Turbulent spectrum of 2D internal gravity waves*

Our work contributes to a nearly 60-year quest to derive the turbulent spectrum of weakly interacting internal gravity waves from first principles.

A promising avenue lies in the kinetic approach; however, the Boussinesq equation both is an anisotropic, non-canonical Hamiltonian equation, making the classical wave turbulence approach almost irrelevant. Previous attempts at weak wave turbulence analysis in 3D have fallen short of providing a definitive prediction for the energy spectrum. While observations emphasize the central role dispersive internal gravity waves play in natural processes like the ocean's climate cycle, decoupling these from the evolution of slow modes - degrees of freedom with vanishing frequency - proves difficult. Slow modes, such as shear modes in 2D and 3D and vortical modes in 3D, non-linearly interact with the waves, and tend to occupy a prominent part of the energy. Here we consider the 2D problem, which compared to 3D, holds a few advantages: it has no vortical modes, it is cheaper for direct numerical simulation and its, recently derived by the authors, kinetic equation takes a particular simple form.

We offer a new approach – we regularize the kinetic equation around the curve of vanishing frequency and look for steady solutions with nonzero energy fluxes. In the limit of a vanishing regulator, we find the turbulent spectrum of weakly interacting internal gravity waves. Our spectrum exactly matches the phenomenological oceanic Garrett-Munk spectrum in the limit of large vertical wave numbers and zero rotation.