Magnetic diffusion (${\partial }_t\mathbf{B}={\eta }_m{\nabla }^2\mathbf{B}$)

Finite electrical conductivity permits resistive smoothing of a magnetic field. With uniform coefficients and no motion the field satisfies a diffusion equation; with motion the diffusion term supplements the curl of $\mathbf{u}\times \mathbf{B}$. The magnetic Reynolds number compares these effects.

A nonuniform angular velocity $f(r)$ winds an azimuthal magnetic mode into phase $m(\varphi -f(r)t)$. Its radial phase gradient grows as $-mt{f}^{\prime }(r)$, creating small scales even when its ideal amplitude does not decay. The corresponding leading resistive damping rate is $\eta m^2{t}^2{f}^{\prime }(r{)}^2$, so a local short-wavelength estimate gives damping factor $\exp [-\eta m^2{f}^{\prime }(r{)}^2{t}^3/3]$. Hence magnetic diffusion can become important at time $[\eta m^2{f}^{\prime 2}{]}^{-1/3}$ although it is initially small. This is a local phase-mixing estimate, not a globally exact solution in a finite radial domain: boundary forcing, field stretching and spatial variation of $f^{\prime }$ must also be retained there.

For azimuthal flow $u_{\varphi }=s({\Omega }_0+\omega s^2)$, a planar magnetic flux mode $s^{m}g(t)e^{im\varphi -im{\Omega }_{0}t-i{s}^{2}f(t)}$ obeys the resistive induction equation if $\dot{f}=m\omega -4i\eta f^2$ and $\dot{g}/g=-4i\eta (m+1)f$. With $f(0)=0$ and $g(0)=1$, their solution is $f=(m\omega /\mu )\tanh \mu t$ and $g=(\cosh \mu t{)}^{-(m+1)}$. Choose $\mathrm{Re}\mu >0$. For positive $\eta ,\omega$, the fixed-position field decays asymptotically at rate $(m+1)\sqrt{2\eta m\omega }$. Differential rotation creates small spatial scales, accelerating magnetic diffusion.

For a solenoidal vector field carrying current only in a conductor contained in a sphere of radius $R$, match the exterior field to a decaying potential field and include exterior magnetic energy. The smallest free-decay eigenvalue of the enclosing sphere is ${\pi }^2/R^2$, from its dipolar poloidal mode. The corresponding variational estimate gives the displayed bound. Continuity and the usual insulating-interface magnetic conditions are part of the admissible field class. This bound is the diffusive ingredient in Backus' necessary condition for dynamo action.

Magnetic diffusivity is the coefficient of magnetic diffusion. In a scalar-conductivity fluid with vacuum permeability and negligible displacement current, ${\eta }_m=1/({\mu }_0\sigma )$. Its dimensions are length squared per time, and a diffusion time on scale $L$ is $L^2/{\eta }_m$.

The magnetic Prandtl number compares kinematic viscosity with magnetic diffusivity. It is the ratio of the viscous and magnetic diffusion coefficients; equivalently it is the magnetic Reynolds number divided by the Reynolds number when the same speed and length scale are used.

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.