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For zero mean velocity and magnetic field in a shearing box, the perturbation energy obeys $\dot{E}\le 2(|A|-4{\pi }^2\eta /L^2)E$ when kinematic viscosity and magnetic diffusivity agree. Bound the work of Reynolds stress and Maxwell stress tensor by $2|A|E$, and use the spectral gap of a shearing-periodic box to bound dissipation below. Thus the displayed bound on magnetic Reynolds number guarantees exponential decay and excludes sustained turbulence.

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.

A shearing box solves the shearing sheet equations in a finite domain with periodic azimuthal and vertical faces and a shearing-periodic boundary condition on its radial faces. It follows a small region rotating at a reference orbital frequency. The model can sustain stress and exchange energy with its imposed differential rotation, but its identified radial faces do not represent a global disk mass sink.

For background velocity $-2Ax{\mathbf{e}}_y$, opposite radial faces of a shearing box are identified after the displayed azimuthal shift, modulo $L_y$. Products and spatial derivatives inherit this condition. Their radial surface integrals cancel because a translation preserves an integral over a periodic azimuthal interval.

For zero-mean fields in a shearing box, Fourier modes have wavevectors $(2\pi n/L_x+2At2\pi m/L_y,2\pi m/L_y,2\pi p/L_z)$. A nonzero azimuthal or vertical index supplies at least $2\pi /L$, where $L$ is the largest box side; otherwise the nonzero radial index does. Parseval identity gives the displayed Poincaré inequality uniformly in time.