Short transverse wavelength limit for magnetic buoyancy ($i{k}_x{\xi }_x=\frac{\rho g{\xi }_z+Ci{k}_y{\xi }_y}{\gamma p+C}-i{k}_y{\xi }_y-{\xi }_z^{\prime }$)

For a horizontal field, set $C=B^2/{\mu }_0$. Static force balance gives $\delta \Pi =\rho g{\xi }_z-(\gamma p+C)\nabla \cdot \xi +Ci{k}_y{\xi }_y$. 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.