Turbulent Spectrum of 2D Internal Gravity Waves

We find the turbulent energy spectrum of weakly interacting 2D internal gravity waves using the full, nonhydrostatic dispersion relation. This spectrum is an exact solution of a regularized kinetic equation, from which the zero-frequency shear modes have been excised by a careful limiting process. This is a new method in wave kinetic theory. The turbulent spectrum agrees with the 2D oceanic Garrett-Munk spectrum for frequencies large compared to the Coriolis frequency and vertical scales small compared to the depth of the ocean. We show that this turbulent spectrum is the unique power law solution to the steady kinetic equation with a nonzero radial flux. Our solution provides an interesting insight into a turbulent energy cascade in an anisotropic system-like isotropic turbulence it is self-similar in scale, but its angular part is peaked along the curve of vanishing frequency and is self-similar in frequency.