Magnetic diffusion explained

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 (also referred to as the induction equation) is $$\backslash frac\; =\; \backslash nabla\; \backslash times\; \backslash left[\backslash vec\{v\}\; \backslash times\; \backslash vec\{B\}\backslash right]\; +\; \backslash frac\backslash nabla^2\; \backslash vec$$where

\mu _{0} is the permeability of free space and \sigma is the electrical conductivity of the material, which is assumed to be constant. \vec{v} 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, R _{m} .

In the case of a non-uniform conductivity the magnetic diffusion equation is$$\backslash frac\; =\; \backslash nabla\; \backslash times\; \backslash left[\backslash vec\{v\}\; \backslash times\; \backslash vec\{B\}\backslash right]\; -\; \backslash frac\; \backslash nabla\; \backslash times\; \backslash left[\backslash frac\{1\}\{\backslash sigma\}\; \backslash nabla\; \backslash times\; \backslash vec\{B\}\; \backslash right]$$however, it becomes significantly harder to solve.

Derivation

Starting from the generalized Ohm's law:^{[1] [2]} $$\backslash vec\; =\; \backslash sigma\; \backslash left(\backslash vec+\backslash vec\backslash times\backslash vec\; \backslash right)$$and the curl equations for small displacement currents (i.e. low frequencies)$$\backslash nabla\backslash times\backslash vec\; =\; \backslash mu\_0\; \backslash vec\; +\; \backslash epsilon\_0\; \backslash mu\_0\; \backslash frac\; \backslash approx\; \backslash mu\_0\; \backslash vec$$$$\backslash nabla\backslash times\backslash vec\; =\; -\backslash frac$$substitute

\vec{J} into the Ampere-Maxwell law to get$$\backslash frac\; \backslash nabla\backslash times\backslash vec\; =\; \backslash vec\; +\; \backslash vec\backslash times\backslash vec\; \backslash quad\backslash Rightarrow\backslash quad\; \backslash vec\; =\; \backslash frac\backslash nabla\backslash times\backslash vec-\backslash vec\backslash times\backslash vec.$$Taking the curl of the above equation and substituting into Faraday's law,$$\backslash nabla\backslash times\backslash vec\; =\; \backslash nabla\backslash times\backslash left(\backslash frac\backslash nabla\backslash times\backslash vec\; -\; \backslash vec\backslash times\backslash vec\backslash right)\; =\; -\backslash frac.$$This expression can be simplified further by writing it in terms of the i-th component of \vec{B} and the Levi-Cevita tensor \varepsilon _{ijk} :Using the identity^[3] \varepsilon _{kij} \varepsilon _{klm} = \delta _{il} \delta _{jm} - \delta _{im} \delta _{jl} and recalling \partial _{j} B _{j} = 0 , the cross products can be eliminated:Written in vector form, the final expression is$$\backslash frac+\backslash left(\backslash vec\backslash cdot\backslash nabla\backslash right)\backslash vec\; =\; \backslash frac\; =\; \backslash left(\backslash vec\backslash cdot\backslash nabla\backslash right)\backslash vec-\backslash vec\backslash left(\backslash nabla\backslash cdot\backslash vec\backslash right)+\backslash frac\backslash nabla^2\; \backslash vec$$where (D)/(= Dt) (\partial)/(\partial t) + \vec{v} ⋅ \nabla is the material derivative. This can be rearranged into a more useful form using vector calculus identities and \nabla ⋅ \vec{B} =0 :$$\backslash frac=\; \backslash nabla\; \backslash times\; [\backslash vec\{v\}\; \backslash times\; \backslash vec\{B\}]\; +\; \backslash frac\backslash nabla^2\; \backslash vec$$In the case \vec{v} =0 , this becomes a diffusion equation for the magnetic field,$$\backslash frac\; =\; \backslash frac\backslash nabla^2\; \backslash vec\; =\; \backslash eta\backslash nabla^2\; \backslash vec$$where η = (1)/(\mu _{0} \sigma) 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$$R\_m\; =\; \backslash frac$$where

η is the diffusivity, v is the magnitude of the plasma's velocity and L is a characteristic length of the plasma.

(R _{m)}	Physical Condition	Dominating Term	Magnetic Diffusion Equation	Examples
\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.	(\partial \vec{B)/() } \approx \nabla \times [\vec{v} \times \vec{B}]	The Sun (R _{m} ≈ 10 ^{6)} or the core of the earth (R _{m} ≈ 10 ^{3)}
\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.	(\partial \vec{B)/() } \approx \frac\nabla^2 \vec	Solar flares or created in laboratories using mercury or other liquid metals.

Relation to Skin Effect

\delta for the penetration of an AC electromagnetic field into a conductor is:$$\backslash delta\; =\; \backslash sqrt$$Comparing with the formula for η , the skin depth is the diffusion length of the field over one period of oscillation:$$\backslash delta\; =\; \backslash sqrt\; =\; \backslash sqrt$$

Examples and Visualization

For the limit

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$$\backslash vec\; =\; v\_0\backslash sin(k\; y)\backslash hat$$with a uniform initial magnetic field \vec{B}\left(\vec{r}, 0 \right) = B _{0\hat{y}} . The equation for this limit, (\partial \vec{B)/() } = \nabla \times [\vec{v} \times \vec{B}] , has the solution^[4] $$\backslash vec\backslash left(\backslash vec,t\backslash right)\; =\; B\_0\; k\; v\_0\; t\backslash cos(k\; y)\backslash hat+B\_0\backslash hat$$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

R _{m} \ll 1 , the magnetic diffusion equation (\partial \vec{B)/() } = \frac \nabla^2 \vec 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

(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.

Notes and References

1. Book: Holt . E. H. . Haskell . R. E. . Foundations of Plasma Dynamics . registration . 1965 . Macmillan . New York . 429-431.
2. Book: Chen . Francis F. . Introduction to Plasma Physics and Controlled Fusion . 2016 . Springer . Heidelberg . 978-3-319-22308-7 . 192–194 . 3rd.
3. Book: Landau . L. D. . Lifshitz . E. M. . The Classical Theory of Fields . 2013 . Elsevier . New York . 9781483293288 . 4th revised.
4. Web site: Longcope . Dana . 2002 . Notes on Magnetohydrodynamics . Montana State University - Department of Physics . 30 April 2019.
5. Book: Brauer . J. R. . Magnetic Actuators and Sensors . Wiley IEEE Press . Hoboken NJ . 2014 . 2nd . 978-1-118-50525-0.