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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{review}

\title{Wave turbulence: a solvable problem applied to the
Navier--Stokes equations}

\alttitle{Turbulence d'ondes : un probl\`eme r\'esoluble appliqu\'e aux
\'equations de Navier--Stokes}

\author{\firstname{S\'ebastien} \lastname{Galtier}\CDRorcid{0000-0001-8685-9497}}
\address{Laboratoire de Physique des Plasmas, Universit\'e
Paris-Saclay, \'Ecole polytechnique, 91128 Palaiseau, France}
\email{sebastien.galtier@lpp.polytechnique.fr}

\begin{abstract}
Wave turbulence and eddy turbulence are the two regimes that we may
encounter in nature. The attention of fluid mechanics being mainly
focused on incompressible hydrodynamics, it is usually the second
regime that is treated in books, whereas waves are often present in
geophysics and astrophysics. In this review, I present the theory of
wave turbulence which is free from the closure problem encountered in
eddy turbulence. Basically, the wave amplitude is introduced in a
multiple time scale method as a small parameter to derive the so-called
kinetic equations from which exact results can be obtained (power-law
spectra, direction of the cascade, Kolmogorov's constant) and compared
with the data. Two hydrodynamic applications are considered with
capillary waves and inertial waves, the first leading to isotropic
turbulence and the second to anisotropic turbulence. 
\end{abstract}

\begin{altabstract}
La turbulence d'ondes et la turbulence de tourbillons sont les deux
r\'egimes que l'on peut rencontrer dans la nature. L'attention des 
m\'ecaniciens des fluides \'etant principalement port\'ee sur
l'hydrodynamique incompressible, c'est g\'en\'eralement le second
r\'egime qui est trait\'e dans les livres, alors que les ondes sont
souvent pr\'esentes en g\'eophysique et en astrophysique. Dans cette
revue, je pr\'esente la th\'eorie de la turbulence d'ondes qui est
exempte du probl\`eme de fermeture rencontr\'e dans le cas classique.
Fondamentalement, l'amplitude de l'onde est introduite dans une
m\'ethode \`a \'echelles de temps multiples comme un petit param\`etre
pour trouver les \'equations dites cin\'etiques \`a partir desquelles
des r\'esultats exacts peuvent \^etre obtenus (spectres en loi de
puissance, direction de la cascade, constante de Kolmogorov) et
compar\'es avec les donn\'ees. Deux applications hydrodynamiques sont
consid\'er\'ees avec des ondes capillaires et des ondes inertielles, la
premi\`ere conduisant \`a une turbulence isotrope et la seconde \`a une
turbulence anisotrope. 
\vspace*{-2pt}
\end{altabstract}

\keywords{\kwd{Hydrodynamics}
\kwd{Turbulence}
\kwd{Wave}}

\altkeywords{\kwd{Hydrodynamique}
\kwd{Onde}
\kwd{Turbulence}}

\thanks{Simons Foundation (Grant No. 651461, PPC)}

\thanks{\textbf{Note}. S\'ebastien Galtier is the 2024 recipient of the
``C\'ecile Dewitt-Morette / \'Ecole de Physique des Houches / Fondation CFM
pour la recherche'' prize of the Acad\'emie des sciences.}

\maketitle

\vspace*{.6pc}

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\xsection{}

Wave turbulence\footnote{This article is a review on wave turbulence
with an application to the Navier--Stokes equations, whose 200th
anniversary we celebrated in 2023. It was an opportunity for me to give
a series of lectures on this subject at the \'Ecole de Physique des
Houches.} offers the possibility of a deep understanding of physical
systems composed of an ensemble of random waves interacting
nonlinearly. The reason for this is, first of all, the possibility of
analytically deriving a set of integro-differential equations for
spectral \mbox{cumulants---the} so-called kinetic equations---which are free
from the closure problem classically encountered in eddy turbulence. To
achieve this natural asymptotic closure, the wave amplitude is used as
a small parameter. Second, exact solutions can be found from the
kinetic equations. In addition to the usual thermodynamic solutions,
the kinetic equations have finite flux solutions that capture the flow
of conserved densities from sources to sinks. Third, these exact
solutions correspond to power-law spectra that can be compared to the
data. The number of experiments, observations, and diagnostics has
increased dramatically over the past two decades, and now, with direct
numerical simulations, wave turbulence has become a leading field in
turbulence from which new fundamental questions have
arisen~\cite{GaltierCUP2023}. 

The Navier--Stokes equations are fundamental in turbulence because it is
from them and from laboratory experiments based on water or air that
the first theoretical advances (concepts, exact laws) have been made.
This is a reason why the literature on turbulence focuses mainly---if
not entirely---on incompressible hydrodynamics, thus excluding a whole
series of applications where waves make a major contribution to the
dynamics. Yet turbulent systems containing waves are very numerous, and
incompressible hydrodynamics (pure eddy turbulence) seems to be quite
singular. In essence, wave turbulence  concerns only systems in which
waves can be excited, which excludes a priori the classical
Navier--Stokes equations, the archetypal system of turbulence. However,
wave turbulence can also be found in incompressible hydrodynamics if
one modifies this system to include, for example, surface (gravity and
capillary) waves, or if one adds the Coriolis force which introduces
inertial waves. 

A priori, we can distinguish two regimes in wave turbulence: the one
where the waves are of weak amplitude and the one where they are not.
The regime that concerns us here is the first case called weak wave
turbulence. I will follow the usual practice and simply write 'wave
turbulence' to avoid thinking that weak (wave) turbulence also means
small Reynolds number: indeed, as for eddy turbulence, the Reynolds
number is not limited or, in other words, it is only limited by the
effect of viscosity and the size of the experiment.  The existence of a
small parameter---the amplitude of the wave---allows a systematic
approach to the problem. The general idea is that the weak amplitude of
the waves makes it possible to distinguish the temporal evolution of
the amplitude from that of the phase, the first evolving more slowly
than the second. This separation of the time scales can be formally
written using a multiple time scale method. This is the condition
applied to obtain a uniform asymptotic development which leads to a
natural closure of the hierarchy of equations and finally to the
so-called kinetic equations. 

In this review, I will introduce the multiple time scale method to an
archetypal wave equation that involves three-wave interactions. I will
show in detail how the kinetic equation can emerge from this
development. This is a technical part that has rarely (if never) been
described in detail since the seminal paper~\cite{Benney1966}.  A brief
history will allow us to understand how the theory of wave turbulence
was built. For pedagogical reasons, the multiple time scale method will
first be applied to a simple example: the Duffing model. We will see
that the main problem is the existence of secular terms proportional to
time which can break the uniformity of the development. This is
precisely the condition applied to eliminate them that leads to the
kinetic equations of wave turbulence. I will examine two examples from
the modified Navier--Stokes equations: capillary wave turbulence and
inertial wave turbulence involving three-wave interactions. The first 
implies isotropy while the second leads to anisotropy, and thus to a
new difficulty. Finally, using a toy model, I will show how the
Kolmogorov--Zakharov spectrum can be derived. 

\section{Introduction to wave turbulence: origin and applications}

It is essentially in the field of oceanography and the study of surface
waves that we find the beginnings of the theory of wave 
turbulence~\cite{Phillips1981}. This work began in the late 1950s, at a
time when the surface waves were treated essentially in a linear
manner, the spectra as a superposition of linear waves and non-linear
effects restricted, for example, to periodic wave distortion phenomena
described for a long time by Stokes~\cite{Stokes1847}.  There are two
kinds of surface waves: gravity waves---that is, waves on the ocean
(see Figure~\ref{Fig0}), and at smaller scale capillary waves. The
Navier--Stokes equations, modified by the force of gravity, are the
theoretical framework from which the first results emerged. To simplify
the problem, the movements are generally assumed to be irrotational:
this corresponds to an air--water interface disturbed by a
unidirectional blowing wind---a typical condition encountered in the
open sea. The problem is then reduced to Bernoulli's equation applied
to the free surface of the fluid to which is added a Lagrangian
equation to describe the deformation of the fluid surface and another
to take into account the additional hypothesis called deep water. 

\begin{figure}
\includegraphics{fig01}
\vspace*{-3pt}
\caption{\label{Fig0}Gravity and capillary waves.}
\end{figure}

\subsection{Resonant wave interactions}

Work in the late 1950s and early 1960s led to a major theoretical
breakthrough with the discovery of the existence of resonant
interactions\footnote{Note that earlier studies on three- and four-wave
resonance had already been carried out, for example, in the field of
thermal conduction in crystals~\cite{Peierls1929}. See
also~\cite{Nordheim1928} who wrote the first kinetic equation: it was
for four-wave interactions in the context of the electron theory of
conductivity.} between non-linear waves of weak amplitude. The waves in
question are of course gravity waves. The idea has its origin in the
work of~\cite{Phillips1960,Longuet1962,Hasselmann1962} who tried to
understand how the non-linear interactions between gravity waves could
redistribute the energy initially present in these modes. The
underlying idea is that in the initial phase of the development of
turbulence it is the resonant interactions that provide the dominant
mechanism for transferring energy from one wave to another. The
ultimate phase of development had already been studied previously
by~\cite{Phillips1958} who, using dimensional analysis, was able to
make a spectral prediction for strong (wave) turbulence which is still
in use today (the so-called critical balance). 

After long calculations based on a classical perturbative development,
it was possible to demonstrate that the energy redistribution was
generally negligible for interactions involving less than four waves,
but that it became important for four waves if the following resonance
relationship was satisfied
{\begin{eqnarray}
\label{R1L1}
\left\{
\begin{array}{l}
\kk_1 + \kk_2 = \kk_3 + \kk,\\ 
\omega_1 + \omega_2 = \omega_3 + \omega, 
\end{array}
\right.
\end{eqnarray}}\unskip
with the dispersion relation $\omega^{2}=g\vert \mathbf{k} \vert$ ($g$
being the acceleration of gravity). The experimental demonstration of
the existence of resonant interactions was then published
by~\cite{Longuet1966,McGoldrick1966}. Although the existence of
resonant wave interactions appeared to be real, their interpretation
was then subject to debate because the energy exchange between resonant
waves also led to a linear growth (in time)---the secular terms---of
the amplitude of the waves ultimately breaking the hypothesis
underlying the existence of resonant interactions, i.e. the presence of
waves of weak amplitude; in this case, the perturbation is said to be
non-uniform. It was therefore not clear whether this energy transfer
could be really effective. 

An answer to this question was first proposed statistically
by~\cite{Hasselmann1962} who considered a random set of gravity waves.
The hypothesis of weak non-linearities was then used to justify another
hypothesis, that of a Gaussian statistics which greatly simplifies the
calculations by eliminating in particular moments of odd order (see
also~\cite{Drummond1962}). 


\subsection{Multiple time scale method}

A second major theoretical breakthrough came with Benney's work which
showed that the problem of wave turbulence is similar to that of weakly
coupled oscillators in mechanics~\cite{2020Akylas}. With a new
mathematical technique for the analysis of dispersive wave packets
involving two time scales, \cite{Benney1962} obtained a relatively
simple form for the (discrete and non-statistical) equations which
govern the temporal evolution of resonant modes. In this approach, the
secular terms disappear: they are somehow absorbed by the slow
variation of the wave amplitude. This work shows, in a simple way, how
the exchange of energy between four gravity waves is carried out: this
exchange takes place while conserving energy. Here we recover a
property of strong turbulence, that of detailed conservation. 

This approach paved the way for the use of a new mathematical technique
called multiple time scale
method~\cite{Poincare1893,Sturrock1957,Nayfeh2004}, which made it
possible to demonstrate that the (continuous) statistical equations of
wave turbulence have a natural closure due to the separation of scales
in time. As shown by~\cite{Benney1966} for quadratic non-linearities
and~\cite{Benney1967} for cubic non-linearities, the equations of wave
turbulence are asymptotically valid over long periods of time and do
not require the statistical assumption of Gaussianity made
by~\cite{Hasselmann1962}. The method of multiple time scales offers a
systematic and consistent theoretical framework in which the procedure
expansion allows in principle to determine the slow rate of change of
the amplitude of the waves at any order in
$\epsilon$~\cite{Benney1967b,Benney1969}. The so-called kinetic
equation of gravity wave turbulence then takes the following
(schematic) integro-differential form in three-dimension
{\begin{eqnarray} 
\frac{\partial N(\kk)}{\partial t} &=&  \epsilon^4
\int_{\mathbb{R}^{9}} S(\kk,\kk_{1},\kk_{2},\kk_{3}) [N_1 N_2 (N+N_3) -
N_3 N (N_1+N_2)]  \delta(\omega_1 + \omega_2 - \omega_3 - \omega)
\nonumber\\
&& \times\, \delta(\kk_1 + \kk_2 - \kk_3 -\kk)  \,\mathrm{d}\kk_1 
\,\mathrm{d}\kk_2\,\mathrm{d}\kk_3,\label{EqKin}
\end{eqnarray}}\unskip
with $N \equiv N(\kk)=E(\kk)/\omega$ the wave action
spectrum\footnote{By analogy with plasma physics, wave action is often
associated with particles. This is a quantity that is sometimes
conserved in four-wave processes. This is the case for gravity wave
turbulence.} and $E(\kk)$ the energy spectrum. The Dirac functions
translate the resonance condition (\ref{R1L1}) discussed above. The
presence of the $\epsilon^4$ factor means that the time scale
(normalized to the wave period ${\sim} 1 / \omega$) on which the spectra
are modified by the non-linear dynamics is of the order of $\mathcal{O}
(1/\epsilon^{4})$.  Therefore, it is a relatively slow process. 

\subsection{Kolmogorov--Zakharov spectrum}

Parallel to the work carried out in the Western world, major
theoretical advances were also made in the East. From the beginning of
the 1960s, the Soviet school became interested in wave turbulence
mainly through plasma physics~\cite{Sagdeev1966,Vedenov67} from which
certain notations and vocabulary were borrowed (one speaks, for
example, of kinetic equation or collision integral). In passing, it is
curious to note that this work was carried out simultaneously by the
two parts of the world without much communication between them. In
particular, by the assumption known as random phase approximation,
kinetic equations of wave turbulence were proposed in a primitive form
by~\cite{Kadomtsev1963} for a problem of plasma physics, then in a
modern form by Zakharov and his
collaborators~\cite{Zakharov1965,Zakharov1966,Zakharov1967,Zakharov1967b}. 
This work is generally based on a Hamiltonian approach to the problem,
whereas it is the Eulerian approach which was mainly used in the West. 

The random phase approximation leads, in practice, to the same kinetic
equations as by the method of multiple time scales where the random
phase occurs naturally dynamically (and therefore is not assumed). From
these integro-differential equations, a major breakthrough was achieved
with the discovery of a conformal transformation to extract from the
non-linear kinetic equations the exact power law solutions. This
transformation---now called the Zakharov transformation---was first
proposed for capillary wave turbulence involving triadic
interactions~\cite{Zakharov1966,Zakharov1967}, then for Langmuir wave
turbulence where the interactions are
quartic~\cite{Zakharov1967b,Kaner1970}. There are two types of
solution: the zero flux solution (which was the regime studied
by~\cite{Nordheim1928}) and the non-zero constant flux solution. The
first case corresponds to the thermodynamic solution (constant entropy)
and the second to the Kolmogorov--Zakharov spectrum, which is the most
interesting solution because it is non-trivial and corresponds to a
cascade. 

\subsection{Applications of wave turbulence}

There are many examples of application of wave turbulence. Below is a
non-exhaustive list of applications with some references.\medskip

\noindent
$\bullet$ {\bf Surface wave:} These are the first applications of wave
turbulence. They include capillary waves and gravity waves. The former
involve three-wave interactions, the theory of which was published in
English by~\cite{Zakharov1967} in the deep water limit. Note that the
shallow water limit is also the subject of studies (see for
example~\cite{Clark2014}). Gravity wave turbulence is a problem that
needs to be addressed at the level of four
waves~\cite{Hasselmann1962,BenneyNewell1967}. This regime has been well
reproduced in the laboratory or by direct numerical simulations (see
e.g.~\cite{Deike2011,Zhang2022}). Its detection at sea is more difficult
but not impossible (see e.g.~\cite{Hwang2000,Lenain2017}). The two
subjects being linked, several experiments deal with the interaction
between gravity and capillary waves. All in all, it is a subject that
is still very much under study (see the review
by~\cite{Falcon2022}).\medskip

\noindent
$\bullet$ {\bf Internal gravity wave:} These waves are a variant of the
previous ones in the sense that we are interested here in gravity waves
in oceans. These waves contribute dynamically to the turbulent
transport of heat, which is important to understand to properly
evaluate the impact of  oceans on the climate~\cite{MacKinnon2017}.
Internal gravity wave turbulence is an anisotropic three-wave problem
for which an Eulerian theory has been developed by~\cite{Caillol2000}
as well as several experiments (see e.g.~\cite{Savaro2020}).\medskip

\noindent
$\bullet$ {\bf Inertial wave:} This is the closest example to the
standard case of eddy turbulence in the sense that the equations are
those of Navier--Stokes which are modified simply by adding the
Coriolis force. Inertial wave turbulence is a three-wave anisotropic
problem whose theory has been published by~\cite{Galtier2003}. Several
experiments have been made to study this regime (see
e.g.~\cite{Monsalve2020}). This example will be discussed later.\medskip

\noindent
$\bullet$ {\bf Rossby wave:} These waves appear in a situation of
differential rotation. They are used in the modeling of planetary
atmospheres, the interiors of gaseous planets and stars; these are
often referred to as planetary waves~\cite{LonguetH1967}. In wave
turbulence, the dynamics is driven by three-wave processes with a
dominance of non-local interactions~\cite{Balk1990,Balk1990b}.\medskip

\noindent
$\bullet$ {\bf Plasma wave:} As mentioned above, we find the beginnings
of wave turbulence in the field of plasma
physics~\cite{Sagdeev1966,Vedenov67}. In this very vast domain, waves
are legion. Incompressible \mbox{magnetohydrodynamics (MHD)} is an
important example of application because it is a model often used in
astrophysics. We talk about Alfv\'en wave turbulence which involves
three-wave interactions. The theory was published
by~\cite{Galtier2000}: it is a case where the anisotropy is so strong
that the cascade is completely inhibited in the direction of the strong
applied magnetic field.\medskip

\noindent
$\bullet$ {\bf Geodynamo wave:} Geodynamo waves are defined as waves
present in the Earth's outer liquid core. These waves take part in the
dynamo effect, i.e. the physical mechanism that maintains the Earth
magnetic field~\cite{Finlay2008}. These are magnetostrophic waves and
inertial waves. The three-wave theory models a homogeneous medium with
a small Rossby number~\cite{Galtier2014}: in this framework, an
anisotropic inverse cascade of hybrid helicity is predicted, which
could be at the origin of the regeneration of the magnetic field on a
large scale~\cite{Menu2019}.\medskip

\noindent
$\bullet$ {\bf Acoustic wave:} Acoustic wave turbulence is a priori
driven by three-wave interactions but these waves are semi-dispersive.
This is a critical situation for the application of wave turbulence
because the uniformity of the development is not guaranteed. The first
works on the subject date back to the early 1970s~\cite{Zakharov1970}.
The way in which the asymptotic is subtly modified is discussed
by~\cite{Newell1971}, then by~\cite{Lvov1997}. It can be shown for this
regime that energy is at best redistributed according to rays (in
three-dimensional Fourier space). Note that a similar situation can be
found in compressible MHD~\cite{Galtier2023}.\medskip

\noindent
$\bullet$ {\bf Elastic wave:} It is a turbulence produced by a thin
elastic plate or by the introduction of long polymer molecules in a 
liquid~\cite{Steinberg2021}. In the former case, we are in a very
different situation from the traditional turbulence produced by a
fluid. The vibrations of a plate can in theory produce weak or strong
wave turbulence, depending on the forcing. The theory of wave
turbulence has been published by~\cite{During2006}: it is a four wave
problem characterized by a direct energy cascade. Although the wave
action is not conserved in this problem, direct numerical simulations
show an inverse cascade which seems to be established in an explosive
way~\cite{During2015}. Laboratory experiments using steel plates have
been carried out in order to reproduce with varied success the theoretical
predictions~\cite{Boudaoud2008,Mordant2008,Cobelli2009,Mordant2010}.\medskip

\noindent
$\bullet$ {\bf Optical wave:} Wave turbulence is also found in the
field of non-linear optics. The multi-dimensional non-linear
Schr\"odinger equation can be used to describe the evolution of
quasi-monochromatic plane wave envelopes~\cite{Sulem1999}. The theory
is explained by~\cite{Dyachenko1992}: this wave turbulence is governed
by four-wave interactions and an intermittency mechanism characterized
by a collapse phenomenon in physical space can occur. In the case of a
reduction of the problem to one dimension, it can be shown that the
dominant resonant interactions imply six waves, involving a much slower
non-linear dynamics~\cite{Laurie2012}. All this work is part of the
study of non-linear effects on the propagation of incoherent optical
beams~\cite{Mitchell1996,Picozzi2014}.\medskip

\noindent
$\bullet$ {\bf Quantum turbulence:} This subject is very close to the
previous one in the sense that the model used is a variant of the
non-linear Schr\"odinger equation: by changing the sign of the
non-linear term, the interaction becomes repulsive and the physics of
turbulence is modified. The associated equation---also called the
Gross-Pitaevskii equation---describes a Bose gas at very low
temperatures. Note that the emergence of quantum turbulence in an
oscillating Bose--Einstein condensate has been experimentally
demonstrated~\cite{Henn2009}. The wave turbulence regime is described
by~\cite{Dyachenko1992}: it is shown that four-wave interactions
conserve energy and wave action. The inverse cascade associated with
the latter leads to the formation of a stable condensate, i.e. the
accumulation of wave action in the $k=0$ mode. Direct numerical
simulations in two~\cite{Nazarenko2006BEC,Nazarenko2007} and
three~\cite{Proment2009,Proment2012} dimensions illustrate this
regime.\medskip

\noindent
$\bullet$ {\bf Kelvin wave:} Kelvin waves are studied in the context of
superfluids whose temperature is close to the absolute zero. These
waves can propagate along filaments of vorticity and modify the
dynamics of turbulence~\cite{Vinen2000,Kivotides2001}. Kelvin's theory
of wave turbulence has been proposed by~\cite{Kozik2004}: it involves
six-wave processes that conserve both energy and wave action. As these
cascade processes are extremely slow, it is particularly interesting to
consider a local model of non-linear
diffusion~\cite{Nazarenko2006}.\medskip

\noindent
$\bullet$ {\bf Gravitational wave:} This example illustrates the wide
range of possible applications of wave turbulence. It is about
cosmology and more precisely the birth of the Universe: primordial
gravitational waves could be at the origin of the inflation phase of
the Universe, a question which is still open to this day and which
touches the limits of our knowledge~\cite{Galtier2020gw}. The theory
has been published by~\cite{Galtier2017,Galtier2021}: the dynamics is
governed by four-wave processes sufficiently symmetric to conserve wave
action. The latter is characterized by an explosive inverse cascade. It
is interesting to note that this regime is close to that of elastic
waves in the strong tension limit as shown by the theoretical,
numerical and experimental work of~\cite{Hassaini2019}. 

\section{Duffing's equation: a pedagogical model}

\subsection{Classical perturbation method}

The aim of this section is to discuss on a pedagogical/toy model the
idea of the multiple time scale method through the Duffing equation 
{\begin{equation}
\frac{\mathrm{d}^2 f}{\mathrm{d}t^2} + f = -\epsilon f^3 ,
\end{equation}}\unskip
where $\epsilon$ is a small parameter ($0<\epsilon\ll 1$) which
measures the intensity of the non-linearity and $f$ is a function of
time only. The main solution to this equation is a harmonic oscillator.
This solution will, however, be slightly modified over time by the
presence of the small non-linear perturbation (right hand side term).
Let us first use the classical perturbation theory and introduce the
following development
{\begin{equation}
f = \sum_{n=0}^{+\infty} \epsilon^{n} f_{n} . 
\end{equation}}\unskip
We then obtain an infinite system of equations which, for the first
three orders, can be written
{\begin{subequations}
\begin{eqnarray}
& \displaystyle \mathcal{O} (\epsilon^0):\quad \frac{\mathrm{d}^2
f_{0}}{\mathrm{d}t^2} + f_{0} = 0 , \Seqnsplit
& \displaystyle \mathcal{O} (\epsilon^1):\quad \frac{\mathrm{d}^2
f_{1}}{\mathrm{d}t^2} + f_{1} = -f^{3}_{0} , \Seqnsplit
& \displaystyle \mathcal{O} (\epsilon^2):\quad \frac{\mathrm{d}^2
f_{2}}{\mathrm{d}t^2} + f_{2} = -3f^{2}_{0}f_{1}.
\end{eqnarray}
\end{subequations}}\unskip
We can see that the solution to a given order will affect the solution
to the higher order. The first solution is trivially $f_{0}=A
\cos(t+\phi)$. By introducing this one in the second equation, we can
deduce the exact solution for $f_{1}$. If we stop at this order, the
solution reads
{\begin{equation}
f(t)= A \cos(t+\phi) + \epsilon A^{3} \left[ -\tfrac{3}{8} t \sin
(t+\phi) + \tfrac{1}{32} \cos (3t+3\phi) \right] + \mathcal{O}
(\epsilon^2) .
\label{19}
\end{equation}}\unskip
As a first approximation, this system behaves well as a harmonic
oscillator. The condition to be checked is that the non-linear
perturbation remains of weak amplitude and the time considered
relatively short. On the other hand, over long periods of the order of
$\mathcal{O} (1/\epsilon)$, this solution is modified in a
non-negligible way. The term which is at the origin of it, in $\epsilon
t$, is said to be secular. For even longer periods of time the
development diverges; this divergence is all the more so as certain
secular terms of a higher order reinforce it. Then, the uniformity of
the development is broken.\looseness=-1 

What information is useful for our problem? This very simple system
illustrates schematically the problems that we can encounter in a
classical perturbation  development. Such a \mbox{development} should
allow us, in principle, to follow the evolution of the dynamics of wave
\mbox{turbulence} at various orders in $\epsilon$. The main problem is
to ensure that the development remains neat (or uniform). In other
words, this means making sure that no secular term of the order $n$
comes to interfere in the dynamics at an order $m$, such as $m < n$, in
order not to find oneself in the same situation
as~\cite{Hasselmann1962}, i.e. in the presence of a secular term. This
type of term constitutes an obstacle to the closure of the hierarchy of
equations. The more sophisticated method of multiple time
scales~\cite{Poincare1893} will allow us to solve this problem. 

\subsection{Multiple time scale method}

Let us take the Duffing equation again and introduce the following
independent time variables
{\begin{equation}
T_{n} \equiv \epsilon^{n} t \quad \text{with}\ n=0, 1, 2, \ldots .
\label{METM1}
\end{equation}}\unskip
The power series development of the variable $f$ is then written ($t
\equiv T_0$)
{\begin{equation}
f = \sum_{n=0}^{+\infty} \epsilon^{n} f_{n}(t, T_{1}, T_{2},\ldots),
\label{METM2}
\end{equation}}\unskip
and the time derivative becomes a sum of partial derivatives
{\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} = \sum_{n=0}^{+\infty} \epsilon^{n}
\frac{\partial}{\partial T_{n}}.
\label{METM3}
\end{equation}}\unskip
The introduction of relations (\ref{METM1})--(\ref{METM3}) into the
Duffing equation gives
{\begin{eqnarray}
&&\left(\frac{\partial}{\partial t} + \epsilon \frac{\partial}{\partial
T_{1}} + \epsilon^{2} \frac{\partial}{\partial T_{2}} + \cdots\right)^{2}
(f_0 + \epsilon f_1 + \epsilon^{2} f_{2} +\cdots)  + (f_0 + \epsilon f_1 +
\epsilon^{2} f_{2} + \cdots) \nonumber\\
&&\quad= -\epsilon (f_0 + \epsilon f_1 + \epsilon^{2} f_{2} +\cdots)^{3}. 
\end{eqnarray}}\unskip
We obtain an infinite system of equations which, for the first two
orders, can be written
{\begin{subequations}
\begin{eqnarray}
& \displaystyle \mathcal{O} (\epsilon^0):\quad \frac{\partial^2
f_{0}}{\partial t^2} + f_{0} = 0 , \Seqnsplit
& \displaystyle \mathcal{O} (\epsilon^1):\quad \frac{\partial^2
f_{1}}{\partial t^2} + f_{1} + 2 \frac{\partial^{2} f_{0}}{\partial t
\partial T_{1}}= -f^{3}_{0} . \label{118}
\end{eqnarray}
\end{subequations}}\unskip
We can see the presence of a new term in the time evolution equation of
$f_{1}$ whose importance emerges over long times. We look for a
solution of the form
{\begin{equation}
f_{0}=A(T_{1},T_{2},\ldots) \cos(t+\phi(T_{1},T_{2},\ldots)) ,
\end{equation}}\unskip
with an amplitude $A$ and a phase $\phi$ which can vary slowly over
time (hence the dependence in $T_1$, $T_2$, etc).  The introduction of
previous expression in (\ref{118}) gives the relation
{\begin{eqnarray}
\frac{\partial^2 f_{1}}{\partial t^2}  + f_{1} = 2 \frac{\partial
A}{\partial T_{1}} \sin(t+\phi)  + \left(2 A \frac{\partial
\phi}{\partial T_{1}} - \frac{3}{4} A^{3}\right) \cos(t+\phi)
-\frac{A^{3}}{4} \cos(3(t+\phi)) . 
\end{eqnarray}}\unskip
To ensure that the $f_{1}$ solution does not include a secular term,
the following conditions must be imposed
{\begin{equation}
\frac{\partial A}{\partial T_{1}} = 0 \quad \text{and} \quad
\frac{\partial \phi}{\partial T_{1}} = \frac{3}{8} A^{2} ,
\end{equation}}\unskip
which gives $\phi=(3/8)A^{2}\epsilon t + \theta$. The solution is then
written
{\begin{equation} \label{duffing2}
f(t) = A \cos\left(t + \frac{3}{8} A^2 \epsilon t + \theta\right) + 
\frac{\epsilon A^3}{32} \cos\left(3t + \frac{9}{8} A^2 \epsilon t + 3
\theta\right) + \mathcal{O} (\epsilon^2) , 
\end{equation}}\unskip
with $A(T_{2},T_{3},\ldots)$ and $\theta(T_{2},T_{3},\ldots)$. In the
order of the truncation, we can therefore consider that $A$ and
$\theta$ are constant.  Note that this development is compatible with
the previous one (\ref{19}) for $t \le \mathcal{O} (\epsilon)$. This
example illustrates the fact that we can achieve a uniform development
in a systematic way using the multiple time scale method. 

In Figure \ref{Fig6-1ab} we show the result of a numerical simulation
of the Duffing equation with $\epsilon=0.01$. The function $f(t)$ is
plotted in solid line. At the top, we see the evolution for $t \in
[0,200]$ and at the bottom, we display an enlargement of the end of the
simulation. The mathematical solutions (\ref{19}) and (\ref{duffing2})
are plotted in dotted and dashed lines, respectively. The divergence
with $t$ of the solution (\ref{19}) whose origin is the secular term
can be seen. As expected, this divergence occurs over a time $t$ of the
order of $\mathcal{O} (1/\epsilon)$ (the reference time is the wave
period). On the other hand, the solution (\ref{duffing2}) remains close
to the real solution until the final time. 

\begin{figure}
\includegraphics{fig02}
\caption{\label{Fig6-1ab}Variation of $f(t)$ (solid line) obtained by a numerical
simulation of the Duffing equation with $\epsilon=0.01$. The solutions
(\ref{19}) and (\ref{duffing2}) are plotted in red dotted and black
dashed lines, respectively.}
\end{figure}

\section{Sequential time closures for three-wave interactions}

\subsection{Fundamental equation}

We present here a detailed derivation of the kinetic equation for
three-wave interactions, a situation of interest for the Navier--Stokes
equations when capillary waves or inertial waves are involved, two
examples that will be used to illustrate the theory of wave turbulence.

Consider the following inviscid model
{\begin{equation}
{\partial u(\mathbf{x},t) \over \partial t} = {L}(u) + \epsilon \, {N}(u,u),
\label{eq1}
\end{equation}}\unskip
where $u$ is a random variable such that $\langle u \rangle = 0$ (with
$\langle \rangle$ an ensemble average), ${ L}$ is a linear operator
which guarantees that the waves are linear solutions to the problem,
and ${ N}$ is a non-linear quadratic operator (thus involving a priori
triadic interactions). The coefficient $\epsilon$ is a small parameter
($0 < \epsilon \ll 1$) which measures the amplitude of the
non-linearities (it is originally a measure of the wave amplitude). The
Fourier transform in $d$-dimension is 
{\begin{equation}
\hat u(\kk,t) \equiv {1 \over (2\rmpi)^d} \int_{\mathbb{R}^{d}}
u(\mathbf{x},t) \mathrm{e}^{-\mathrm{i} \kk \cdot \mathbf{x}}
\,\mathrm{d}\mathbf{x}.
\end{equation}}\unskip
Note that we consider a continuous medium which can lead to
mathematical difficulties connected with  infinite dimensional phase
spaces. For this reason, it is preferable to assume a variable
spatially periodic over a box of finite size $L$. However, in the
derivation of the kinetic equation, the limit $L \to +\infty$ is
finally taken (before the long time limit, or equivalently the limit
$\epsilon \to 0$). As both approaches lead to the same kinetic
equation, for simplicity, we anticipate this result and follow the
original approach of~\cite{Benney1966}. 

The Fourier transform of Equation~(\ref{eq1}) takes the following
schematic form when the canonical variables are introduced
{\begin{equation} \label{21e}
\left( {\partial \over \partial t} + \mathrm{i} s_k \omega_k \right) A^{s_k}
(\kk,t) =  \epsilon \sum_{s_{p} s_{q}}  \int_{\mathbb{R}^{2d}}
\HH_{-\kk \pp \qq} A^{s_p}(\pp,t) A^{s_q}(\qq,t) \delta(\kk-\pp-\qq)
\,\mathrm{d}\pp \,\mathrm{d}\qq ,
\end{equation}}\unskip
with $s_i=\pm$ the directional polarity. On the left hand side, we find
$\omega_k$ which is fixed by the dispersion relation (we assume $0 <
\omega_k \propto \vert \kk \vert^x$ with $x>1$) while on the right hand
side $\HH_{\kk \pp \qq}$ is an operator which depends on the shape of
the non-linearities and is symmetrical in $\pp$ and $\qq$ (i.e.\ 
$\HH_{\kk \qq \pp} = \HH_{\kk \pp \qq}$). We recall that the presence
of the Dirac function originates in the convolution product, and that
the quadratic nature of the non-linearities leads to triadic
interactions. 

Equation~(\ref{21e}) is simplified by making the following change of
variables (the writing is simplified in passing)
{\begin{equation}
A^{s_k}(\kk,t) = a^{s_k}(\kk,t) \mathrm{e}^{-\mathrm{i} s_k\omega_k t}
= a^{s_k}_{k} \mathrm{e}^{-\mathrm{i} s_k\omega_k t} .
\label{phaseamplitude}
\end{equation}}\unskip
In the interaction representation, we obtain the fundamental equation
{\begin{equation}
{\partial a^{s_k}_{k} \over \partial t} = \epsilon \sum_{s_{p} s_{q}}
\int_{\mathbb{R}^{2d}} \HH_{-\kk \pp \qq} a^{s_{p}}_{p} a^{s_{q}}_{q} 
\mathrm{e}^{\mathrm{i} \Omega_{k,pq}t} \delta_{k,pq}\, \mathrm{d}\pp
\,\mathrm{d}\qq ,
\label{eq6}
\end{equation}}\unskip
with by definition $\delta_{k,pq} \equiv \delta(\kk-\pp-\qq)$ and
$\Omega_{k,pq} \equiv s_k\omega_k - s_{p}\omega_p -s_{q}\omega_q$. 
Additional properties can be established if first we use the relation
$A^{s_k}(\kk)=(A^{-s_k}(-\kk))^*$, with $*$ the complex conjugate:
then, we have necessarily $\HH_{\kk \pp \qq} = \HH^*_{-\kk -\pp -\qq}$.
As mentioned above, we consider only zero mean fluctuations in the
physical space, which means with our notation $a^{s_k}(\kk=0,t) = 0$
(below, we will assume  statistical spatial homogeneity) and therefore,
$\HH_{\mathbf{0} \pp \qq} = 0$. Finally, if $\HH_{\kk \pp \qq}$ is purely
imaginary, we have $\HH^*_{\kk \pp \qq}=-\HH_{\kk \pp \qq}$. This
property may be used to simplify the kinetic equations. 

Equation~(\ref{eq6}) highlights the temporal evolution of the wave
amplitude: this evolution is a priori relatively slow since it induces
a non-linear term proportional to $\epsilon$. We will find this
property with the multiple time scale method. The presence of the
complex exponential is fundamental for the asymptotic closure: since we
are interested in the dynamics over a long time with respect to the
wave period, the contribution of this exponential is essentially zero.
Only some terms will survive: those for which $\Omega_{k,pq}=0$.  With
the condition imposed by the Dirac, we obtain the resonance condition
{\begin{eqnarray} \label{cr2425z}
\left\{
\begin{array}{l}
\kk =  \pp+\qq , \\
s_k \omega_k =  s_{p}\omega_p +s_{q}\omega_q .
\end{array}
\right.
\end{eqnarray}}\unskip

\subsection{Dispersion relation and resonance}

The triadic resonance condition (\ref{cr2425z}) does not always have a
solution. To realize this, we can represent this condition
geometrically in the two-dimensional and isotropic case: 
Figure~\ref{Fig6-2} shows a (axisymmetric) dispersion relation of the
type $\omega_{k} \sim \vert \kk \vert^{x}$, with $x>1$ (left) and
$0<x<1$ (right). The solutions of the resonance condition
(\ref{cr2425z}) correspond to the intersection between the surface
$\omega(\pp)$ (which is identified with that of $\omega(\kk)$) and the
surface $\omega(\qq)$. We find that this intersection exists only for a
convex dispersion relation ($x>1$). The concave case corresponding to
$0<x<1$ is the one encountered a priori with gravity waves for which
$x=1/2$. Note that for anisotropic problems like inertial wave
turbulence this simple picture does not work. 

\begin{figure}
\includegraphics{fig03}
\caption{\label{Fig6-2}Resonance condition for a dispersion relation of the type
$\omega_{k} \sim \vert \kk \vert^{x}$ with $x>1$ (left) and $0<x<1$
(right).}
\end{figure}

The special case $x=1$ is that of non-dispersive (or semi-dispersive)
waves. We can easily see that the only possible solution is that where
all three wave vectors are aligned. Although a solution to the triadic
resonance condition exists, an asymptotic closure of the hierarchy of
equations is not guaranteed. Physically, we can easily understand this
problem: two semi-dispersive waves propagate at the same speed,
therefore, if they initially overlap their interactions will be strong,
otherwise they will never interact. Using the multiple time scale
method presented in the next section, it can be shown that, in one
dimension, a contribution from non-linear terms is possible on a time
scale of the order of $T_{1}$ while the closure is made on a longer
time scale in $T_{2}$~\cite{Benney1966}. The appearance of these
secular contributions requires a new analysis to suppress them and
justify the application of the wave turbulence theory. This situation
can be found in acoustic wave turbulence or in fast magneto-acoustic
wave turbulence in three dimensions~\cite{Galtier2023}. 

Note that for four-wave interactions, the situation is less
constraining because new possibilities exist. For example, gravity
waves can be analyzed at this order. In the case of non-dispersive
waves, it may be possible to develop a theory of wave turbulence
because the associated four wave vectors are not necessarily collinear.
An example is given by gravitational waves for which the dispersion
relation is $\omega_{k}=c k$, with $c$ the speed of
light~\cite{Galtier2017}. 

\subsection{Multiple time scale method}

Following our analysis of the Duffing equation, we wish to make a
development of Equation~(\ref{eq6}) following the method of multiple
time scales~\cite{Poincare1893}. We introduce a sequence of time
scales, $T_{0}, T_{1}, T_{2}, \ldots,$ which will be treated as
independent variables, with
{\begin{equation}
T_{0} \equiv t, \quad T_{1}\equiv \epsilon t, \quad T_{2}\equiv
\epsilon^{2} t, \ldots .
\end{equation}}\unskip
The variation of the wave amplitude with $T_1$ and $T_2$ represents the
slow variation that we wish to extract.  Note that it is because of the
weak dependence in $t$ of $T_{1}, T_{2}, \ldots,$ that all these
variables can be treated (it is an approximation) as independent.  (The
smaller $\epsilon$ is, the better the approximation.) We obtain ($T_0$
being replaced by $t$)
{\begin{eqnarray}
\left({\partial \over \partial t} + \epsilon {\partial \over \partial
T_{1}} + \epsilon^{2} {\partial \over \partial T_{2}} + \cdots\right)
a^{s_k}_{k} =  \epsilon \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}}
\HH_{-\kk \pp \qq} a^{s_{p}}_{p} a^{s_{q}}_{q} \mathrm{e}^{\mathrm{i}
\Omega_{k,pq}t} \delta_{k,pq}  \mathrm{d}\pp \mathrm{d}\qq . 
\label{eq6bis}
\end{eqnarray}}\unskip
The variable $a^{s_k}_{k}$ must also be expanded to the power of
$\epsilon$ and make the various scales appear in time
{\begin{equation} \label{adevl}
a^{s_k}_{k} = \sum_{n=0}^{+\infty} \epsilon^n
a^{s_k}_{k,n}(t,T_{1},T_{2},\ldots) = a^{s_k}_{k,0} + \epsilon
a^{s_k}_{k,1} + \epsilon^{2} a^{s_k}_{k,2} + \cdots .
\end{equation}}\unskip
Expression (\ref{adevl}) is then introduced into the fundamental
equation (\ref{eq6bis}). We obtain for the first three terms
{\begin{subequations}
\begin{eqnarray}
& \displaystyle {\partial a^{s_k}_{k,0} \over \partial t} = 0,\Seqnsplit
& \displaystyle {\partial a^{s_k}_{k,1} \over \partial t} = - {\partial
a^{s_k}_{k,0} \over \partial T_{1}} +  \sum_{s_{p} s_{q}}
\int_{\mathbb{R}^{2d}} \HH_{-\kk \pp \qq} a_{p,0}^{s_{p}}
a_{q,0}^{s_{q}} \mathrm{e}^{\mathrm{i} \Omega_{k,pq}t} \delta_{k,pq}
\mathrm{d}\pp \mathrm{d}\qq , \Seqnsplit
& \displaystyle {\partial a^{s_k}_{k,2} \over \partial t} = - {\partial
a^{s_k}_{k,1} \over \partial T_{1}} - {\partial a^{s_k}_{k,0} \over
\partial T_{2}} + \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk
\pp \qq} \left[ a_{p,1}^{s_{p}} a_{q,0}^{s_{q}} + a_{p,0}^{s_{p}}
a_{q,1}^{s_{q}} \right] \mathrm{e}^{\mathrm{i} \Omega_{k,pq}t}
\delta_{k,pq} \mathrm{d}\pp \mathrm{d}\qq . 
\end{eqnarray}
\end{subequations}}\unskip
To lighten the writing, the time dependency of the variables has been
omitted. After integration on $t$, one finds
{\begin{subequations}
\begin{eqnarray}
& \displaystyle 
a^{s_k}_{k,0} = a^{s_k}_{k,0}(T_1,T_2,\ldots) ,
\label{137} \Seqnsplit
& \displaystyle 
a^{s_k}_{k,1} = -t {\partial a^{s_k}_{k,0} \over \partial T_{1}} +
b^{s_k}_{k,1} , \label{138} \Seqnsplit
& \displaystyle 
a^{s_k}_{k,2} = \frac{t^2}{2} {\partial^2 a^{s_k}_{k,0} \over
\partial T_{1}^2} -t {\partial a^{s_k}_{k,0} \over \partial T_{2}}  -
\int_0^t {\partial b^{s_k}_{k,1} \over \partial T_{1}} \mathrm{d}t +
\tilde b^{s_k}_{k,2} , \label{139} 
\end{eqnarray}
\end{subequations}}\unskip
with by definition
{\begin{eqnarray} \label{bdelta}
& \displaystyle
b^{s_k}_{k,1} \equiv \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}}
\HH_{-\kk \pp \qq} a_{p,0}^{s_{p}} a_{q,0}^{s_{q}}
\Delta(\Omega_{k,pq}) \delta_{k,pq} \,\mathrm{d}\pp \,\mathrm{d}\qq,\Seqnsplit 
& \displaystyle
\Delta(X) \equiv \int_{0}^{t} \mathrm{e}^{\mathrm{i} Xt} \mathrm{d}t  =
\frac{\mathrm{e}^{\mathrm{i} Xt} -1}{\mathrm{i} X} ,
\end{eqnarray}}\unskip
and
{\begin{equation} \label{bdelta2a}
\tilde b^{s_k}_{k,2} \equiv \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}}
\HH_{-\kk \pp \qq} \int_{0}^{t}  \left[ a_{p,1}^{s_{p}} a_{q,0}^{s_{q}}
+ a_{p,0}^{s_{p}} a_{q,1}^{s_{q}}\right] \mathrm{e}^{\mathrm{i}
\Omega_{k,pq}t} \mathrm{d}t \delta_{k,pq} \,\mathrm{d}\pp \,\mathrm{d}\qq .
\end{equation}}\unskip
The previous equation is modified when expression (\ref{138}) is
introduced; one finds
{\begin{eqnarray}
\tilde b^{s_k}_{k,2} &=& -\sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}}
\HH_{-\kk \pp \qq}  \frac{\partial (a^{s_p}_{p,0}
a_{q,0}^{s_{q}})}{\partial T_{1}} \left( \int_0^t t
\mathrm{e}^{\mathrm{i} \Omega_{k,pq}t} \mathrm{d}t \right)
\delta_{k,pq} \mathrm{d}\pp
\mathrm{d}\qq \nonumber\\
&& +\, \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk \pp \qq} 
\left( \int_{0}^{t} (b^{s_p}_{p,1} a_{q,0}^{s_{q}} + b^{s_q}_{q,1}
a_{p,0}^{s_{p}} ) \mathrm{e}^{\mathrm{i} \Omega_{k,pq}t} \mathrm{d}t
\right)  \delta_{k,pq} \,\mathrm{d}\pp \,\mathrm{d}\qq .
\end{eqnarray}}\unskip
Expression (\ref{139}) becomes
{\begin{equation} \label{34e}
a^{s_k}_{k,2} = \frac{t^2}{2} {\partial^2 a^{s_k}_{k,0} \over \partial
T_{1}^2} -t {\partial a^{s_k}_{k,0} \over \partial T_{2}} +
b^{s_k}_{k,2} , 
\end{equation}}\unskip
with
{\begin{eqnarray*}
b^{s_k}_{k,2} &=& -\sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk
\pp \qq} \frac{\partial (a^{s_p}_{p,0} a_{q,0}^{s_{q}})}{\partial
T_{1}} \left( \int_0^t \left[\Delta( \Omega_{k,pq}) + t
\mathrm{e}^{\mathrm{i} \Omega_{k,pq}t} \right] \mathrm{d}t \right) 
\delta_{k,pq} \,\mathrm{d}\pp \,\mathrm{d}\qq \\
&&+\, 2 \sum_{s_{p} s_{q} s_{p'} s_{q'} } \int_{\mathbb{R}^{4d}}
\HH_{-\kk \pp \qq} \HH_{-\pp \pp' \qq'} a_{p',0}^{s_{p'}}
a_{q',0}^{s_{q'}}  a_{q,0}^{s_{q}} \nonumber\\
&&\times\,\left( \int_{0}^{t} \Delta( \Omega_{p,p'q'})
\mathrm{e}^{\mathrm{i} \Omega_{k,pq}t} \mathrm{d}t \right)
\delta_{k,pq} \delta_{p,p'q'} \,\mathrm{d}\pp\, \mathrm{d}\qq\,
\mathrm{d}\pp' \,\mathrm{d}\qq' .
\end{eqnarray*}}\unskip
The time integrals give the relations
{\begin{eqnarray}
& \displaystyle
\int_0^t \left[\Delta( \Omega_{k,pq}) + t \mathrm{e}^{\mathrm{i}
\Omega_{k,pq}t} \right] \mathrm{d}t = t \Delta( \Omega_{k,pq}) ,
\Seqnsplit
& \displaystyle
\int_{0}^{t} \Delta( \Omega_{p,p'q'}) \mathrm{e}^{\mathrm{i}
\Omega_{k,pq}t} \mathrm{d}t = \frac{\Delta(\Omega_{k,p'q'q}) -
\Delta(\Omega_{k,pq})}{\mathrm{i} (\Omega_{k,p'q'q}-\Omega_{k,pq})} ,
\end{eqnarray}}\unskip
that will be used below in the long time limit. 

\subsection{First asymptotic closure at time $T_1$}

With the previous definitions, the perturbative expansion of the
second-order moment writes
{\begin{eqnarray} 
\langle a^{s_k}_{k} a^{s_{k'}}_{k'} \rangle &=&  \langle (a^{s_k}_{k,0}
+ \epsilon a^{s_k}_{k,1} + \epsilon^2 a^{s_k}_{k,2} +
\cdots)(a^{s_{k'}}_{k',0} + \epsilon a^{s_{k'}}_{k',1} + \epsilon^2
a^{s_{k'}}_{k',2} + \cdots) \rangle \nonumber\\
&=& \langle a^{s_k}_{k,0} a^{s_{k'}}_{k',0} \rangle + \epsilon \langle
a^{s_k}_{k,0} a^{s_{k'}}_{k',1} + a^{s_k}_{k,1} a^{s_{k'}}_{k',0}
\rangle  + \epsilon^2 \langle a^{s_k}_{k,0} a^{s_{k'}}_{k',2} +
a^{s_k}_{k,1} a^{s_{k'}}_{k',1} + a^{s_k}_{k,2} a^{s_{k'}}_{k',0} 
\rangle + \cdots \label{37ez}
\end{eqnarray}}\unskip
We shall assume that this turbulence is statistically homogeneous. In
this case, the second-order moment can be written in term of
second-order cumulant $q^{s_ks_{k'}}(\kk,\kk') \equiv
q_{k}^{s_ks_{k'}}$ such that 
{\begin{equation}
\langle a^{s_k}_{k} a^{s_{k'}}_{k'} \rangle = q_{k}^{s_ks_{k'}} \delta
(\kk+\kk') ,
\end{equation}}\unskip
where the presence of $\delta (\kk+\kk')$ is the consequence of the
statistical homogeneity~\cite{GaltierCUP2023}.  We also assume---this
is the basic idea of the method---that the second-order moment (in
fact, the coefficient $q_{k}^{s_ks_{k'}}$ in front of the Dirac
function) in the left hand side remains bounded at all
time~\cite{Benney1966}.  As an example, we can think of the energy
spectrum which, as we know, remains physically bounded. Therefore, the
contributions on the right hand side must also be bounded (in the sense
introduced above). We will see that secular terms can appear at
different orders in $\epsilon$; this leads to certain conditions to
cancel their contributions in order to keep the development uniform in
time. As we shall see, at order $\mathcal{O} (\epsilon^2)$ this
condition leads to the so-called kinetic equations. 

At order $\mathcal{O} (\epsilon^0)$, we have the contribution of
$\langle a^{s_k}_{k,0} a^{s_{k'}}_{k',0} \rangle$ which will therefore
be assumed to be bounded at all times. 

At order $\mathcal{O} (\epsilon^1)$, we have the contribution
{\begin{eqnarray} 
\langle a^{s_k}_{k,0} a^{s_{k'}}_{k',1} + a^{s_k}_{k,1}
a^{s_{k'}}_{k',0} \rangle &=&  \left\langle a^{s_k}_{k,0} \left(-t
{\partial a^{s_{k'}}_{k',0} \over \partial T_{1}} +
b^{s_{k'}}_{k',1}\right) + \left(-t {\partial a^{s_k}_{k,0} \over
\partial T_{1}} + b^{s_k}_{k,1}\right) a^{s_{k'}}_{k',0} \right\rangle
\nonumber\\
&=& -t \frac{\partial}{\partial T_{1}} \langle a^{s_k}_{k,0}
a^{s_{k'}}_{k',0} \rangle + \langle a^{s_k}_{k,0} b^{s_{k'}}_{k',1}  +
b^{s_k}_{k,1} a^{s_{k'}}_{k',0} \rangle .\label{38e}
\end{eqnarray}}\unskip
The first term on the right hand side gives a secular contribution
proportional to $t$. For the second term, we have
{\begin{equation} \label{39e}
\sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk \pp \qq} \langle
a^{s_k}_{k,0} a_{p,0}^{s_{p}} a_{q,0}^{s_{q}} \rangle
\Delta(\Omega_{k,pq}) \delta_{k,pq} \,\mathrm{d}\pp \,\mathrm{d}\qq .
\end{equation}}\unskip
The long time limit ($t \gg 1/\omega$) of this oscillating integral
will be given by the Riemann--Lebesgue lemma (the proof requires to use
of generalized functions)
{\begin{eqnarray}
\Delta(X) = \frac{\mathrm{e}^{\mathrm{i} Xt} -1}{\mathrm{i} X}
\xrightarrow{t \to +\infty} \uppi \delta(X) + \mathrm{i} { P}
\left(\frac{1}{X}\right) ,
\label{RiemannL}
\end{eqnarray}}\unskip
where ${ P}$ is the Cauchy principal value of the integral. Therefore,
the long time limit of expression (\ref{39e}) gives no secular
contribution. The same conclusion is obtained for the third term of
expression (\ref{38e}). Also, the condition to cancel the unique
secular term is
{\begin{equation}
\frac{\partial \langle a^{s_k}_{k,0} a^{s_{k'}}_{k',0} \rangle
}{\partial T_{1}} = 0 ,
\end{equation}}\unskip
which means that the second-order moment does not evolve over a time
scale $T_1$. As will be seen later, a turbulent cascade is only
expected on a time scale $T_2$. 

\subsection{Second asymptotic closure at time $T_2$}

The analysis continues at order $\mathcal{O} (\epsilon^2)$. With
expression (\ref{34e}), the next contribution reads
{\begin{eqnarray}
&&\left\langle a^{s_k}_{k,1} a^{s_{k'}}_{k',1} + a^{s_k}_{k,0}
a^{s_{k'}}_{k',2} + a^{s}_{k,2} a^{s_{k'}}_{k',0} \right\rangle = 
\left\langle \left(-t {\partial a^{s_k}_{k,0} \over \partial T_{1}} +
b^{s_k}_{k,1}\right)\left(-t {\partial a^{s_{k'}}_{k',0} \over \partial
T_{1}} + b^{s_{k'}}_{k',1}\right) \right\rangle \nonumber\\
&& \quad + \left\langle a^{s_k}_{k,0} \left(\frac{t^2}{2} {\partial^2
a^{s_{k'}}_{k',0} \over \partial T_{1}^2} -t {\partial
a^{s_{k'}}_{k',0} \over \partial T_{2}}  + b^{s_{k'}}_{k',2} \right) +
a^{s_{k'}}_{k',0} \left(\frac{t^2}{2} {\partial^2 a^{s_k}_{k,0} \over
\partial T_{1}^2} -t {\partial a^{s_k}_{k,0} \over \partial T_{2}} +
b^{s_k}_{k,2} \right) \right\rangle ,  
\end{eqnarray}}\unskip
which gives after development and simplifications
{\begin{eqnarray} 
\left\langle a^{s_k}_{k,1} a^{s_{k'}}_{k',1} + a^{s_k}_{k,0}
a^{s_{k'}}_{k',2} + a^{s_k}_{k,2} a^{s_{k'}}_{k',0} \right\rangle &=& 
\frac{t^2}{2} \frac{\partial^2 \langle a^{s_k}_{k,0} a^{s_{k'}}_{k',0}
\rangle }{\partial T_{1}^2}  - t \frac{\partial \langle a^{s_k}_{k,0}
a^{s_{k'}}_{k',0} \rangle }{\partial T_{2}}  + \left\langle
b^{s_k}_{k,1} b^{s_{k'}}_{k',1} \right\rangle \nonumber\\
&&-\, t \left\langle {\partial a^{s_k}_{k,0} \over \partial T_{1}}
b^{s_{k'}}_{k',1} + b^{s_k}_{k,1} {\partial a^{s_{k'}}_{k',0} \over
\partial T_{1}} \right\rangle  + \left\langle a^{s_k}_{k,0}
b^{s_{k'}}_{k',2} + a^{s_{k'}}_{k',0} b^{s_k}_{k,2} \right\rangle .\quad
\label{eq44e}
\end{eqnarray}}\unskip
The first term on the right hand side cancels over the long time as
required by the first asymptotic closure. The second term gives a
secular contribution. The other three terms can potentially give a
secular contribution: it is obvious for the fourth term and non-trivial
for the third and fifth terms which require further development. 

The third term on the right hand side writes
{\begin{eqnarray} 
\left\langle b^{s_k}_{k,1} b^{s_{k'}}_{k',1} \right\rangle &=& 
\sum_{s_{p} s_{q} s_{p'} s_{q'}} \int_{\mathbb{R}^{4d}} \HH_{-\kk \pp
\qq} \HH_{-\kk' \pp' \qq'}  \langle a_{p,0}^{s_{p}} a_{q,0}^{s_{q}}
a_{p',0}^{s_{p'}} a_{q',0}^{s_{q'}} \rangle \Delta(\Omega_{k,pq})
\Delta(\Omega_{k',p'q'}) \nonumber\\
&&\times\,\delta_{k,pq} \delta_{k',p'q'} \,\mathrm{d}\pp \,\mathrm{d}\qq
\,\mathrm{d}\pp' \,\mathrm{d}\qq' , \label{46e}
\end{eqnarray}}\unskip
Here again, the theory of generalized functions gives us the long time
behavior of this oscillating integral (with the Poincar\'e--Bertrand
formula) 
{\begin{eqnarray}
\Delta(X) \Delta(-X) \xrightarrow{t \to +\infty} 2 \uppi t
\delta(X) + 2 { P} \left(\frac{1}{X}\right) \frac{\partial}{\partial X}.
\end{eqnarray}}\unskip
Therefore, a secular contribution involving a Dirac function is
possible. The fourth-order moment in expression (\ref{46e}) can be
decomposed into products of second-order cumulant plus a fourth-order
cumulant such that (the statistical homogeneity is used as well as
$\langle a_{k,0}^{s_k}\rangle = 0$)
{\begin{eqnarray}  
\langle a_{p,0}^{s_{p}} a_{q,0}^{s_{q}} a_{p',0}^{s_{p'}}
a_{q',0}^{s_{q'}} \rangle &=&   q^{s_{p}s_{q}s_{p'}s_{q'}}_{pqp',0}
\delta(\pp+\qq+\pp'+\qq')  + q^{s_{p} s_q}_{p,0} q^{s_{p'}
s_{q'}}_{p',0} \delta(\pp+\qq) \delta(\pp'+\qq') \nonumber\\
&&+\, q^{s_{p} s_{p'}}_{p,0} q^{s_{q} s_{q'}}_{q,0} \delta(\pp+\pp')
\delta(\qq+\qq')  + q^{s_{p} s_{q'}}_{p,0} q^{s_{q} s_{p'}}_{q,0}
\delta(\pp+\qq') \delta(\qq+\pp') . \label{stat33}
\end{eqnarray}}\unskip
Note that according to expression (\ref{37ez}) by homogeneity we also
have the relation $\kk=-\kk'$. We obtain 
{\begin{eqnarray} 
\left\langle b^{s_k}_{k,1} b^{s_{k'}}_{k',1} \right\rangle &=& 
\sum_{s_{p} s_{q} s_{p'} s_{q'}} \int_{\mathbb{R}^{4d}} \HH_{-\kk \pp
\qq} \HH_{-\kk' \pp' \qq'}  \left[ q^{s_{p}s_{q}s_{p'}s_{q'}}_{pqp',0}
\delta(\pp+\qq+\pp'+\qq') \right.  \nonumber\\
&& +\, q^{s_{p} s_q}_{p,0} q^{s_{p'} s_{q'}}_{p',0} \delta(\pp+\qq)
\delta(\pp'+\qq')  + q^{s_{p} s_{p'}}_{p,0} q^{s_{q} s_{q'}}_{q,0}
\delta(\pp+\pp') \delta(\qq+\qq') \nonumber\\
&&\left. +\, q^{s_{p} s_{q'}}_{p,0} q^{s_{q} s_{p'}}_{q,0}
\delta(\pp+\qq') \delta(\qq+\pp') \right]  \Delta(\Omega_{k,pq})
\Delta(\Omega_{k',p'q'}) \delta_{k,pq} \delta_{k',p'q'} \,\mathrm{d}\pp
\,\mathrm{d}\qq \,\mathrm{d}\pp' \,\mathrm{d}\qq'. \quad\quad\label{48e}
\end{eqnarray}}\unskip
We are looking for secular contributions. In the second line, the first
term does not contribute since it imposes $\kk=\mathbf{0}$ which cancels
$\HH_{\kk \pp \qq}$ but the second term can contribute when the
conditions $s_p=-s_{p'}$ and $s_q=-s_{q'}$ are satisfied. Likewise, in
the third line a contribution is possible when $s_p=-s_{q'}$ and
$s_q=-s_{p'}$. There is no contribution from the first line, which
means that the situation is the same as if the distribution were
Gaussian (however, we do not make this assumption). Finally, in the
long time limit, the secular contribution ${ C}_t \langle
b^{s_k}_{k,1} b^{s_{k'}}_{k',1} \rangle$ is 
{\begin{eqnarray} 
{ C}_t  \left\langle b^{s_k}_{k,1} b^{s_{k'}}_{k',1} \right\rangle &=& 
4 \uppi t \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk \pp \qq}
\HH_{\kk -\pp -\qq}  q^{s_{p} -s_{p}}_{p,0} q^{s_{q} -s_{q}}_{q,0}
\delta (\Omega_{k,pq}) \delta_{k,pq} \delta_{kk'}\, \mathrm{d}\pp\,
\mathrm{d}\qq \nonumber\\
&=& 4 \uppi t \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \vert \HH_{-\kk
\pp \qq} \vert^2 q^{s_{p} -s_{p}}_{p,0} q^{s_{q} -s_{q}}_{q,0} \delta
(\Omega_{k,pq}) \delta_{k,pq} \delta_{kk'} \,\mathrm{d}\pp \,\mathrm{d}
\qq.\label{49e}
\end{eqnarray}}\unskip

The fourth term on the right hand side of Equation~(\ref{eq44e}) does
not contribute over the long time because it depends on the $T_1$
derivative.  The proof is given by a new relation involving the
$n$-order moments which can be written
{\begin{eqnarray}
\langle a_k^{s_k} a_{k'}^{s_{k'}} a_{k''}^{s_{k''}} \ldots \rangle &=&
\langle a_{k,0}^{s_k} a_{k',0}^{s_{k'}} a_{k'',0}^{s_{k''}} \ldots
\rangle \nonumber\\
&&+\, \epsilon \langle a_{k,1}^{s_k} a_{k',0}^{s_{k'}}
a_{k'',0}^{s_{k''}} \cdots + a_{k,0}^{s_k} a_{k',1}^{s_{k'}}
a_{k'',0}^{s_{k''}} \cdots  + a_{k,0}^{s_k} a_{k',0}^{s_{k'}}
a_{k'',1}^{s_{k''}} \cdots + \cdots\rangle \nonumber\\
&&+\, \epsilon^2 \langle \ldots \rangle + \cdots  
\end{eqnarray}}\unskip
As for the second-order moment, we demand that the moments of order $n$
are bounded. At order $\mathcal{O} (\epsilon)$, we obtain the relation
{\begin{eqnarray}
&&\langle a_{k,1}^{s_k} a_{k',0}^{s_{k'}} a_{k'',0}^{s_{k''}} \cdots +
a_{k,0}^{s_k} a_{k',1}^{s_{k'}} a_{k'',0}^{s_{k''}} \cdots +
a_{k,0}^{s_k} a_{k',0}^{s_{k'}} a_{k'',1}^{s_{k''}} \cdots + \cdots\rangle
\nonumber\\
&&\quad = \left\langle \left(-t {\partial a^{s_k}_{k,0} \over \partial
T_{1}} + b^{s_k}_{k,1}\right) a_{k',0}^{s_{k'}} a_{k'',0}^{s_{k''}}
\cdots+ a_{k,0}^{s_k} \left(-t {\partial a^{s_{k'}}_{k',0} \over
\partial T_{1}} + b^{s_{k'}}_{k',1}\right) a_{k'',0}^{s_{k''}} \cdots +
\cdots \right\rangle \nonumber\\
&&\quad = -t {\partial \langle a^{s_k}_{k,0} a^{s_{k'}}_{k',0}
a_{k'',0}^{s_{k''}} \ldots \rangle \over \partial T_{1}}  + \langle
b^{s_k}_{k,1}a_{k',0}^{s_{k'}} a_{k'',0}^{s_{k''}} \ldots \rangle +
\langle a^{s_k}_{k,0} b^{s_{k'}}_{k',1} a_{k'',0}^{s_{k''}} \ldots
\rangle + \cdots 
\end{eqnarray}}\unskip
We see that only the first term of the last line gives a secular
contribution over the long time, which means that we have to satisfy
the condition
{\begin{equation}
{\partial \langle a^{s_k}_{k,0} a^{s_{k'}}_{k',0} a_{k'',0}^{s_{k''}}
\ldots \rangle \over \partial T_{1}} = 0 ,
\end{equation}}\unskip
at any order n. Therefore, the probability density function does not
depend on $T_1$ and we can assume that the variable itself does not
depend on $T_1$.   It is a mild hypothesis because it is difficult to
imagine a turbulent system where everything fluctuates, and in which it
would be possible to have a $T_1$ dependence for the amplitude whereas
the probability density function has no such dependence.  This shows
that the fourth term on the right hand side of Equation~(\ref{eq44e})
does not contribute in the long time limit. 

The last term of Equation~(\ref{eq44e}) writes 
{\begin{eqnarray}
\left\langle a^{s_k}_{k,0} b^{s_{k'}}_{k',2} + a^{s_{k'}}_{k',0}
b^{s_k}_{k,2} \right\rangle 
&=&\sum_{s_{p} s_{q} s_{p'} s_{q'} } \int_{\mathbb{R}^{4d}} 2 \HH_{-\kk'
\pp \qq} \HH_{-\pp \pp' \qq'}  \left\langle a^{s_k}_{k,0}
a_{p',0}^{s_{p'}} a_{q',0}^{s_{q'}}  a_{q,0}^{s_{q}} \right\rangle\nonumber\\
&&\times\,\left( \frac{\Delta(\Omega_{k',p'q'q}) - \Delta(\Omega_{k',pq})}{i
(\Omega_{k',p'q'q}-\Omega_{k',pq})} \right)  \delta_{k',pq}
\delta_{p,p'q'} \,\mathrm{d}\pp \,\mathrm{d}\qq \,\mathrm{d}\pp'\,
\mathrm{d}\qq' \nonumber\\
&&+\,\sum_{s_{p} s_{q} s_{p'} s_{q'} } \int_{\mathbb{R}^{4d}} 2
\HH_{-\kk \pp \qq} \HH_{-\pp \pp' \qq'}  \left\langle a^{s_{k'}}_{k',0}
a_{p',0}^{s_{p'}} a_{q',0}^{s_{q'}}  a_{q,0}^{s_{q}} \right\rangle\nonumber\\
&&\times\,\left( \frac{\Delta(\Omega_{k,p'q'q}) - \Delta(\Omega_{k,pq})}{i
(\Omega_{k,p'q'q}-\Omega_{k,pq})} \right)  \delta_{k,pq}
\delta_{p,p'q'} \,\mathrm{d}\pp \,\mathrm{d}\qq \,\mathrm{d}\pp' \,\mathrm{d}\qq' .
\end{eqnarray}}\unskip
The secular contributions of these oscillating integrals will be given
by the theory of generalized functions with the relation
{\begin{equation}
\frac{\Delta(X) - \Delta(0)}{iX} \xrightarrow{t \to +\infty}
\uppi t \delta(X) + \mathrm{i}t { P} \left(\frac{1}{X}\right) .
\end{equation}}\unskip
We also need to use the following development (and its symmetric in
$\kk'$)
{\begin{eqnarray}  
\langle a_{k,0}^{s_k} a_{p',0}^{s_{p'}} a_{q',0}^{s_{q'}}
a_{q,0}^{s_{q}} \rangle &=&   q^{s_{k}s_{p'}s_{q'}s_{q}}_{kp'q',0}
\delta(\kk+\pp'+\qq'+\qq)  + q^{s_k s_{q}}_{k,0} q^{s_{p'}
s_{q'}}_{p',0} \delta(\kk+\qq) \delta(\pp'+\qq') \nonumber\\
&&+\, q^{s_k s_{q'}}_{k,0} q^{s_{p'} s_{q}}_{p',0} \delta(\kk+\qq')
\delta(\pp'+\qq)  + q^{s_k s_{p'}}_{k,0} q^{s_{q'} s_{q}}_{q',0}
\delta(\kk+\pp') \delta(\qq'+\qq)  . \label{55e}
\end{eqnarray}}\unskip
In the right hand side of expression (\ref{55e}), the first two terms
do not give a secular contribution, however, the last two terms give a
contribution when the following conditions are satisfied, namely
$s_k=-s_{q'}$, $s_{p'}=-s_q$ and $s_k=-s_{p'}$, $s_{q'}=-s_q$,
respectively. After substitution and simplification, we obtain the
secular contributions in the long time limit
{\begin{eqnarray} 
&&{ C}_t \left\langle a^{s_k}_{k,0} b^{s_{k'}}_{k',2} +
a^{s_{k'}}_{k',0} b^{s_k}_{k,2} \right\rangle  \nonumber\\
&&\quad = +4t \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk' \pp \qq}
\HH_{-\pp -\qq \kk'}  q^{s_k s_{k'}}_{k,0} q^{s_{q} -s_{q}}_{q,0}
\left( \uppi \delta(\Omega_{k',pq}) + \mathrm{i} { P}
\left(\frac{1}{\Omega_{k',pq}}\right) \right)  \delta_{k',pq}
\delta_{kk'} \,\mathrm{d}\pp \,\mathrm{d}\qq \nonumber\\
&&\quad +\,4t \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk \pp
\qq} \HH_{-\pp -\qq \kk}  q^{s_k s_{k'}}_{k,0} q^{s_{q} -s_{q}}_{q,0}
\left( \uppi \delta(\Omega_{k,pq}) + \mathrm{i} { P}
\left(\frac{1}{\Omega_{k,pq}}\right) \right)  \delta_{k,pq}
\delta_{kk'} \,\mathrm{d}\pp \,\mathrm{d}\qq \nonumber\\
&&\quad = 8 \uppi t \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk
\pp \qq} \HH_{-\pp -\qq \kk}  q^{s_k s_{k'}}_{k,0} q^{s_{q}
-s_{q}}_{q,0} \delta(\Omega_{k,pq}) \delta_{k,pq} \delta_{kk'} 
\,\mathrm{d}\pp \,\mathrm{d}\qq.
\end{eqnarray}}\unskip
The last writing is obtained by using the property $\HH^*_{-\kk \pp
\qq} = \HH_{-\kk \pp \qq}$ and the symmetry in $\pp \to -\pp$ and $\qq
\to -\qq$. 

If we impose the nullity of the sum of the different secular
contributions at time $T_2$, we find the condition
{\begin{eqnarray}
\frac{\partial \langle a^{s_k}_{k,0} a^{s_{k'}}_{k',0} \rangle
}{\partial T_{2}} &=& \frac{\partial q^{s_ks_{k'}}_{k,0}}{\partial
T_{2}} \delta_{kk'} \nonumber\\
&=& 4 \uppi \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \vert \HH_{-\kk \pp
\qq} \vert^2 q^{s_{p} -s_{p}}_{p,0} q^{s_{q} -s_{q}}_{q,0} \delta
(\Omega_{k,pq}) \delta_{k,pq} \delta_{kk'} \,\mathrm{d}\pp \,\mathrm{d}\qq
\nonumber\\
&&+\,8 \uppi \sum_{s_{p} s_{q}} \int_{\mathbb{R}^{2d}} \HH_{-\kk \pp \qq}
\HH_{-\pp -\qq \kk}  q^{s_k s_{k'}}_{k,0} q^{s_{q} -s_{q}}_{q,0}
\delta(\Omega_{k,pq}) \delta_{k,pq} \delta_{kk'} \,\mathrm{d}\pp
\,\mathrm{d}\qq . 
\end{eqnarray}}\unskip
After integrating over $\kk'$, using the property that
$q^{s_ks_{k'}}_{k,0}$ is real and thus $s_k=-s_{k'}$ and introducing
$N^{s_k}_k \equiv q^{s_k -s_{k}}_{k,0}$, we end up with (some simple
manipulations are also used to symmetrize the equation)
{\begin{eqnarray} \label{ke58}
\frac{\partial N_k^{s_k}}{\partial t} =  4 \uppi \epsilon^2 \sum_{s_{p}
s_{q}} \int_{\mathbb{R}^{2d}} \HH_{\kk \pp \qq} \left[ \frac{\HH_{-\kk
-\pp -\qq}}{N_k^{s_k}} + \frac{\HH_{-\pp -\qq -\kk}}{N_p^{s_p}} +
\frac{\HH_{-\qq -\pp -\kk}}{N_q^{s_q}} \right] N_k^{s_k} N_p^{s_p}
N_q^{s_q} \delta (\Omega_{kpq}) \delta_{kpq} \,\mathrm{d}\pp \,
\mathrm{d}\qq , \nonumber\\
\end{eqnarray}}\unskip
which is the kinetic equation for three-wave interactions. 

\subsection{Conclusion}

The derivation of the kinetic equation (\ref{ke58}) does not involve
any closure assumption such as the quasi-Gaussianity and the random
phase appears naturally dynamically in wave turbulence. The kinetic
equation is obtained by using a sequential asymptotic closures at times
$T_1$ and $T_2$. This method consists in imposing the nullity of
secular terms (proportional to $t$) which emerge over long times in
order to guarantee a bounded value for the associated moments. These
secular terms do not involve fourth-order cumulants but only products
of second-order cumulants.  Note, however, that the derivation made,
although systematic, says nothing about the remaining terms (not used
to obtain the kinetic equation) in the small $\epsilon$ perturbation
expansion, the implicit conjecture being that they are subdominant. 
Although proving this conjecture in general remains a mathematical
challenge, a first step has recently been taken with the rigorous
derivation of the kinetic equation in the special case of the
non-linear cubic Schrodinger\unskip\break 
equation~\cite{Deng2021,Deng2023}.

In summary, it can be said that wave turbulence for three-wave
interactions is characterized by a dynamics on two time scales. On
short timescales, of the order of the wave period, there is a phase
mixing which leads, due to the dispersive nature of the waves, to the
decoupling of the correlations if initially present and to a statistics
close to Gaussianity, as expected from the central limit theorem. On a
longer time scale, the non-linear coupling---weak at short 
times---becomes non-negligible because of the resonance mechanism. This
coupling leads to a regeneration of the cumulants via the product of
lower order cumulants. It is these terms that are at the origin of the
energy transfer mechanism. 

In this derivation, the system studied is assumed to be of infinite
size and can therefore be treated as continuous. Note that numerical
simulation with its grid of points escapes this description. Effects
(freezing of the cascade) linked to the discretization of the Fourier
space can appear because the resonance conditions are a priori more
difficult to satisfy (see for example~\cite{Connaughton2001} for
capillary waves). In theory, the weaker the non-linearities, the
stronger these effects are. For example, in inertial wave
turbulence~\cite{Bouroudia2008} has shown that discretization effects
become preponderant when the Rossby number, $R_{o}$, is smaller than
$10^{-3}$. Beyond this value, but still for a small $R_{o} \ll 1$,
these effects are negligible because of the quasi-resonances which,
together with the resonances, contribute to the transfer of energy. 

In nature we often encounter three-wave problems. As mentioned above,
sometimes higher-level problems (up to six waves) can occur. Despite
the diversity of situations, the kinetic equation takes a universal
form in the sense that it is written (at leading order)
{\begin{equation}
\frac{\partial N^{s_k}_k}{\partial t} = \epsilon^{2n-4} T(\kk) ,
\end{equation}}\unskip
for $n$-wave interactions, with $T$ the transfer function (or collision
integral). The higher the degree of interaction, the longer the
transfer time  $\tau_{\mathrm{tr}}$. This time is characterized by the relation
$\omega \tau_{\mathrm{tr}} \sim \mathcal{O} (1/\epsilon^{2n-4})$. Since the
characteristic time of the waves is $\tau_{\omega} \sim 1/\omega$ and
the small parameter is none other than the ratio between this time and
the non-linear time $\tau_{\mathrm{NL}}$, we arrive at the following
phenomenological expression for the transfer time
{\begin{equation}
\tau_{\mathrm{tr}} \sim \frac{\tau_{\omega}}{\epsilon^{2n-4}} \sim
\frac{\tau_{\mathrm{NL}}^{2n-4}}{\tau_{\omega}^{2n-5}} .
\end{equation}}\unskip
Therefore, for three-wave interactions, we can use the expression:
$\tau_{\mathrm{tr}} \sim \omega \tau^{2}_{\mathrm{NL}}$. 

\section{Applications to Navier--Stokes equations}

\subsection{Capillary wave turbulence}

Together with gravity waves, capillary waves are the main surface waves
encountered in nature. The latter have an advantage over the former in
that they are easier to treat analytically in the non-linear regime
since they involves three-wave interactions. Following the multiple
time scale method, we obtain the following kinetic equation for
capillary wave turbulence~\cite{Zakharov1967,Galtier2020}
{\begin{equation} \label{KEcw}
\frac{\partial N_k^{s_k}}{\partial t} =  4 \uppi \epsilon^{2} \sum_{s_{p}
s_{q}} \int_{\RR^{4}} \vert \HH_{kpq} \vert^2 s_{p}s_{q}   \left[
\frac{s_{p}s_q}{N_k^{s_k}} + \frac{s_k s_q}{N_p^{s_p}} + \frac{s_k
s_p}{N_q^{s_q}} \right]  N_k^{s_k} N_p^{s_p} N_q^{s_q}
\delta(\Omega_{kpq}) \delta_{kpq} \,\mathrm{d}\pp \,\mathrm{d}\qq ,  
\end{equation}}\unskip
with
{\begin{eqnarray}
\vert \HH_{kpq} \vert^{2} = \frac{\sqrt{\sigma}}{8} \left[s_k (\pp
\cdot \qq + pq) \left(\frac{pq}{k}\right)^{1/4}  + s_{p} (\kk \cdot \qq
+ kq) \left(\frac{qk}{p}\right)^{1/4} + s_{q} (\kk \cdot \pp + kp)
\left(\frac{pk}{q}\right)^{1/4} \right]^{2} ,
\end{eqnarray}}\unskip
where $\omega_{k} = \sqrt{\sigma} k^{3/2}$ and $\sigma=\gamma /
\rho_{\mathrm{water}}$, with $\gamma$ the surface tension coefficient
(for the air--water interface $\gamma \simeq 0.07$~N/m) and
$\rho_{\mathrm{water}}$ the mass density of the water. 

\subsection{Inertial wave turbulence}

Inertial wave turbulence is the regime reached by an incompressible
fluid subjected to a rapid and uniform rotation within the limit of a
large Reynolds number.  At first sight, this situation seems similar to
eddy turbulence since it simply involves adding a well-known force to
the Navier--Stokes equations, namely the Coriolis force. However, there
are two major differences: waves exist and the spherical symmetry is
broken by the presence of a privileged axis, that of rotation. This
anisotropy makes the study of inertial wave turbulence more complex
than that of capillary waves. Nevertheless, the multiple time scale
method can be applied and the following anisotropic kinetic equation
can be obtained in the limit $k_\perp \gg
k_\parallel$~\cite{Galtier2003}
{\begin{eqnarray} 
\frac{\partial N_k^{s}}{\partial t} &=& \frac{\uppi \epsilon^2}{4 \,
s\omega_k} \sum_{s_{p} s_{q}} \int   \left( \frac{\sin \theta_k}{\kpn}
\right)^2 (s_p \ppn - s_q \qpn)^2 ( s \kpn + s_p \ppn + s_q \qpn)^2
\nonumber\\
&& \times \left[ s\omega_k N_p^{s_p} N_q^{s_q} + s_p \omega_p N_k^{s}
N_q^{s_q} + s_q \omega_q N_k^{s} N_p^{s_p} \right]  \delta
(\Omega_{kpq}) \delta_{kpq} \,\mathrm{d}\pp \,\mathrm{d}\qq ,\label{KEIW}
\end{eqnarray}}\unskip
with $\omega_{k} = 2 \Omega_0 k_\parallel / k_\perp$, ${\bf
e_\parallel}$ ($\vert \mathbf{e}_\parallel \vert = 1$) the axis of
rotation, $\Omega_0$ the constant rotating rate and $\theta_k$ the
opposite angle to $\kk_\perp$ in the triangle $\kk_\perp = \pp_\perp +
\qq_\perp$. 

\section{Kolmogorov--Zakharov spectrum}

\subsection{A toy model}

Exact stationary solutions with a non-zero flux can be found from the
previous kinetic equations. We usually called these solutions the
Kolmogorov--Zakharov spectra. To expose the technique to get such a
solution, which implies the Zakharov transformation, I will introduce a
simple kinetic equation that must be seen as a toy model because it
does not correspond to any physical system. This kinetic equation is
the following
{\begin{equation} \label{KEtoy}
\frac{\partial E_k}{\partial t} = \sum_{s_k s_{p} s_{q}} \int_{\Delta}
s_{k} \omega_k  \left[ s_k E_p E_q + s_p E_k E_q + s_q E_k E_p \right]
\delta(s_k \omega_k + s_p \omega_p + s_q \omega_q) \,\mathrm{d}p
\,\mathrm{d}q ,  
\end{equation}}\unskip
with $\omega_k = k^2$. The kinetic equation is isotropic and $E_k$ is
the one dimensional energy spectrum. Also, the integration is limited
to a two-dimensional region ($\Delta$) where the triadic relation is
satisfied (see Figure~\ref{Fig3}). Note that Equation~(\ref{KEtoy})
verifies the energy conservation. 

\begin{figure}
\includegraphics{fig04}
\caption{\label{Fig3}The isotropic kinetic equation (\ref{KEtoy}) involves triadic
interactions limited to the infinitely long gray band. The Zakharov
transformations (TZ1 and TZ2 are in the right and left panels,
respectively) consists in the exchange of the four regions separated by
dashes.}
\end{figure}

The first obvious solution that appears is the thermodynamic spectrum,
$E_k = A k^{-2}$. Indeed, the substitution of this solution into
(\ref{KEtoy}) gives trivially
{\begin{equation} \label{KEtoy2}
\frac{\partial E_k}{\partial t} = \sum_{s_k s_{p} s_{q}} \int_{\Delta}
s_{k} \omega_k  \left[ s_k k^2 + s_p p^2 + s_q q^2 \right]
\frac{A^2}{(kpq)^{3}} \delta(s_k \omega_k + s_p \omega_p + s_q
\omega_q)\, \mathrm{d}p \,\mathrm{d}q = 0 . 
\end{equation}}\unskip
The second solution is less trivial to obtain. We need to apply the
Zakharov transformation.  First, we need to write the previous equation
in terms of adimensional wavenumbers. Let us introduce $\xi_p \equiv
p/k$ and $\xi_q \equiv q/k$; then, we obtain
{\begin{equation} \label{KEtoy3}
\frac{\partial E_k}{\partial t} = A^2 k^{2+2x} \sum_{s_k s_{p} s_{q}}
\int_{\Delta} s_{k}  \left[ s_k \xi_p^x \xi_q^x + s_p \xi_q^x + s_q
\xi_p^x \right] \delta(s_k  + s_p \xi^2_p + s_q \xi^2_q)
\,\mathrm{d}\xi_p \,\mathrm{d}\xi_q,  
\end{equation}}\unskip
where we have also introduced $E_k = A k^x$ with $A$ est positive
quantity. The Zakharov transformations correspond to the following
change of variables
{\begin{equation}
\xi_{p} \to \frac{1}{\xi_{p}} , \quad \xi_{q} \to
\frac{\xi_{q}}{\xi_{p}} , \quad \text{(TZ1)} \label{TZ1}
\end{equation}}\unskip
and
{\begin{equation}
\xi_{p} \to \frac{\xi_{p}}{\xi_{q}} , \quad \xi_{q} \to
\frac{1}{\xi_{q}} . \quad \text{(TZ2)} \label{TZ2}
\end{equation}}\unskip
The meaning of these transformations is illustrated in
Figure~\ref{Fig3}. The method consists of splitting the integral
(\ref{KEtoy3}) into three equal contributions,  applying the Zakharov
transformation TZ1 to the second and TZ2 to the third, leaving the
first integral intact. We then combine the different contributions and,
after a few manipulations, obtain the following expression
{\begin{eqnarray} 
\frac{\partial E_k}{\partial t} &=& \frac{A^2}{3} k^{2+2x} \sum_{s_k
s_{p} s_{q}} \int_{\Delta} \xi_q^x \xi_q^x \left( s_k + s_p \xi_p^{-x}
+ s_q \xi_q^{-x} \right) \left( s_k + s_p \xi^{-2x-1}_p + s_q
\xi^{-2x-1}_q \right) \nonumber\\
&&\times\,\delta(s_k + s_p \xi^2_p + s_q \xi^2_q) \mathrm{d}\xi_p \,
\,\mathrm{d}\xi_q .\label{KEtoy4}
\end{eqnarray}}\unskip
Two solutions appear, $x=-2$ and $x=-3/2$. The first corresponds to the
thermodynamic spectrum (as mentioned above) for which the energy flux
is zero and the second to the Kolmogorov--Zakharov spectrum with a
non-zero energy flux. These properties can be obtained by introducing
the isotropic energy flux $\Pi_k$ as follows
{\begin{equation}
\frac{\partial E_k}{\partial t} = - \frac{\partial \Pi_k}{\partial k} =
\frac{A^2}{3} k^{2+2x} I(x) .
\end{equation}}\unskip
This leads to the expression
{\begin{equation}
\Pi_k (x) = - \frac{A^2}{3} \frac{k^{3+2x}}{3+2x} I(x) .
\end{equation}}\unskip
We can immediately see that $x=-2$ corresponds to a zero flux since
$I(-2)=0$. On the other hand, the value $x=-3/2$ requires a little
calculation since $I$ and the denominator are both zero. Using
L'Hospital's rule, we find
{\begin{equation}
\varepsilon \equiv \Pi_k (-3/2) = - \frac{A^2}{3} \lim_{x \to -3/2}
\frac{I(x)}{3+2x} = \frac{A^2}{3} J ,
\end{equation}}\unskip
with
{\begin{eqnarray} 
J \equiv - \lim_{x \to -3/2} \frac{\partial I(x)}{\partial x} &=&
\sum_{s_k s_{p} s_{q}} \int_{\Delta} \xi_q^{-3/2} \xi_q^{-3/2}
\left( s_k + s_p \xi_p^{3/2} + s_q \xi_q^{3/2} \right) \left( s_p
\xi^{2}_p \ln (\xi_p) + s_q \xi^{2}_q \ln (\xi_q) \right) \nonumber\\
&&\times\,\delta(s_k + s_p \xi^2_p + s_q \xi^2_q) \,\mathrm{d}\xi_p 
\,\mathrm{d}\xi_q .\label{fluxE}
\end{eqnarray}}\unskip
The direction of the cascade can be obtained numerically from this
analytical expression. A positive value for $J$ means a direct energy
cascade. The value of $J$ can finally lead to an estimate of the
Kolmogorov constant $C_{\mathrm{K}}$ with
{\begin{equation}
E_k = C_{\mathrm{K}} \sqrt{\varepsilon} k^{-3/2} \quad \text{and} \quad C_{\mathrm{K}} =
\sqrt{\frac{3}{J}} .
\end{equation}}\unskip
The last task is to check the validity of the Kolmogorov--Zakharov
spectrum. We require this solution (the inertial range) to be
independent of the largest and smallest scales. This is a question of
convergence of the collisional integral (\ref{KEtoy3}). Solving this
exercise gives a condition $x_1 < x < x_2$ which can prove that our
solution $x=-3/2$ is physical if it is inside the domain of
convergence.\looseness=-1

In conclusion, we can say that wave turbulence is a solvable problem.
Indeed, we have seen from this toy model that in principle we can
determine all the properties of wave turbulence by giving the exact
solutions, the direction of the cascade, evaluating the Kolmogorov
constant and showing the existence of an inertial range. All these
properties cannot be found in eddy turbulence, even using the exact
Kolmogorov laws. 

\subsection{Applications to Navier--Stokes}

For capillary wave turbulence the isotropic Kolmogorov--Zakharov
spectrum is~\cite{Zakharov1967}
{\begin{equation}
E_k = A k^{-7/4} .
\end{equation}}\unskip
It is an isotropic spectrum well observed experimentally (see the
review~\cite{Falcon2022}). 

For inertial wave turbulence the problem is more complex because
turbulence is anisotropic with an energy cascade mainly in the
direction transverse to the rotation axis. In this case, we have to use
a generalization of the Zakharov transformation. This leads to the
axisymmetric Kolmogorov--Zakharov spectrum~\cite{Galtier2003}
{\begin{equation}
E(k_\perp,k_\parallel) = A k_\perp^{-5/2} k_\parallel^{-1/2} .
\end{equation}}\unskip
Note that the convergence properties and the Kolmogorov constant have
been studied only recently~\cite{David2023}. This spectrum is observed
experimentally and
numerically~\cite{Monsalve2020,LeReun2020,Yokoyama2021}. 

{\vspace*{-2pt}}

\section*{Declaration of interests}

{\vspace*{-2pt}}

The authors do not work for, advise, own shares in, or receive funds
from any organization that  could benefit from this article, and have
declared no affiliations other than their research organizations.

{\vspace*{-2pt}}

\section*{Funding}

{\vspace*{-2pt}}

The author is supported by the Simons Foundation (Grant No. 651461,
PPC).

\CDRGrant[SF]{651461}

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