A weak wave turbulence theory is established for incompressible fluids under rapid rotation using a helicity decomposition, and the kinetic equations for energy E and helicity H are derived for three-wave coupling. As expected, nonlinear interactions of inertial waves lead to two-dimensional behavior of the turbulence with a transfer of energy and helicity mainly in the direction perpendicular to the rotation axis. For such a turbulence, we find, analytically, the anisotropic spectra $E\sim k_{\perp }^{-5/2}{k}_{\Vert }^{-1/2},$ $H\sim k_{\perp }^{-3/2}{k}_{\Vert }^{-1/2},$ and we prove that the energy cascade is to small scales. At lowest order, the wave theory does not describe the dynamics of two-dimensional (2D) modes which decouples from 3D waves.

• Received 21 October 2002

DOI:https://doi.org/10.1103/PhysRevE.68.015301