= Thermal-diffusion scaling of planar magnetoconvection
Use depth $d$, time $d^2/\kappa$, stream-function scale $\kappa$, temperature perturbation scale $\Delta T$, and magnetic flux-function scale $B_0d$. With $\mathbf u=(-\psi_z,0,\psi_x)$, $\mathbf B=(-A_z,0,A_x)$ and $A=x+\chi$, advection is $J(\psi,f)=\psi_xf_z-\psi_zf_x$. The temperature background contributes $\psi_x$ to the heat equation and the imposed vertical field contributes $\psi_z$ to the perturbation induction equation. Taking the curl of the <magnetohydrodynamic momentum equation> gives a magnetic term $\sigma\zeta QJ(A,\nabla^2A)$, where $\sigma=\nu/\kappa$, $\zeta=\eta/\kappa$, $R=g\alpha\Delta T d^3/(\nu\kappa)$ and $Q=B_0^2d^2/(\mu_0\rho\nu\eta)$.
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