The first realistic three-dimensional 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, large-scale 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 scale-interactions in solar
convection^{9}.
On the other hand, the surface features of granulation-scale convection are much better understood with
realistic simulations, which are precisely tailored to this specific purpose.

Readers interested in the particular problem of granulation-scale 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 larger-scale 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.

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 10^{10}. 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
Rayleigh–Bénard 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 large-scale parts of
it.

Overall, it therefore remains to be demonstrated whether or not large-scale 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.

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