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\begin{document}
\title{Oceanic Internal Wave Field: Theory of Scaleinvariant 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 6100394 Japan}
\thanks{Author to whom should be addressed}
\email{nyokoyam@mail.doshisha.ac.jp}
\date{\today}
\pacs{47.35.Bb}
\begin{abstract}
Steady scaleinvariant 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 scaleinvariant collision integral in the kinetic equation
diverges for almost all spectral powerlaw exponents.
These divergences come from resonant
interactions with infrared and/or ultraviolet wavenumbers
with extreme scaleseparations.
We identify a small domain in which the scaleinvariant collision
integral converges and numerically find
a convergent powerlaw solution. This numerical solution is close to the
GarrettMunk spectrum.
Powerlaw exponents which potentially permit
a balance between the infrared and ultraviolet
divergences are also discussed.
The balanced exponents are generalizations of an exact solution of the
scaleinvariant
kinetic equation, the PelinovskyRaevsky 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 selfconsistency
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}
Wavewave 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 internalwave 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 threewave 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,mccomas197782,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 threedimensional 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 NavierStokes turbulence,
one may look for statistically stationary states using scaleinvariant 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 scaleinvariant. It is natural, therefore, in this restricted domain, to
look for selfsimilar 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 righthand 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{polzin2004}). 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 internalwave collision integral diverges
for {\em almost\/} all values of $a$ and $b$. In particular, the collision integral has an
infrared (IR) divergence
at zero, i.e. $\bm{k}_1$ or $\bm{k}_2$ $\to 0$ and an ultraviolet (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 largewavenumber form of the GarrettMunk (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 PelinovskyRaevsky (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 oppositesigned 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 selfsimilar
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 selfsimilarity 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 cutoffs 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 $92a3b=0$ or $b=0$.
These are the induced diffusion lines of steady state solutions, found
originally in Ref.~\onlinecite{mccomas1981}. 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 powerlaw solution that may account for the GM spectrum.
In Sec.~\ref{Balance} we introduce
possible quasisteady 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 powerlaw 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, mccomas197782, pomphrey1980dni}.
Various formulations have been developed for characterizing
wavewave 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,mccomas197782,milder,MO75,O74,olbers1976net,pelinovsky1977wti,pomphrey1980dni,voronovich1979hfi}
for details).
We briefly discuss the derivation of the kinetic equation and wavewave interaction matrix elements
below
in Eq.~(\ref{MatrixElement}).
The starting point for the most extensive investigations has been a noncanonical Hamiltonian formulation in Lagrangian coordinates \citep{mccomas1981} 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 Clebshlike variables \citep{pelinovsky1977wti} and a nonHamiltonian 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{mccomas197782} and reviewed by Ref.~\onlinecite{muller1986nia}. These mechanisms
represent extreme scaleseparated limits.
%
One of these mechanisms represents the
interaction of two small vertical scale, high frequency waves with a
large vertical scale, nearinertial (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 inertialrange 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 PelinovskyRaevsky spectrum.
This spectrum was latter found in Refs.~\onlinecite{caillol2000kea,gm_lvov}.
It follows from applying the ZakharovKuznetsov conformal transformation, which effectively establishes a map between the neighborhoods of zero and infinity. Making these two contributions cancel pointwisely 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: powerlaw 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 powerlaw 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 scaleseparated 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 selfconsistency
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 powerlaw
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 AirSea 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 SiteD\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 powerlaw exponents $(a,b)$ are plotted against each other.
That study comes with many caveats. The most important here is that the powerlaw exponents, $(a,b)$, are not typically derived from twodimensional
horizontal wavenumber, vertical wavenumber spectra. The power laws were fit to onedimensional frequency and onedimensional 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 twodimensional 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 powerlaw 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 PelinovskyRaevsky (PR) spectrum, the convergent numerical solution determined in Sec.~\ref{NovelSolution} (C) and the GM spectrum. Circles with stars represent estimates based upon onedimensional 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 (SiteD), the Arctic (AIWEX) and the eastern North Atlantic (NATRE$^1$ and NATRE$^2$). There are two estimates obtained from twodimensional 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 selfconsistent 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 pancakelike 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
threewave 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 threewave 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 smallamplitude 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 threewave Hamiltonian (\ref{HAM}), the kinetic equation
is the one in
Eq.~(\ref{KineticEquation}),
describing general internal waves
interacting in both rotating and nonrotating 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_{mm_1m_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 horizontalmomentum 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 scaleinvariant solutions in the next section.
\section{Scaleinvariant kinetic equation}
\label{sec:SI}
\subsection{Reduction of Kinetic Equation to the Resonant Manifold\label{Reduction}}
In the highfrequency 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 azimuthallyintegrated 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}{mm_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}
k3 , \label{IR4}
\\
\mathrm{or} \nonumber\\
&
a5<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/24>0 \;\; \mathrm{and} \;\; 20 \;\; \mathrm{and} \;\; b=0 , \label{UV2}
\\
&
a3>0 \;\; \mathrm{and} \;\; b>2 , \label{UV3}
\\
\mathrm{or} \nonumber\\
&
a+b3>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 twodimensional
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(b1)$ prefactor
to the divergent contributions to the collision integral~(\ref{ZeroContribution}). Similarly, for the UV limit, in addition to
the twodimensional 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 scaleinvariant 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 powerlaw ansatz~(\ref{PowerLawSpectrum}) with $b=0$
into the azimuthallyintegrated 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 scaleinvariant 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
agreedupon 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 scaleseparated 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 $32$,
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 powerlaw 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 ZakharovKuznetsov conformal
mapping for systems with cylindrical symmetry~\citep{zak1968,zakharov1967iwn,kuznetsov1972tis}.
This ZakharovKolmogorov conformal mapping
effectively establishes a map between the
neighborhoods of zero and infinity. Making these two contributions
cancel pointwisely 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 oppositesigned 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 KuznetsovZakharov transformation.
Such {\em generalized\/} KuznetsovZakharov 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
powerlaw exponents of the quasisteady 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 nonuniversality.
\section{Regularization by the Coriolis effect}
\label{CorRegularization}
Physically, ocean does not perform
generalized KuznetsovZakharov 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 subdominant 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_1E_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_1E_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_10, \; \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_1m$.
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 powerlaw 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}=ab.$$
Equation~(\ref{KEinternalAveragedAngles}) transforms into
\begin{widetext}
\begin{eqnarray}
\frac{\partial}{\partial t} n(k(\omega,m),m) &=&\frac{1}{k}\int d\omega_1 d m_1 J
\frac{V^0_{12}^2}{S^0_{1,2}}
\left(n_1 n_2  n(n_1+n_2)\right)
\vline_{\small { \omega_2=\omega\omega_1, m_2 = m  m_1}}
\nonumber\\
&&\frac{2}{k}\int d\omega_1 d m_1 J\frac{V^1_{02}^2}{S^1_{2,0}}
\left(n n_2  n_1 ( n + n_2)\right)
\vline_{\omega_2=\omega_1\omega, m_2 = m_1  m}
\label{KE3}
\end{eqnarray}
\end{widetext}
%
We have used the dispersion relation
%
$
k_i=m_i\sqrt{\omega_i^2  f^2},
$
%
and defined $J$ as the Jacobian of the transformation from $(k_1,k_2)$ into
$(\omega_1,\omega_2)$, times the $k k_1 k_2 $ factor,
%
$$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.$$
\item ID2 region. In this region, $\omega_1$ is slightly bigger than $\omega+f$ where
$\omega_2=\omega_1\omega$ is small, and $m_2=m_1m$ is negative and small. This is
the region (\ref{Region4}). Also
%
$$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.$$
Note that this region can be obtained from the region ID1 by flipping indices
$1$ and $2$. Consequently, only one of the ID1, ID2 should be taken into account,
with a factor of two multiplying the respective contribution.
\end{enumerate}
Making these simplification, and taking into account the areas of
integration in the kinematic box, we obtain
%
\begin{widetext}
\begin{eqnarray}
\frac{\partial}{\partial t} n(k(\omega,m),m) = \frac{2}{k}
\int _f^{f+\omega_{\mathrm{s}}} d \omega_1 \int_{E_3(\omega_1)}^{E_1(\omega_1)}
d m_1
J\frac{V^0_{1,2}^2}{S^0_{1,2}} n_1 (n_2  n)
 {\frac{2}{k}} \int _{\omegaf\omega_{\mathrm{s}}}^{\omegaf} d \omega_1
\int_{E_3(\omega_1)}^{E_1(\omega_1)} d m_1
J\frac{V^2_{1,0}^2}{S^2_{1,0}} n_2 (n  n_1)
,\nonumber \\
\label{KE4}
\end{eqnarray}
\end{widetext}
%
where the small parameter $\omega_{\mathrm{s}}$ is introduced to restrict the integration
to a neighborhood of the ID corners. The arbitrariness of the small parameter will not affect
the end result below.
To quantify the contribution of nearinertial waves to a $(\omega,m)$ mode, we write
%
$$\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}
92a3b=0
\quad \mathrm{or} \quad
b=0.
\label{NewIDcurve}
\end{eqnarray}
This is the family of powerlaw steadystate solutions to the kinetic
equations dominated by infrared ID interactions. These steady states
are identical to the ID stationary states identified by \textcite{mccomas197782},
who derived a diffusive approximation to
their collision integral in the infrared ID limit.
%
Note that \textcite{mccomas1981} interpreted $b=0$ as a no action flux in
vertical wavenumber domain, while $92a3b=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 (nonrotating) solution is in
close proximity to the experimental points determined from the
twodimensional 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 PelinovskyRaevsky (PR) spectrum, the convergent numerical solution determined in Section 4 and the GM spectrum. Circles with stars represent powerlaw estimates based upon onedimensional spectra. Circles with cross hairs represent estimates based upon twodimensional data sets. See Fig. \ref{fig:onlydata} for the identification of the field programs. Light grey shading represents regions of the powerlaw domain for which the collision integral converges in either the IR or UV limit. The dark grey shading represents the region of the powerlaw domain for which neither the IR or UV limits converge. The region of black shading represents the subdomain 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 scaleinvariant 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 scaleseparated 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 largewavenumber asymptotic form of the GarrettMunk spectrum, $(a,b)=(4,0)$.
We have argued that there exist two regions of powerlaw exponents
which can yield quasisteady solutions of the kinetic equation.
For these ranges of exponents,
the contribution of the scaleseparated interactions
due to the IR and UV wavenumbers
can be made to balance each other.
The PelinovskyRaevsky spectrum is a special case of this scenario.
None of the observational powerlaw exponents lie within the region of samesigned
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 powerlaw 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 steadysolution 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 internalwave spectrum. To the degree that the large
scales are location dependent and not universal, the highfrequency,
high verticalwavenumber internalwave spectrum ought to be affected
by this variability. Consequently, the internalwave 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 infrared and ultraviolet 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{Infrared 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 3.
\end{eqnarray*}
\subsection{Ultraviolet asymptotics}
\begin{table*}
\caption{Asymptotics as $k_1 \to \infty$.
PSI (\ref{eq:sol1}, \ref{eq:sol2}) gives $\epsilon^{a+b3}$.
ES (\ref{eq:sol3}, \ref{eq:sol5}) gives $\epsilon^{a3}$.
ID (\ref{eq:sol4}, \ref{eq:sol6}) gives $\epsilon^{a+b/24}$ ($\epsilon^{a7/2}$).
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+b2}$
\\
(\ref{eq:sol2}) & $\epsilon^{1}$ & $\epsilon^{1}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{a+b}$ & $\epsilon^{1}$ & $\epsilon^{a+b2}$
\\
(\ref{eq:sol3}) & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{1}$ & $\epsilon^{1}$ & $\epsilon^{1}$ & $\epsilon^{a}$ & $\epsilon^{1}$ & $\epsilon^{a2}$
\\
(\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/23}$ ($\epsilon^{a5/2}$)
\\
(\ref{eq:sol5}) & $\epsilon^{0}$ & $\epsilon^{0}$ & $\epsilon^{1}$ & $\epsilon^{1}$ & $\epsilon^{1}$ & $\epsilon^{a}$ & $\epsilon^{1}$ & $\epsilon^{a2}$
\\
(\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/23}$ ($\epsilon^{a5/2}$)
\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^{42a} m^{12b}
b \int_0 x^{a+b/25} d x .
\label{InfinityContribution}
\end{eqnarray}
It has $O(\epsilon^{a+b/24})$.
%
Therefore, the integral converges if
\begin{eqnarray*}
a+b/24>0 \;\;\mathrm{and}\;\; 22$.
Consequently,
the integral converges also if
\begin{eqnarray*}
a3 > 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{Frequencyverticalwavenumber and horizontalverticalwavenumber
spectrum}
\label{OmegaMKMtransform}
The theoretical work presented below addresses the asymptotic power
laws of a threedimensional action spectrum. In order to connect with
that work, note that a horizontally isotropic powerlaw form of the
threedimensional wave action $n(\bm{k},m)$ is given by
Eq.~(\ref{PowerLawSpectrum}).
The corresponding vertical wavenumberfrequency spectrum of energy
is obtained by transforming $ n_{\bm{k},m}$ from wavenumber space
$(\bm{k},m)$ to the vertical wavenumberfrequency space $(\omega,m)$ and
multiplying by frequency. In the highfrequency largewavenumber
limit,
$$ E(m,\omega)\propto \omega^{2a} m^{2ab} ~~ . $$
The total energy density of the wave field is then $$ 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}=ab.$$
\section{Asymptotic expansion for small $f$ values.\label{Appendixf}}
In this section we perform the small $f$ calculations of Sec.~\ref{CorRegularization}.
We start from the kinetic equation written as Eq.~(\ref{KE4}). There
we change variables in the first line of
Eq.~(\ref{KE4}) as
%
$$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})).$$
Then the Eq.~(\ref{KE4}) becomes the following form:
\begin{eqnarray}
\frac{\partial}{\partial t} n(k(\omega,m),m) &=&\frac{2}{k}
\int _f^{f+\omega_{\mathrm{s}}} d \omega_1 \int_0^1 d\mu {\cal P}_1
\nonumber\\
&&
 {\frac{2}{k}} \int_{\omegaf\omega_{\mathrm{s}}}^{\omegaf} d \omega_1
\int_0^1 d \mu {\cal P}_2
.
\label{KE5}
\end{eqnarray}
%
Here we introduced integrand ${\cal P}_1$ and ${\cal P}_2$ to be
%
\begin{eqnarray}
{\cal P}_1 =
J\frac{V^0_{1,2}^2}{S^0_{1,2}} n_1 (n_2  n)
(E_1(\omega_1)  E_3(\omega_1)), \nonumber \\
{\cal P}_2 =
J\frac{V^0_{1,2}^2}{S^0_{1,2}} n_2 (n  n_1)
(E_2(\omega_1)E_3(\omega_1)).
\label{IntegrandParts}
\end{eqnarray}
%
Before proceeding, note the following symmetry:
%
$$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}).$$
%
To quantify the contribution of nearinertial waves to a $(\omega,m)$ mode, we write
%
$$\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}
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\end{document}