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Weak turbulence theory for rotating magnetohydrodynamics and planetary flows

Published online by Cambridge University Press:  19 September 2014

Sébastien Galtier
Sébastien Galtier*
Affiliation:
Laboratoire de Physique des Plasmas, École Polytechnique, F-91128 Palaiseau CEDEX, France
*
†Email address for correspondence: sebastien.galtier@lpp.polytechnique.fr
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Abstract

A weak turbulence theory is derived for magnetohydrodynamics (MHD) under rapid rotation and in the presence of a uniform large-scale magnetic field which is associated with a constant Alfvén velocity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\boldsymbol {b}}_{{0}}$ . The angular velocity ${\boldsymbol{\Omega}}_{{0}}$ is assumed to be uniform and parallel to ${\boldsymbol {b}}_{{0}}$ . Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length $d \equiv b_0/\varOmega _0$ . In the large-scale limit ( $kd \to 0$ , with $k$ being the wavenumber) the left- and right-handed waves tend to the inertial and magnetostrophic waves, respectively, 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 behaviour of weakly dispersive interacting waves via three-wave interaction processes. It is shown that the nonlinear dynamics is mainly anisotropic, with a stronger transfer perpendicular than parallel to the rotation axis. The general theory may converge to pure weak inertial/magnetostrophic or Alfvén wave turbulence when the large- or small-scale 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, families 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 to the magnetostrophic dynamo, whose main applications are the Earth and the giant planets, such as Jupiter and Saturn, for which a small ( ${\sim }10^{-6}$ ) Rossby number is expected.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory MHD and Electrohydrodynamics: MHD turbulence

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Journal of Fluid Mechanics , Volume 757 , 25 October 2014 , pp. 114 - 154
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© 2014 Cambridge University Press 

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