6.2 Numerical simulations of solar convection

After this presentation of the state-of-the-art numerical modelling of Rayleigh–Bénard 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 (1975Jump To The Next Citation Point) who revealed an asymmetry of up and downflows in two-dimensional 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. (1982Jump To The Next Citation Point). Massaguer and Zahn (1980Jump To The Next Citation Point) 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 (1982Jump To The Next Citation Point). 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 set-up tailored for the purpose of studying specific physical processes. In general, a given set-up 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 (1975Jump To The Next Citation Point) 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 well-suited for the numerical simulation of incompressible homogeneous turbulence (Vincent and Meneguzzi, 1991Ishihara 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 high-order finite differences or compact finite differences with spectral-like precision (e.g., Rincon et al., 2005Jump To The Next Citation Point). 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., 1999Jump To The Next Citation Point).

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 (1982Jump To The Next Citation Point), 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, solar-like 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 blow-up. 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 set-ups. This might not be crucial in some specific cases but, as has been already briefly mentioned in Section 6.1, probing the large-scale 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., 2007Jump To The Next Citation Point), which might be responsible for the generation of internetwork fields (Vögler and Schüssler, 2007Jump To The Next Citation PointPietarila Graham et al., 2009), and for the problem of angular momentum transport mediated by magneto-rotational turbulence in accretion discs (Lesur and Longaretti, 2007Fromang 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 Rayleigh–Bénard simulations presented in Section 6.1, which as we have seen are not themselves asymptotic in several respects either.

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