\documentclass[prb,%
%twocolumn,%
amsfonts,amsmath,amssymb,%
showpacs,%
floatfix,%
]{revtex4}

\usepackage[sort&compress]{natbib} 
\usepackage{graphicx}
\usepackage{bm}
\usepackage{longtable}


\begin{document}
\title{Oceanic Internal Wave Field: Theory of Scale-invariant Spectra}

\author{Yuri V. \surname{Lvov}}
\affiliation{Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180 USA}

\author{Kurt L. \surname{Polzin}}
\affiliation{Woods Hole Oceanographic Institution, MS\#21, Woods Hole, MA 02543 USA}

\author{Esteban G. \surname{Tabak}}
\affiliation{Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 USA}

\author{Naoto \surname{Yokoyama}}
\affiliation{Department of Mechanical Engineering, Doshisha University, Kyotanabe, Kyoto 610-0394 Japan}
\thanks{Author to whom should be addressed}
\email{nyokoyam@mail.doshisha.ac.jp}

\date{\today}
\pacs{47.35.Bb}

\begin{abstract}
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. 
\end{abstract}

\maketitle

\section{Introduction}

Wave-wave interactions in continuously stratified flows have been a
subject of intensive research in the last few decades. Of particular
importance is the observation of a nearly universal internal-wave energy spectrum
in the ocean, first described by Garrett and Munk. It is generally
thought that the existence of a universal spectrum is at least partly
the result of nonlinear interactions
among internal waves. Due to the quadratic nonlinearity
of the underlying fluid equations and dispersion relation allowing three-wave resonances, 
internal waves interact through triads.
In the weakly nonlinear regime, the nonlinear
interactions among internal waves concentrate on their {\em resonant\/} set, and 
can be described by a kinetic equation, which assumes the
familiar form~\citep{caillol2000kea,H66,K66,K68,gm_lvov,mccomas-1977-82,milder,MO75,O74,olbers1976net,pelinovsky1977wti,pomphrey1980dni,voronovich1979hfi,zak_book}:

\begin{eqnarray}
\frac{\partial n_{\bm{p}}}{\partial t} \! &=& \!
 4\pi \int d \bm{p}_{12}
\left(
 |V_{\bm{p}_1,\bm{p}_2}^{\bm{p}}|^2 \, f^{\bm{p}}_{\bm{p}_1\bm{p}_2} \,
\delta_{{\bm{p} - \bm{p}_1-\bm{p}_2}} \, \delta_{\omega_{\bm{p}}
-\omega_{{\bm{p}_1}}-\omega_{{\bm{p}_2}}}
\right.
\nonumber \\
&&
\left.
- |V_{\bm{p}_2,\bm{p}}^{\bm{p}_1}|^2\, f^{\bm{p}_1}_{\bm{p}_2\bm{p}} \, \delta_{{\bm{p}_1 - \bm{p}_2-\bm{p}}} \,
  \delta_{{\omega_{\bm{p}_1} -\omega_{\bm{p}_2}-\omega_{\bm{p}}}}
\right.
\nonumber \\
&&
\left.
-
 \, |V_{\bm{p},\bm{p}_1}^{\bm{p}_2}|^2\, f^{\bm{p}_2}_{\bm{p}\bm{p}_1}\, \delta_{{\bm{p}_2 - \bm{p}-\bm{p}_1}} \,
  \delta_{{\omega_{\bm{p}_2} -\omega_{\bm{p}}-\omega_{\bm{p}_1}}}
\right)
 \, ,\nonumber\\
&&
\mathrm{with} \quad f^{\bm{p}}_{\bm{p}_1\bm{p}_2} = n_{\bm{p}_1}n_{\bm{p}_2} -
n_{\bm{p}}(n_{\bm{p}_1}+n_{\bm{p}_2}) \, .
\label{KineticEquation}
\end{eqnarray}
%
Here $n_{\bm{p}} = n(\bm{p})$ is a three-dimensional action spectrum (see Eq.~(\ref{WaveAction})) with wavenumber
%
$\bm{p} = (\bm{k}, m),$
%
i.e.\ $\bm{k}$ and $m$ are the horizontal and vertical components of $\bm{p}$.
The frequency $\omega$ is given by a linear dispersion relation.
The wavenumbers are signed variables, while
the wave frequencies are always positive.  The factor
$V^{\bm{p}}_{\bm{p}_1\bm{p}_2}$ is the interaction matrix 
describing the transfer of wave action among the members of a resonant triad. 

Following Kolmogorov's viewpoint of energy cascades
in isotropic Navier--Stokes turbulence,
one may look for statistically stationary states using scale-invariant solutions to the 
kinetic equation~(\ref{KineticEquation}). The solution may occur in an inertial
subrange of wavenumbers and frequencies that are far from those where forcing and
dissipation act, and also far
from characteristic scales of the system, including  
the Coriolis frequency 
resulting from the rotation of the Earth, the buoyancy frequency due
to stratification and the ocean depth.
Under these assumptions, the dispersion relation and the interaction matrix elements are 
locally scale-invariant.  It is natural, therefore, in this restricted domain, to
look for self-similar solutions of Eq.~(\ref{KineticEquation}), which 
take the form
%
\begin{eqnarray}
   n(\bm{k}, m)=  |\bm{k}|^{-a}  |m|^{-b}.
\label{PowerLawSpectrum}
\end{eqnarray}
%
Values of $a$
and $b$ in the right-hand side of
Eq.~(\ref{KineticEquation}) vanishes identically
correspond to steady solutions of the kinetic
equation, and hopefully also to statistically steady states of the ocean's
wave field. 
Unlike Kolmogorov turbulence, 
the exponents which give steady solutions
can not be determined by the dimensional analysis alone (see, for example, Ref.~\onlinecite{polzin-2004}). This is the case 
owing to multiple characteristic length scales in anisotropic systems. 

Before seeking steady solutions, however,
one should find out whether the
improper integrals in the kinetic equation~(\ref{KineticEquation}) converge. 
This is related to the question of {\em locality\/} of the
interactions: a convergent integral characterizes the physical scenario where 
interactions of neighboring wavenumbers dominate the evolution of the wave spectrum,
while a divergent one 
implies that distant, nonlocal interactions in the wavenumber space dominate. 


It turns out 
that the internal-wave collision integral diverges
for {\em almost\/} all values of $a$ and $b$. In particular, the collision integral has an 
infra-red (IR) divergence 
at zero, i.e. $|\bm{k}_1|$ or $|\bm{k}_2|$ $\to 0$ and an ultra-violet (UV) divergence
at infinity, i.e. $|\bm{k}_1|$ and $|\bm{k}_2|$ $\to \infty$.
There is only one exception where the integral converges:
the segment with $b=0$ and $7/2 < a < 4$.
This segment corresponds to wave action independent of vertical wavenumbers, 
$\partial n / \partial m = 0$.
Within this segment we numerically determine a new 
steady {\em convergent} solution to
Eq.~(\ref{KineticEquation}), with
\begin{eqnarray} n(\bm{k},m) \propto  |\bm{k}|^{-3.7}.
\label{NewPoint}
\end{eqnarray}
This solution is not far from the large-wavenumber form of the Garrett--Munk (GM) spectrum:  
\begin{equation}
n(\bm{k},m) \propto  |\bm{k}|^{-4}~.
\label{GM}
\end{equation}


Alternatively, one 
can explore
the physical interpretation of divergent solutions. 
We find a region in $(a,b)$ 
space where there are both IR and UV divergences having
opposite signs.
This suggests a possible scenario where the
two divergent contributions may cancel each other, yielding a steady
state. An example of such a case 
is provided by the Pelinovsky--Raevsky (PR) spectrum,
\begin{eqnarray}
 n_{\bm{k},m} \propto |\bm{k}|^{-7/2}
 |m|^{-1/2}
.
\label{LT}
\end{eqnarray}

This solution, however, is only one among infinitely many. 
The problem at hand is a generalization of the
concept of principal value integrals:
for $a$ and $b$ which give opposite signs of the divergences at zero and infinity,
one can {\em regularize\/} the
integral by cutting out small neighborhoods of the two singularities
in such a way that the divergences cancel each other
and the remaining contributions are small.
Hence all
the exponents which yield opposite-signed divergence at the both ends
can be steady solutions of Eq.~(\ref{KineticEquation}).
As we will see below this general statement helps 
to describe the experimental oceanographic data which are available to us.
The nature of such  steady solutions 
depends on the  particular truncation of the 
divergent integrals. 

So far, we have kept the formalism at the level of the self-similar
limit of the kinetic equation~(\ref{KineticEquation}).
However,
once one considers energy transfer mechanisms dominated by 
interactions with extreme modes of the system, one can no longer neglect
the deviations from self-similarity near the spectral boundaries:
the inertial frequency due to the rotation of the Earth at the IR end, and the buoyancy frequency and/or
dissipative cut-offs at the UV end.  

For example, we may consider a scenario in
which interactions with the smallest horizontal wavenumbers dominate 
the energy transfer within the inertial subrange, either because
the collision integral at infinity converges or
because the system is more heavily truncated at the large wavenumbers by
wave breaking or dissipation.
We will demonstrate
that the IR divergence of the collision integral
has a simple physical interpretation:
the evolution of each wave is dominated
by the interaction with its nearest neighboring vertical wavenumbers, mediated by
the smallest horizontal wavenumbers of the system, a mechanism denoted Induced Diffusion
in the oceanographic literature.

To bring back the effects of the rotation of the Earth
in Eq.~(\ref{KineticEquation}), one introduces the Coriolis parameter $f$
there and in the linear dispersion relation.  
Since we are considering the evolution of
waves with frequency $\omega$ much larger than 
$f$, $f$ can be considered to be small.  However, since
the interaction with waves near $f$ dominates the energy transfer,
one needs to invert the
order in which the limits are taken, postponing making 
$f$ zero to the end.
This gives rise to an integral
that diverges like $f$ raised to a negative power smaller than $-1$,  
but multiplied by a prefactor that
vanishes if either $9-2a-3b=0$ or $b=0$.
These are the induced diffusion lines of steady state solutions, found
originally in Ref.~\onlinecite{mccomas-1981}.  This family of stationary states does a reasonable job of explaining the gamut of observed variability.  

The paper is organized as follows.
Wave turbulence theory for the internal wave field and
the corresponding kinetic equation 
are briefly summarized in Sec.~\ref{sec:WTKE}  along with the
motivating observations.
We analyze the divergence of the kinetic equation in Sec.~\ref{sec:SI}.  
Section \ref{NovelSolution}
includes a special, convergent power-law solution that may account for the GM spectrum.
In Sec.~\ref{Balance} we introduce 
possible quasi-steady solutions  of the kinetic equation which are based on cancellations 
of two singularities.
Section~\ref{CorRegularization} shows that the IR divergence is dominated by induced diffusion,
and computes the family of power-law solutions
which arises from taking it into account.
We conclude in Sec.~\ref{sec:conclusion}.


\section{Wave turbulence theory for internal waves}
\label{sec:WTKE}

\subsection{Background and history}

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.~\onlinecite{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.~\onlinecite{elementLPY} for 
a brief review, and Refs.~\onlinecite{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.~(\ref{MatrixElement}). 

The starting point for the most extensive investigations has been a non-canonical Hamiltonian formulation in Lagrangian coordinates \citep{mccomas-1981} 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 \citep{pelinovsky1977wti} and a non-Hamiltonian formulation in Eulerian coordinates \citep{caillol2000kea}.  Here we employ a canonical Hamiltonian representation in isopycnal coordinates \citep{gm_lvov} 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.~\onlinecite{mccomas-1977-82} and reviewed by Ref.~\onlinecite{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 $f$) 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.~(\ref{PowerLawSpectrum}).  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, \textcite{pelinovsky1977wti} derived a kinetic equation
for oceanic internal waves. They also have found the statistically steady state spectrum of internal waves, Eq.~(\ref{LT}),
which we propose to call Pelinovsky-Raevsky spectrum.
This spectrum was latter found in Refs.~\onlinecite{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~(\ref{LT}).


Both \textcite{pelinovsky1977wti} and \textcite{caillol2000kea} noted that
the solution~(\ref{LT}) 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
(\ref{PowerLawSpectrum}) 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 $(a,b)$ in Eq.~(\ref{PowerLawSpectrum}) 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. 


\subsection{Experimental motivation}
\label{sec:PRL}
Power laws provide a simple and intuitive physical description of
complicated wave fields. Therefore we assumed that the
spectral energy density can be represented as
Eq.~(\ref{PowerLawSpectrum}), and 
undertook a systematic study of published observational programs. In doing so we were fitting the experimental data available to us by power-law
spectra. We were presuming that the power laws offer a good fit of the data. We were not assuming that spectra are given by Garrett and Munk 
spectrum. That effort is reported in detail on elsewhere\citep{PLT}, here we just give  a brief synopsis to motivate present theoretical study.   

We reviewed the following observational programs:
\begin{itemize}
\item  The Internal Wave Experiment (IWEX)\cite{M78}; 
\item The Arctic Internal Wave Experiment (AIWEX)\cite{Letal87,dasaro1991iwa};
\item The Frontal Air-Sea Interaction Experiment
(FASINEX)~\cite{weller1991for,eriksen1991ofv};
\item  The Patches Experiment (PATCHEX)\cite{sherman21evw};
\item  The Surface Wave Process Program (SWAPP) experiment\cite{anderson1992ssa};
\item North Atlantic Tracer Release Experiment (NATRE)\cite{P03} / Subduction\cite{Weller04};
\item Salt Finger Tracer Release Experiment (SFTRE)\cite{Schmitt05} / Polymode IIIc (PMIII)\cite{Keffer83}; 
\item Site-D\cite{Foff69};
\item Storm Transfer and Response Experiment (STREX)\cite{DA84} / Ocean Storms Experiment (OS)\cite{DA95}.
\end{itemize}

Here we extract the summary of that study in the form of Fig.~\ref{fig:onlydata}, in which estimates of power-law exponents $(a,b)$ are plotted against each other.  

That study comes with many caveats.  The most important here is that the power-law exponents, $(a,b)$, are not typically derived from two-dimensional
horizontal wavenumber, vertical wavenumber spectra.  The power laws were fit to one-dimensional frequency and one-dimensional vertical wavenumber spectra of energy, $e(\omega) \propto \omega^{-c}$ and $e(m) \propto m^{-d}$, with conversion to $(a,b)$ requiring that:
$$
a = c + 2   {\rm ~~and~~   }  b = d - c .  
$$
This procedure explicitly assumes that the two-dimensional spectra are separable.
That is to say that the $\omega$--$m$ spectrum can be represented as a product of function of $\omega$ and function of $m$
alone. 
It is clear that this assumption is not warranted in all cases.  This is particularly true of the NATRE data, for which two power-law estimates are provided.  
%
Nevertheless, this is the best data that is available to us at a present time. 
We see that these points are not randomly scattered, but have some pattern.
%
Explaining the location of the experimental points and making sense out of this pattern is the main physical motivation for this study.


\begin{figure}
 \begin{center}
\includegraphics[scale=0.45]{1.eps}
 \end{center}
\caption{The observational points.  The filled circles represent the Pelinovsky--Raevsky (PR) spectrum, the convergent numerical solution determined in Sec.~\ref{NovelSolution} (C) and the GM spectrum.  Circles with stars represent estimates based upon one-dimensional spectra from the western North Atlantic south of the Gulf Stream (IWEX, FASINEX and SFTRE/PMIII), the eastern North Pacific (STREX/OS and PATCHEX$^1$), the western North Atlantic north of the Gulf Stream (Site-D), the Arctic (AIWEX) and the eastern North Atlantic (NATRE$^1$ and NATRE$^2$).  There are two estimates obtained from two-dimensional data sets from the eastern North Pacific (SWAPP and PATCHEX$^2$) represented as circles with cross hairs.  NATRE$^1$ represents a fit over frequencies of $1 < \omega < 6 $cpd and NATRE$^2$ a fit over higher frequencies.  }
\label{fig:onlydata}
\end{figure}


\subsection{Hamiltonian Structure and Wave Turbulence Theory}
\label{HamiltonianSection}

This subsection briefly summarizes the derivation in 
Ref.~\onlinecite{gm_lvov}; it is included 
here only for completeness and to allow references from
the core of the paper.

The equations of motion satisfied by an
incompressible stratified rotating flow in hydrostatic balance
under the Boussinesq approximation are:
%
\begin{eqnarray}
\frac{\partial}{\partial t}\frac{\partial z}{\partial \rho} + \nabla \cdot \left(\frac{\partial z}{\partial \rho} \bm{u} \right) &=& 0 , \nonumber \\
\frac{\partial \bm{u}}{\partial t} +f \bm{u}^\perp+ \bm{u} \cdot
\nabla \bm{u} + \frac{\nabla M}{\rho_0} &=& 0 ,
\nonumber
\\
\frac{\partial M}{\partial \rho} - g z &=& 0 .
\label{PrimitiveEquations}
\end{eqnarray}
%
These equations result from mass convervation, horizontal momentum convervation and hydrostatic balance.
The equations are written in isopycnal coordinates with the density
$\rho$ replacing the height $z$ in its role as independent vertical
variable.  Here $\bm{u} = (u,v)$ is the horizontal component of the
velocity field, $\bm{u}^{\perp} = (-v, u)$, $\nabla = (\partial/\partial x, \partial/\partial y)$ is the gradient
operator along isopycnals,
$M$ is the Montgomery potential
 $$M=P+g\,\rho\,z \, , $$ 
%
$f$ is the Coriolis parameter, 
and $\rho_0$ is a reference density in its role as
inertia, considered
constant under the Boussinesq approximation.

The potential vorticity is given by
%
\begin{equation}
 q = \frac{f+\partial v/\partial y - \partial u/\partial x}{\Pi} ,
\label{PVorig}
\end{equation}
%
where
$\Pi = \rho / g \partial^2 M/\partial \rho^2 = \rho \partial z/\partial \rho $
%
is a normalized differential layer thickness. Since both the potential vorticity
and the fluid density
are conserved along particle trajectories,
an initial profile of the potential
vorticity that is a function of the density will be preserved by the flow.
Hence it is self-consistent to assume that
%
\begin{equation}
  q(\rho) = q_0(\rho) = \frac{f}{\Pi_0(\rho)} \, ,
 \label{PV}
\end{equation}
%
where $\Pi_0(\rho) = -g / N(\rho)^2$ is a reference stratification profile
with the constant background buoyancy frequency, $N = (-g/(\rho \partial z/\partial\rho|_{\mathrm{bg}}))^{1/2}$. This assumption
is not unrealistic: it represents a pancake-like distribution of potential vorticity,
the result of its comparatively faster homogenization along than across isopycnal surfaces.



It is shown in Ref.~\onlinecite{gm_lvov} that the
primitive equations of motion (\ref{PrimitiveEquations})
under the assumption ({\ref{PV}) 
can be written as a pair of
canonical Hamiltonian equations,
\begin{equation}
 \frac{\partial \Pi}{\partial t}  = - \frac{\delta {\cal H}}{\delta \phi} \, ,
  \qquad
\frac{\partial \phi}{\partial t}  = \frac{\delta {\cal H}}{\delta \Pi} \, ,
  \label{Canonical}
\end{equation}
where $\phi$ is the isopycnal velocity potential, and
the Hamiltonian is the sum of kinetic and potential energies,
%
\begin{widetext}
\begin{eqnarray}
 {\cal H} = \int d \bm{x} d \rho
 \left(
 - \frac{1}{2} \left(
\Pi_0+\Pi(\bm{x}, \rho) \right) \,
  \left|\nabla \phi(\bm{x}, \rho) + \frac{f}{\Pi_0} \nabla^{{\perp}}\Delta^{-1} \Pi(\bm{x}, \rho)
  \right|^2
+
  \frac{g}{2} \left|\int^{\rho} d\rho^{\prime} \frac{\Pi(\bm{x}, \rho^{\prime})}{\rho^{\prime}} \right|^2
  \right) \, .
  \label{HTL2}
\end{eqnarray}
\end{widetext}
%
Here, $\nabla^{\perp}=(-\partial / \partial y, \partial / \partial x)$
and $\Delta^{-1}$ is the inverse Laplacian.

Switching to Fourier space, and
introducing a complex field variable $a_{\bm{p}}$ through the transformation
%
\begin{eqnarray}
 \phi_{\bm{p}} &=& \frac{i N \sqrt{\omega_{\bm{p}}}}{\sqrt{2 g} |\bm{k}|} \left(a_{\bm{p}}-
a^{\ast}_{-{\bm{p}}}\right)\, ,\nonumber\\
\Pi_{\bm{p}} &=& \Pi_0-\frac{N\, \Pi_0\,
|\bm{k}|}{\sqrt{2\, g \omega_{\bm{p}}}}\left(a_{\bm{p}}+a^{\ast}_{-{\bm{p}}}\right)\, ,
\label{transformationToSingleEquation2}
\end{eqnarray}
%
where the frequency $\omega$ satisfies the linear dispersion relation
%
\begin{eqnarray}
\omega_{\bm{p}}=\sqrt{ f^2 + \frac{g^2}{\rho_0^2 N^2} \frac{|\bm{k}|^2 }{m^2}},
\label{InternalWavesDispersion}
\end{eqnarray}
%
the equations of motion
(\ref{PrimitiveEquations}) adopt the canonical form
%
\begin{equation}
i\frac{\partial}{\partial t} a_{\bm{p}} = \frac{\delta {\cal H}}{\delta
a_{\bm{p}}^{\ast}} \, ,\label{fieldequation}
\end{equation}
%
with Hamiltonian:
\begin{eqnarray}
&& {\cal H} = \int d\bm{p} \, \omega_{\bm{p}} |a_{\bm{p}}|^2
\nonumber \\
&& 
\quad
+ \int d\bm{p}_{012}
\left(
 \delta_{\bm{p}+\bm{p}_1+\bm{p}_2} (U_{\bm{p},\bm{p}_1,\bm{p}_2} a_{\bm{p}}^{\ast} a_{\bm{p}_1}^{\ast} a_{\bm{p}_2}^{\ast} + \mathrm{c.c.})
\right.
\nonumber \\
&&
\quad
\left.
+ \delta_{-\bm{p}+\bm{p}_1+\bm{p}_2} (V_{\bm{p}_1,\bm{p}_2}^{\bm{p}} a_{\bm{p}}^{\ast} a_{\bm{p}_1} a_{\bm{p}_2} + \mathrm{c.c.})
\right)
.
\label{HAM}
\end{eqnarray}
%
This is the standard form of the Hamiltonian of a system dominated by 
three-wave interactions\cite{zak_book}.
Calculations of interaction coefficients are tedious but straightforward task, completed in Ref.~\onlinecite{gm_lvov}.
These coefficients are  given by 
%
\begin{subequations}
 \begin{eqnarray}
&&  \!\!\!\!
V_{\bm{p}_1,\bm{p}_2}^{\bm{p}} = \frac{N}{4 \sqrt{2g}} 
  \frac{1}{k k_1 k_2}
  \left(
  I_{\bm{p},\bm{p}_1,\bm{p}_2} + J_{\bm{p}_1,\bm{p}_2}^{\bm{p}} + K_{\bm{p},\bm{p}_1,\bm{p}_2}
  \right) ,
\nonumber\\
  \label{eq:V}
 \\
&&  \!\!\!\!
U_{\bm{p},\bm{p}_1,\bm{p}_2} = \frac{N}{4 \sqrt{2g}} \frac{1}{3}
  \frac{1}{k k_1 k_2}
  \left(
  I_{\bm{p},\bm{p}_1,\bm{p}_2} + J_{\bm{p}_1,\bm{p}_2}^{-{\bm{p}}} + K_{\bm{p},\bm{p}_1,\bm{p}_2}
  \right),
\nonumber\\
\label{eq:U}
  \\
&& \!\!\!\!
I_{\bm{p},\bm{p}_1,\bm{p}_2} =
 - \sqrt{\frac{\omega_1 \omega_2}{\omega}}
  k^2 \bm{k}_1 \cdot \bm{k}_2
\nonumber\\
&&
\quad
  - \left( (0,1,2) \rightarrow (1,2,0) \right)
  - \left( (0,1,2) \rightarrow (2,0,1) \right)
,
\label{eq:I}
\\
&& \!\!\!\!
J_{\bm{p}_1,\bm{p}_2}^{\bm{p}} = \frac{f^2}{\sqrt{\omega \omega_1 \omega_2}}
  \big(k^2 \bm{k}_1 \cdot \bm{k}_2
\nonumber\\
&&
\quad
  - \left( (0,1,2) \rightarrow (1,2,0) \right)
  - \left( (0,1,2) \rightarrow (2,0,1) \right)
\big)
,
\label{eq:J}
\\
&& \!\!\!\!
K_{\bm{p},\bm{p}_1,\bm{p}_2} = -i f
\bigg(\bigg(
  \sqrt{\frac{\omega}{\omega_1 \omega_2}}
  (k_1^2 - k_2^2)
  \bm{k}_1 \cdot \bm{k}_2^{\perp}
\nonumber\\
&&
\quad
  + \big( (0,1,2) \rightarrow (1,2,0) \big)
  + \big( (0,1,2) \rightarrow (2,0,1) \big)
\bigg)\bigg)
,
\nonumber \\
\label{eq:K}%
 \end{eqnarray}%
\label{MatrixElement}%
\end{subequations}
%
where $((0,1,2) \to (1,2,0))$ and $((0,1,2) \to (2,0,1))$ denote exchanges of suffixes.
%
We stress that the field equation
(\ref{fieldequation}) with the three-wave Hamiltonian
(\ref{InternalWavesDispersion}, \ref{HAM}, \ref{MatrixElement}) is {\em
equivalent\/} to the primitive equations of motion for internal waves
(\ref{PrimitiveEquations}).  Other approaches, in particular the Lagrangian approach, is based on small-amplitude expansion to arrive to this type of equations. 

In wave turbulence theory, one proposes a perturbation
expansion in the amplitude of the nonlinearity, yielding 
linear waves at the leading order. Wave amplitudes are
modulated by the nonlinear interactions,  
and the modulation is statistically described by an kinetic equation\citep{zak_book} for
the wave action $n_{\bm{p}}$ defined by
%
\begin{eqnarray}
n_{\bm{p}} \delta(\bm{p} - \bm{p}^{\prime}) = \langle a_{\bm{p}}^{\ast} a_{\bm{p}^{\prime}}\rangle.
\label{WaveAction}
\end{eqnarray}
%
The derivation of this kinetic equation is well studied and understood. 
For the three-wave Hamiltonian (\ref{HAM}), the kinetic equation
is the one in
Eq.~(\ref{KineticEquation}),
describing general internal waves
interacting in both rotating and non-rotating environments. 

The delta functions in the kinetic equation ensures that spectral transfer happens on the 
{\em resonant manifold}, defined as 
\begin{subequations}
\begin{eqnarray}
%a)~~
\left\{
\begin{array}{l}
 \bm{p} = \bm{p}_1 + \bm{p}_2 \\
 \omega = \omega_1 + \omega_2
\end{array}
\right. ,
\label{RESONANCESa}
\\
\left\{
\begin{array}{l}
 \bm{p}_1 = \bm{p}_2 + \bm{p} \\
 \omega_1 = \omega_2 + \omega
\end{array}
\right. ,
\label{RESONANCESb}
\\
\left\{
\begin{array}{l}
 \bm{p}_2 = \bm{p} + \bm{p}_1 \\
 \omega_2 = \omega + \omega_1
\end{array}
\right. .
\label{RESONANCESc}
\end{eqnarray}
\label{RESONANCES}
\end{subequations}

Now let us assume that the wave action is independent of the direction of the 
horizontal wavenumber, 
%
$$n_{\bm{p}} = n(|\bm{k}|, |m|).$$ Note  that value of the interaction matrix element is independent of horizontal azimuth as it depends only on the magnitude of interacting wavenumbers.  
Therefore one can integrate
the kinetic equation (\ref{KineticEquation}) 
over horizontal azimuth \citep{zak_book}, yielding
%
\begin{eqnarray}
 \frac{\partial n_{\bm{p}}}{\partial t}
= \frac{2}{k}\int
\left(R^{0}_{12} - R^{1}_{20} - R^{2}_{01} \right) \,
d k_1 d k_2 d m_1 d m_2 \, , \nonumber \\
R^{0}_{12}=
   f^{\bm{p}}_{\bm{p}_1\bm{p}_2} \, |V^{\bm{p}}_{\bm{p}_1\bm{p}_2}|^2 \, \delta_{m-m_1-m_2} 
\delta_{\omega_{\bm{p}}-\omega_{\bm{p}_1}-\omega_{\bm{p}_2}}
k k_1 k_2
/ S^0_{1,2}
\, .
\nonumber\\
 \label{KEinternalAveragedAngles}
\end{eqnarray}
%
Here $ S^0_{1,2} $ appears as the result of integration of the horizontal-momentum conservative delta function over all 
possible orientations and is equal to the area of the triangle with sides with the length
of the horizontal wavenumbers $k = |\bm{k}|$, $k_1 = |\bm{k}_1|$ and $k_2  = |\bm{k}_2|$.
%
%
This is the form of the kinetic equation which will be used to find scale-invariant solutions in the next section.


\section{Scale-invariant kinetic equation}
\label{sec:SI}

\subsection{Reduction of Kinetic Equation to the Resonant Manifold\label{Reduction}}

In the high-frequency limit $\omega\gg f $, 
one could conceivably neglect the
effects of the rotation of the Earth.
The dispersion relation (\ref{InternalWavesDispersion}) then becomes
\begin{equation}
\omega_{\bm{p}} \equiv \omega_{\bm{k},m} \simeq
 \frac{g}{\rho_0 N} \frac{|\bm{k}|}{|m|} \, ,
\label{HighFrequencyDispersion}
\end{equation}
%
and, to the leading order, the matrix element
(\ref{MatrixElement}) retains only its first term, $I_{\bm{p},\bm{p}_1,\bm{p}_2}$.

The azimuthally-integrated kinetic equation~(\ref{KEinternalAveragedAngles}) includes integration over $k_1$ and $k_2$ 
since the integrations over $m_1$ and $m_2$ can be done by using delta functions. To use delta functions, we need to perform what is called reduction to the resonant manifold. 
Consider, for example, resonances of type (\ref{RESONANCESa}). Given
$k$, $k_1$, $k_2$
and $m$, one can find $m_1$ and $m_2$ satisfying
the resonant condition by solving simultaneous equations
%
\begin{eqnarray}
m = m_1 + m_2 ,
\qquad
\frac{k}{|m|}=\frac{k_1}{|m_1|} +\frac{k_2}{|m-m_1|} .
\label{ResonantConditionM2}
\end{eqnarray}
%
The solutions of this quadratic equation are given by
\begin{subequations}
 \begin{eqnarray}
%&& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
%\left\{
%\begin{array}{l}
 m_1 &=& \displaystyle\frac{m}{2 k} \left(k + k_1 + k_2 + \sqrt{(k + k_1 + k_2)^2 - 4 k k_1}\right)
 \nonumber\\
 m_2 &=& m - m_1.
,
\label{eq:sol1}
\\
\mathrm{and} \nonumber \\
m_1 &=& \displaystyle\frac{m}{2k} \left(k - k_1 - k_2 - \sqrt{(k - k_1 - k_2)^2 + 4 k k_1}\right)
\nonumber\\
m_2 &=& m - m_1.
.
\label{eq:sol2}
\end{eqnarray}
\label{eq:sol12}
\end{subequations}
Note that Eq.~(\ref{eq:sol1}) translates into Eq.~(\ref{eq:sol2}) if the indices $1$ and $2$ are exchanged.
Similarly, resonances of type (\ref{RESONANCESb}) yield
\begin{subequations}
 \begin{eqnarray}
m_2 &=& - \displaystyle\frac{m}{2 k} \left(k - k_1 - k_2 + \sqrt{(k - k_1 - k_2)^2 + 4 k k_2}\right)
\nonumber\\
m_1 &=& m + m_2.
,
\label{eq:sol3}
\\
\mathrm{and} \nonumber \\
m_2 &=& - \displaystyle\frac{m}{2k} \left(k + k_1 - k_2 + \sqrt{(k + k_1 - k_2)^2 + 4 k k_2}\right)
\nonumber\\
m_1 &=& m + m_2
.
\label{eq:sol4}
 \end{eqnarray}
\label{eq:sol134}
\end{subequations}
%
and resonances of type (\ref{RESONANCESc}) yield
%
\begin{subequations}
 \begin{eqnarray}
m_1 &=& - \displaystyle\frac{m}{2k} \left(k - k_1 - k_2 + \sqrt{(k - k_1 - k_2)^2 + 4 k k_1}\right)
\nonumber\\
m_2 &=& m + m_1.
,
\label{eq:sol5}
\\
\mathrm{and} \nonumber \\
m_1 &=& - \displaystyle\frac{m}{2k} \left(k - k_1 + k_2 + \sqrt{(k - k_1 + k_2)^2 + 4 k k_1}\right)
\nonumber\\
m_2 &=& m + m_1.
.
\label{eq:sol6}
 \end{eqnarray}
\label{eq:sol56}
\end{subequations}

After this reduction, a double integral
over $k_1$ and $k_2$ is left. The domain of integration is further restricted by the
triangle inequalities
%
\begin{equation}
 k<k_1+k_2, \quad k_1<k+k_2, \quad \mathrm{and} \quad k_2<k+k_1.
\label{KinematicBoxK}
\end{equation}
%
These conditions ensure that one can construct a triangle out of the wavenumbers with lengths $k$, $k_1$ and $k_2$ and determine the domain
in the $(k_1,k_2)$ plane called the {\em kinematic box\/} in the oceanographic literature.

Numerical evaluation of the collision integral is a complicated yet
straightforward task.  Interpretation of the results, though, is more
difficult, mostly due to the complexity of the interaction matrix element
and the nontrivial nature of the resonant set.  Starting with
\textcite{mccomas-1977-82}, therefore, predictions were made based on a
further simplification. This simplification is based on the assertion that
it is interactions between wavenumbers with extreme scale separation
that contribute mostly to the 
nonlinear dynamics.
Three main  classes of such resonant triads appear, characterized by extreme
scale separation. These three main classes are 
\begin{itemize}
\item the vertical backscattering of a high-frequency wave
      by a low-frequency wave of twice the vertical wavenumber
      into a second high-frequency wave of oppositely signed vertical wavenumber.
      This type of scattering, as in 
      Eqs.~(\ref{eq:ES1}, \ref{eq:ES2}, \ref{eq:ES1inf}, \ref{eq:ES2inf}) below, 
      is called elastic scattering (ES).
\item The scattering of a high-frequency wave by a low-frequency, small-wavenumber wave into a second, nearly identical, high-frequency large-wavenumber wave.
      This type of scattering, as in 
      Eqs.~(\ref{eq:ID1}, \ref{eq:ID2}, \ref{eq:ID1inf}, \ref{eq:ID2inf}) below,
      is called induced diffusion (ID).
\item The decay of a small-wavenumber wave into two large vertical-wavenumber waves of approximately one-half the frequency.
      This type of scattering, as in 
      Eqs.~(\ref{eq:PSI1}, \ref{eq:PSI2}, \ref{eq:PSI1inf}, \ref{eq:PSI2inf}) below,
      is called parametric subharmonic instability (PSI).
\end{itemize}


To see how this classification appears analytically, we
perform the limit of $k_1 \to 0$ and the limit $k_1\to\infty$ in
Eqs~(\ref{eq:sol12}--\ref{eq:sol56}). We will refer to the $k_1$ {\em or\/} $k_2 \to 0$ limits as
IR limits, while the $k_1$ {\em and\/} $k_2 \to \infty$ limit
will be referred as an UV limit.  Since the integrals in the kinetic
equation for power-law solutions will be dominated by the scale-separated
interaction, this will help us analyze possible solutions to the kinetic equation.  


The results of the $k_1\to  0 $ limit of 
Eqs.~(\ref{eq:sol12}--\ref{eq:sol56}) are given by
%
\begin{subequations}
 \begin{eqnarray}
&
\left\{
\begin{array}{l}
m_1 \to 2m, m_2 \to -m
\\
\omega_1 \ll \omega, \omega_2 \sim \omega
\end{array}
\right.
,
\label{eq:ES1}
\\
&
\left\{
\begin{array}{l}
-m_1 \ll m, m_2 \sim m
\\
\omega_1 \ll \omega, \omega_2 \sim \omega
\end{array}
\right.
,
\label{eq:ID1}
 \end{eqnarray}
\end{subequations}
\begin{subequations}
 \begin{eqnarray}
&
\left\{
\begin{array}{l}
m_1 \ll m, m_2 \sim -m
\\
\omega_1 \sim 2\omega, \omega_2 \sim \omega
\end{array}
\right.
,
\label{eq:PSI1}
\\
&
\left\{
\begin{array}{l}
-m_1 \ll m, m_2 \sim -m
\\
\omega_1 \sim 2\omega, \omega_2 \sim \omega
\end{array}
\right.
,
\label{eq:PSI2}
\end{eqnarray}
\end{subequations}
\begin{subequations}
 \begin{eqnarray}
&
\left\{
\begin{array}{l}
-m_1 \ll m, m_2 \sim m
\\
\omega_1 \ll \omega, \omega_2 \sim \omega
\end{array}
\right.
,
\label{eq:ID2}
\\
&
\left\{
\begin{array}{l}
m_1 \to -2m, m_2 \to -m
\\
\omega_1 \ll \omega, \omega_2 \sim \omega
\end{array}
.
\right.
\label{eq:ES2}
\end{eqnarray}
\end{subequations}
%
We now see that 
the interactions (\ref{eq:ES1}, \ref{eq:ES2}) correspond
to the elastic scattering (ES) mechanism,  
the interactions (\ref{eq:ID1}, \ref{eq:ID2},
correspond to the induced diffusion (ID).  The interactions (\ref{eq:PSI1}, \ref{eq:PSI2}),
correspond to the  parametric subharmonic instability (PSI).


Similarly, taking the $k_1$ and $k_2 \to \infty$ limits, of 
Eqs.~(\ref{eq:sol12}--\ref{eq:sol56}) we obtain
%
\begin{subequations}
 \begin{eqnarray}
&
\left\{
\begin{array}{l}
m_1 \gg m, -m_2 \gg m
\\
\omega_1, \omega_2 \sim \omega/2
\end{array}
\right.
,
\label{eq:PSI1inf}
\\
&
\left\{
\begin{array}{l}
-m_1 \gg m, m_2 \gg m
\\
\omega_1, \omega_2 \sim \omega/2
\end{array}
\right.
,
\label{eq:PSI2inf}
 \end{eqnarray}
\end{subequations}
\begin{subequations}
 \begin{eqnarray}
&
\left\{
\begin{array}{l}
m_1 \sim m/2, m_2 \sim -m/2
\\
\omega_1,\omega_2 \gg \omega
\end{array}
\right.
,
\label{eq:ES1inf}
\\
&
\left\{
\begin{array}{l}
-m_1 \gg m, -m_2 \gg m
\\
\omega_1,\omega_2 \gg \omega
\end{array}
\right.
,
\label{eq:ID1inf}
\end{eqnarray}
\end{subequations}
\begin{subequations}
 \begin{eqnarray}
&
\left\{
\begin{array}{l}
m_1 \sim -m/2, m_2 \sim m/2
\\
\omega_1, \omega_2 \gg \omega
\end{array}
\right.
,
\label{eq:ES2inf}
\\
&
\left\{
\begin{array}{l}
-m_1 \gg m, -m_2 \gg m
\\
\omega_1, \omega_2 \gg \omega
\end{array}
.
\right.
\label{eq:ID2inf}
\end{eqnarray}
\end{subequations}
We now can identify the interactions~(\ref{eq:PSI1inf}, \ref{eq:PSI2inf}) to be PSI, 
the interactions~(\ref{eq:ES1inf}, \ref{eq:ES2inf}) to be ES,
and finally the interactions~(\ref{eq:ID1inf}, \ref{eq:ID2inf}) as being ID.

This classification provides an easy and intuitive tool for describing extremely scale-separated interactions. We will see below that one of these interactions, namely ID, explain reasonably well the experimental data that is available to us. 


\subsection{Convergences and divergences  of the kinetic equation. }
\label{ssec:tw}

Neglecting the effects of the rotation of the Earth
yields a scale-invariant system
with dispersion relation given by Eq.~(\ref{HighFrequencyDispersion})
and matrix element given only by the $I_{\bm{p},\bm{p}_1,\bm{p}_2}$ in Eq.~(\ref{MatrixElement}).
%
This is the kinetic equation of Refs.~\onlinecite{gm_lvov,lvov-2004-92}, describing
internal waves in hydrostatic balance in a non-rotating environment.

Proposing a self-similar separable spectrum of the form
(\ref{PowerLawSpectrum}), it is clear from
the bi-homogeneous nature of the azimuthally-integrated kinetic equation (\ref{KEinternalAveragedAngles}) that
%
\begin{equation}
\frac{\partial n(\alpha \bm{k}, \beta m)}{\partial t} = \alpha^{4 - 2 a} \beta^{1 - 2b} \frac{\partial n(\bm{k} ,m)}{\partial t}.
\label{Rescaling}
\end{equation}
%
In order to 
find a steady scale-invariant solution for all values of $k$
and $m$, it is therefore sufficient to find exponents that give zero collision integral for one wavenumber.
One can fix $k$ and $m$, adopting for instance $k=m=1$, and seek zeros of the collision integral as a 
function of $a$ and $b$:
%
\begin{equation}
 \frac{\partial n(k=1,m=1)}{\partial t}\equiv  {\cal C}(a,b) .
\label{ScaleInvariantKineticEquation}
\end{equation}
%

\begin{figure}
 \begin{center}
  \includegraphics[scale=0.8]{2.eps}
 \end{center}
 \caption{Divergence/convergence due to IR wavenumbers (top)
and due to UV wavenumbers (bottom).
The integral converge for the exponents in the shaded regions
or on the segments $(b=0, 7/2 < a < 4$) and $(b=1, 3 < a < 7/2)$.  Dashed lines distinguish the 
domains where the indicated named triads dominate the singularity.}
\label{DivergenceConvergence}
\end{figure}

Before embarking on numerical or analytical integration of the kinetic equation 
(\ref{KEinternalAveragedAngles}) with scale-invariant solutions~(\ref{PowerLawSpectrum}), it is 
necessary to check whether or not the collision integral converges.
Appendix~\ref{CalculationsOfDivergences}
outlines these calculations. The condition for the scale-invariant collision integral
(\ref{ScaleInvariantKineticEquation}) 
to converge at the IR end, $k_1 \;\mathrm{or}\; k_2 \to 0$ is given by
%
\begin{subequations}
 \begin{eqnarray}
 &
a+b/2-7/2<0 \;\; \mathrm{and} \;\; -3 < b < 3, \label{IR1}
\\
&
a-4<0 \;\; \mathrm{and} \;\; b=0,              \label{IR2}
\\
&
a-7/2<0 \;\; \mathrm{and} \;\; b=1 ,           \label{IR3}
\\
&
a+b-5<0 \;\; \mathrm{and} \;\; b>3 ,            \label{IR4}
\\
\mathrm{or} \nonumber\\
&
a-5<0 \;\; \mathrm{and} \;\; b<-3 .                \label{IR5}
\end{eqnarray}
\label{InfraRed}
\end{subequations}
%


Similarly, UV convergence as $k_1 \;\mathrm{and}\; k_2 \to \infty$ implies that
%
\begin{subequations}
 \begin{eqnarray}
 &
a+b/2-4>0 \;\; \mathrm{and} \;\; -2<b<2, \label{UV1}
\\
&
a-7/2>0 \;\; \mathrm{and} \;\; b=0 ,      \label{UV2}
\\
&
a-3>0 \;\; \mathrm{and} \;\; b>2 ,         \label{UV3}
\\
\mathrm{or} \nonumber\\
&
a+b-3>0 \;\; \mathrm{and} \;\; b<-2 .      \label{UV4}
\end{eqnarray}
\label{UltraViolet}
\end{subequations}
%
The domains of divergence and convergence
are shown in Fig.~\ref{DivergenceConvergence}.

Figure~\ref{DivergenceConvergence} 
also displays the classes of 
triads dominating the interactions.
%
Knowing the classes of interactions that lead to the divergences of the kinetic
equation allows us to find possible physical scenarios of the 
convergent solutions or to find a possible physical regularization of the 
divergences.

Note that in addition to the two-dimensional
domain of IR convergence [the regions~(\ref{IR1}, \ref{IR4}, \ref{IR5})] there are
two additional IR convergent line segments given by Eqs.~(\ref{IR2}) and
(\ref{IR3}}). These two special line segments appear because 
of the $b(b-1)$ prefactor
to the divergent contributions to the collision integral~(\ref{ZeroContribution}).  Similarly, for the UV limit, in addition to 
the two-dimensional region of convergence~(\ref{UV1}, \ref{UV3}, \ref{UV4})
there is an additional special line segment of $b=0$~(\ref{UV2}).

We see that these domains of convergence overlap only on the segment
%
\begin{equation}
 7/2 < a < 4 \;\; \mathrm{and} \;\; b = 0 \, . \label{ConvergentSolution}
\end{equation}
%
Note that  $b=0$ corresponds to wave action independent of vertical wavenumbers, 
$\partial n / \partial m = 0$.
Existence of the $b=0$ line will allow us to find novel convergent solution 
in Sec.~\ref{NovelSolution}. 
We also note that the IR segment on $b=0$ coincides with one of the ID 
solution determined in Sec.~\ref{CorRegularization}.  
The other segment on $b=1$ do not coincides with the ID solution in Sec.~\ref{CorRegularization} since the scale-invariant system has higher symmetry
than the system with Coriolis effect.


\section{A novel convergent solution \label{NovelSolution}}
To find out whether there is a steady solution
of the kinetic equation along the convergent segment~(\ref{ConvergentSolution}), we
substitute the power-law ansatz~(\ref{PowerLawSpectrum}) with $b=0$
into the azimuthally-integrated kinetic equation~(\ref{KEinternalAveragedAngles}).
We then compute numerically the
collision integral as a function of $a$ for $b=0$. To this end, 
we fix $k=m=1$ and perform a numerical integration over
the kinematic box (\ref{KinematicBoxK}), reducing the integral to the resonant
manifold as described in Sec.~\ref{Reduction}.

The result of this numerical integration is shown in 
Fig.~\ref{ThreeAndSeven}. The figure clearly  shows the  
existence of a steady solution
of the kinetic equation~(\ref{KEinternalAveragedAngles})
near $a \cong 3.7$ and $b=0$.

%
\begin{figure}
\begin{center}
 \includegraphics[scale=0.85]{3.eps}
\end{center}
\caption{Value of the collision integral as a function of $a$
on the convergent segment, $b=0$.}
\label{ThreeAndSeven}
\end{figure}
%
This is, therefore, {\em the only convergent steady solution\/} to the scale-invariant kinetic
equation for the internal wave field. It is highly suggestive that
it should exist so close to the GM spectrum, $a=4$ and $b=0$ for large wavenumbers, the most
agreed-upon fit to the spectra observed throughout the ocean. It remains to be 
seen whether how this solution is modified by inclusion of background rotation.

\section{Balance between divergences\label{Balance}}

The fact that the collision integral ${\cal C}$
diverges for almost
all values of $a$ and $b$
can be viewed both as a challenge and as a blessing. On the one hand,
it makes the prediction of steady spectral slopes more difficult,
since it now depends on the details of the truncation of the domain of
the integration.
Fortunately,
it provides a powerful tool for quantifying
the effects of fundamental players in ocean dynamics, most of which
live on the fringes of the inertial subrange of the internal wave field:
the Coriolis effect, as well as tides and storms at the IR end
of the spectrum, and wave breaking and dissipation at the UV
end. The sensitive response of the inertial subrange to the detailed
modeling of these scale-separated mechanisms permits, in principle, building
simple models in which these are the only players, bypassing the need to
consider the long range of wave scales in between.

\begin{figure}
 \begin{center}
  \includegraphics[scale=0.8]{4.eps}
 \end{center}
 \caption{Signs of the divergences
 where the both IR and UV contributions diverge.
 The left symbols show the signs due to IR wavenumbers and
 the right symbols show the signs due to UV wavenumbers.}
\label{SignContributions}
\end{figure}

At the IR end,
the resonant interactions are dominated by the ID singularity for $-3<b<3$, 
Fig.~\ref{DivergenceConvergence}.  
The sign of the divergences is given by $-b(b-1)$
(Eq.~(\ref{ZeroContribution})).
Similarly,
at the UV end resonant interactions are dominated by 
the ID singularity for $-2<b<2$ (Fig.~\ref{DivergenceConvergence}).
The sign of the divergence is given by $b$ (Eq.~(\ref{InfinityContribution})).
At the UV end for $b>2$,
where ES determines the divergences,
the sign of the singularity is given by $-b$, i.e. the sign is negative
(Eq.~(\ref{InfinityContributionES})).
Figure~\ref{SignContributions} shows 
the signs of the divergences
where both the IR and UV contributions diverge:
the left sign corresponds to the IR contribution,
and the right sign to the UV contribution.

Hence,
in the regions,
\begin{subequations}
 \begin{eqnarray}
&
 7 < 2a+b < 8, \;\;\mathrm{and}\;\; -2 < b < 1
,
\\
\mathrm{or} \nonumber\\
&
7 < 2a+b, a<3, \;\;\mathrm{and}\;\; b>2
,
\end{eqnarray}
\end{subequations}
the divergences of the collision integral at the
IR and UV ends have opposite signs. 
Then formal solutions can be constructed by 
having these two divergences cancel each other out. 

This observation justifies the existence of the PR solution (\ref{LT}). Indeed, the  PR spectrum
has divergent power-law exponents at the both ends.
One can prove that
the PR spectrum is an exact steady solution of the kinetic equation~(\ref{KineticEquation})
by applying the Zakharov--Kuznetsov conformal
mapping for systems with cylindrical symmetry~\citep{zak1968,zakharov1967iwn,kuznetsov1972tis}.
This Zakharov--Kolmogorov conformal mapping
effectively establishes a map between the
neighborhoods of zero and infinity.  Making these two contributions
cancel point-wisely yields the solution~(\ref{LT}).  For this transformation to be mathematically 
applicable the integrals have to converge.  This transformation leads only to a formal solution for divergent integrals.  

The PR spectrum was first found by \textcite{pelinovsky1977wti}.
However,
they realized that it was only a formal solution.  
The solution was found again in Ref.~\onlinecite{caillol2000kea} through a
renormalization argument, and in Ref.~\onlinecite{gm_lvov}
within an isopycnal formulation of the wave field.

The idea of a formal solution, such as PR, can be generalized 
quite widely: in fact,
{\em any\/} point in the regions with opposite-signed divergences
can be made into a
steady solution
under a suitable mapping that makes the divergences at zero and infinity cancel each
other, as does the Kuznetsov--Zakharov transformation. 
Such {\em generalized\/} Kuznetsov--Zakharov transformation 
is an extension of the idea of principle value for a divergent integral,
whereby two divergent contributions are made to cancel each other through a
specific relation between their respective contributions.  

This scenario also suggests an possible explanation for the variability of the 
power-law exponents of the quasi-steady spectra:
inertial subrange spectral variability is to be expected
when it is driven by the nonlocal interactions. The natural local
variability of players outside the inertial range, such as tides
and storms, translates into a certain degree of non-universality.

\section{Regularization by the Coriolis effect}
\label{CorRegularization}

Physically, ocean does not perform  
generalized Kuznetsov--Zakharov transformation.  
However, in the ocean there are finite boundaries in frequency 
domain. In particular, 
the inertial frequency, $f$, provides a truncation
for the IR part of the spectrum, while the UV truncation is provided by the buoyancy frequency, $N$.  
These two frequencies vary from place to place, giving grounds for
spectral variability.
Consequently, the
integrals are not truly divergent, rather they have a large
numerical value dictated by the location of the IR cutoff. 


All the experimental points are located in regions of the $(a,b)$ domain $a > (-b + 7)/2$ 
for which the integral diverges in the IR region.  
For most of the experimental points, the UV divergence is not an issue, as $a > -b/2 + 4$.   
The UV region is therefore assumed to be 
either sub-dominant or convergent in this section, where 
we study the regularization resulting from a finite value of $f$. 

Since the IR cutoff is given by $f$, a frequency,
it is easier to analyze the resulting integral in $(\omega_1,m_1)$
rather than in the traditional $(k_1,m_1)$ domain.  Thus we need to 
express both the kinetic equation and the kinematic box in 
terms of frequency and vertical wavenumber.
For this, we use the dispersion relation (\ref{InternalWavesDispersion}) 
to express $k$ in terms of $\omega$ in the description of the kinematic box
(\ref{KinematicBoxK}): 
%
\begin{subequations}
 \begin{eqnarray}
&& \omega_1<E_3(m_1), \; \omega_1>E_4(m_1), \; \omega_1>E_1(m_1)
\nonumber\\
&& \quad \mathrm{if\ } m_1<0, \; \omega_1>\omega,
 \label{Region1}\\
&& \omega_1>E_3(m_1), \; \omega_1<E_4(m_1), \; \omega_1<E_1(m_1)
\nonumber\\
&& \quad \mathrm{if\ } m_1<0, \; \omega_1<\omega,
 \label{Region2}\\
&& \omega_1<E_2(m_1), \; \omega_1>E_2(m_1), \; \omega_1>E_4(m_1)
\nonumber\\
&& \quad \mathrm{if\ } m_1>0, \; \omega_1<\omega,
 \label{Region3}\\
&& \omega_1>E_3(m_1), \; \omega_1<E_1(m_1), \; \omega_1<E_2(m_1)
\nonumber\\
&& \quad \mathrm{if\ } m_1>0, \; \omega_1>\omega,
 \label{Region4}%
\end{eqnarray}%
 \label{Region}%
\end{subequations}
%
where we have introduced the four curves in the $(\omega_1,m_1)$ domain that
parameterize the kinematic box:
\begin{eqnarray}
E_1(\omega_1) &=& m \frac{
 -\sqrt{-f^2+\omega^2} +    \sqrt{-f^2+ (\omega-\omega_1)^2}}
{ \sqrt{-f^2+\omega_1^2} +  \sqrt{-f^2+(\omega-\omega_1)^2}} , \nonumber\\
E_2(\omega_1) &=& m \frac{
  \sqrt{-f^2+\omega^2} +    \sqrt{-f^2+ (\omega-\omega_1)^2}}
{ \sqrt{-f^2+\omega_1^2} +  \sqrt{-f^2+(\omega-\omega_1)^2}}, \nonumber\\
E_3(\omega_1) &=& m \frac{
 -\sqrt{-f^2+\omega^2} +    \sqrt{-f^2+ (\omega-\omega_1)^2}}
{-\sqrt{-f^2+\omega_1^2} +  \sqrt{-f^2+(\omega-\omega_1)^2}} , \nonumber\\
E_4(\omega_1) &=& m \frac{
  \sqrt{-f^2+\omega^2} +    \sqrt{-f^2+ (\omega-\omega_1)^2}}
{ - \sqrt{-f^2+\omega_1^2} +  \sqrt{-f^2+(\omega-\omega_1)^2}} . \nonumber
 \label{KinematicBoxOM}
\end{eqnarray}
%

\begin{figure}
\begin{center}
\includegraphics[scale=0.8]{5.eps}
\end{center}
\caption{The kinematic box in the $(\omega_1,m_1)$ domain. 
Two disconnected regions where $\omega_1 < \omega$ depict regions with
``sum'' interactions, namely
%
$\omega=\omega_1+\omega_2,$
$m=m_1+m_2,$
%
type of the resonances.
%
The connected regions where $\omega_1 > \omega$ depict ``difference'' resonances,
$\omega_2=\omega_1-\omega,$,
$m_2=m_1-m$.
The parameters are chosen so that  $f/\omega=0.1$.}
\label{pic:OmegaMKinematicBox}
\end{figure}

The {\em kinematic box} in the $(\omega,m)$ domain is shown in Fig.~\ref{pic:OmegaMKinematicBox}. 
To help in the transition from the traditional kinematic box to the kinematic box in $(\omega,m)$ domain, 
the following limits were identified:
%
\begin{itemize}
\item  ID1  is the ID  limit of Eq.~(\ref{eq:ID1}) with indices 1 and 2 being flipped
\item  ID2  is the ID  limit of Eq.~(\ref{eq:ID2}) with indices 1 and 2 being flipped
\item  ID3  is the ID  limit of Eq.~(\ref{eq:ID1}) 
\item  ID4  is the ID  limit of Eqs.~(\ref{eq:ID1inf}, \ref{eq:ID2inf})
\item  PSI1 is the PSI limit of Eq.~(\ref{eq:PSI1inf})
\item  PSI2 is the PSI limit of Eq.~(\ref{eq:PSI2inf})
\item  PSI3 is the PSI limit of Eq.~(\ref{eq:PSI1})
\item  PSI4 is the PSI limit of Eq.~(\ref{eq:PSI2})
\item  ES1  is the ES  limit of Eq.~(\ref{eq:ES1})
\item  ES2  is the ES  limit of Eq.~(\ref{eq:ES2}) with indices 1 and 2 being flipped
\item  ES3  is the ES  limit of Eq.~(\ref{eq:ES2}) with indices 1 and 2 being flipped
\item  ES4  is the ES  limit of Eq.~(\ref{eq:ES1inf})
\end{itemize}
%
An advantage of the $(\omega,m)$ presentation for the kinematic box is that 
it allows a transparent  reduction to the resonant manifold. A disadvantage is the 
curvilinear boundaries of the box, requiring more sophisticated analytical treatment. 
To proceed, we assume a power-law spectrum, similar to Eq.~(\ref{PowerLawSpectrum}), but in the
$(\omega, m$ space:
%
\begin{eqnarray}
n_{\omega,m} \propto \omega^{\widetilde{a}} m^{\widetilde{b}}.
\label{PowerLawSpectrumFrequencyVertical}
\end{eqnarray}
%
We need to transform the wave action as a function of $k$ and $m$
to a function of $\omega$ and $m$.
This is done in Appendix~\ref{OmegaMKMtransform}. The relation between $a$, $b$ and
$\widetilde{a}$, $\widetilde{b}$ reads:
$$\widetilde{a}=-a, \quad \widetilde{b}=-a-b.$$

Equation~(\ref{KEinternalAveragedAngles}) transforms into
\begin{widetext}
\begin{eqnarray}
\frac{\partial}{\partial t} n(k(\omega,m),m) &=&\frac{1}{k}\int d\omega_1 d m_1 J
              \frac{|V^0_{12}|^2}{S^0_{1,2}}
\left(n_1 n_2 - n(n_1+n_2)\right) 
\vline_{\small { \omega_2=\omega-\omega_1, m_2 = m - m_1}}
\nonumber\\
&-&\frac{2}{k}\int d\omega_1 d m_1 J\frac{|V^1_{02}|^2}{S^1_{2,0}}
\left(n n_2 - n_1 ( n + n_2)\right)
\vline_{\omega_2=\omega_1-\omega, m_2 = m_1 - m}
\label{KE3}
\end{eqnarray}
\end{widetext}
%
We have used the dispersion relation 
%
$
k_i=m_i\sqrt{\omega_i^2 - f^2},
$ 
%
and defined $J$ as the Jacobian of the transformation from $(k_1,k_2)$ into 
$(\omega_1,\omega_2)$, times the $k k_1 k_2 $ factor,
%
$$J= k k_1 k_2 \frac{d k_1}{d \omega_1} \frac{d k_2}{d \omega_2}.$$
%

In Fig.~\ref{pic:OmegaMKinematicBox}, there are three ID corners with significant
contribution to the collision integral:
\begin{enumerate}
%
\item ID1 region. In this region, $m_1$ is slightly bigger than $m$, $\omega_1$ slightly smaller than
$\omega$, and $\omega_2$ and $m_2$ are both very small. 
This region can be obtained from
the region (\ref{Region2}) above by interchanging indices $1$ and $2$. 
%
In this region, 
$$n_2\gg n, \, n_1.$$  
\item ID2 region. In this region, $\omega_1$ is slightly bigger than $\omega+f$ where
$\omega_2=\omega_1-\omega$ is small, and $m_2=m_1-m$ is negative and small. This is 
the region (\ref{Region4}). Also
%
$$n_2\gg n, \, n_1.$$  
%
\item ID3 region. Small $\omega_1$, small negative $m_1$. This corresponds
to the  region (\ref{Region2}), where 
$$n_1\gg n, \, n_2.$$
Note that this region can be obtained from the region ID1 by flipping indices 
$1$ and $2$. Consequently, only one of the ID1, ID2 should be taken into account,
with a factor of two multiplying the respective contribution. 
\end{enumerate} 

Making these simplification, and taking into account the areas of
integration in the kinematic box, we obtain
%
\begin{widetext}
\begin{eqnarray}
\frac{\partial}{\partial t} n(k(\omega,m),m) = \frac{2}{k}
  \int _f^{f+\omega_{\mathrm{s}}} d \omega_1 \int_{E_3(\omega_1)}^{E_1(\omega_1)}
d m_1
J\frac{|V^0_{1,2}|^2}{S^0_{1,2}} n_1 (n_2 - n)
- {\frac{2}{k}} \int _{\omega-f-\omega_{\mathrm{s}}}^{\omega-f} d \omega_1
\int_{E_3(\omega_1)}^{E_1(\omega_1)} d m_1
J\frac{|V^2_{1,0}|^2}{S^2_{1,0}} n_2 (n - n_1)
,\nonumber \\
\label{KE4}
\end{eqnarray}
\end{widetext}
%
where the small parameter $\omega_{\mathrm{s}}$ is introduced to restrict the integration
to a neighborhood of the ID corners.  The arbitrariness of the small parameter will not affect 
the end result below.

To quantify the contribution of near-inertial waves to a $(\omega,m)$ mode, we write
%
$$\epsilon \sim f \ll \omega=1. $$
%
Subsequently, near the region ID3 of the kinematic box, we write
%%
$$\omega_1 = f + \epsilon,$$
% 
while near the ID2 corners of the kinematic box we write
%
$$\omega_1 = \omega+f+\epsilon.$$
%
%
We then expand the resulting analytical  expression (\ref{KE4})  in powers of
$\epsilon$ and $f$ without making any assumptions on their relative size.
%
These calculations, including the integration over vertical wavenumbers  $m_1$,
are presented in Appendix~\ref{Appendixf}.  
The resulting expression for the kinetic equation is   given by
%
\begin{eqnarray}
&& \!\!\!\!\!
\frac{\partial}{\partial t} n(k(\omega,m),m) = \frac{\pi}{4k}
\left( \widetilde{a} - \widetilde{b} \right) 
 \left( \widetilde{a} - 3 \left( 3 + \widetilde{b} \right)  \right)
\nonumber\\
&& \!
\times
m^{5 + 2 \widetilde{b}} \omega^{-3 + \widetilde{a} - \widetilde{b}}
  \int_0^{\mu} d \epsilon 
{\left( \epsilon + f \right) }^{4 + \widetilde{a} + \widetilde{b}} 
(\epsilon^2+ 2 \epsilon f + 17 f^2) ~.
\nonumber \\
\label{KE6}
\end{eqnarray}
%
The integral over $\epsilon$ diverges at $\epsilon=0$, 
if $f=0$ and if $6+\widetilde{a} +\widetilde{b} > -1$.\footnote{Naturally this condition coincides with (\ref{IR1}).}

However, if we postpone taking $f=0$ limit, we see that 
the integral is zero to leading order if 
%
\begin{eqnarray}
 \widetilde{a} - 3\,\left( 3 + \widetilde{b} \right) =0
\quad \mathrm{or} \quad
 \widetilde{a} - \widetilde{b} = 0
\end{eqnarray}
%
or, in terms of
$a$ and $b$, 
\begin{eqnarray}
9-2a-3b=0
\quad \mathrm{or} \quad
b=0.
\label{NewIDcurve}
\end{eqnarray}


This is the family of power-law steady-state solutions to the kinetic
equations dominated by infra-red ID interactions. These steady states
are identical to the ID stationary states identified by \textcite{mccomas-1977-82},
 who derived a diffusive approximation to
their collision integral in the infra-red ID limit.
%
Note that \textcite{mccomas-1981} interpreted $b=0$ as a no action flux in
vertical wavenumber domain, while $9-2a-3b=0$ is a constant action
flux solution. 
%
What is presented in this section is a rigorous 
asymptotic derivation of this result.
These ID solutions helps us to
interpret observational data of Fig.~\ref{fig:onlydata} that is currently available to us.

\section{Conclusions}
\label{sec:conclusion}

The results in this paper provide an interpretation of the
variability in the observed spectral power laws. Combining Figs.~\ref{fig:onlydata}, \ref{DivergenceConvergence} and
\ref{SignContributions} with Eqs.~(\ref{GM}), (\ref{LT}) and
(\ref{NewIDcurve}), produces the results shown in Fig \ref{fig:everything}.  

Notice that most of the observational points are UV convergent and IR
divergent and 
that regularization
of the collision integral by $f$, as in Sec.~\ref{CorRegularization},
produces a family of stationary states that collapses much of the
variability.  Four points land in the
region where the spectra are both IR and UV divergent, with all but
one point close to the ID lines.  None of
the observations lies in a region where both IR and UV divergences have
the same sign.  Furthermore, the novel convergent (non-rotating) solution is in
close proximity to the experimental points determined from the
two-dimensional spectra.

\begin{figure}
 \begin{center}
\includegraphics[scale=0.45]{6.eps}
 \end{center}
\caption{The observational points and the theories.  The filled circles represent the Pelinovsky--Raevsky (PR) spectrum, the convergent numerical solution determined in Section 4 and the GM spectrum.  Circles with stars represent power-law estimates based upon one-dimensional spectra.  Circles with cross hairs represent estimates based upon two-dimensional data sets.  See Fig. \ref{fig:onlydata} for the identification of the field programs.  Light grey shading represents regions of the power-law domain for which the collision integral converges in either the IR or UV limit.  The dark grey shading represents the region of the power-law domain for which neither the IR or UV limits converge.  The region of black shading represents the sub-domain for which the IR and UV divergences have the same sign.  Overlain as solid white lines are the induced diffusion stationary states.  }
\label{fig:everything}
\end{figure}


Summarizing the paper,
we have analyzed the scale-invariant kinetic equation for internal gravity waves,
and shown that its collision integral diverges
for almost all spectral exponents.
Figure~\ref{fig:everything} shows that
the integral nearly always diverges, either
at zero or at infinity.
This means that, in  the wave turbulence kinetic equation framework, 
the energy transfer is dominated by the scale-separated interactions 
with either large or small scales.

The only exception where the integral converges
is a segment of a line, $7/2 < a < 4$, with $b=0$.
On this convergent segment, we found a special solution, $(a,b) = (3.7, 0)$.
This new solution is not 
far from the large-wavenumber asymptotic form of the Garrett--Munk spectrum, $(a,b)=(4,0)$.

We have argued that there exist two regions of power-law exponents
which can yield quasi-steady solutions of the kinetic equation.
For these ranges of exponents,
the contribution of the scale-separated interactions
due to the IR and UV wavenumbers 
can be made to balance each other.
The Pelinovsky--Raevsky spectrum is a special case of this scenario.
None of the observational power-law exponents lie within the region of same-signed 
UV and IR divergences.  

This scenario, in which the energy spectrum in the inertial subrange is determined 
by the nonlocal interactions, provides an explanation
for the variability of the power-law exponents of the observed spectra:
they are a reflection of the variability of dominant players outside
of the inertial range, such as
the Coriolis effect, tides and storms.
can not be universal.

We also rederive ID steady-solution lines,
which are also shown as white lines in Fig.~\ref{fig:everything}.

All of theory, experimental data, and the results of numerical simulation in
Ref.~\onlinecite{numericsLY} hint at the
importance of the IR contribution to the collision integral.  The nonlocal
interactions with large scales will therefore play a dominant role in
forming the internal-wave spectrum.  To the degree that the large
scales are location dependent and not universal, the high-frequency,
high vertical-wavenumber internal-wave spectrum ought to be affected
by this variability. Consequently, the internal-wave spectrum should be strongly dependent on the regional
characteristics of the ocean.

\begin{appendix}

\begin{acknowledgments}
This research is supported by NSF CMG grants
0417724, 0417732 and 0417466.

We are grateful to YITP in Kyoto University for allowing us to use their facility.
\end{acknowledgments}

\section{Asymptotics of collision integral in infra-red and ultra-violet limits}
\label{CalculationsOfDivergences}
 
Let us integrate Eq.~(\ref{KEinternalAveragedAngles}) over $m_1$ and $m_2$.
\begin{eqnarray}
 \frac{\partial n_{\bm{p}}}{\partial t} &=&
\frac{1}{k}
  \int \left(T^0_{1,2} - T^1_{2,0} - T^2_{0,1}\right) dk_1 dk_2
,
\nonumber\\
 T^0_{1,2} &=& k k_1 k_2 |V^{\bm{p}}_{\bm{p}_1\bm{p}_2}|^2 f^{\bm{p}}_{\bm{p}_1\bm{p}_2}
  / (|g^{0 \prime}_{1,2}| S^0_{1,2})
,
\nonumber \\
 g^{0 \prime}_{1,2}(k_1, k_2) &=& \left.\frac{d g^0_{1,2}(m_1)}{d m_1} \right|_{m_1 = m_1^{\ast}(k_1, k_2)},
\nonumber\\
 g^0_{1,2}(m_1) &=& \frac{k}{m} - \frac{k_1}{|m_1|} - \frac{k_2}{|m - m_1|} \, ,
\label{eq:keap}
\end{eqnarray}
where $g^{0 \prime}_{1,2}$ appears owing to $\delta_{\omega_{\bm{p}}
-\omega_{{\bm{p}_1}}-\omega_{{\bm{p}_2}}}$ and $m_1^{\ast}(k_1, k_2)$
is given by the resonant conditions (\ref{eq:sol12})--(\ref{eq:sol56}).

\subsection{Infra-red asymptotics}
We consider the asymptotics of the integral in Eq.~(\ref{eq:keap}) as $k_1 \to 0$.
We employ the independent variables $x$ and $y$,
where $k_1 = k x$, $k_2 = k (1 + y)$, $x,y = O(\epsilon)$,
$x>0$ and $-x<y<x$.
In this limit of $\epsilon \to 0$, $n_1 \gg n, n_2$.
In this limit,
Eqs.~(\ref{eq:sol1}) and (\ref{eq:sol6}),
Eqs.~(\ref{eq:sol2}) and (\ref{eq:sol5}),
and
Eqs.~(\ref{eq:sol3}) and (\ref{eq:sol4})
correspond to
ES, Eqs.~(\ref{eq:ES1}, \ref{eq:ES2}) 
ID, Eqs.~(\ref{eq:ID1}, \ref{eq:ID2}),
PSI, Eqs.~(\ref{eq:ES1}, \ref{eq:ES2}),
respectively.
Without loss of generality $m$ is set to be positive.

\begin{table*}
  \caption{Asymptotics as $k_1 \to 0$.
 ES (\ref{eq:sol1}, \ref{eq:sol6}) gives $\epsilon^{-a+5}$ owing to the symmetry of $y$.
 ID (\ref{eq:sol2}, \ref{eq:sol5}) gives $\epsilon^{-a-(b-7)/2}$ ($\epsilon^{-a+4}$) owing to the second cancellation.
PSI (\ref{eq:sol3}, \ref{eq:sol4}) gives $\epsilon^{-a-b+5}$.
The asymptotics for $b=0$ appear in parentheses.
}
 \label{table:zero}
 \begin{center}
\begin{ruledtabular}
\begin{longtable}{ccccccccc}
 & $m_1$ & $m_2$ & $\omega_1$ & $\omega_2$ & $V^{\bm{p}_i}_{\bm{p}_j\bm{p}_k}$ & $f^{\bm{p}_i}_{\bm{p}_j\bm{p}_k}$ & $g^{i \prime}_{j, k}$ & $T^i_{j,k}$
\\
\hline
(\ref{eq:sol1}) & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{1}$ & $\epsilon^{0}$ & $\epsilon^{1/2}$ & $\epsilon^{-a+1}$ & $\epsilon^{0}$ & $\epsilon^{-a+2}$
\\
(\ref{eq:sol2}) & $\epsilon^{1/2}$ & $\epsilon^{0}$ & $\epsilon^{1/2}$ & $\epsilon^{0}$ & $\epsilon^{1/4}$ & $\epsilon^{-a-(b-1)/2}$ ($\epsilon^{-a+1}$) & $\epsilon^{0}$ & $\epsilon^{-a-b/2+1}$ ($\epsilon^{-a+3/2}$)
\\
(\ref{eq:sol3}) & $\epsilon^{1}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{1}$ & $\epsilon^{-a-b}$ & $\epsilon^{-1}$ & $\epsilon^{-a-b+3}$
\\
(\ref{eq:sol4}) & $\epsilon^{1}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{1}$ & $\epsilon^{-a-b}$ & $\epsilon^{-1}$ & $\epsilon^{-a-b+3}$
\\
(\ref{eq:sol5}) & $\epsilon^{1/2}$ & $\epsilon^{0}$ & $\epsilon^{1/2}$ & $\epsilon^{0}$ & $\epsilon^{1/4}$ & $\epsilon^{-a-(b-1)/2}$ ($\epsilon^{-a+1}$) & $\epsilon^{0}$ & $\epsilon^{-a-b/2+1}$  ($\epsilon^{-a+3/2}$)
\\
(\ref{eq:sol6}) & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{1}$ & $\epsilon^{0}$ & $\epsilon^{1/2}$ & $\epsilon^{-a+1}$ & $\epsilon^{0}$ & $\epsilon^{-a+2}$
\end{longtable}
\end{ruledtabular}
\end{center}
\end{table*}
Assuming the power-law spectrum, $n(\bm{k}, m) = |\bm{k}|^{-a} |m|^{-b}$,
we make Taylor expansion for the integrand of the kinetic equation (\ref{eq:keap})
as powers of $\epsilon$, that is $x$ and $y$.
Then, we get Table~\ref{table:zero} which shows
the leading orders of the each terms according to the asymptotics.
The leading order of the collision integral is given by ID when $-3<b<3$.
Therefore, we are going to show the procedure to get the leading order of ID (\ref{eq:sol2}) and (\ref{eq:sol5}) below.

As $\epsilon \to 0$,
$n_2 \to n$ for ID solutions.
Therefore,
the leading orders of
$f^0_{1,2} \sim n_1 (n_2 - n)$ and
$f^2_{0,1} \sim n_1 (n - n_2)$
is $O(\epsilon^{-a-(b-1)/2})$.
The order $O(\epsilon^{-a-b/2})$ is canceled as $\epsilon \to 0$.
This is called the {\em first cancellation\/}.
It must be noted that the leading order when $b=0$ is $1/2$ larger than that when $b \neq 0$
since $\partial n/\partial m = 0$.
The leading orders when $b=0$ are written in parentheses in Table~\ref{table:zero}.

The leading order of the integrand in Eq.~(\ref{eq:keap}) is written as
\begin{eqnarray}
&& \!\!\!\!\!\!\!\!
T^0_{1,2} - T^1_{2,0} - T^2_{0,1} \propto k^{-2a + 3} m^{-2b+1}
\frac{x^{-a - (b+1)/2} y}{\sqrt{(x+y)(x-y)}}
\nonumber\\
&& \!\!\!\!\!\!\!\!
\times \!
\left(
-2 a y^2 \! - b \left((1-b) y (x+y) -2x(x-y)\right)  + b (b+1) xy
\right) .
\nonumber\\
\end{eqnarray}
Therefore,
the integrand has $O(\epsilon^{-a - (b - 5)/2})$.
The term which has $O(\epsilon^{-a -b/2 + 2})$ is canceled
since $T^2_{0,1} \to T^0_{1,2}$ (and $T^1_{2,0} \to 0$) as $\epsilon \to 0$.
This is the {\em second cancellation\/}.

Finally, we get the leading order of the kinetic equation after integration over $y$ from $-x$ to $x$:
\begin{eqnarray}
  \frac{\partial n_{\bm{p}}}{\partial t} &\propto&
- b (1-b) k^{4 - 2 a} m^{1 - 2b}
 \int_0  x^{-a - (b-5)/2} dx.
\label{ZeroContribution}
\end{eqnarray}
The integral has $O(\epsilon^{-a - (b - 7)/2})$.
%
Consequently integral converges if 
\begin{eqnarray*}
a + (b - 7)/2 <0 \;\;\mathrm{and}\;\; -3 < b < 3.
\end{eqnarray*}


The integral for the PR spectrum, which gives $O(\epsilon^{-1/4})$,
diverges as $k_1 \to 0$.
However, the integral for the GM spectrum converges
because $b=0$ and the next order is $O(1)$.

It should be noted that the leading order when $b=1$ is $1/2$ larger than that when $b \neq 0,1$
since a balance between first- and second-order derivative is made.
The leading orders when $b=1$ are $O(\epsilon^{-a+7/2})$.
%
It is also helpful to note that
$T^0_{1,2} - T^1_{2,0} - T^2_{1,0} = O(\epsilon^{-a+2})$ for ES
because of no second cancellation.
However, the collision integral has $O(\epsilon^{-a+5})$
because of symmetry of $y$.
Therefore, the integral which is dominated by ES converges
\begin{eqnarray*}
 a - 5 < 0 \;\;\mathrm{and}\;\; b < -3.
\end{eqnarray*}

Similarly, the integral which is dominated by PSI converges
\begin{eqnarray*}
 a + b - 5 < 0 \;\;\mathrm{and}\;\; b > 3.
\end{eqnarray*}

\subsection{Ultra-violet asymptotics}

\begin{table*}
  \caption{Asymptotics as $k_1 \to \infty$.
 PSI (\ref{eq:sol1}, \ref{eq:sol2}) gives $\epsilon^{a+b-3}$.
 ES (\ref{eq:sol3}, \ref{eq:sol5}) gives $\epsilon^{a-3}$.
 ID (\ref{eq:sol4}, \ref{eq:sol6}) gives $\epsilon^{a+b/2-4}$ ($\epsilon^{a-7/2}$).
The asymptotics for $b=0$ appear in parentheses.
}
  \label{table:infinity}
 \begin{center}
\begin{ruledtabular}
\begin{longtable}{ccccccccc}
 & $m_1$ & $m_2$ & $\omega_1$ & $\omega_2$ & $V^{\bm{p}_i}_{\bm{p}_j\bm{p}_k}$ & $f^{\bm{p}_i}_{\bm{p}_j\bm{p}_k}$ & $g^{i \prime}_{j, k}$ & $T^i_{j,k}$
\\
\hline
(\ref{eq:sol1}) & $\epsilon^{-1}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{a+b}$ & $\epsilon^{1}$ & $\epsilon^{a+b-2}$
\\
(\ref{eq:sol2}) & $\epsilon^{-1}$ & $\epsilon^{-1}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{a+b}$ & $\epsilon^{1}$ & $\epsilon^{a+b-2}$
\\
(\ref{eq:sol3}) & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{-1}$ & $\epsilon^{-1}$ & $\epsilon^{-1}$ & $\epsilon^{a}$ & $\epsilon^{-1}$ & $\epsilon^{a-2}$
\\
(\ref{eq:sol4}) & $\epsilon^{-1/2}$ & $\epsilon^{-1/2}$ & $\epsilon^{-1/2}$ & $\epsilon^{-1/2}$ & $\epsilon^{-1}$ & $\epsilon^{a+b/2+1}$ ($\epsilon^{a+3/2}$) & $\epsilon^{1/2}$ & $\epsilon^{a+b/2-3}$ ($\epsilon^{a-5/2}$)
\\
(\ref{eq:sol5}) & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{-1}$ & $\epsilon^{-1}$ & $\epsilon^{-1}$ & $\epsilon^{a}$ & $\epsilon^{-1}$ & $\epsilon^{a-2}$
\\
(\ref{eq:sol6}) & $\epsilon^{-1/2}$ & $\epsilon^{-1/2}$ & $\epsilon^{-1/2}$ & $\epsilon^{-1/2}$ & $\epsilon^{-1}$ & $\epsilon^{a+b/2+1}$ ($\epsilon^{a+3/2}$) & $\epsilon^{1/2}$ & $\epsilon^{a+b/2-3}$ ($\epsilon^{a-5/2}$)
\end{longtable}
\end{ruledtabular}
\end{center}
\end{table*}

Next, we consider the limit $k_1 \to \infty$.
In this case, $k_2$ also approaches to infinity.
We employ the independent variables $x$ and $y$
as $k_1 = k/2 (1 + 1/x + y)$ and $k_2 = k/2 (1 + 1/x - y)$,
where $x = O(\epsilon)$ and $-1 < y < 1$.
Again, $m>0$ is assumed.

The leading orders are obtained by the similar manner used in the IR asymptotic
and are summarized in Table~\ref{table:infinity}.
The leading order of the integral is given by ID,
whose wavenumbers are given by Eqs.~(\ref{eq:sol4}, \ref{eq:sol6}),
when $-2 < b < 2$.
In this limit, no second cancellation is made.

As the result of the perturbation theory,
we get the leading order,
\begin{eqnarray}
 \frac{\partial n_{\bm{p}}}{\partial t} &\propto&
k^{4-2a} m^{1-2b}
b \int_0 x^{a+b/2-5} d x .
\label{InfinityContribution}
\end{eqnarray}
It has $O(\epsilon^{a+b/2-4})$.
%
Therefore, the integral converges if
\begin{eqnarray*}
a+b/2-4>0 \;\;\mathrm{and}\;\; -2<b<2 .
\end{eqnarray*}

The integral for the PR spectrum, which gives $O(\epsilon^{-1/4})$,
diverges as $k_1 \to \infty$.
and that for the Garrett--Munk spectrum, which gives $O(\epsilon^0)$,
converges owing that $b=0$.

Similarly,
\begin{eqnarray}
 \frac{\partial n_{\bm{p}}}{\partial t} &\propto&
- k^{4-2a} m^{1-2b}
b \int_0 x^{a-2}  d x
\label{InfinityContributionES}
\end{eqnarray}
for ES, which is dominant for $b>2$.
Consequently,
the integral converges also if
\begin{eqnarray*}
 a-3 > 0 \;\;\mathrm{and}\;\; b>2.
\end{eqnarray*}

In the same manner,
the convergent domain of the integral for PSI is given by
\begin{eqnarray*}
 a + b - 3 > 0 \;\;\mathrm{and}\;\; b < -2. 
\end{eqnarray*}

\section{Frequency--vertical-wavenumber and horizontal--vertical-wavenumber
spectrum}
\label{OmegaMKMtransform}

The theoretical work presented below addresses the asymptotic power
laws of a three-dimensional action spectrum.  In order to connect with
that work, note that a horizontally isotropic power-law form of the
three-dimensional wave action $n(\bm{k},m)$ is given by
Eq.~(\ref{PowerLawSpectrum}).


The corresponding vertical wavenumber-frequency spectrum of energy
is obtained by transforming $ n_{\bm{k},m}$ from wavenumber space
$(\bm{k},m)$ to the vertical wavenumber-frequency space $(\omega,m)$ and
multiplying by frequency.  In the high-frequency large-wavenumber
limit,
$$ E(m,\omega)\propto \omega^{2-a} m^{2-a-b} ~~ .  $$

The total energy density of the wave field is then $$ E=\int \omega(\bm{k},m) n(\bm{k},m)\  d\bm{k} d m  = \int  E(\omega,m) \ d\omega d m  \label{TotalEnergy}.  $$
%
Thus, we also it convenient to work with the wave action spectrum
expressed as a function of $\omega$ and $m$
Therefore we also introduced (\ref{PowerLawSpectrumFrequencyVertical}).
%
The relation between $a$, $b$ and $\widetilde{a}$, $\widetilde{b}$ reads:
$$\widetilde{a}=-a, \quad \widetilde{b}=-a-b.$$


\section{Asymptotic expansion for small $f$ values.\label{Appendixf}}
In this section we perform the small $f$ calculations of Sec.~\ref{CorRegularization}.
We start from the kinetic equation written as Eq.~(\ref{KE4}). There 
we  change variables in the first line of 
Eq.~(\ref{KE4}) as
%
$$m_1 = E_3(\omega_1)+ \mu(E_1(\omega_1) - E_3(\omega_1) ), $$
%
and in the second line of Eq.~(\ref{KE4}) as
$$m_1=E_3(\omega_1)+\mu (E_2({\omega_1}) - E_3({\omega_1})).$$ 
Then the  Eq.~(\ref{KE4}) becomes the following form:
\begin{eqnarray}
\frac{\partial}{\partial t} n(k(\omega,m),m) &=&\frac{2}{k}
  \int _f^{f+\omega_{\mathrm{s}}} d \omega_1 \int_0^1 d\mu {\cal P}_1
\nonumber\\
&&
- {\frac{2}{k}} \int_{\omega-f-\omega_{\mathrm{s}}}^{\omega-f} d \omega_1
\int_0^1 d \mu {\cal P}_2
.
\label{KE5}
\end{eqnarray}
%
Here we introduced integrand ${\cal P}_1$ and ${\cal P}_2$ to be
%
\begin{eqnarray}
{\cal P}_1 = 
J\frac{|V^0_{1,2}|^2}{S^0_{1,2}} n_1 (n_2 - n) 
(E_1(\omega_1) - E_3(\omega_1)), \nonumber \\
{\cal P}_2 = 
J\frac{|V^0_{1,2}|^2}{S^0_{1,2}} n_2 (n - n_1) 
(E_2(\omega_1)-E_3(\omega_1)).
\label{IntegrandParts}
\end{eqnarray}
%

Before proceeding, note the following symmetry:
%
$$E_1(\omega_1 = \omega -  \omega_1^{\prime}) = m - E_2(\omega_1^{\prime}),$$
%
and
%
$$E_3(\omega_1 = \omega - \omega_1^{\prime}) = m  - E_3(\omega_1^{\prime}).$$
%
To quantify the contribution of near-inertial waves to a $(\omega,m)$ mode, we write
%
$$\epsilon \sim f \ll \omega=1. $$
%
Subsequently, in the domain (a) we  write
%%
$$\omega_1 = f + \epsilon,$$
% 
in ${\cal P}_1$, and 
%
$$\omega_1 = \omega+f+\epsilon$$
%
in ${\cal P}_2$.
%
Furthermore, we expand ${\cal P}_1$ and ${\cal P}_2$ in powers of
$\epsilon$ and $f$ without making any assumptions of the relative
smallness of $f$ and $\epsilon$.
We use the facts that
$$
m>0, \quad \epsilon >0, \quad f>0, \quad 0<\mu<1.
$$
%
Define 
%
$${\cal P}_1 = P_1+P_2,$$
%
and 
%
$${\cal P}_2 = P_3+P_4.$$

This allows us to expand $P_1$, $P_2$, $P_3$ and $P_4$ in powers of $f$ and $\epsilon$.
We perform these calculations on
Mathematica using {\em Series\/} command, and extensively using {\em
Assumptions\/} field in the {\em FullSimplify\/} command.

Mathematica was then able to 
perform the integrals of ${\cal P}_1$ and ${\cal P}_2$ over $\mu$
from $0$ to $1$ in (\ref{KE5}) analytically. The result is given by
Eq.~(\ref{KE6}). 


%

\end{appendix}

\begin{thebibliography}{99}

\bibitem[Caillol and Zeitlin(2000)Caillol and Zeitlin]{caillol2000kea}
P.~Caillol and V.~Zeitlin,
\newblock ``Kinetic equations and stationary energy spectra of weakly nonlinear
  internal gravity waves,''
\newblock Dyn. Atmos. Oceans {\bf 32}, 81 (2000).

\bibitem{H66}
K.~Hasselmann,
\newblock ``{F}eynman diagrams and interaction rules of wave-wave scattering
  processes,''
\newblock Rev. Geophys. {\bf 4}, 1 (1966).

\bibitem{K66}
K.~E. Kenyon,
\newblock {\em Wave-wave scattering for gravity waves and rossby waves},
\newblock PhD thesis, UCSD, 1966.

\bibitem{K68}
K.~E. Kenyon,
\newblock ``Wave-wave interactions of surface and internal waves,''
\newblock J. Mar. Res. {\bf 26}, 208 (1968).

\bibitem{gm_lvov}
Y.~V. Lvov and E.~G. Tabak,
\newblock ``Hamiltonian formalism and the {G}arrett--{M}unk spectrum of internal
  waves in the ocean,''
\newblock Phys. Rev. Lett. {\bf 87}, 168501 (2001).

\bibitem[Mc{C}omas and Bretherton(1977)Mc{C}omas and Bretherton]{mccomas-1977-82}
C.~H. Mc{C}omas and F.~P. Bretherton,
\newblock ``Resonant interaction of oceanic internal waves,''
\newblock J. Geophys. Res. {\bf 82}, 1397 (1977).

\bibitem{milder}
D.~M. Milder,
\newblock ``The effects of truncation on surface-wave {H}amiltonian,''
\newblock J. Fluid Mech. {\bf 216}, 249 (1990).

\bibitem{MO75}
P.~M. D.~J. Olbers,
\newblock ``On the dynamics of internal waves in the deep ocean,''
\newblock J. Geophys. Res. {\bf 80}, 3848 (1975).

\bibitem{O74}
D.~J. Olbers,
\newblock ``On the energy balance of small scale internal waves in the deep sea,''
\newblock Hamburg, Geophys. Einzelschriftern {\bf 27} (1974).

\bibitem{olbers1976net}
D.~J. Olbers,
\newblock ``Nonlinear energy transfer and the energy balance of the internal wave
  field in the deep ocean,''
\newblock J. Fluid Mech. {\bf 74}, 375 (1976).

\bibitem[Pelinovsky and Raevsky(1977)Pelinovsky and Raevsky]{pelinovsky1977wti}
E.~N. Pelinovsky and M.~A. Raevsky,
\newblock ``Weak turbulence of the internal waves of the ocean,''
\newblock Atm. Ocean Phys.-Izvestija {\bf 13}, 187 (1977).

\bibitem{pomphrey1980dni}
N.~Pomphrey, J.~D. Meiss, and K.~M. Watson,
\newblock ``Description of nonlinear internal wave interactions using {L}angevin
  methods,''
\newblock J. Geophys. Res {\bf 85}, 1085 (1980).

\bibitem{voronovich1979hfi}
A.~G. Voronovich,
\newblock ``{H}amiltonian formalism for internal waves in the ocean,''
\newblock Izvestiya, Atmospheric and Oceanic Physics {\bf 15}, 52 (1979).

\bibitem{zak_book}
V.~E. Zakharov, V.~S. L'vov, and G.~Falkovich,
\newblock {\em {K}olmogorov Spectra of Turbulence I: Wave Turbulence},
\newblock Springer-Verlag, Berlin, 1992.

\bibitem{polzin-2004}
K.~Polzin,
\newblock ``A heuristic description of internal wave dynamics,''
\newblock J. Phys. Oceanogr. {\bf 34}, 214 (2004).

\bibitem[McComas and M\"{u}ller(1981)McComas and M\"{u}ller]{mccomas-1981}
C.~H. McComas and P.~M\"{u}ller,
\newblock ``Time scales of resonant interactions among oceanic internal waves,''
\newblock J. Phys. Oceanogr. {\bf 11}, 139 (1981).

\bibitem{elementLPY}
Y.~V. Lvov, K.~L. Polzin, and N.~Yokoyama,
\newblock ``Wave-wave interactions in stratified fluids: A comparison of
  approaches,''
\newblock (submitted to J. Phys. Oceanogr.).

\bibitem{muller1986nia}
P.~G. M\"uller, G.~Holloway, F.~Henyey, and N.~Pomphrey,
\newblock ``Nonlinear interactions among internal gravity waves,''
\newblock Reviews of geophysics {\bf 24}, 493 (1986).

\bibitem{PLT}
K.~L. Polzin, Y.~V. Lvov, and E.~G. Tabak,
\newblock ``Towards regional characterization of internal wave fields,''
\newblock (in preparation).

\bibitem{M78}
P.~M\"{u}ller, D.~J. Olbers, and J.~Willebrand,
\newblock ``The {IWEX} spectrum,''
\newblock J. Geophys. Res. {\bf 83}, 479 (1978).

\bibitem{Letal87}
M.~D. Levine, C.~A. Paulson, and J.~H. Morison,
\newblock ``Observations of internal gravity waves under the arctic pack ice,''
\newblock J. Geophys. Res. {\bf 92}, 779 (1987).

\bibitem{dasaro1991iwa}
M.~D. D'Asaro, E.~A. \&~Morehead,
\newblock ``Internal waves and velocity fine structure in the {A}rctic {O}cean,''
\newblock J. Geophys. Res. {\bf 96}, 12725 (1991).

\bibitem{weller1991for}
R.~A. Weller, K.~L. Polzin, D.~L. Rundnick, C.~C. Eriksen, and N.~S. Oakey,
\newblock ``Forced ocean response during the {F}rontal {A}ir-{S}ea {I}nteraction
  {E}xperiment,''
\newblock J. Geophys. Res. {\bf 96}, 8611 (1991).

\bibitem{eriksen1991ofv}
C.~Eriksen, D.~Rudnick, R.~Weller, R.~T. Pollard, and L.~Regier,
\newblock ``Ocean frontal variability in the {F}rontal {A}ir-{S}ea {I}nteraction
  {E}xperiment,''
\newblock J. Geophys. Res. {\bf 96}, 8569 (1991).

\bibitem{sherman21evw}
J.~T. Sherman and R.~Pinkel,
\newblock ``Estimates of the vertical wavenumber--frequency spectra of vertical
  shear and strain,''
\newblock J. Phys. Oceanogr. {\bf 21}, 292 (1991).

\bibitem{anderson1992ssa}
S.~Anderson,
\newblock {\em Shear, strain, and thermohaline vertical fine structure in the
  upper ocean},
\newblock PhD thesis, UCSD, 1992.

\bibitem{P03}
K.~L. Polzin, E.~Kunze, J.~M. Toole, and R.~W. Schmitt,
\newblock ``The partition of finescale energy into internal waves and subinertial
  motions,''
\newblock J. Phys. Oceanogr. {\bf 33}, 234 (2003).

\bibitem{Weller04}
R.~A. Weller, P.~W. Furey, M.~A. Spall, and R.~E. Davis,
\newblock ``The large-scale cotext for oceanic subduction in the northeast
  atlantic,''
\newblock Deep-Sea Res. I {\bf 51}, 665 (2004).

\bibitem{Schmitt05}
R.~W. Schmitt, J.~R. Ledwell, E.~T. Montgomery, K.~L. Polzin, and J.~M. Toole,
\newblock ``Enhanced diapycnal mixing by salt fingers in the thermocline of the
  tropical atlantic,''
\newblock Science {\bf 308}, 685 (2005).

\bibitem{Keffer83}
T.~Keffer,
\newblock ``The baroclinic stability of the atlantic north equatorial current,''
\newblock J. Phys. Oceanogr. {\bf 13}, 624 (1983).

\bibitem{Foff69}
N.~P. Foffonoff,
\newblock ``Spectral characteristics of internal waves in the ocean,''
\newblock Deep Sea Res. {\it Suppl.} to {\bf 16}, 58 (1969).

\bibitem{DA84}
E.~A. D'Asaro,
\newblock ``Wind forced internal waves in the {N}orth {P}acific and {S}argasso {S}ea,''
\newblock J. Phys. Oceanogr.  {\bf 14}, 781 (1984).

\bibitem{DA95}
E.~A. D'Asaro,
\newblock ``A collection of papers on the ocean storms experiment,''
\newblock J. Phys. Oceanogr. {\bf 25}, 2817 (1995).

\bibitem{lvov-2004-92}
Y.~V. Lvov, K.~L. Polzin, and E.~G. Tabak,
\newblock ``Energy spectra of the ocean's internal wave field: theory and
  observations,''
\newblock Phys. Rev. Lett. {\bf 92}, 128501 (2004).

\bibitem{zak1968}
V.~E. Zakharov,
\newblock ``Stability of periodic waves of finite amplitude on the surface of a
  deep fluid,''
\newblock J. Appl. Mech. Tech. Phys. {\bf 2}, 190 (1968).

\bibitem{zakharov1967iwn}
V.~E. Zakharov,
\newblock ``The instability of waves in nonlinear dispersive media,''
\newblock Sov. Phys. JETP {\bf 24}, 740 (1967).

\bibitem{kuznetsov1972tis}
E.~A. Kuznetsov,
\newblock ``On turbulence of ion sound in plasma in a magnetic field,''
\newblock Zh. Eksper. Teoret. Fiz {\bf 62}, 584 (1972).

\bibitem{numericsLY}
Y.~V. Lvov and N.~Yokoyama,
\newblock ``Nonlinear wave-wave interactions in stratified flows: Direct
  numerical simulations,''
\newblock (submitted to Phys. Fluids).

\end{thebibliography}
\end{document}