Magnetic diffusion

Short description: Type of motion of magnetic fields

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.

## Contents

* 1 Equation
* 2 Derivation
* 3 Limiting Cases
* 3.1 Relation to Skin Effect
* 4 Examples and Visualization
* 5 Diffusion Times for Stationary Conductors
* 6 References

## Equation

The magnetic diffusion equation (also referred to as the induction equation) is ∂𝐁∂t=∇×(𝐯×𝐁)+1μ0σ∇2𝐁 where μ0 is the permeability of free space and σ is the electrical conductivity of the material, which is assumed to be constant. 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, Rm.

In the case of a non-uniform conductivity the magnetic diffusion equation is ∂𝐁∂t=∇×(𝐯×𝐁)−1μ0∇×(1σ∇×𝐁) however, it becomes significantly harder to solve.

## Derivation

Starting from the generalized Ohm's law:[1][2] 𝐉=σ(𝐄+𝐯×𝐁) and the curl equations for small displacement currents (i.e. low frequencies) ∇×𝐁=μ0𝐉+1c2∂𝐄∂t≈μ0𝐉∇×𝐄=−∂𝐁∂t substitute 𝐉 into the Ampere–Maxwell law to get 1μ0σ∇×𝐁=𝐄+𝐯×𝐁⇒𝐄=1μ0σ∇×𝐁−𝐯×𝐁. Taking the curl of the above equation and substituting into Faraday's law, ∇×𝐄=∇×(1μ0σ∇×𝐁−𝐯×𝐁)=−∂𝐁∂t. This expression can be simplified further by writing it in terms of the i-th component of 𝐁 and the Levi-Civita tensor εijk: −∂Bi∂t=εijk∂j(1μ0σεklm∂lBm−εklmvlBm)=εkijεklm(1μ0σ∂j∂lBm−(vl∂jBm+Bm∂jvl)) Using the identity[3] εkijεklm=δilδjm−δimδjl and recalling ∇⋅𝐁=∂jBj=0, the cross products can be eliminated: −∂Bi∂t=1μ0σ(∂i∂jBj−∂j∂jBi)−(vi∂jBj−vj∂jBi)−(Bj∂jvi−Bi∂jvj)=−1μ0σ∂j∂jBi+vj∂jBi−(Bj∂jvi−Bi∂jvj) Written in vector form, the final expression is ∂𝐁∂t+(𝐯⋅∇)𝐁=(𝐁⋅∇)𝐯−𝐁(∇⋅𝐯)+1μ0σ∇2𝐁⟶D𝐁Dt=(𝐁⋅∇)𝐯−𝐁(∇⋅𝐯)+1μ0σ∇2𝐁 where DDt=∂∂t+𝐯⋅∇ is the material derivative. This can be rearranged into a more useful form using vector calculus identities and ∇⋅𝐁=0: ∂𝐁∂t=∇×(𝐯×𝐁)+1μ0σ∇2𝐁 In the case 𝐯=0, this becomes a diffusion equation for the magnetic field, ∂𝐁∂t=1μ0σ∇2𝐁=η∇2𝐁 where η=1μ0σ 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 Rm=vLη where η is the diffusivity, v is the magnitude of the plasma's velocity and L is a characteristic length of the plasma.

(Rm) | Physical Condition | Dominating Term | Magnetic Diffusion Equation | Examples

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≫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.  |  ∂𝐁∂t≈∇×(𝐯×𝐁) | The Sun (Rm≈106) or the core of the earth (Rm≈103)
≪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. |  ∂𝐁∂t≈1μ0σ∇2𝐁 | Solar flares or created in laboratories using mercury or other liquid metals.

### Relation to Skin Effect

At low frequencies, the skin depth δ for the penetration of an AC electromagnetic field into a conductor is: δ=2μσω Comparing with the formula for η, the skin depth is the diffusion length of the field over one period of oscillation: δ=2ηω=ηTπ

## Examples and Visualization

Example of magnetic field frozen into fluid flow.

For the limit Rm≫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 𝐯=v0sin⁡(ky)𝐱^ with a uniform initial magnetic field 𝐁(𝐫,0)=B0𝐲^. The equation for this limit, ∂𝐁∂t=∇×(𝐯×𝐁), has the solution[4] 𝐁(𝐫,t)=B0(kv0tcos⁡(ky)𝐱^+𝐲^) 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 Rm≪1, the magnetic diffusion equation ∂𝐁∂t=1μ0σ∇2𝐁 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.

## Diffusion Times for Stationary Conductors

For stationary conductors (Rm=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.

## References

1. ↑ Holt, E. H.; Haskell, R. E. (1965). Foundations of Plasma Dynamics. New York: Macmillan. pp. 429-431. https://archive.org/details/foundationsofpla0000holt.
2. ↑ Chen, Francis F. (2016). Introduction to Plasma Physics and Controlled Fusion (3rd ed.). Heidelberg: Springer. pp. 192–194. ISBN 978-3-319-22308-7.
3. ↑ Landau, L. D.; Lifshitz, E. M. (2013). The Classical Theory of Fields (4th revised ed.). New York: Elsevier. ISBN 9781483293288.
4. ↑ Longcope, Dana (2002). "Notes on Magnetohydrodynamics". http://solar.physics.montana.edu/dana/mhd_notes.pdf. Retrieved 30 April 2019.
5. ↑ Brauer, J. R. (2014). Magnetic Actuators and Sensors (2nd ed.). Hoboken NJ: Wiley IEEE Press. ISBN 978-1-118-50525-0.

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