- Thermal convection scenarios, in which buoyancy is the main driver of the supergranulation flow (Section 5.2). Various effects (magnetic fields, shear, rotation, effective boundary conditions) have been explored within the framework of linear and weakly nonlinear theory to explain the size of of the supergranulation pattern, its weak thermal signature, its oscillations and propagation.
- Collective interaction scenarios, whereby supergranulation emerges as a large-scale coherent pattern triggered by nonlinear interactions of vigorous smaller-scale structures like granulation (see Rieutord et al., 2000 and Section 5.3). These scenarios have mostly been explored quantitatively through “toy model” simulations (Rast, 2003b; Crouch et al., 2007) that do not incorporate the full complexity of dynamical MHD equations. In our view, direct numerical simulations provide the most promising way of making progress on this side in the future.

A shared property of all models is the looseness of the approximations on which they rely (e.g., linear theory with turbulent viscosity parametrisation, or purely phenomenological arguments on the nature of dynamical interactions between granules and their potential large-scale instabilities). Completely distinct theoretical arguments can easily be tuned to produce results that are all broadly consistent with observations. This degeneracy makes it impossible to discriminate between various scenarios and to come up with a proper theoretical explanation for the origin of supergranulation that could be unambiguously validated by observations.

Finally, it is possible but certainly not obvious that supergranulation can be explained quantitatively by a simple mathematical theoretical model. In any case, one of the most urgent tasks to overcome some of the previously mentioned shortcomings is to figure out if the basic assumptions and arguments on which current theoretical models rely (linear theory, effective boundary conditions, convection in uniform magnetic fields, etc.) are justified, and to test them quantitatively with the help of large-scale numerical simulations.

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