Magnetohydrodynamic energy principle

For a static smooth ideal magnetohydrodynamics equilibrium with conservative external forces and surface terms eliminated by the boundary conditions, the linear displacement equation is $\rho \ddot{\mathbf{\xi }}=\mathcal{F}\mathbf{\xi }$, where the force operator is a self-adjoint operator in the appropriate integral pairing. Its restoring quadratic form $\delta W=-\frac{1}{2}\int {\mathbf{\xi }}^{\ast }\cdot \mathcal{F}\mathbf{\xi }dV$ is real. Multiplying the normal mode equation by the complex conjugate of the displacement and integrating gives ${\omega }^2\int \rho |\mathbf{\xi }{|}^{2}dV=2\delta W$. Hence squared frequencies are real, and a negative restoring form gives an unstable mode through the variational Rayleigh quotient, subject to the operator domain and spectral hypotheses. Background fluid flow adds terms to this argument; the static hypothesis is essential.

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.

For a static equilibrium and a fluid Lagrangian displacement $\xi$, conservation of mass, adiabatic entropy and magnetic flux freezing give $\delta p=-\xi \cdot \nabla p-\gamma p\nabla \cdot \xi$ together with the displayed relations. Linearizing the pressure/tension form of the momentum equation yields $\rho \ddot{\xi }=-\nabla \delta \Pi -g\delta \rho e_z+{\mu }_0^{-1}(\delta B\cdot \nabla B+B\cdot \nabla \delta B)$, where $\delta \Pi =\delta p+B\cdot \delta B/{\mu }_0$. Static force balance removes the zeroth-order terms; the density multiplying acceleration is the equilibrium density.