 Plane-parallel magnetohydrodynamic flow with imposed shear

A useful ideal magnetohydrodynamics reduction has $\mathbf{u}=\mathbf{v}(z,t)+ax{\mathbf{e}}_y$, $\mathbf{B}=\mathbf{B}(z,t)$ and $\rho =\rho (z,t)$. The constant background shear flow supplies $a{v}_x$ in the axial momentum equation and $a{B}_x$ in axial magnetic induction. The solenoidal constraint and MHD induction equation together make $B_z$ constant.