Planar ideal-MHD Bernoulli invariant ($Q=u^2/2+\Phi +w-U{B}_z/({\mu }_{0}k)$)

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.