternative scenarios. The suggestion that supergranulation could be a largescale dynamical feature of MHD turbulence in the quiet photosphere is particularly appealing in this respect. It is notably supported by recent observations of the disappearance of the spectral bump of supergranulation during the emergence of a pore (Rieutord et al., 2010).
8.1.2 Supergranulation, network and internetwork fields
Supergranulation and the magnetic network are strongly correlated observationally (see Simon and Leighton, and Section 4.6). The process of magnetic network formation has often been described in simple kinematic terms by assuming that smallscale magnetic fields are locally intensified and stochastically advected to larger scales, being eventually concentrated at the boundaries of a preexisting supergranulation flow field. Here, we would like to take on a somewhat more dynamical and less causal point of view, by simply suggesting that the magnetic network and supergranulation may originate in a coupled, undistinguishable dynamical way. This suggestion notably echoes the conclusions of Crouch et al. (2007), who describe supergranulation as an “emergent length scale” based on the results of numerical nbody toy models mimicking MHD effects.
Also, even though internetwork and network fields have been historically separated into two families based on their different strength, distribution, and orientation properties, there is no a priori reason to believe that their dynamics are physically disconnected, so we wish to keep both types of fields in the following discussion of the dynamics at large scales (see also Section 4.6.2 and the findings of Meunier et al., 2007aregarding the correlation between internetwork field strengths and supergranulation scales). From an MHD turbulence perspective, they can all be considered as part of a continuous hierarchy of magnetic structures whose energy distribution and geometrical properties vary with scale in some yet to be understood way (for a similar argument, see Stenflo and Holzreuter, 2003a). Note that this statement is not quite as extreme as saying that the structure of the magnetic field is selfsimilar (Section 4.6.3).
8.2 Nonlinear MHD at large scales
Understanding the dynamical interactions of network and internetwork magnetic fields with largescale flows is a difficult task, as the physical origin of these fields is itself rather uncertain. They may result from the emergence of magnetic fields generated far into the SCZ, or from smallscale turbulent dynamo action in the surface layers (Durney et al., 1993; Petrovay and Szakaly, 1993; ünd, 1999; Vögler and Schüussler, 2007; Danilovic et al., ^{2010}; Pietarila Graham et al., 2010), or from some reprocessing of the field of decaying active regions (Spruit et al., 1987), or from a mixture of all these processes.
8.2.1 Comparing energy spectra
8.2.2 Breaking the largescale similarity of solar surface flows
Without yet going into the details of dynamical magnetic feedback, note that the previous suggestion is simply that MHD turbulence in extended domains (in comparison to the typical injection scale of the turbulence) exhibits dynamical magnetic effects specifically enabled by the large extent of the domain. These nonlinearities very likely add up to more familiar MHD nonlinearities (quenching of specific flow scales, density evacuation) affecting scales in between the injection and dissipation scales of the turbulence (see, for instance, Schekochihin et al., 2004 for a detailed account of saturation in turbulent dynamo simulations at large to moderate Pm). In this context, supergranulation would probably be better interpreted as a byproduct of the nonlinear saturation of MHD turbulence in the quiet Sun, not as its main cause.
8.2.3 Dynamical magnetic feedback: a tricky question
The main theoretical challenges to enforce the credibility of this scenario are to understand physically how the magnetic field feeds back on the largescale flow and how the magnetic and kinetic power spectra form consistently. These two questions are currently almost completely open.
The most intuitive feedback mechanism that can be thought of in the light of our current knowledge of the dynamics of the quiet photosphere is that a largescale distribution of strong magnetic flux tubes emerging from smallerscale dynamics (like for instance in the nbody model of Crouch et al., 2007) collectively reinforces the flow at supergranulation scales, very much like strong sunspot fields support circulations towards the umbra in the far field (see Section 4.6.4, Figure 8 and Wang, 1988 for a similar suggestion). This picture qualitatively complies with the remark of Ustyugov (2009) that strong magnetic flux concentrations seem to play an important role in the scaleselection process in simulations of network formation. A very interesting physical and mathematical argument along this line was made by Longcope et al. (2003), whose calculations suggest that the dynamical feedback of a distribution of magnetic fibrils embedded into the solar plasma physically translates into a largescale viscoelasticity of the plasma. We note that a central question in this problem is to determine whether one should expect a depletion or an increase in the kinetic energy at the supergranulation scale, as a result of the magnetic feedback.
Yet another possible magnetic feedback mechanism is through the interactions between magnetic fields and radiation^{13}. Observations, theory, and simulations all suggest that magnetic concentrations tend to depress the opacity surfaces of the photosphere, which in turn is thought to channel radiation outwards (Spruit, 1976; Vögler, 2005). Strong magnetic concentrations at network scales may thereby alter the convection process at supergranulation scale and consequently single this scale out in the energy spectrum.
Overall, we note that the difficulty to understand the physical nature of dynamical magnetic feedback in this problem is in no way an exception. Nonlinear MHD phenomena are notorious for defying simple handwaving arguments. For instance, current observations seem to rule out the possibility of a simple scalebyscale equipartition of magnetic and kinetic energy down to the smallest observable scales. The most advanced simulations of the smallscale solar surface dynamo (Vögler and Schüssler, 2007) cannot answer the question of nonlinear dynamo saturation in a definite, asymptotic way as yet^{14}. Finally, the scalelocality of dynamical interactions is not guaranteed in nonlinear MHD flows, including the smallscale dynamo (Schekochihin et al., 2004; Yousef et al., 2007), so the simple observation of an equipartition of energy at some scale is probably not sufficient to understand the physics of magnetic feedback fully consistently.
8.2.4 Comments on the equipartition argument
This result mostly suggests that supergranulationscale motions cannot themselves generate these flux tubes. Partial evacuation of density and vigorous localized motions such as granulationscale motions seem to be required to obtain superequipartition fields (Webb and Roberts, 1978; Spruit, 1979; Spruit and Zweibel, 1979; Unno and Ando, 1979; Proctor, 1983; Hughes and Proctor, 1988; Bushby et al., 2008). This does not imply, however, that the supergranulation and network scales are not selected by nonlinear magnetic feedback processes. Actually, the existence of localized magnetic concentrations exceeding equipartition with the supergranulation flow certainly hints that magnetic effects cannot be bypassed to explain the dynamics of supergranulation.
Note finally that the energetics of the field is not the only important parameter of the problem. The curvature of magnetic field lines (the variation of the field along itself) is equally important to understand their dynamical role. In this respect, strong but straight localized flux tubes may not be particularly effective at interacting with the flow in comparison to weaker but significantly more tangled fields. Understanding the effective largescale magnetic response at the surface of the quiet Sun therefore very likely requires considering the integrated dynamical contribution of the whole multiscale distribution of surface fields instead of the simple magnetic pressure estimate of individual magnetic elements populating the magnetic network.
8.3 Suggestions for future research

What is the magnetic power spectrum of the quiet Sun in the six decades spanning the 100 m – 100 Mm range?

How does the multiscale magnetic field distribution of the quiet Sun originate?

How does the dynamical magnetic feedback operate, and at which scales?
Some of these questions are already at least partially answerable with nowadays observational and numerical facilities, or they will become so in the near future.
From the observational point of view, we emphasise the need for a statistical description of solar surface MHD turbulence, as opposed to a description in terms of individual “structures” such as flux tubes or magnetic “elements”. The distributions and geometries of magnetic and velocity fields in the quiet photosphere appear to be so different that their largescale interactions can probably only be understood in statistical terms. A determination of the magnetic energy spectrum of the quiet Sun over a very wide range of scales would notably be extremely useful to understand the physics of MHD turbulence in the quiet Sun and to put constraints on the physical processes at the origin of network and internetwork fields  and consequently on the supergranulation problem.
Our theoretical and numerical understanding of supergranulationscale MHD convection is scarce. We do have numerical (Ustyugov, 2009; Stein et al., 2009b) hints that advection of weak, smallscale fields and their subsequent clustering can lead to the formation of increasingly energetic magnetic features distributed on larger scales. This phenomenology was already discussed many years ago by Parker (1963) and is included in the nbody model of Crouch et al. 2007. But is it possible to gain a better understanding of this process, starting from the MHD equations?
An essential issue from the point of view of largescale MHD turbulence is to decipher the preliminary stages of production of quiet Sun magnetic fields up to supergranulation scales. A possible theoretical approach to this problem would be to study the largescale structure of magnetic eigenmodes in simplified models of turbulent dynamo action, such as the Kraichnan.Kazantsev model (Kazantsev, 1968; Kraichnan, 1968). Such an approach has recently been taken on from a generic perspective by Malyshkin and Boldyrev (2009). Another interesting exercise would be to study the outcome of turbulent induction by a high Rm compressible flow in an extended domain threaded by a weak uniform mean field. This process is distinct from the fluctuation dynamo and could be responsible for the generation of a smallscale magnetic imprint of the global solar dynamo (e.g., Brandenburg and Subramanian, 2005). In the incompressible limit, it produces a simple k^{−1} magnetic energy spectrum (Ruzmaikin and Shukurov, 1982; Schekochihin et al., 2007), not that far from the solar magnetic power spectrum in the 1 – 10 Mm range (Section 4.6.2).
Finally, in order to explain why the supergranulation scale appears to be special in the quiet photosphere, one needs to better understand the statistically steady state of turbulent MHD convection in this region. The main problem is that we do not currently know what simple nonhelical incompressible low Pm MHD turbulence looks like in dynamical regimes — even in small spatial domains — both when the magnetic field is produced by local turbulent dynamo action (e.g., Cattaneo, 1999; Schekochihin et al., 2004, 2007; Vögler and Schüssler, 2007) and in the presence of a net magnetic flux (Cattaneo et al., 2003; Stein et al., 2009b; Ustyugov, 2009). Addressing this question in the context of supergranulationscale simulations therefore represents a daunting task. On this side, we currently have no choice but to perform mildly nonlinear simulations in Pm ∼ 1 regimes with imposed magnetic flux. To gain some insight into the highly nonlinear behaviour of turbulent MHD flows, these efforts should be complemented by dynamical simulations of specific MHD processes, such as the smallscale dynamo, in smaller domains but more extreme parameter regimes (higher Re and Rm, low Pm). By combining both approaches, it may eventually be possible to understand nonlinear MHD physics at supergranulation scales.
The general message that we tried to convey in this section is that the supergranulation puzzle may turn out to be a very challenging MHD turbulence problem, the solution to which will certainly require simultaneous progress on MHD theory and observational and numerical solar physics. This dual fundamental physics and astrophysics perspective of the supergranulation problem, we believe, makes it a particularly exciting challenge for the future.
Footnotes
 1.
^{1} In solar physics, an appropriate length unit is the megametre (Mm), also 1000 km.
 2.
^{2} Hinode is led by the Japanese Aerospace Exploration Agency (JAXA) in collaboration with NASA, the Science and Technology Facilities Council (STFC), and the European Space Agency (ESA). Hinode is a Japanese mission developed, launched, and operated by ISAS/JAXA, in partnership with NAOJ, NASA, and STFC (UK). Additional operational support is provided by ESA and NSC (Norway). The project website can be found at http://solarb. msfc.nasa.gov/
 3.
^{3} l_{ v } can be understood physically as the scale at which viscous effects start to dominate over inertial effects, so that the Reynolds number at this scale Rel_{ v } = l_{ v }Vl_{ v }/v, where Vl_{ v } ∼ ^{1/4}V is the typical velocity at scale l_{ v } in the framework of the Kolmogorov theory.
 4.
^{4} l_{ n } is the scale at which resistive effects take over magnetic field stretching, corresponding to a scaledefined magnetic Reynolds number \(Rm_{\ell _\eta } = \ell _\eta V_{\ell _\eta } /\eta = 1\).
 5.
^{5} Similarly to l_{ v } and l_{ n }, this scale corresponds to the scaledefined Péclet number \(Pe_{\ell _\kappa } = V_{\ell _\kappa } \ell _\kappa /\kappa = 1\).
 6.
^{6} At granulation scales, the spectral power density is less than 300 km^{3} s^{−2}. We recall that granules have a much larger typical velocity than supergranules though, of the order 1 – 2 km s^{−1} (Section 3.1). The difference comes from the definition of the spectral power density at wavenumber k, E(k) ∼ k^{−1}V _{ k } ^{2} , which introduces an extra k factor.
 7.
^{7} The Roberts number q = 1/ζ is also used in the context of convection in planetary cores. In the Earth’s core, q ≪ 1, see, e.g., Zhang and Jones (1996).
 8.
^{8} Note that a previous work of Simon et al. (1991) already used a purely kinematic model to model mesogranular flows and exploding granules.
 9.
^{9} Stratified simulations with a bottom wall tend to exhibit more smallscale turbulent activity in deep layers that their openwall counterparts, though (see Figure 12). This behaviour may be related to the enhanced shear and recirculations generated at the bottom wall, or with the fact that most of these simulations use a constant dynamical viscosity which, combined with density stratification, enhances the local Reynolds number as one moves deeper down.
 10.
^{10} Idealised simulations evolve nondimensional equations such as (2), so their results are not given in solar units. However, even in this kind of idealised simulations, granulationlike cells clearly appear in a thin thermal boundary layer at the upper boundary (Figure 12 provides horizontal temperature maps extracted from a similar simulation exhibiting this phenomenon). This simple observation usually serves to “calibrate” the size of the dynamical structures present in the simulations with respect to the size of granules.
 11.
^{11} Amongst the significant differences between idealised simulations and the realistic ones, Nordlund et al. (1994) pointed out that using “walltype” boundary conditions, as is standard in idealised simulations, can alter significantly the shape of the convective pattern. Indeed, this type of boundary conditions allows for a return flow after plumes smash down onto the bottom wall, which of course does not occur in the Sun until the very deep layers of the solar convection zone are reached by descending plumes. Nordlund et al. (1994) suggested to use stronger stratifications in simulations in closed domains to attenuate this effect. Finally, upflows seem to play a much more important role in the process of vertical heat transport in idealised simulations than in realistic ones (Stein and Nordlund, 1994). This may have some important consequences regarding the most energetic scales of the flows, which correspond to the mesoscales in idealised simulations.
 12.
^{13} Available magnetic power spectra are derived from polarimetric measurements which mostly track the vertical component of the magnetic field at disc centre.
 13.
^{14} We are grateful to one of the referees for pointing this out to us.
 14.
^{15} All such simulations (Cattaneo, 1999; Vögler and Schüssler, 2007) are for Pm ∼ 1 or larger and asymptotically not large Re. How saturation takes place and whether equipartition should be expected in both (Pm ≫ 1, ≫ Re ≫ 1) and (Pm ≪ 1, Re ≫≫ 1) limits is unknown (Schekochihin et al., 2004; Yousef et al., 2007; Schekochihin et al., 2007; Tilgner and Brandenburg, 2008; Cattaneo and Tobias, 2009), so it is currently very difficult and potentially risky to predict how and at which level of magnetic energy the putative solar surface dynamo saturates.
Notes
Acknowledgements
We wish to thank our many collaborators and colleagues who have contributed in various ways to trigger and sustain our interest in the supergranulation problem over the years. We are particularly grateful to Thierry Roudier, François Lignières and Nadège Meunier for sharing their insights into solar convection and magnetism with us and for their reading of the manuscript. We also thank Laurent Gizon for providing us with some information on local helioseismology techniques.
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