Magnetic diffusion

Magnetic diffusion refers to the motion of magnetic field lines, typically in the presence of a conducting solid or fluid such as a plasma. The motion of magnetic fields is described by the magnetic diffusion equation and is due primarily to induction and diffusion of magnetic fields through the material. The magnetic diffusion equation is a partial differential equation commonly used in physics. Understanding the phenomenon is essential to magnetohydrodynamics and has important consequences in astrophysics, geophysics, and electrical engineering.
Equation.
The magnetic diffusion equation is 
where formula_2 is the permeability of free space and formula_3 is the electrical conductivity of the material, which is assumed to be constant. formula_4 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 later 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, formula_5.
In the case of a non-uniform conductivity the magnetic diffusion equation is
however, it becomes significantly harder to solve.
Derivation.
Starting from the generalized Ohm's law:
and the curl equations for small displacement currents (i.e. low frequencies)
substitute formula_10 into the Ampere-Maxwell law to get
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 formula_13 and the Levi-Cevita tensor formula_14:
Using the identity formula_16 and recalling formula_17, the cross products can be eliminated:
Written in vector form, the final expression is
where formula_20 is the material derivative. This can be rearranged into a more useful form using vector calculus identities and formula_21:
In the case formula_23, this becomes a diffusion equation for the magnetic field,
where formula_25 is the magnetic diffusivity.
Limiting Cases.
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
where formula_27 is the diffusivity, formula_28 is the magnitude of the plasma's velocity and formula_29 is a characteristic length of the plasma.
Relation to Skin Effect.
At low frequencies, the skin depth formula_30 for the penetration of an AC electromagnetic field into a conductor is:
Comparing with the formula for formula_32, the skin depth is the diffusion length of the field over one period of oscillation:
Examples and Visualization.
For the limit formula_34, 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 formula_36. The equation for this limit, formula_37, has the solution
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.
For the limit formula_39, the magnetic diffusion equation formula_40 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. This behavior is illustrated in the figure below.
Diffusion Times for Stationary Conductors.
For stationary conductors formula_41 with simple geometries a time constant called magnetic diffusion time can be derived <ref>. 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.