Generation of momentum transport in weakly turbulent $β$-plane magnetohydrodynamics
Magnetohydrodynamic (MHD) turbulence on a $β$-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 $β$ and thereby show that the momentum transport in the system is $O(β^2)$, thus quantifying the transition away from Alfvénized turbulence. Finally, we verify our theoretical results by performing direct numerical simulations of the system over a broad range of $β$.