Energy decay criterion for a viscous-resistive shearing box ($\mathrm{Rm}<4{\pi }^2(\nu =\eta )$)

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.