All simulations of the 1980s and 1990s could only address either granulation-scale 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 large-scale 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 global-scale 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 set-up. 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 large-scale 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 large-scale dynamics of convection and magnetic dynamics in the quiet Sun.

Global spherical simulations of turbulent convection appeared thirty years ago. Gilman (1975) devised the first numerical model of 2D Boussinesq convection in a spherical shell and used it to study the influence of rotation on convection, the problem of large-scale circulations in the solar convection zone and that of the interactions between supergranulation and rotation (e.g., Gilman and Foukal, 1979). Gilman and Glatzmaier (1981) and Glatzmaier (1984, 1985) extended this work to the anelastic approximation, while Valdettaro and Meneguzzi (1991) devised a fully compressible model. As a result of computer limitations at that time, these simulations were restricted to fairly laminar regimes and very large solar scales. Most simulations in spherical geometry use the expansions of the fields on spherical harmonics up to a given resolution (the order of the smallest scale spherical harmonic). In those terms, the resolution of the early simulations was approximately . In solar units, this means that the smallest resolved horizontal scale in these simulations is

much larger than the scale of supergranulation (36 Mm, corresponding to the spherical harmonic , see Section 4.2).

In recent years, one of the most popular codes for high-resolution three-dimensional spherical simulations of stellar convection has been the ASH (Anelastic Spherical Harmonics) code (Clune et al., 1999). An interesting attempt to study supergranulation with ASH is that by DeRosa (2001), DeRosa and Toomre (2001) and DeRosa et al. (2002), who carried out “idealised” three-dimensional hydrodynamic simulations in thin spherical shells with a horizontal resolution of , corresponding to a smallest resolved horizontal scale of 13 Mm. Their simulations exhibit structures at scales comparable to that of supergranulation (see Figure 11). However, the physical origin of these structures is rather uncertain because the grid scale (13 Mm) is not small compared to the supergranulation scale. It is also difficult to spot why supergranulation scales would play a special role (except for being in the dissipative range) in their set-up. Finally, as mentioned earlier, the granulation dynamics, which dominates the power spectrum of solar surface convection, is not included in the model. The conclusions of this precise set of simulations are therefore unfortunately limited, but the next generations of experiments of this kind may enable significant progress on the problem.

New simulations of solar-like convective shells by Miesch et al. (2008) at higher spherical harmonics resolution () 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 large-scale 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 supergranulation-scale 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 set-up 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 three-dimensional 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 supergranulation-like pattern in their simulations but reported the existence of a
slowly evolving granule-advecting 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 granulation-scale 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).

The next three-dimensional experiment in the series was done by Rieutord et al. (2002).
They performed a “realistic” hydrodynamic simulation of turbulent convection at aspect ratio
10. They did observe a growth of the typical size of convective structures with depth but did
not find any evidence for the formation of dynamical scales larger than that of granulation
at the surface. They pointed out that the turbulence was not very vigorous in this kind of
simulations, which might explain why no supergranulation-scale dynamics is present. Another
possibility is that the numerical domain was not wide or deep enough to accommodate this kind of
large-scale dynamics. An important point to note is that their simulation, unlike that of Cattaneo
et al. (2001), was designed with an open bottom boundary condition and a strongly stratified
atmosphere^{11}.

Motivated by these various results, Rincon et al. (2005) extended three-dimensional local hydrodynamic simulations to a very wide aspect ratio 42, using a fully compressible polytropic set-up with modest density stratification, Rayleigh numbers comparable to those of Cattaneo et al. (2001) and wall-type boundary conditions at the top and bottom of the numerical domain. They did not find any trace of a “supergranulation bump” in the large-scale end of the velocity power spectrum either, even though their set-up allowed for such scales, but confirmed the existence of long-lived 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 large-scale simulations to date.

Another result of the study by Rincon et al. (2005) is that, from the strict scale-by-scale 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 large-scale 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.

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 three-dimensional 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 pop-up 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 three-dimensional box, using a subgrid scale model to emulate the unresolved small-scale 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 large-scale 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.

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, ). 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 travelling-wave 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 divergence-vorticity correlations obtained from simulations of rotating convection and those inferred from observations of the supergranulation flow field (discussed in Section 4.5).

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 supergranulation-scale MHD, we restrict attention to three-dimensional 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 “large-scale” 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 . In the simulations with largest aspect ratio (equivalently largest horizontal extent in their set-up), they observed that magnetic fields tended to separate from the convective motions (flux separation). Strong-field 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).

Large-scale 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 set-up mentioned at the beginning of the paragraph), they found that small-scale disordered magnetic fields generated by turbulent dynamo action organise into larger-scale “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 point-like flux concentrations using mesoscale simulations (accommodating for just a few granules). The first group followed the idealised approach of simulations of three-dimensional 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.

The specific problem of supergranulation-scale MHD was only attacked in recent years. Ustyugov (2006, 2007, 2009) performed several realistic magnetoconvection simulations in an imposed 50 G vertical field, the largest of these simulations being for a 20 Mm deep and 60 Mm wide box. He observed the formation of a magnetic network at scales in the meso-supergranulation range, with magnetic elements being either point-like or organised in flux sheets or magnetic ribbons. Some snapshots of his simulations are reproduced in Figure 13.

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 supergranular-like structures leads to the formation of a magnetic network, very much like in the simulations of Ustyugov (2009).

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