Magnetic diffusion

Magnetic diffusion refers to the motion of magnetic fields, typically in the presence of a conducting solid or fluid such as a plasma.

Understanding the phenomenon is essential to magnetohydrodynamics and has important consequences in astrophysics, geophysics, and electrical engineering.

is the electrical conductivity of the material, which is assumed to be constant.

denotes the (non-relativistic) velocity of the plasma.

The first term on the right hand side accounts for effects from induction of the plasma, while the second accounts for diffusion.

The latter acts as a dissipation term, resulting in a loss of magnetic field energy to heat.

The relative importance of the two terms is characterized by the magnetic Reynolds number,

In the case of a non-uniform conductivity the magnetic diffusion equation is

Starting from the generalized Ohm's law:[1][2]

and the curl equations for small displacement currents (i.e. low frequencies)

Taking the curl of the above equation and substituting into Faraday's law,

This expression can be simplified further by writing it in terms of the i-th component of

{\displaystyle \varepsilon _{kij}\varepsilon _{klm}=\delta _{il}\delta _{jm}-\delta _{im}\delta _{jl}}

Written in vector form, the final expression is

This can be rearranged into a more useful form using vector calculus identities and

, this becomes a diffusion equation for the magnetic field,

In some cases it is possible to neglect one of the terms in the magnetic diffusion equation.

This is done by estimating the magnetic Reynolds number

At low frequencies, the skin depth

for the penetration of an AC electromagnetic field into a conductor is:

, the skin depth is the diffusion length of the field over one period of oscillation:

, the magnetic field lines become "frozen in" to the motion of the conducting fluid.

A simple example illustrating this behavior has a sinusoidally-varying shear flow

with a uniform initial magnetic field

As can be seen in the figure to the right, the fluid drags the magnetic field lines so that they obtain the sinusoidal character of the flow field.

is just a vector-valued form of the heat equation.

For a localized initial magnetic field (e.g. Gaussian distribution) within a conducting material, the maxima and minima will asymptotically decay to a value consistent with Laplace's equation for the given boundary conditions.

with simple geometries a time constant called magnetic diffusion time can be derived.

[5] Different one-dimensional equations apply for conducting slabs and conducting cylinders with constant magnetic permeability.

Also, different diffusion time equations can be derived for nonlinear saturable materials such as steel.