6.1 Numerical simulations of turbulent Rayleigh–Bénard convection

Even if the hydrodynamic non-rotating Rayleigh–Bénard 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 Rayleigh–Bénard and Navier–Stokes

In the non-rotating hydrodynamic case (T a = Q = 0, no induction), the Boussinesq equations (2View Equation) are very similar to the forced Navier–Stokes equations, except that the forcing term is not an external body force but is determined self-consistently from the time-evolution of the temperature fluctuations. Based on both experimental and numerical evidence at order one Prandtl number, several authors (Rincon, 2006Lohse 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 Navier–Stokes turbulence in the inertial-range (i.e., Kolmogorov turbulence plus possible intermittency corrections). Hence, at a very good first approximation, the numerical issues and requirements to simulate the Rayleigh–Bénard problem at very high Rayleigh numbers are the same as those pertaining to the simulation of forced Navier–Stokes turbulence at high Reynolds numbers.

6.1.2 State-of-the-art modelling

The performances of turbulent convection simulations are often measured by the ratio of the imposed Rayleigh number to its critical value Ra∕Racrit 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 Rayleigh–Bénard problem are approximately Ra ∕Racrit ∼ 1011 (Verzicco and Camussi, 2003Jump To The Next Citation PointAmati et al., 2005), but most “routine” simulations of the problem are in the much softer range Ra ∕Racrit ∼ 105– 107. Achieving highly turbulent regimes first requires using high-order 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 ∼ 5123 to ∼ 10003) on a non-uniform 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 long-time, large-scale, mean-field dynamics.

View Image

Figure 10: Snapshots of temperature fluctuations in a vertical plane, from numerical simulations of Rayleigh–Bénard convection in a slender cylindrical cell at Pr = 0.7 and Rayleigh-numbers (a) 2 × 107, (b) 2 × 109, and (c) 2 × 1011 (from Verzicco and Camussi, 2003).

Moreover, simulations at very high Rayleigh numbers are restricted to fairly low aspect ratio domains (the ratio between the horizontal and vertical extents of the domain), typically 1/2 or 1, so they do not as yet allow to probe the very large-scale dynamics of turbulent convection. Finally, they are limited to Prandtl numbers of order unity, so it is currently not possible to investigate the effect of scale separation between the various dissipation scales of the problem while preserving the highly nonlinear character of the simulations. An example of such a high-performance simulation is provided by Figure 10View Image, which shows temperature snapshots in vertical planes of simulations of turbulent Rayleigh–Bénard convection in a slender cylindrical cell at P r = 0.7. Two very important features can be seen on the figure:

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 large-scale dynamical evolution of the system.

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