# Magnetic Reynolds number

The Magnetic Reynolds number (Rm) is a dimensionless group that occurs in magnetohydrodynamics. It gives an estimate of the effects of magnetic advection to magnetic diffusion, and is typically defined by:

$\mathrm{R}_\mathrm{m} = \frac{U L}{\eta}$

where

• $U$ is a typical velocity scale of the flow
• $L$ is a typical length scale of the flow
• $\eta$ is the magnetic diffusivity

## General Characteristics for Large and Small Rm

For $\mathrm{R}_\mathrm{m} \ll 1$, 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 $\mathrm{R}_\mathrm{m} \gg 1$, 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.

## Relationship to the 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, $R_m$, is also used in cases where there is no physical fluid involved.

$R_m = \mu \sigma$ × (characteristic length) × (characteristic velocity)
where
$\mu$ is the magnetic permeability
$\sigma$ is the electrical conductivity.

For $R_m < 1$ the skin effect is negligible and the eddy current braking torque follows the theoretical curve of an induction motor.

For $R_m > 30$ the skin effect dominates and the braking torque decreases much slower with increasing speed than predicted by the induction motor model.[1]