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

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 $(B_p^2/2{\mu }_0){\tau }_{\mathrm{corr}}$, 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.

arXiv comment16 pages, 8 figures

References in corpus (12)

• Constraints on Deep-seated Zonal Winds Inside Jupiter and Saturn
• Magnetic Drag on Hot Jupiter Atmospheric Winds
• Saturn's deep atmospheric flows revealed by the Cassini Grand Finale gravity measurements
• Astrophysical Fluid Dynamics via Direct Statistical Simulation
• Cassini Ring Seismology as a Probe of Saturn's Interior I: Rigid Rotation
• Zonal Flow Magnetic Field Interaction in the Semi-Conducting Region of Giant Planets
• Zonal Flow as Pattern Formation
• Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory
• Anelastic spherical dynamos with radially variable electrical conductivity
• Dynamics of Zonal Flows: Failure of Wave-Kinetic Theory, and New Geometrical Optics Approximations
• S3T stability of the homogeneous state of barotropic beta-plane turbulence
• Magnetic suppression of zonal flows on a beta plane

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