Magnetic Reynolds Number Calculator

The magnetic Reynolds number, Rm, is a dimensionless number that compares how quickly a magnetic field is advected (carried along) by a moving conducting fluid versus how quickly it diffuses through that fluid due to electrical resistivity. It is the magnetohydrodynamic analogue of the ordinary fluid Reynolds number, which compares inertial to viscous forces.

Rm = μ₀σLV, where μ₀ is the vacuum permeability, σ is the fluid's electrical conductivity, L is a characteristic length scale of the flow, and V is a characteristic flow velocity. It can equivalently be written as Rm = LV/η_m, where η_m = 1/(μ₀σ) is the magnetic diffusivity.

When Rm ≫ 1, magnetic field lines behave as if they are 'frozen' into the conducting fluid, moving and deforming exactly with the fluid's motion, a result known as Alfvén's frozen-in flux theorem. This is an excellent approximation for most astrophysical and space plasmas, where enormous length scales push Rm to huge values even for modest conductivity.

When Rm ≪ 1, magnetic diffusion dominates over advection, the field essentially slips through the fluid, largely unaffected by the fluid's motion, similar to how heat diffuses through a poor conductor regardless of any gentle stirring. This regime is typical of small-scale laboratory experiments and resistive processes like magnetic reconnection.

Rm scales directly with the characteristic length L, and astrophysical systems (stars, galaxies, the solar wind) have length scales of thousands to billions of kilometres, dwarfing anything achievable in a laboratory. Even with modest plasma conductivity, these vast length scales push Rm to values of 10¹⁰ or higher, making the frozen-in approximation essentially exact on large scales.

Laboratory dynamo experiments (using liquid sodium or gallium to mimic planetary core dynamics) are limited to metres-scale devices, so achieving Rm comparable to a planetary core (which spans thousands of kilometres) requires extremely high fluid velocities and conductivities, right at the edge of engineering feasibility. This is why successful laboratory dynamo experiments are rare and celebrated results.

Magnetic reconnection, the topological rearrangement of field lines that releases energy in solar flares and Earth's magnetotail, happens in thin regions where the local Rm drops low enough for diffusion to compete with advection, even though the surrounding plasma has an enormous global Rm. This local breakdown of the frozen-in approximation is essential to how reconnection can occur at all.

They are mathematical analogues (both compare an advective effect to a diffusive one) but describe different physics, the ordinary Reynolds number compares inertia to viscosity in a fluid flow, while the magnetic Reynolds number compares advection to resistive diffusion of a magnetic field. A flow can independently be laminar or turbulent (low or high ordinary Reynolds number) while also having a low or high magnetic Reynolds number.

Magnetic diffusivity η_m = 1/(μ₀σ) depends only on the fluid's electrical conductivity, higher conductivity means lower diffusivity, so the field diffuses more slowly and Rm is correspondingly larger for the same length and velocity scales.

A characteristic (typical, order-of-magnitude) velocity and length for the flow of interest, such as a plasma's bulk flow speed and the size of the region being studied, are standard choices. Since Rm depends on both linearly, the exact choice of 'characteristic' scale matters less than getting the right order of magnitude, which is usually all that is needed to determine the dominant regime.