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| Magnetic Reynolds number | |
|---|---|
| Name | Magnetic Reynolds number |
| Dimension | dimensionless |
| Typical values | 10^-6–10^12 |
| Related | Reynolds number, Peclet number, Lundquist number |
Magnetic Reynolds number The Magnetic Reynolds number is a dimensionless quantity that characterizes the relative importance of advection of magnetic field by a conducting fluid to diffusion of that field by electrical resistivity. It plays a central role in the dynamics of magnetohydrodynamic flows in contexts ranging from laboratory plasmas to planetary cores and astrophysical plasmas. The quantity provides a criterion for magnetic flux freezing and for the onset of dynamo action in conducting media.
The Magnetic Reynolds number measures the competition between magnetic advection and magnetic diffusion in a conducting fluid interacting with a magnetic field; high values indicate that field lines are effectively carried with the flow, low values indicate diffusion dominates. In the context of the Solar dynamo and Earth's core processes, R_m helps discriminate regimes where the frozen-flux approximation used in models of the Magnetosphere of the Earth or of the Solar corona is valid. It informs interpretations of observations from missions such as Voyager 1, Parker Solar Probe, and experiments at facilities like ITER and the Princeton Plasma Physics Laboratory.
R_m is defined as R_m = U L / η, where U is a characteristic velocity, L is a characteristic length scale, and η is the magnetic diffusivity (η = 1/(μ σ), with μ the magnetic permeability and σ the electrical conductivity). In analyses of the Navier–Stokes equations coupled to the induction equation used in models of the Magnetohydrodynamics of the Sun or Jupiter, U and L are chosen to reflect dominant flow structures such as differential rotation in the Sun's convective zone or zonal jets in the Galilean moons environment. Dimensional analysis linking R_m to the Magnetorotational instability treatments in accretion disc theory often pairs it with characteristic shear rates derived from models of Keplerian rotation.
R_m is commonly considered alongside the Reynolds number Re and the Magnetic Prandtl number Pm = ν/η, where ν is kinematic viscosity; combinations determine regimes addressed in studies of the Taylor–Couette flow or the Kolmogorov spectrum in MHD turbulence. The Lundquist number S, Alfvé n number A, and Peclet number Pe provide complementary measures: S ~ B L/(η √(μ ρ)) and A links magnetic tension to inertial forces used in treatments of the Kelvin–Helmholtz instability in sheared magnetized plasmas. In geodynamo modeling for the Geodynamo or in stellar convection simulations for the Hertzsprung–Russell diagram classification, ratios among these dimensionless groups set the asymptotic scaling.
In astrophysics, R_m governs magnetic field evolution in objects such as accretion discs around black holes, the Interstellar medium, the Solar wind, and stellar interiors including the Red Giant Branch envelopes. High R_m in galaxies and galaxy clusters supports large-scale dynamo theories for the Milky Way and systems observed by the Hubble Space Telescope and the Chandra X-ray Observatory. In geophysics, R_m in the Earth's outer core is central to models explaining geomagnetic reversals and secular variation studied by institutions like the US Geological Survey and the British Geological Survey; it also informs paleomagnetic interpretations of data from the Kola Peninsula and the Canadian Shield.
In laboratory plasmas and engineering, R_m informs design and interpretation of experiments at facilities such as the Alfvénic Laboratory, the Rutherford Appleton Laboratory, and liquid-metal dynamo experiments performed in places like Perm and Garching. In liquid-metal cooling systems for fission reactors and in metallurgy at industrial sites including Argonne National Laboratory-supported programs, controlling R_m affects electromagnetic stirring and braking. In magnetic confinement fusion research at JET and ITER, R_m helps distinguish regimes for resistive instabilities and magnetic reconnection investigated with diagnostics developed at Culham Centre for Fusion Energy.
Estimating R_m requires measurements or models of velocity U, length scale L, and conductivity σ; in situ spacecraft observations from Ulysses or remote sensing with instruments on SOHO inform estimates for the solar wind, while borehole and seismological constraints combined with mineral physics from groups at the Max Planck Institute for Solar System Research and the Lamont–Doherty Earth Observatory constrain conductivity profiles in planetary interiors. Scaling laws used to extrapolate laboratory results to planetary or stellar conditions employ asymptotic arguments developed in literature connected to the Prandtl number and turbulence theories by researchers associated with institutes like the Institute of Physics and the Royal Society.
R_m is a control parameter in dynamo theory: when it exceeds critical thresholds in models motivated by the Parker dynamo or the α–Ω dynamo, self-excited magnetic fields can be sustained against diffusion. Stability analyses of resistive instabilities, reconnection events inspired by the Sweet–Parker model and the Petschek reconnection framework, and nonlinear saturation in simulations performed with codes developed at institutions like Los Alamos National Laboratory and Princeton University rely on specifying R_m alongside Pm and Re. Large-scale numerical models for the Solar dynamo and the Earth's geodynamo use adaptive mesh refinement and subgrid models to handle the extreme R_m regimes inferred from astronomical and geophysical observations.