= Solution
In the rotating <shearing sheet>, write the background orbital shear as $\mathbf U_0=-q\Omega x\mathbf e_y$. The rotating momentum equation contains <Coriolis acceleration> $2\boldsymbol\Omega\times\mathbf u$ and the <shearing-sheet tidal potential>; the background shear balances the radial tidal acceleration.
Set
$$
\mathbf u=\mathbf U_0+(v_x(z,t),v_y(z,t),0),\qquad
\mathbf B=(B_x(z,t),B_y(z,t),B_z).
$$
<Gauss's law for magnetism> gives $\partial_zB_z=0$, and the vertical <ideal magnetohydrodynamic induction equation> gives $\partial_tB_z=0$. Horizontal invariance and $v_z=0$ remove the nonlinear horizontal advection. The remaining background-shear term is $(\mathbf v\mathbin\cdot\nabla)\mathbf U_0=-q\Omega v_x\mathbf e_y$. The horizontal <magnetic tension> is $B_z\partial_z\mathbf B_h/\mu_0$, while horizontal pressure gradients vanish. Subtracting background balance yields
$$
\boxed{\partial_tv_x-2\Omega v_y=\frac{B_z}{\mu_0\rho}\partial_zB_x},\qquad
\boxed{\partial_tv_y+(2-q)\Omega v_x=\frac{B_z}{\mu_0\rho}\partial_zB_y}.
$$
For <ideal magnetohydrodynamics>, the <ideal magnetohydrodynamic induction equation> is $\partial_t\mathbf B+\mathbf u\mathbin\cdot\nabla\mathbf B=\mathbf B\mathbin\cdot\nabla\mathbf u$. Its horizontal components give
$$
\boxed{\partial_tB_x=B_z\partial_zv_x},\qquad
\boxed{\partial_tB_y+q\Omega B_x=B_z\partial_zv_y}.
$$
These <horizontally invariant magnetized shearing-sheet equations> are exact within the stated local, incompressible ansatz, even for finite horizontal amplitudes. The vertical equation determines the pressure needed to balance vertical gravity and <magnetic pressure>; it does not add another horizontal evolution equation.
Back to article page
