  Zonal flows are mean flows in the east-west direction, which are ubiquitous
on planets, and can be formed through 'zonostrophic instability': within
turbulence or random waves, a weak large-scale zonal flow can grow
exponentially to become prominent. In this paper, we study the statistical
behaviour of the zonostrophic instability and the effect of magnetic fields. We
use a stochastic white noise forcing to drive random waves, and study the
growth of a mean flow in this random system. The dispersion relation for the
growth rate of the expectation of the mean flow is derived, and properties of
the instability are discussed. In the limits of weak and strong magnetic
diffusivity, the dispersion relation reduces to manageable expressions, which
provide clear insights into the effect of the magnetic field and scaling laws
for the threshold of instability. The magnetic field mainly plays a stabilising
role and thus impedes the formation of the zonal flow, but under certain
conditions it can also have destabilising effects. Numerical simulation of the
stochastic flow is performed to confirm the theory. Results indicate that the
magnetic field can significantly increase the randomness of the zonal flow. It
is found that the zonal flow of an individual realisation may behave very
differently from the expectation. For weak magnetic diffusivity and moderate
magnetic field strengths, this leads to considerable variation of the outcome,
that is whether zonostrophic instability takes place or not in individual
realisations.
