= Steady planar ideal-MHD field-line invariants
{title2=$\mathbf u=k\mathbf B/\rho+U\mathbf e_z$}
For a stationary, $z$-independent <ideal magnetohydrodynamics> flow with $E_z=0$, nonzero horizontal <magnetic field> and nonzero <magnetohydrodynamic mass loading> $k$, the displayed decomposition has $\mathbf B\cdot\nabla k=\mathbf B\cdot\nabla U=0$. The <continuity equation> gives the first invariant, and the <ideal magnetohydrodynamic induction equation> gives the second. Entropy and $V=u_z-B_z/(\mu_0k)$ are also constant on each <magnetic field line>. Stationarity alone permits constant nonzero $E_z$, so it does not imply this decomposition.
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