The idea of using wave turbulence formalism to describe internal waves is certainly not new; it dates to Kenyon, with calculations of the kinetic equations for oceanic spectra presented in Refs. olbers1976net, mccomas-1977-82, pomphrey1980dni.
Various formulations have been developed for characterizing wave-wave interactions in the stratified wave turbulence in the last four decades (see Ref. elementLPY for a brief review, and Refs. caillol2000kea,H66,K66,K68,gm_lvov,mccomas-1977-82,milder,MO75,O74,olbers1976net,pelinovsky1977wti,pomphrey1980dni,voronovich1979hfi for details).
We briefly discuss the derivation of the kinetic equation and wave-wave interaction matrix elements below in Eq. (15).
The starting point for the most extensive investigations has been a non-canonical Hamiltonian formulation in Lagrangian coordinates (16) that requires an unconstrained approximation in smallness of wave amplitude in addition to the assumption that nonlinear transfers take place on much longer time scales than the underlying linear dynamics. Other work has as its basis a formulation in Clebsh-like variables (11) and a non-Hamiltonian formulation in Eulerian coordinates (1). Here we employ a canonical Hamiltonian representation in isopycnal coordinates (5) which, as a canonical representation, preserves the original symmetries and hence conservation properties of the original equations of motion.
Energy transfers in the kinetic equation are characterized by three simple
mechanisms identified by
Ref. mccomas-1977-82 and reviewed by Ref. muller1986nia. These mechanisms
represent extreme scale-separated limits.
One of these mechanisms represents the
interaction of two small vertical scale, high frequency waves with a
large vertical scale, near-inertial (frequency near 
) wave and has
received the name Induced Diffusion (ID). The ID mechanism exhibits a
family of stationary states, i.e. a family of solutions to
Eq. (2). A comprehensive inertial-range theory with
constant downscale transfer of energy can be obtained by patching
these mechanisms together in a solution that closely mimics the
empirical universal spectrum (GM). A fundamental caveat from this work
is that the interaction time scales of high
frequency waves are sufficiently small at small spatial scales as to
violate the assumption of weak nonlinearity.
In parallel work, pelinovsky1977wti derived a kinetic equation for oceanic internal waves. They also have found the statistically steady state spectrum of internal waves, Eq. (5), which we propose to call Pelinovsky-Raevsky spectrum. This spectrum was latter found in Refs. caillol2000kea,gm_lvov. It follows from applying the Zakharov-Kuznetsov conformal transformation, which effectively establishes a map between the neighborhoods of zero and infinity. Making these two contributions cancel point-wisely yields the solution (5).
Both pelinovsky1977wti and caillol2000kea noted that the solution (5) comes through a cancellation between oppositely signed divergent contributions in their respective collision integrals. A fundamental caveat is that one can not use conformal mapping for divergent integrals. Therefore, the existence of such a solution is fortuitous.
Here we demonstrate that our canonical Hamiltonian structure admits to a similar characterization: power-law solutions of the form
(2) return collision integrals that are, in general, divergent. Regularization of the integral allows us to examine the
conditions under which it is possible to rigorously determine the power-law exponents 
in Eq. (2) that lead to stationary states. In doing so we obtain the ID family.
The situation is somewhat peculiar: We have assumed weak nonlinearity to derive the kinetic equation. The kinetic equation then predicts that nonlocal, strongly scale-separated interactions dominate the dynamics. These interactions have a less chance to be weakly nonlinear than regular, ``local'' interactions. Thus the derivation of the kinetic equation and its self-consistency is at risk. As we will see below, despite this caveat, the weakly nonlinear theory is consistent with much of the observational evidence.