Steady planar ideal-MHD field-line invariants ($\mathbf{u}=k\mathbf{B}/\rho +U{\mathbf{e}}_z$)

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 $V=u_z-B_z/({\mu }_{0}k)$ are also constant on each magnetic field line. Stationarity alone permits constant nonzero $E_z$, so it does not imply this decomposition.

The two axial invariants in steady planar ideal-MHD field-line invariants give the displayed identity. When $V-U\ne 0$, a finite smooth flow cannot reach horizontal Alfvén speed. When $V=U$, an everywhere Alfvénic branch is possible even with $B_z\ne 0$: constant $\rho$ and $\mathbf{u}=\mathbf{B}/\sqrt{{\mu }_0\rho }$ give an example. The nondegenerate exclusion must not be applied to this branch.

Under the hypotheses of steady planar ideal-MHD field-line invariants, the specific enthalpy relation and the velocity projection of momentum give $\mathbf{B}\cdot \nabla Q=0$. The horizontal total energy flux, including the Poynting vector, is $k{\mathbf{B}}_{h}Q$. The magnetic correction accounts for work exchanged between the fluid and the field.