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.

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 $L^2/{\eta }_m$.