 Solution

Use the same length, thermal-time, velocity and temperature scales as in thermal convection. Write the imposed magnetic field as $B_0\hat{\mathbf{z}}$ and measure its perturbation in units of $B_0$. The physical linear magnetohydrodynamic momentum equation contains magnetic tension $(B_0^2/{\mu }_0){\partial }_z\mathbf{b}$; the gradient part of the Lorentz force density is absorbed into magnetohydrodynamic total pressure.

Divide the momentum equation by the viscous force scale ${\rho }_0\nu \kappa /d^3$. Its acceleration coefficient is $\kappa /\nu =1/\sigma$ and its magnetic coefficient isHere $Q$ is the Chandrasekhar number, comparing magnetic and viscous-diffusive effects, and $\zeta$ is the ratio of magnetic diffusivity to thermal diffusivity. The induction scale is $B_0\kappa /d^2$, so linearization givesComplete the linear system with $\dot{\theta }=w+{\nabla }^2\theta$, $\nabla \cdot \mathbf{u}=0$ and the solenoidal magnetic-field constraint $\nabla \cdot \mathbf{b}=0$. The omitted advective and perturbed-field tension terms are quadratic in the disturbances. Magnetic tension resists bending of the vertical field, while diffusion allows that bending to relax.