Incompressible magnetic displacement energy identity (${\sigma }^2\int \rho |\xi {|}^2=\int {\xi }_i^{\ast }{\xi }_j({\Pi }_{,ij}+\rho {\Phi }_{,ij})+{\mu }_0^{-1}\int |(B\cdot \nabla )\xi {|}^2$)

For a static ideal magnetohydrodynamics equilibrium in fixed gravity, let $\nabla \cdot \xi =0$ and $\Pi =p+B^2/(2{\mu }_0)$. Frozen flux gives $b=(B\cdot \nabla )\xi -(\xi \cdot \nabla )B$ and continuity gives ${\rho }^{\prime }=-\xi \cdot \nabla \rho$. Multiplying the linear momentum equation by the conjugate displacement and integrating by parts gives the displayed identity when surface terms vanish. The two magnetic cross terms cancel; the remaining magnetic equilibrium derivative combines with the mass density-gradient force to give the two Hessian matrices. Real symmetric Hessian matrices and the nonnegative magnetic tension term make ${\sigma }^2$ real.

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.