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.

At $w^2=v_{az}^2$, the transverse steady equations require $w{B}_x{w}^{\prime }=0$ and $w{B}_y{w}^{\prime }=a(w{B}_x-B_z{v}_x)$. These conditions can be lost by division when eliminating transverse magnetic derivatives. A scalar equation displaying the magnetosonic critical speeds need not display every compatibility condition at an Alfvén speed.

The energy of the residual flow and magnetic field in a maintained shear flow has source $S=-a{T}_{xy}$, where $T_{xy}=\rho v_x{v}_y-B_x{B}_y/{\mu }_0$. The kinetic and magnetic stresses can exchange energy with the imposed background. The source vanishes when their off-diagonal contributions cancel.