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
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.
Yuri V. Lvov
Kurt L. Polzin
Esteban G. Tabak
July 8, 2008