  Zonal flows in rotating systems have been previously shown to be suppressed
by the imposition of a background magnetic field aligned with the direction of
rotation. Understanding the physics behind the suppression may be important in
systems found in astrophysical fluid dynamics, such as stellar interiors.
However, the mechanism of suppression has not yet been explained. In the
idealized setting of a magnetized beta plane, we provide a theoretical
explanation that shows how magnetic fluctuations directly counteract the growth
of weak zonal flows. Two distinct calculations yield consistent conclusions.
The first, which is simpler and more physically transparent, extends the
Kelvin-Orr shearing wave to include magnetic fields and shows that weak,
long-wavelength shear flow organizes magnetic fluctuations to absorb energy
from the mean flow. The second calculation, based on the quasilinear,
statistical CE2 framework, is valid for arbitrary wavelength zonal flow and
predicts a self-consistent growth rate of the zonal flow. We find that a
background magnetic field suppresses zonal flow if the bare Alfven frequency is
comparable to or larger than the bare Rossby frequency. However, suppression
can occur for even smaller magnetic fields if the resistivity is sufficiently
small enough to allow sizable magnetic fluctuations. Our calculations reproduce
the $\eta/B_0^2 = \text{const.}$ scaling that describes the boundary of
zonation, as found in previous work, and we explicitly link this scaling to the
amplitude of magnetic fluctuations.
