Dimensionless measures of turbulent magnetohydrodynamic dissipation rates

The magnetic Reynolds number, R-M, is defined as the product of a characteristic scale and associated flow speed divided by the microphysical magnetic diffusivity. For laminar flows, R-M also approximates the ratio of advective to dissipative terms in the total magnetic energy equation, but for turbulent flows this latter ratio depends on the energy spectra and approaches unity in a steady state. To generalize for flows of arbitrary spectra we define an effective magnetic dissipation number, R-M,R-e, as the ratio of the advection to microphysical dissipation terms in the total magnetic energy equation, incorporating the full spectrum of scales, arbitrary magnetic Prandtl numbers, and distinct pairs of inner and outer scales for magnetic and kinetic spectra. As expected, for a substantial parameter range R-M,R-e similar to O(1)<< R-M. We also distinguish R-M,R-e from (R-M,R-e) over tilde where the latter is an effective magnetic Reynolds number for the mean magnetic field equation when a turbulent diffusivity is explicitly imposed as a closure. That R-M,R-e and approach unity even if R-M >> 1 highlights that, just as in hydrodynamic turbulence, energy dissipation of large-scale structures in turbulent flows via a cascade can be much faster than the dissipation of large-scale structures in laminar flows. This illustrates that the rate of energy dissipation by magnetic reconnection is much faster in turbulent flows, and much less sensitive to microphysical reconnection rates compared to laminar flows.