  Physical experiments and numerical simulations have observed a remarkable
stabilizing phenomenon: a background magnetic field stabilizes and damps
electrically conducting fluids. This paper intends to establish this phenomenon
as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with
anisotropic dissipation in $\mathbb R^3$. The velocity equation in this system
is the 3D Navier-Stokes equation with dissipation only in the $x_1$-direction
while the magnetic field obeys the induction equation with magnetic diffusion
in two horizontal directions. We establish that any perturbation near the
background magnetic field $(0,1,0)$ is globally stable in the Sobolev setting
$H^3(\mathbb R^3)$. In addition, explicit decay rates in $H^2(\mathbb R^3)$ are
also obtained. When there is no presence of the magnetic field, the 3D
anisotropic Navier-Stokes equation in $\mathbb R^3$ is not well understood and
the small data global well-posedness remains an intriguing open problem. This
paper reveals the mechanism of how the magnetic field generates enhanced
dissipation and helps stabilize the fluid.
