Localized interchange criterion for a magnetized atmosphere ($-\frac{d\ln \rho }{dz}<\frac{\rho g}{\gamma p+B^2/{\mu }_0}$)

For disturbances constant along the horizontal magnetic field, the minimized energy density is $\rho g[-(\ln \rho {)}^{\prime }-\rho g/(\gamma p+C)]|{\xi }_z{|}^2$. If its coefficient is nonnegative everywhere, the full energy is nonnegative. If it is strictly negative at a point of a smooth equilibrium, a compactly supported vertical displacement in a neighbouring interval gives negative energy. These interchange motions compress gas and magnetic pressure together.