  Magnetohydrodynamic (MHD) turbulence on a $\beta$-plane with an in-plane mean
field, a system which serves as a simple model for the solar tachocline, is
investigated analytically and computationally. We first derive two useful
analytic constraints: we express the mean turbulent cross-helicity in terms of
the mean turbulent magnetic energy, and then show that (for weak turbulence)
the time-averaged momentum transport in the system can be expressed in terms of
the cross-helicity spectrum. We then complete a closure of the system using
weak turbulence theory, appropriately extended to a system with multiple
interacting eigenmodes. We use this closure to perturbatively solve for the
spectra at lowest order in the Rossby parameter $\beta$ and thereby show that
the momentum transport in the system is $O(\beta^2)$, thus quantifying the
transition away from Alfv\'enized turbulence. Finally, we verify our
theoretical results by performing direct numerical simulations of the system
over a broad range of $\beta$.
