Numerical simulations dedicated to the supergranulation problem are still in their infancy though, mostly because they remain awfully expensive in terms of computing time. The latest generation of numerical experiments, summarised in Table 1, barely accommodates for the scale of supergranulation. The main results obtained so far are summarised below.

- Global spherical simulations (DeRosa et al., 2002) exhibit a supergranulation-like pattern, but the scale of this pattern is dangerously close to the grid scale of the simulations (Section 6.4.2).
- In local large-scale 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 larger-scale, 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 in-depth discussion).
- Local large-scale 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 large-scale 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 scale-selection 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 Rayleigh–Bénard simulations described in Section 6.1, for instance). Local large-scale 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 non-helical turbulent velocity fields. A related point is that uncovering the full dynamical physics of large scales and avoiding spurious finite-box 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 supergranulation-scale 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. Large-scale 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 ten-years future.

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