Generation of momentum transport in weakly turbulent $\beta$-plane magnetohydrodynamics

TL;DR

This paper analytically and computationally investigates how weak turbulence on a $eta$-plane influences momentum transport in magnetohydrodynamics, revealing it scales with the square of the Rossby parameter and validating results with simulations.

Contribution

It derives new analytic constraints linking cross-helicity and magnetic energy, and extends weak turbulence theory to systems with multiple eigenmodes for the first time.

Findings

Momentum transport scales as $eta^2$, indicating a quadratic dependence.

Analytic constraints relate cross-helicity to magnetic energy.

Numerical simulations confirm the theoretical $eta^2$ scaling.

Abstract

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 $\beta$.