6 Numerical Modelling

Our understanding of turbulent convection in general and turbulent convection at the solar surface in particular has improved significantly with the advent of large-scale computing resources. As far as supergranulation is concerned, even though numerical modelling has been and is still confronted with several major computing limitations, it is now on the verge of making very interesting progress towards discriminating between various physical scenarios and ideas such as those presented in Section 5. In this section, we review the evolution of numerical simulations of solar surface convection over the last thirty years and how they have helped us make progress on the specific problem of supergranulation. We notably discuss the advantages and limitations of the various types of numerical models used to study solar surface convection, in order to identify what numerical simulations have really told us (or not told us) on the problem of supergranulation and to provide the reader with a (hopefully) clear understanding of the current important modelling issues.

The section starts with two introductory paragraphs on the numerical simulation of turbulent thermal convection in the Rayleigh–Bénard framework (Section 6.1) and in the solar context (Section 6.2). We then recall the main results obtained from granulation-scale simulations (6.3), before describing in detail the history and current status of “large-scale” hydrodynamic and MHD simulations (6.4) aiming at understanding the dynamics at scales comparable to that of supergranulation.

 6.1 Numerical simulations of turbulent Rayleigh–Bénard convection
  6.1.1 Rayleigh–Bénard and Navier–Stokes
  6.1.2 State-of-the-art modelling
 6.2 Numerical simulations of solar convection
  6.2.1 Idealised simulations
  6.2.2 Realistic simulations
  6.2.3 Current limitations
 6.3 Simulations at granulation scales
  6.3.1 Stratified convection and flows
  6.3.2 Main successes and caveats
 6.4 Large-scale simulations
  6.4.1 Introduction
  6.4.2 Global spherical simulations
  6.4.3 Local hydrodynamic Cartesian simulations: mesoscale dynamics
  6.4.4 State-of-the-art local hydrodynamic Cartesian simulations
  6.4.5 Local hydrodynamic Cartesian simulations with rotation
  6.4.6 Local MHD Cartesian simulations
 6.5 Conclusions

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