The magnetic Reynolds number (Rm) is the magnetic analogue of the Reynolds number, a fundamental dimensionless group that occurs in magnetohydrodynamics. It gives an estimate of the relative effects of advection or induction of a magnetic field by the motion of a conducting medium, often a fluid, to magnetic diffusion. It is typically defined by:
Contents
- General characteristics for large and small Rm
- Range of values
- Bounds
- Relationship to Reynolds number and Pclet number
- Relationship to eddy current braking
- References
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.
General characteristics for large and small Rm
For
For
Range of values
The Sun is huge and has a large
For the Earth,
The human length scale is very small so that typically
Bounds
In situations where permanent magnetisation is not possible, e.g. above the Curie temperature, to maintain a magnetic field
where S is a measure of strain. One of the most well known results is due to Backus which states that the minimum
where
Many studies of the generation of magnetic field by a flow consider the computationally-convenient periodic cube. In this case the minimum is found to be
where
Relationship to Reynolds number and Péclet number
The magnetic Reynolds number has a similar form to both the Péclet number and the Reynolds number. All three can be regarded as giving the ratio of advective to diffusive effects for a particular physical field, and have a similar form of a velocity times a length divided by a diffusivity. The magnetic Reynolds number is related to the magnetic field in an MHD flow, while the Reynolds number is related to the fluid velocity itself, and the Péclet number a related to heat. The dimensionless groups arise in the non-dimensionalization of the respective governing equations, the induction equation, the momentum equation, and the heat equation.
Relationship to eddy current braking
The dimensionless magnetic Reynolds number,
For
For