On the role of cross-helicity in $\beta$-plane magnetohydrodynamic turbulence.

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 Alfvenic turbulence.