  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.
