It is the magnetic analogue of the Reynolds number in fluid mechanics and is typically defined by: where The mechanism by which the motion of a conducting fluid generates a magnetic field is the subject of dynamo theory.
When the magnetic Reynolds number is very large, however, diffusion and the dynamo are less of a concern, and in this case focus instead often rests on the influence of the magnetic field on the flow.
In the theory of magnetohydrodynamics, the magnetic Reynolds number can be derived from the induction equation: where The first term on the right hand side accounts for effects from magnetic induction in the plasma and the second term accounts for effects from magnetic diffusion.
The relative importance of these two terms can be found by taking their ratio, the magnetic Reynolds number
, advection is relatively unimportant, and so the magnetic field will tend to relax towards a purely diffusive state, determined by the boundary conditions rather than the flow.
, diffusion is relatively unimportant on the length scale L. Flux lines of the magnetic field are then advected with the fluid flow, until such time as gradients are concentrated into regions of short enough length scale that diffusion can balance advection.
[citation needed] Dissipative affects are generally small, and there is no difficulty in maintaining a magnetic field against diffusion.
[1] Dissipation is more significant, but a magnetic field is supported by motion in the liquid iron outer core.
There are other bodies in the solar system that have working dynamos, e.g. Jupiter, Saturn, and Mercury, and others that do not, e.g. Mars, Venus and the Moon.
The generation of magnetic field by the motion of a conducting fluid has been achieved in only a handful of large experiments using mercury or liquid sodium.
[2][3][4] In situations where permanent magnetisation is not possible, e.g. above the Curie temperature, to maintain a magnetic field
It is not the absolute magnitude of velocity that is important for induction, but rather the relative differences and shearing in the flow, which stretch and fold magnetic field lines.
[5] A more appropriate form for the magnetic Reynolds number in this case is therefore where S is a measure of strain.
[7] Many studies of the generation of magnetic field by a flow consider the computationally-convenient periodic cube.
is the root-mean-square strain over a scaled domain with sides of length
If shearing over small length scales in the cube is ruled out, then
All three can be regarded as giving the ratio of advective to diffusive effects for a particular physical field and have the form of the product of a velocity and a length divided by a diffusivity.
the skin effect is negligible and the eddy current braking torque follows the theoretical curve of an induction motor.
the skin effect dominates and the braking torque decreases much slower with increasing speed than predicted by the induction motor model.