Magnetic Eddy Viscosity of Mean Shear Flows in Magnetohydrodynamics

Abstract:

There has been significant interest in how magnetic fields interact with differential rotation in astrophysical and geophysical domains, such as in dynamos, the solar interior, and accretion disks. In the interior of gas giants, magnetic fields are likely to be relevant in explaining the recent measurements by Juno and Cassini regarding the depth that zonal flows reach inside Jupiter and Saturn. We study the interaction of shear and magnetic field in an idealized model with a magnetohydrodynamic (MHD) fluid. We show that when the magnetic Reynolds number is greater than 1, 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_p²/2μ₀)τ_c, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible MHD. 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. Our findings have potentially broad implications for the understanding of geophysical fluid turbulence where magnetic fields are important, and additionally offers hints for closures in reduced models.

[1] J. B. Parker and N. C. Constantinou, Phys. Rev. Fluids 4, 083701 (2019)