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 $B_{0}d$. With $\mathbf{u}=(-{\psi }_z,0,{\psi }_x)$, $\mathbf{B}=(-A_z,0,A_x)$ and $A=x+\chi$, advection is $J(\psi ,f)={\psi }_x{f}_z-{\psi }_z{f}_x$. 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 $\sigma \zeta QJ(A,{\nabla }^{2}A)$, where $\sigma =\nu /\kappa$, $\zeta =\eta /\kappa$, $R=g\alpha \Delta T{d}^3/(\nu \kappa )$ and $Q=B_0^2{d}^2/({\mu }_0\rho \nu \eta )$.