Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180 USA

Woods Hole Oceanographic Institution, MS#21, Woods Hole, MA 02543 USA

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 USA

Department of Mechanical Engineering, Doshisha University, Kyotanabe, Kyoto 610-0394 Japan
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47.35.Bb

Steady scale-invariant solutions of a kinetic equation
describing the statistics of oceanic internal gravity waves
based on wave turbulence theory are investigated.
It is shown that
the scale-invariant collision integral in the kinetic equation
diverges for almost all spectral power-law exponents.
These divergences come from resonant
interactions with infra-red and/or ultra-violet wavenumbers
with extreme scale-separations.

We identify a small domain in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett-Munk spectrum. Power-law exponents which potentially permit a balance between the infra-red and ultra-violet divergences are also discussed. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky-Raevsky spectrum.

Ways to regularize this divergence are considered. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime.

The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. Yet these weakly nonlinear stationary states are consistent with much of the observational evidence.

**July 8, 2008**

- Introduction
- Wave turbulence theory for internal waves

- Scale-invariant kinetic equation
- Reduction of Kinetic Equation to the Resonant Manifold
- Convergences and divergences of the kinetic equation.

- A novel convergent solution
- Balance between divergences
- Regularization by the Coriolis effect
- Conclusions
- Asymptotics of collision integral in infra-red and ultra-violet limits

- Frequency-vertical-wavenumber and horizontal-vertical-wavenumber spectrum
- Asymptotic expansion for small values.
- Bibliography
- About this document ...