Incorporating Turbulence into Dimensionless Measures of Magnetohydrodynamic Dissipation Rates

The magnetic Reynolds number RM , is defined as the product of a characteristic scale and associated flow speed divided by the microphysical magnetic diffusivity. For laminar flows, RM also approximates the ratio of advective to dissipative terms in the total magnetic energy equation. However, for turbulent flows this latter ratio depends on the energy spectra and approaches unity in a steady state. The physical implication of RM >> 1 for laminar flows (i.e. negligible magnetic dissipation of magnetic energy on dynamical times) is therefore incorrect for turbulent flows. To correctly capture the importance of dissipation for flows of arbitrary spectra we define an effective magnetic dissipation number, RM,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. For a substantial parameter range, RM,e ∼ O(1) << RM . We also distinguish RM,e from R̃M,e 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 RM,e and R̃M,e can approach unity even if RM >> 1 captures why 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. The latter is more sensitively dependent on microphysical reconnection rates than the former. This highlights the importance of distinguishing the study of magnetic reconnection in laminar versus turbulent flows.