Turbulent magnetic diffusivity (${\eta }_t$)

Turbulent magnetic diffusivity is an effective transport coefficient produced by correlated small-scale velocity and magnetic field. It supplements microscopic magnetic diffusivity in a mean-field dynamo description. Its identification requires a specified mean-field geometry and averaging procedure; in a planar flux problem, the correlation $\overline{w{a}^{\prime }}=-{\eta }_t{B}_{0}f(z)$ describes transport relative to a prescribed uniform field.

For $\overline{w{a}^{\prime }}=-{\eta }_t{B}_{0}f$, $⟨f⟩=1$ and positive diffusivities, the Zeldovich planar flux balance gives $G=({\eta }_t/\eta )B_0^2[1-({\eta }_t/\eta )\mathrm{Var}(f)]$, where $G=⟨|\nabla a^{\prime }{|}^2⟩$. The Cauchy-Schwarz inequality and a Poincaré inequality $⟨a^{\prime 2}⟩\le d^{2}G/{\pi }^2$ give ${\eta }_t^2\mathrm{Var}(f)\le ⟨w^2⟩d^{2}s(1-s)/{\pi }^2$, with $s=({\eta }_t/\eta )\mathrm{Var}(f)\in [0,1]$. Maximizing $s(1-s)$ gives the displayed bound. Large turbulent-to-microscopic diffusivity therefore requires $f$ to be close to one in mean square, without guaranteeing pointwise uniformity.