 Solution

Choose a reference circular orbit at $r_0$, and use local coordinates $x=r-r_0$, $y=r_0[\varphi -{\Omega }_{0}t]$, with the frame rotating at $\Omega ={\Omega }_0=\Omega (r_0)$. Linearizing differential rotation in the shearing sheet gives the background flowThus the velocity perturbation is measured relative to this background. The local orbital shear parameter is $q=2A/\Omega =-d\ln \Omega /d\ln r$; Keplerian rotation has $q=3/2$ and $A=3\Omega /4$. Rotation supplies the Coriolis force, while the linear shear supplies the terms proportional to $A$.

For constant density, $\psi$ is the total-pressure potential after background force balance has been subtracted:up to a function of time and any prescribed perturbation potential that is absorbed into it. The magnetic-pressure gradient has been combined with the gas-pressure gradient, leaving magnetic tension as $\mathbf{B}\cdot \nabla \mathbf{B}/({\mu }_0\rho )$. The pressure also enforces incompressibility. In the unstratified shearing box here, the tidal/background terms have already been removed in deriving the perturbation equations; no independent vertical stratification is being evolved.