Weak turbulence theory for rotating magnetohydrodynamics and planetary flows
Résumé
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 . The angular velocity is assumed to be uniform and parallel to . Such a system exhibits left and right circularly polarized waves which can be obtained by introducing the magneto-inertial length . In the large-scale limit ( , with being the wavenumber) the left- and right-handed waves tend to the inertial and magnetostrophic waves, respectively, whereas in the small-scale limit ( ) 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 at which the helicity flux is injected, with an inverse cascade if 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 ( ) Rossby number is expected.