Magnetic Reynolds number

In magnetohydrodynamics, the magnetic Reynolds number (R_m) 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:

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

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

In the theory of magnetohydrodynamics, the magnetic Reynolds number can be derived from the induction equation:

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

where

• ${\displaystyle \mathbf {B} }$ is the magnetic field,
• ${\displaystyle \mathbf {u} }$ is the fluid velocity,
• ${\displaystyle \eta }$ 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 ${\displaystyle \mathrm {R} _{\mathrm {m} }}$. If it is assumed that both terms share the scale length ${\displaystyle L}$ such that ${\displaystyle \nabla \sim 1/L}$ and the scale velocity ${\displaystyle U}$ such that ${\displaystyle \mathbf {u} \sim U}$, the induction term can be written as

${\displaystyle \nabla \times (\mathbf {u} \times \mathbf {B} )\sim {\frac {UB}{L}}}$

and the diffusion term as

${\displaystyle \eta \nabla ^{2}\mathbf {B} \sim {\frac {\eta B}{L^{2}}}.}$

The ratio of the two terms is therefore

${\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}.}$

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.^{[ citation needed ]} 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 the form of the product of a velocity and a length divided by a diffusivity. While the magnetic Reynolds number is related to the magnetic field in an magnetohydrodynamic flow, the Reynolds number is related to the fluid velocity itself and the Péclet number is related to heat. The dimensionless groups arise in the non-dimensionalization of the respective governing equations: the induction equation, the Navier–Stokes equations, 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 (9): 678–685. 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): 3024–3027. arXiv: physics/0010047 . Bibcode:2001PhRvL..86.3024G. doi:10.1103/PhysRevLett.86.3024. PMID   11290098. S2CID   638748.
3. Steiglitz, R.; U. Muller (2001). "Experimental demonstration of a homogeneous two-scale dynamo". Physics of Fluids. 13 (3): 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 (4): 044502. arXiv: physics/0701075 . Bibcode:2007PhRvL..98d4502M. doi:10.1103/PhysRevLett.98.044502. PMID   17358779. S2CID   21114816.
5. Moffatt, K. (2000). "Reflections on Magnetohydrodynamics" (PDF): 347–391.
6. Backus, G. (1958). "A class of self-sustaining dissipative spherical dynamos". Ann. Phys. 4 (4): 372–447. 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 (1): 89–93. Bibcode:1977GApFD...9...89P. doi:10.1080/03091927708242317.
8. Willis, A. (2012). "Optimization of the Magnetic Dynamo". Physical Review Letters. 109 (25): 251101. arXiv: 1209.1559 . Bibcode:2012PhRvL.109y1101W. doi:10.1103/PhysRevLett.109.251101. PMID   23368443. S2CID   23466555.
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" Archived 2007-09-29 at the Wayback Machine . 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.