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Wave turbulence of inertia--gravity waves: a theory for the oceanic spectrum

We present a derivation using kinetic wave theory of the two-dimensional empirical Garrett--Munk spectrum for ocean internal waves, valid at all frequencies including near-inertial frequencies. This is based directly on the governing equations for a two-dimensional Boussinesq system with constant stratification and rotation. Our results improve on previous work by side-stepping the use of canonical variables, by taking full account of the Coriolis parameter in a non-hydrostatic dispersion relation, by filtering the balanced flow component from the dynamics, by using the conservation laws for energy and two components of pseudomomentum to bring the collision integral into a very simple form, by giving precise convergence conditions for the collision integral, and by finding the unique scale-invariant turbulent wave spectrum that corresponds to turbulent fluxes from small to large wavenumbers. The last step was achieved in the limit of small but nonzero Coriolis parameter. Key results are that any nonzero Coriolis parameter regularizes the singular nature of the non-rotating problem and that the homogeneity properties of the dispersion relation and of the interaction coefficients alone already imply that the spectrum is separable in vertical wavenumber and frequency. Within the restrictions of two-dimensional dynamics, this provides a theoretical framework for internal-wave turbulence consistent with oceanic observations.