Long-wavelength undular magnetic buoyancy criterion ($-\frac{d\ln \rho }{dz}<\frac{\rho g}{\gamma p}$)

At nonzero along-field wavenumber, put $q=i{k}_y{\xi }_y$, $P=\gamma p$ and $C=B^2/{\mu }_0$. After pressure minimization the energy density completes to $CP/(P+C)|q-\rho g{\xi }_z/P{|}^2+[C{k}_y^2-g{\rho }^{\prime }-{\rho }^2{g}^2/P]|{\xi }_z{|}^2$. Choosing the square to vanish and then taking a sufficiently small nonzero $k_y$ gives negative energy exactly when the displayed strict criterion holds somewhere. A fixed prescribed wavenumber retains the stabilizing $C{k}_y^2$ tension term; the criterion asserts existence of an allowed nonzero wavenumber. The associated along-field displacement lets material drain while magnetic flux bends.