= Linear displacement equations for a magnetized atmosphere
{title2=$\delta\rho=-\nabla\cdot(\rho\xi),\quad\delta B=\nabla\times(\xi\times B)$}
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>.
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