Weak turbulence theory for rotating magnetohydrodynamics and planetary dynamos

TL;DR

This paper develops a weak turbulence theory for rotating magnetohydrodynamics, revealing anisotropic wave interactions and energy cascades relevant to planetary dynamos like Earth's core.

Contribution

It introduces a novel weak turbulence framework for rotating MHD with magnetic fields, analyzing wave interactions and energy transfer mechanisms across scales.

Findings

Identifies anisotropic energy transfer predominantly perpendicular to the rotation axis.

Derives exact spectral solutions for different wave regimes.

Shows the cascade direction of hybrid helicity depends on the injection scale.

Abstract

A weak turbulence theory is derived for magnetohydrodynamics under rapid rotation and in the presence of a large-scale magnetic field. The angular velocity $\Omega_0$ is assumed to be uniform and parallel to the constant Alfv\'en speed ${\bf b_0}$. Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length $d \equiv b_0/\Omega_0$. In the large-scale limit ($kd \to 0$; $k$ being the wave number), the left- and right-handed waves tend respectively to the inertial and magnetostrophic waves whereas in the small-scale limit ($kd \to + \infty$) pure Alfv\'en waves are recovered. By using a complex helicity decomposition, the asymptotic weak turbulence equations are derived which describe the long-time behavior of weakly dispersive interacting waves {\it via} three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic with a stronger transfer perpendicular ($\perp$) than parallel ($\parallel$) to the rotating axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfv\'en wave turbulence when the large or small-scales limits are taken respectively. Inertial wave turbulence is asymptotically dominated by the kinetic energy/helicity whereas the magnetostrophic wave turbulence is dominated by the magnetic energy/helicity. For both regimes a family of exact solutions are found for the spectra which do not correspond necessarily to a maximal helicity state. It is shown that the hybrid helicity exhibits a cascade whose direction may vary according to the scale $k_f$ at which the helicity flux is injected with an inverse cascade if $k_fd < 1$ and a direct cascade otherwise. The theory is relevant for the magnetostrophic dynamo whose main applications are the Earth and giant planets for which a small ($\sim 10^{-6}$) Rossby number is expected.