5.1 The rotating MHD Rayleigh–Bénard convection problem

5.1.1 Formulation

The simplest formulation of the problem of thermal convection of a fluid is called the Rayleigh–Bénard problem. It describes convection of a liquid enclosed between two differentially heated horizontal plates, each held at a fixed temperature. The mathematical model is derived under the Boussinesq approximation, which amounts to assuming that the flow is highly subsonic and that density perturbations δρ to a uniform and constant background density ρ o are negligible everywhere except in the buoyancy term δρ⃗g, where ⃗g = − g⃗ez stands for the gravity (Chandrasekhar, 1961Jump To The Next Citation Point). The equilibrium background state is a linear temperature profile with temperature decreasing from the bottom to the top of the layer. This case is in many respects different and simpler than the strongly stratified SCZ case, which treatment requires using more general compressible fluid and energy equations than those given below (Nordlund, 1982Jump To The Next Citation Point), but is sufficient to discuss many of the important physical (Section 5.2) and numerical (Section 6.1) issues pertaining to supergranulation.

Anticipating upcoming discussions on the origin of supergranulation, we extend the simplest hydrodynamic formulation of the Rayleigh–Bénard problem to the case of an electrically conducting liquid threaded by a mean vertical magnetic field denoted by B⃗ = B ⃗e o oz and rotating around a vertical axis, with a rotation rate ⃗ Ω = Ω ⃗ez. This set-up is shown on Figure 9View Image.

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Figure 9: The rotating MHD Rayleigh–Bénard convection problem.

In nondimensional form, the equations for momentum and energy conservation, the induction equation, the equations for mass conservation and magnetic field solenoidality read

∂ ⃗u √ --- P r2 --- + ⃗u ⋅ ⃗∇ ⃗u + TaP r ⃗ez × ⃗u = −∇⃗p + RaP r𝜃⃗ez + Q ----(⃗∇ × ⃗B) × ⃗B + Pr Δ⃗u , ∂ τ P m ∂-𝜃 + ⃗u ⋅ ⃗∇ 𝜃 − u = Δ 𝜃 , ∂ τ z (2 ) ∂ ⃗B P r ----+ ⃗u ⋅∇⃗B⃗ = ⃗B ⋅∇⃗⃗u + ----ΔB⃗ , ∂τ Pm ⃗∇ ⋅ ⃗u = 0 , ⃗∇ ⋅ ⃗B = 0 ,
where the momentum equation has been written in the rotating frame, lengths are measured in terms of the thickness of the convection layer d, times are defined with respect to the thermal diffusion time τκ = d2∕ κ (κ is the thermal diffusivity), the total magnetic field B⃗ is expressed in terms of the Alfvén speed √ ----- VA = Bo ∕ ρoμo, and temperature deviations 𝜃 to the initial linear temperature profile are measured in terms of the background temperature difference ΔT = T − T top bot between the two horizontal plates enclosing the fluid in the vertical direction. Nondimensional velocity and pressure fluctuations are denoted by ⃗u and p, respectively. This set of equations must be complemented by appropriate boundary conditions, most commonly fixed temperature or fixed thermal flux conditions on the temperature, no-slip or stress-free conditions on velocity perturbations, and perfectly conducting or insulating boundaries for the magnetic field.

Several important numbers appear in the equations above, starting with the Rayleigh number

3 Ra = α|ΔT-|gd--= |N 2|τντκ , (3 ) νκ
where α is the thermal expansion coefficient of the fluid defined according to δρ ∕ρo = − α 𝜃. Here, N 2 = α ΔT g∕d < 0 is the square of the Brunt–Väisälä frequency (negative for a convectively unstable layer) and 2 τν = d ∕ν is the viscous diffusion time, so the Rayleigh number measures the relative effects of the convection “engine”, buoyancy, and of the “brakes”, namely viscous friction and heat diffusion. The second important parameter above is the Chandrasekhar number
B2 d2 τντη Q = --o----= --2- , (4 ) ρoμoνη τA
which is a measure of the relative importance of magnetic tension (τA = d ∕VA is the Alfvén crossing time) on the flow in comparison to magnetic diffusion (η is the magnetic diffusivity, τ = d2∕η η is the typical magnetic diffusion time) and viscous friction. The relative importance of the Coriolis force in comparison to viscous friction is measured by the Taylor number,
4Ω2d4- 2 2 T a = ν2 = (2Ω )τν . (5 )
Finally, P r = ν∕κ and P m = ν∕η, where η is the magnetic diffusivity, stand for the thermal and magnetic Prandtl numbers (see Section 2.2).

5.1.2 Linear instability and the solar regime

In the simplest non-rotating hydrodynamic case (T a = Q = 0, no induction), when the Rayleigh number is less than a critical value Racrit that depends on the particular choice of boundary conditions, diffusive processes dominate over buoyancy: the hydrostatic solution is stable, i.e., any velocity or temperature perturbations decays. For Ra > Racrit, convection sets in as a linear instability and perturbations grow exponentially in the form of convection rolls or hexagons with a horizontal spatial periodicity comparable to the convective layer depth d in most cases. The effects of magnetic fields and rotation on the linear stability analysis are discussed in the next paragraphs.

It should be noted that Ra, Q, and T a are all extremely large numbers in the Sun, if they are computed from microscopic transport coefficients (Section 2.2). So, in principle, there is no reason why solar convection should be close to the instability threshold. However, theoretical studies of large-scale convection (such as supergranulation) commonly assume that viscous, thermal, and magnetic diffusion at such scales are determined by turbulent transport, not microscopic transport. This leads to much larger transport coefficients (which can be estimated, for instance, using the typical scale and velocity of the granulation pattern) and much smaller “effective” Ra, Q, and Ta, so the “large-scale” system is generally considered not too far away from criticality. Making this (strong) mean-field assumption serves to legitimate using the standard toolkits of linear and weakly nonlinear analysis to understand the large-scale behaviour of solar convection.

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