  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 show that conservation of
squared magnetic potential in this system leads to a global increase in the
cross-helicity, which is otherwise conserved in pure MHD (for which the Rossby
parameter $\beta$ is zero). We perform a closure using weak turbulence theory
and show that the cross-helicity spectrum entirely determines momentum
transport. We also note that perturbation theory for small Rossby parameter is
impossible in weak turbulence, since it changes the topology of surfaces on
which resonant interactions occur. We support our results with numerical
simulations, which clearly indicate that the cross-helicity is most important
in the transitional regime between Rossby and Alfv\'enic turbulence.
