Magnetic diffusion refers to the motion of magnetic fields, 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

$${\displaystyle {\frac {\partial {\vec {B))}{\partial t))=\nabla \times \left[{\vec {v))\times {\vec {B))\right]+{\frac {1}{\mu _{0}\sigma ))\nabla ^{2}{\vec {B))}$$

${\displaystyle \mu _{0))$

${\displaystyle \sigma }$

${\displaystyle {\vec {v))}$

${\displaystyle R_{m))$

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

$${\displaystyle {\frac {\partial {\vec {B))}{\partial t))=\nabla \times \left[{\vec {v))\times {\vec {B))\right]-{\frac {1}{\mu _{0))}\nabla \times \left[{\frac {1}{\sigma ))\nabla \times {\vec {B))\right]}$$

Derivation

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

$${\displaystyle {\vec {J))=\sigma \left({\vec {E))+{\vec {v))\times {\vec {B))\right)}$$

$${\displaystyle \nabla \times {\vec {B))=\mu _{0}{\vec {J))+\epsilon _{0}\mu _{0}{\frac {\partial {\vec {E))}{\partial t))\approx \mu _{0}{\vec {J))}$$

$${\displaystyle \nabla \times {\vec {E))=-{\frac {\partial {\vec {B))}{\partial t))}$$

${\displaystyle {\vec {J))}$

$${\displaystyle {\frac {1}{\mu _{0}\sigma ))\nabla \times {\vec {B))={\vec {E))+{\vec {v))\times {\vec {B))\quad \Rightarrow \quad {\vec {E))={\frac {1}{\mu _{0}\sigma ))\nabla \times {\vec {B))-{\vec {v))\times {\vec {B)).}$$

$${\displaystyle \nabla \times {\vec {E))=\nabla \times \left({\frac {1}{\mu _{0}\sigma ))\nabla \times {\vec {B))-{\vec {v))\times {\vec {B))\right)=-{\frac {\partial {\vec {B))}{\partial t)).}$$

${\displaystyle {\vec {B))}$

${\displaystyle \varepsilon _{ijk))$

$${\displaystyle {\begin{aligned}-{\frac {\partial B_{i)){\partial t))&=\varepsilon _{ijk}\partial _{j}\left({\frac {1}{\mu _{0}\sigma ))\varepsilon _{klm}\partial _{l}B_{m}-\varepsilon _{klm}v_{l}B_{m}\right)\\&=\varepsilon _{kij}\varepsilon _{klm}\left({\frac {1}{\mu _{0}\sigma ))\partial _{j}\partial _{l}B_{m}-\left(v_{l}\partial _{j}B_{m}+B_{m}\partial _{j}v_{l}\right)\right)\end{aligned))}$$

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

${\displaystyle \partial _{j}B_{j}=0}$

$${\displaystyle {\begin{aligned}-{\frac {\partial B_{i)){\partial t))&={\frac {1}{\mu _{0}\sigma ))\left(\partial _{i}\partial _{j}B_{j}-\partial _{j}\partial _{j}B_{i}\right)-\left(v_{i}\partial _{j}B_{j}-v_{j}\partial _{j}B_{i}\right)-\left(B_{j}\partial _{j}v_{i}-B_{i}\partial _{j}v_{j}\right)\\&=-{\frac {1}{\mu _{0}\sigma ))\partial _{j}\partial _{j}B_{i}+v_{j}\partial _{j}B_{i}-\left(B_{j}\partial _{j}v_{i}-B_{i}\partial _{j}v_{j}\right)\end{aligned))}$$

$${\displaystyle {\frac {\partial {\vec {B))}{\partial t))+\left({\vec {v))\cdot \nabla \right){\vec {B))={\frac {D{\vec {B))}{Dt))=\left({\vec {B))\cdot \nabla \right){\vec {v))-{\vec {B))\left(\nabla \cdot {\vec {v))\right)+{\frac {1}{\mu _{0}\sigma ))\nabla ^{2}{\vec {B))}$$

${\displaystyle {\frac {D}{Dt))={\frac {\partial }{\partial t))+{\vec {v))\cdot \nabla }$

${\displaystyle \nabla \cdot {\vec {B))=0}$

$${\displaystyle {\frac {\partial {\vec {B))}{\partial t))=\nabla \times [{\vec {v))\times {\vec {B))]+{\frac {1}{\mu _{0}\sigma ))\nabla ^{2}{\vec {B))}$$

${\displaystyle {\vec {v))=0}$

$${\displaystyle {\frac {\partial {\vec {B))}{\partial t))={\frac {1}{\mu _{0}\sigma ))\nabla ^{2}{\vec {B))=\eta \nabla ^{2}{\vec {B))}$$

${\displaystyle \eta ={\frac {1}{\mu _{0}\sigma ))}$

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

$${\displaystyle R_{m}={\frac {vL}{\eta ))}$$

${\displaystyle \eta }$

${\displaystyle v}$

${\displaystyle L}$

${\displaystyle (R_{m})}$	Physical Condition	Dominating Term	Magnetic Diffusion Equation	Examples
${\displaystyle \gg 1}$	Large electrical conductivity, large length scales or high plasma velocity.	The inductive term dominates in this case. The motion of magnetic fields is determined by the flow of the plasma. This is the case for most naturally occurring plasmas in the universe.	${\displaystyle {\frac {\partial {\vec {B))}{\partial t))\approx \nabla \times [{\vec {v))\times {\vec {B))]}$	The Sun ${\displaystyle (R_{m}\approx 10^{6})}$ or the core of the earth ${\displaystyle (R_{m}\approx 10^{3})}$
${\displaystyle \ll 1}$	Small electrical conductivity, small length scales or low plasma velocity.	The diffusive term dominates in this case. The motion of the magnetic field obeys the typical (nonconducting) fluid diffusion equation.	${\displaystyle {\frac {\partial {\vec {B))}{\partial t))\approx {\frac {1}{\mu _{0}\sigma ))\nabla ^{2}{\vec {B))}$	Solar flares or created in laboratories using mercury or other liquid metals.

Relation to Skin Effect

At low frequencies, the skin depth ${\displaystyle \delta }$ for the penetration of an AC electromagnetic field into a conductor is:

$${\displaystyle \delta ={\sqrt {\frac {2}{\mu \sigma \omega ))))$$

${\displaystyle \eta }$

$${\displaystyle \delta ={\sqrt {\frac {2\eta }{\omega ))}={\sqrt {\frac {\eta T}{\pi ))))$$

Examples and Visualization

For the limit ${\displaystyle R_{m}\gg 1}$, 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

$${\displaystyle {\vec {v))=v_{0}\sin(ky){\hat {x))}$$

${\displaystyle {\vec {B))\left({\vec {r)),0\right)=B_{0}{\hat {y))}$

${\displaystyle {\frac {\partial {\vec {B))}{\partial t))=\nabla \times [{\vec {v))\times {\vec {B))]}$

$${\displaystyle {\vec {B))\left({\vec {r)),t\right)=B_{0}kv_{0}t\cos(ky){\hat {x))+B_{0}{\hat {y))}$$

For the limit ${\displaystyle R_{m}\ll 1}$, the magnetic diffusion equation ${\displaystyle {\frac {\partial {\vec {B))}{\partial t))={\frac {1}{\mu _{0}\sigma ))\nabla ^{2}{\vec {B))}$ 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 ${\displaystyle (R_{m}=0)}$ 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.