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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.

Define the perturbation energy per unit mass by $E=⟨v^2/2+B^2/(2{\mu }_0\rho )⟩$. Dot the momentum equation with $\mathbf{v}$ and the induction equation with $\mathbf{B}/({\mu }_0\rho )$, then take volume averages. The Coriolis force does no work. Pressure work and nonlinear advection are divergences and integrate to zero, as does the background $x{\partial }_y$ transport. The two magnetic exchange terms add toThe velocity shear term supplies $2A⟨v_x{v}_y⟩$, while magnetic shear supplies $-2A⟨B_x{B}_y⟩/({\mu }_0\rho )$. Integration by parts gives the viscous and resistive losses as $-\nu ⟨|\nabla \mathbf{v}{|}^2⟩$ and $-\eta ⟨|\nabla \mathbf{B}{|}^2⟩/({\mu }_0\rho )$. For a solenoidal field $\mathbf{F}$, another integration by parts showsAll boundary terms cancel by the shearing-periodic identities. The resulting balance isThe source is work extracted from maintained differential rotation by Reynolds stress and the magnetic part of the Maxwell stress tensor; internal magnetic/kinetic exchange has canceled.

For the decay bound, the elementary inequality $2|ab|\le a^2+b^2$ givesA uniform spectral gap of a shearing-periodic box is needed; ordinary radial periodicity cannot simply be assumed. At each time the compatible Fourier modes areThey are orthogonal over the box. Their wavevectors areFor every nonconstant mode, $|\mathbf{k}|\ge 2\pi /L$, where $L=\max (L_x,L_y,L_z)$: if $m$ or $p$ is nonzero, its corresponding unsheared component supplies the bound; otherwise $n\ne 0$ and the radial component does. Zero mean excludes precisely the constant mode. The Parseval identity therefore proves the time-uniform Poincaré inequalityApply it componentwise to $\mathbf{v}$ and $\mathbf{B}$. With $\nu =\eta$, the energy decay criterion for a viscous-resistive shearing box becomesHenceThis excludes sustained finite-amplitude turbulence as well as infinitesimal growth. It is a sufficient stability condition, not a claim that every box above the threshold must become turbulent. The zero-mean restrictions matter: an imposed uniform magnetic field or bulk epicycle is not controlled by this spatial spectral-gap inequality.