# Magnetic eddy viscosity of mean shear flows in two-dimensional magnetohydrodynamics

## TL;DR

This paper introduces the concept of magnetic eddy viscosity in high magnetic Reynolds number regimes of 2D magnetohydrodynamics, showing it causes angular momentum transport and influences zonal flows in astrophysical objects.

## Contribution

It derives and confirms the magnetic eddy viscosity in 2D MHD flows, extending the understanding of magnetic effects in high Rm regimes and generalizing Ferraro's law of isorotation.

## Key findings

- Magnetic eddy viscosity arises in high Rm, hydrodynamic-dominated MHD flows.
- Numerical simulations confirm the theoretical magnetic viscosity.
- The results suggest magnetic viscosity influences angular momentum transport in astrophysical contexts.

## Abstract

Magnetic induction in magnetohydrodynamic fluids at magnetic Reynolds number (Rm) less than~1 has long been known to cause magnetic drag. Here, we show that when $\mathrm{Rm} \gg 1$ and the fluid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than the kinetic energy, induction due to a mean shear flow leads to a magnetic eddy viscosity. The magnetic viscosity is derived from simple physical arguments, where a coherent response due to shear flow builds up in the magnetic field until decorrelated by turbulent motion. The dynamic viscosity coefficient is approximately $(B_p^2/2\mu_0) \tau_{\rm corr}$, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional case, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero viscosity whenever there is differential rotation. Hence, these results serve as a dynamical generalization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular momentum and may be of importance to zonal flows in astrophysical domains such as the interior of some gas giants.