= Short transverse wavelength limit for magnetic buoyancy
{title2=$ik_x\xi_x=\frac{\rho g\xi_z+Cik_y\xi_y}{\gamma p+C}-ik_y\xi_y-\xi_z'$}
For a horizontal field, set $C=B^2/\mu_0$. Static force balance gives $\delta\Pi=\rho g\xi_z-(\gamma p+C)\nabla\cdot\xi+Cik_y\xi_y$. Given smooth compactly supported vertical and along-field displacements, the displayed choice makes total-pressure perturbation zero while $\xi_x=O(k_x^{-1})$. The nonnegative <pressure> and transverse-tension terms vanish as $|k_x|\to\infty$. This constructs the infimum needed for the <magnetohydrodynamic energy principle>; a strictly negative limit is already negative at some finite wavenumber.
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