Magnetic Reynolds number

The magnetic Reynolds number (R_m) 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:

${\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}~~\sim {\frac {\mathrm {induction} }{\mathrm {diffusion} }}}$

where

• ${\displaystyle U}$ is a typical velocity scale of the flow
• ${\displaystyle L}$ is a typical length scale of the flow
• ${\displaystyle \eta }$ 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.

Derivation

${\displaystyle \mathrm {R} _{\mathrm {m} }}$ is widely used in plasma physics, where two types of SI units (Gaussian cgs and SI mks) are common, because the Gaussian cgs units often allow cleaner derivations from which the physical reasoning is more clear, so it is worthwhile to write down the derivation in both sets of units. In the theory of magnetohydrodynamics, the transport equation for magnetic field, ${\displaystyle \mathbf {B} }$, is

${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {u} \times \mathbf {B} )+{\frac {\rho _{e}}{\mu _{o}}}\nabla ^{2}\mathbf {B} }$

in SI mks units and

${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {u} \times \mathbf {B} )+{\frac {\rho _{e}c^{2}}{4\pi }}\nabla ^{2}\mathbf {B} }$

in Gaussian cgs units, for permeability of free space ${\displaystyle \mu _{o}}$, speed of light ${\displaystyle c}$, fluid velocity ${\displaystyle \mathbf {u} }$, and resistivity ${\displaystyle \rho _{e}}$. The units of ${\displaystyle \rho _{e}}$ are Ohm-m in SI mks and seconds in Gaussian cgs. The final term in each of these equations is a diffusion term, with the kinematic diffusion coefficient, ${\displaystyle \eta }$ having units of distance squared per unit time, being the factor that multiplies the ${\displaystyle \nabla ^{2}\mathbf {B} }$. Thus, the units-independent form of these two equations is

${\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {u} \times \mathbf {B} )+\eta \nabla ^{2}\mathbf {B} .}$

${\displaystyle \mathrm {R} _{\mathrm {m} }}$ is the ratio of the two terms on the right hand side, on the assumption that they share the scale length ${\displaystyle L}$ such that ${\displaystyle \nabla \sim 1/L}$ in both terms, and that the scale of ${\displaystyle \mathbf {u} }$ is ${\displaystyle U}$. Thus one finds

${\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}={\frac {UL\mu _{o}}{\rho _{e}}}}$

in SI mks units and

${\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}={\frac {4\pi UL}{\rho _{e}c^{2}}}}$

in Gaussian cgs units.

Some confusion often arises because ${\displaystyle \eta }$ is commonly used both for the magnetic diffusivity and for the resistivity of a plasma, with the relation in SI mks units being that ${\displaystyle \eta =\rho _{e}/\mu _{o}}$.

General characteristics for large and small R_m

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

Range of values

The Sun is huge and has a large ${\displaystyle \mathrm {R} _{\mathrm {m} }}$, of order 10⁶. Dissipative affects are generally small, and there is no difficulty in maintaining a magnetic field against diffusion.

For the Earth, ${\displaystyle \mathrm {R} _{\mathrm {m} }}$ is estimated to be of order 10³ .[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 human length scale is very small so that typically ${\displaystyle \mathrm {R} _{\mathrm {m} }\ll 1}$. 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]

Bounds

In situations where permanent magnetisation is not possible, e.g. above the Curie temperature, to maintain a magnetic field ${\displaystyle \mathrm {R} _{\mathrm {m} }}$ must be large enough such that induction outweighs diffusion. 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

${\displaystyle \mathrm {\hat {R}} _{\mathrm {m} }={\frac {L^{2}S}{\eta }}}$

where S is a measure of strain. One of the most well known results is due to Backus [6] which states that the minimum ${\displaystyle \mathrm {R} _{\mathrm {m} }}$ for generation of a magnetic field by flow in a sphere is such that

${\displaystyle \mathrm {\hat {R}} _{\mathrm {m} }\geq \pi ^{2}}$

where ${\displaystyle L=a}$ is the radius of the sphere and ${\displaystyle S=e_{max}}$ is the maximum strain rate. This bound has since been improved by approximately 25% by Proctor.[7]

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[8]

${\displaystyle \mathrm {\hat {R}} _{\mathrm {m} }=2.48}$

where ${\displaystyle S}$ is the root-mean-square strain over a scaled domain with sides of length ${\displaystyle 2\pi }$. If shearing over small length scales in the cube is ruled out, then ${\displaystyle \mathrm {R} _{\mathrm {m} }=1.73}$ is the minimum, where ${\displaystyle U}$ is the root-mean-square value.

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, ${\displaystyle R_{m}}$, is also used in cases where there is no physical fluid involved.

${\displaystyle R_{m}=\mu \sigma }$ × (characteristic length) × (characteristic velocity)

where

${\displaystyle \mu }$ is the magnetic permeability

${\displaystyle \sigma }$ is the electrical conductivity.

For ${\displaystyle R_{m}<1}$ the skin effect is negligible and the eddy current braking torque follows the theoretical curve of an induction motor.

For ${\displaystyle R_{m}>30}$ the skin effect dominates and the braking torque decreases much slower with increasing speed than predicted by the induction motor model.[9]

See also

• Lundquist number
• Magnetohydrodynamics
• Reynolds number
• Péclet number

References

1. Davies, C.; et al. (2015). "Constraints from material properties on the dynamics and evolution of Earth's core" (PDF). Nature Geoscience. 8: 678. Bibcode:2015NatGe...8..678D. doi:10.1038/ngeo2492.
2. Gailitis, A.; et al. (2001). "Magnetic field saturation in the Riga dynamo experiment". Physical Review Letters. 86 (14): 3024. arXiv:physics/0010047. Bibcode:2001PhRvL..86.3024G. doi:10.1103/PhysRevLett.86.3024. PMID 11290098.
3. Steiglitz, R.; U. Muller (2001). "Experimental demonstration of a homogeneous two-scale dynamo". Physics of Fluids. 13: 561–564. Bibcode:2001PhFl...13..561S. doi:10.1063/1.1331315.
4. Moncheaux, R.; et al. (2007). "Generation of a Magnetic Field by Dynamo Action in a Turbulent Flow of Liquid Sodium". Physical Review Letters. 98: 044502. arXiv:physics/0701075. Bibcode:2007PhRvL..98d4502M. doi:10.1103/PhysRevLett.98.044502.
5. Moffatt, K. (2000). "Reflections on Magnetohydrodynamics" (PDF): 347–391. Cite journal requires |journal= (help)
6. Backus, G. (1958). "A class of self-sustaining dissipative spherical dynamos". Ann. Phys. 4: 372. Bibcode:1958AnPhy...4..372B. doi:10.1016/0003-4916(58)90054-X.
7. Proctor, M. (1977). "On Backus' necessary condition for dynamo action in a conducting sphere". Geophysical & Astrophysical Fluid Dynamics. 9: 177. Bibcode:1977GApFD...9...89P. doi:10.1080/03091927708242317.
8. Willis, A. (2012). "Optimization of the Magnetic Dynamo". Physical Review Letters. 109: 251101. arXiv:1209.1559. Bibcode:2012PhRvL.109y1101W. doi:10.1103/PhysRevLett.109.251101. PMID 23368443.
9. Ripper, M.D; Endean, V.G (Mar 1975). "Eddy-Current Braking-Torque Measurements on a Thick Copper Disc". Proc IEE. 122 (3): 301–302. doi:10.1049/piee.1975.0080.

Further reading

• Moffatt, H. Keith, 2000, "Reflections on Magnetohydrodynamics". In: Perspectives in Fluid Dynamics (ISBN 0-521-53169-1) (Ed. G.K. Batchelor, H.K. Moffatt & M.G. Worster) Cambridge University Press, p 347–391.
• P. A. Davidson, 2001, An Introduction to Magnetohydrodynamics (ISBN 0-521-79487-0), Cambridge University Press.