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Concept
Magnetic Reynolds number
Summary
In magnetohydrodynamics, the magnetic Reynolds number (Rm) is a dimensionless quantity that estimates the relative effects of advection or induction of a magnetic field by the motion of a conducting medium to the magnetic diffusion. It is the magnetic analogue of the Reynolds number in fluid mechanics and is typically defined by:
where
is a typical velocity scale of the flow,
is a typical length scale of the flow,
is the magnetic diffusivity.
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
is the magnetic field,
is the fluid velocity,
is the magnetic diffusivity.
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 . If it is assumed that both terms share the scale length such that and the scale velocity such that , the induction term can be written as
and the diffusion term as
The ratio of the two terms is therefore
For , 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.
For , 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.
The Sun has a large , of order 106. Dissipative affects are generally small, and there is no difficulty in maintaining a magnetic field against diffusion.
Official source
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