= Orthogonal-field magnetic Rayleigh-Taylor dispersion relation
{title2=$( \rho_++\rho_-)s^2=g(\rho_+-\rho_-)K-(B_+^2k^2+B_-^2m^2)/\mu_0$}
Consider uniform fields $B_-\hat{\mathbf y}$ below and $B_+\hat{\mathbf x}$ above an interface, with local constant densities $\rho_-,\rho_+$. For ideal <magnetohydrodynamics> and local incompressible <potential flow>, the interface displacement $h e^{st+ikx+imy}$ has displacement potentials $\chi_-=h e^{Kz}/K$ and $\chi_+=-h e^{-Kz}/K$, $K=\sqrt{k^2+m^2}$. <Magnetic flux freezing> gives $\mathbf b_i=(\mathbf B_i\cdot\nabla)\nabla\chi_i$. The linear <magnetohydrodynamic total pressure> is $-[\rho_i s^2+(\mathbf k_h\cdot\mathbf B_i)^2/\mu_0]\chi_i$. Matching Lagrangian total pressure gives the displayed <dispersion relation>. For $\rho_+>\rho_-$ the most unstable orientation has its wavevector along the weaker field.
// Target: continuum-mechanics.bigb
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