We now turn to the description of existing theoretical models of supergranulation. These models are basically of two types: those that postulate that supergranulation has a convective origin (i.e., it is driven by thermal buoyancy), and those that do not. In order to set the stage for upcoming discussions, we start with a brief description of the rotating MHD Rayleigh–Bénard problem (Section 5.1), which provides the simplest mathematical description of rotating magnetoconvection in a fluid. We then review various thermal convection models of supergranulation (Section 5.2) and discuss other possible physical mechanisms involving collective “turbulent” dynamics of smaller-scale convection (Section 5.3). A few concluding remarks follow.

Before we start, it is perhaps useful to mention that most of these models are unfortunately only very qualitative, in the sense that they either rely on extremely simplified theoretical frameworks (like linear or weakly nonlinear theory in two dimensions, or simple energetic arguments) or on simple dynamical toy models designed after phenomenological considerations. The looseness of theoretical models, combined with the incompleteness of observational constraints and shortcomings of numerical simulations, has made it difficult to either validate or invalidate any theoretical argument so far. What numerical simulations tell us and how the theoretical models described below fit with numerical results and observations will be discussed in detail in Section 6.

5.1 The rotating MHD Rayleigh–Bénard convection problem

5.1.1 Formulation

5.1.2 Linear instability and the solar regime

5.2 Convective origin of supergranulation

5.2.1 Multiple mode convection

5.2.2 Effects of temperature boundary conditions

5.2.3 Oscillatory convection and the role of dissipative processes

5.2.4 Oscillatory convection, rotation, and shear

5.2.5 Oscillatory convection and magnetic fields

5.2.6 Other effects

5.2.7 Shortcomings of simple convection models

5.3 Large-scale instabilities and collective interactions

5.3.1 Rip currents and large-scale instabilities

5.3.2 Plume interactions

5.4 Conclusions

5.1.1 Formulation

5.1.2 Linear instability and the solar regime

5.2 Convective origin of supergranulation

5.2.1 Multiple mode convection

5.2.2 Effects of temperature boundary conditions

5.2.3 Oscillatory convection and the role of dissipative processes

5.2.4 Oscillatory convection, rotation, and shear

5.2.5 Oscillatory convection and magnetic fields

5.2.6 Other effects

5.2.7 Shortcomings of simple convection models

5.3 Large-scale instabilities and collective interactions

5.3.1 Rip currents and large-scale instabilities

5.3.2 Plume interactions

5.4 Conclusions

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