Weak turbulence theory for rotating magnetohydrodynamics and planetary dynamos

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 Alfvén speed ${\mathbf{b}}_0$. 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 Alfvén 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 Alfvén 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 $k_{f}d<1$ 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.

4 figures, 33 pages

References in corpus (8)

• Wave turbulence in incompressible Hall MHD
• Helicity cascades in rotating turbulence
• On the magnetic fields generated by experimental dynamos
• Development of Anisotropy in Incompressible Magnetohydrodynamic Turbulence
• Quasi-Adiabatic Decay of Capillary Turbulence on the Charged Surface of Liquid Hydrogen
• Magnetic Helicity Conservation and Inverse Energy Cascade in Electron Magnetohydrodynamic Wave Packets
• On the excitation of inertial modes in an experimental spherical Couette flow
• On the effect of rotation on magnetohydrodynamic turbulence at high magnetic Reynolds number

Cited by in corpus (3)

• Direct Evidence of a Dual Cascade in Gravitational Wave Turbulence
• Wave Turbulence in Inertial Electron Magnetohydrodynamics
• Inverse cascade of hybrid helicity in $B\Omega$-MHD turbulence