 Linear displacement equations for a magnetized atmosphere

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.