= Interchange stability of an incompressible magnetized atmosphere
{title2=$Q=-g\int\rho_z|\xi_z|^2+\mu_0^{-1}\int B^2|\partial_x\xi|^2$}
For a horizontal field $B(z)e_x$ in constant downward gravity, equilibrium obeys $(p+B^2/(2\mu_0))_z=-\rho g$. The displacement energy reduces to the displayed <quadratic form>. Perturbations independent of the field direction do not bend <magnetic field lines> and therefore have no <magnetic tension> penalty: an increasing <mass density> with height permits instability. A nonincreasing <mass density> gives nonnegative energy for all admitted incompressible perturbations. Along-field variation adds <magnetic tension> proportional to <wavenumber> squared, stabilizing sufficiently short waves when buoyancy is unfavorable.
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