Weak inertial-wave turbulence theory.

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 approximately k(-5/2)(perpendicular)k(-1/2)(parallel), H approximately k(-3/2)(perpendicular)k(-1/2)(parallel), 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.