= Energy decay criterion for a viscous-resistive shearing box
{title2=$\mathrm{Rm}<4\pi^2\quad(\nu=\eta)$}
For zero mean velocity and <magnetic field> in a <shearing box>, the perturbation energy obeys $\dot E\le2(|A|-4\pi^2\eta/L^2)E$ 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.
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