= Horizontally invariant magnetized shearing-sheet equations
For <ideal magnetohydrodynamics> at constant <mass density> $\rho$, a <shearing sheet> with velocity $-q\Omega x\mathbf e_y+\mathbf v_h(z,t)$ and <magnetic field> $\mathbf B_h(z,t)+B_z\mathbf e_z$ obeys
$$
\partial_tv_x-2\Omega v_y=\frac{B_z}{\mu_0\rho}\partial_zB_x,\qquad
\partial_tv_y+(2-q)\Omega v_x=\frac{B_z}{\mu_0\rho}\partial_zB_y,
$$
$$
\partial_tB_x=B_z\partial_zv_x,\qquad
\partial_tB_y+q\Omega B_x=B_z\partial_zv_y.
$$
Horizontal invariance and vanishing vertical velocity eliminate nonlinear <advection>. These equations retain <Coriolis acceleration>, orbital shear and <magnetic tension>. The vertical pressure adjusts to balance gravity and <magnetic pressure>.
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