# A multiple time scale approach for anisotropic inertial wave turbulence

Sébastien Galtier  Université Paris-Saclay, Institut universitaire de France,
Laboratoire de Physique des Plasmas, Ecole polytechnique, 91128 Palaiseau,
France

###### Abstract

Wave turbulence is the study of the long-time statistical behaviour of
equations describing a set of weakly non-linear interacting waves. Such a
theory, which has a natural asymptotic closure, allows us to probe the nature
of turbulence more deeply than the exact Kolmogorov laws by rigorously proving
the direction of the cascade and the existence of an inertial range,
predicting stationary spectra for conserved quantities, or evaluating the
Kolmogorov constant. An emblematic example is given by fast rotating fluids
for which a wave turbulence theory has been derived by Galtier (2003). This
work involves nontrivial analytical developments for a problem that is
anisotropic by nature. We propose here a new path for the derivation of the
kinetic equation by using the anisotropy at the beginning of the analysis. We
show that the helicity basis is not necessary to obtain the wave amplitude
equation for the canonical variables that involve a combination of poloidal
and toroidal fields. The multiple time scale method adapted to this
anisotropic problem is then used to derive the kinetic equation which is the
same as the original work when anisotropy is eventually taken into account.
This result proves the commutativity between asymptotic closure and
anisotropy. In addition, the multiple time scale method informs us that the
kinetic equation can be derived without imposing restrictions on the
probability distribution of the wave amplitude such as quasi-Gaussianity, or
on the phase such as random phase approximation which naturally occurs
dynamically.

##  I Introduction

As we celebrate two hundred years of the Navier-Stokes equations, it is
remarkable to note that hydrodynamic turbulence is still a much studied
subject. Among the different fields of study, there is wave turbulence which
has become important in geophysics and astrophysics where waves are
omnipresent Galtier (2023a). The strength of the (weak) wave turbulence theory
is that it offers the possibility of a deep understanding of physical systems
composed of a set 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 second-order spectral cumulants – the so-called
kinetic equations – which are free from the closure problem classically
encountered in eddy turbulence. Indeed, in wave turbulence there is a natural
asymptotic closure rooted on the existence of a small parameter, the wave
amplitude. Secondly, exact solutions (Kolmogorov-Zakharov spectra) 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. Thirdly, these solutions
correspond to power law spectra that can be compared to the data. The number
of experiments, observations, and diagnostics has increased considerably over
the past two decades and today, thanks also to direct numerical simulations
(DNS), wave turbulence has become a leading field in turbulence where new
fundamental questions are being raised (Galtier and Nazarenko, 2017; Hassaini
and Mordant, 2017; Hassaini et al., 2019; Savaro et al., 2020; Galtier and
Nazarenko, 2021; Ricard and Falcon, 2021; David and Galtier, 2022; Falcon and
Mordant, 2022; Griffin et al., 2022; Hrabski and Pan, 2022; Kochurin and
Kuznetsov, 2022; Onorato et al., 2022; Rodda et al., 2022; Zhang and Pan,
2022; Dematteis and Lvov, 2023; Galtier, 2023b; Lanchon et al., 2023; Novkoski
et al., 2023; Zhu et al., 2023).

Rotating fluids are one of the most studied examples in (strong/weak) wave
turbulence as it involves inertial waves that are easy to excite
experimentally (Hopfinger et al., 1982; Jacquin et al., 1990; Morize et al.,
2005). This regime is of interest in a number of fields such as geophysics
where the Coriolis force is felt, for example, through large-scale atmospheric
motions. Very early, it was recognised that rotating turbulence behaves
differently from classical eddy turbulence with a reduction of the cascade
along the axis of rotation ${\bf\Omega_{0}}$ with possibly a steeper energy
spectrum than the well-known Kolmogorov spectrum (Hossain, 1994; Zeman, 1994;
Zhou, 1995; Cambon et al., 1997; Smith and Waleffe, 1999; Baroud et al., 2002;
Godeferd and Moisy, 2015). Another remarkable feature, still not understood,
is the self-similarity found in the scaling of velocity structure functions in
the direction transverse to ${\bm{\Omega}}_{0}=\Omega_{0}\bm{e}_{\parallel}$
(Baroud et al., 2002; van Bokhoven et al., 2009), which is in strong contrast
with intermittency observed in hydrodynamic turbulence. In recent years, the
(weak) inertial wave turbulence regime has been specifically studied
experimentally (Yarom and Sharon, 2014; Monsalve et al., 2020). For example,
it was shown that the energy spectrum is concentrated along the dispersion
relation, as expected in wave turbulence, with a scaling in agreement with the
theoretical prediction. Numerical simulations (including DNS) have also been
carried out, notably to study the spectral properties (Bellet et al., 2006;
Scott, 2014; Clark di Leoni and Mininni, 2016; Le Reun et al., 2017; Sharma et
al., 2018; Galtier and David, 2020; Le Reun et al., 2020; Yokoyama and
Takaoka, 2021). They confirm the previous experimental and theoretical results
and reveal, for example, the existence of a non-stationary solution different
from the stationary Kolmogorov-Zakharov spectrum.

The theory of inertial wave turbulence has been derived by Galtier (2003). The
kinetic equation obtained is valid in the most general case, i.e. without
making the assumption of anisotropy. However, a simple argument based on the
resonance condition shows that the cascade is anisotropic with a transfer
mainly in the perpendicular ($\perp$) direction to ${\bm{\Omega}}_{0}$. Using
this feature, the kinetic equation was eventually reduced to the axisymmetric
case from which the exact (Kolmogorov-Zakharov) energy spectrum was derived.
This solution takes the form $E(k_{\perp},k_{\parallel})\sim
k_{\perp}^{-5/2}k_{\parallel}^{-1/2}$, with
$\bm{k}=\bm{k}_{\perp}+k_{\parallel}\bm{e}_{\parallel}$ the wavevector. As
recently proved by David and Galtier (2023), this energy spectrum corresponds
to a local turbulence (with an inertial range independent of the largest and
smallest scale) for which we can also estimate the Kolmogorov constant. The
derivation of the kinetic equation of inertial wave turbulence in the general
case (without the axisymmetry assumption) is cumbersome and the use of the
Hamiltonian formalism does not drastically simplify the calculation (Gelash et
al., 2017). Here, it is proposed to take another path for the derivation of
such an equation by using the anisotropy assumption ($k_{\perp}\gg
k_{\parallel}$) at the beginning of the analysis. As shown in §II, in this
case the wave amplitude equation can be obtained without the introduction of a
complex helicity basis, which is an interesting simplification: the velocity
field is decomposed into poloidal and toroidal fields from which we can define
the canonical variables. In §III, the multiple time scale method introduced by
Benney and Saffman (1966) is adapted (a few points are also clarified) to this
anisotropic problem and then used to derive such a kinetic equation which is
the same as the original work when anisotropy is finally taken into account.
This result shows the commutativity between asymptotic closure and anisotropy.
Furthermore, the multiple time scale method informs us that the kinetic
equation (of weak wave turbulence) can be derived without imposing
restrictions on the probability distribution of the wave amplitude such as
quasi-Gaussianity, or on the phase such as random phase approximation which
naturally occurs dynamically. Finally, we conclude with a discussion in §IV,

##  II Wave amplitude equation

###  II.1 Canonical variables

The Navier-Stokes equations with the Coriolis force read

$\frac{\partial{\bm{w}}}{\partial
t}-2({\bm{\Omega}}_{0}\cdot{\boldsymbol{\nabla}}){\bf
u}=({\bm{w}}\cdot{\boldsymbol{\nabla}}){\bf u}-({\bf
u}\cdot{\boldsymbol{\nabla}}){\bm{w}}+\nu\,\nabla^{2}{\bm{w}},$ (1)

where ${\bf u}$ is a solenoidal velocity (${\boldsymbol{\nabla}}\cdot{\bf
u}=0$), ${\bm{w}}={\boldsymbol{\nabla}}\times{\bf u}$ the vorticity and
${\bm{\Omega}}_{0}$ a constant rotation rate. Hereafter, we will neglect the
viscosity. We introduce the toroidal ($\psi$) and poloidal ($\phi$) scalar
fields in the following manner

${\bf
u}=\nabla\times(\psi\bm{e}_{\parallel})+\nabla\times(\nabla\times(\phi\bm{e}_{\parallel})),$
(2)

whose Fourier transform writes

$\hat{\bf
u}_{k}=i\hat{\psi}_{k}\bm{k}\times\bm{e}_{\parallel}-\hat{\phi}_{k}\bm{k}\times(\bm{k}\times\bm{e}_{\parallel})=i\hat{\psi}_{k}\bm{k}\times\bm{e}_{\parallel}+\hat{\phi}_{k}(k^{2}\bm{e}_{\parallel}-k_{\parallel}\bm{k}),$
(3)

from which we deduce the vorticity vector ($|\bm{k}|=k$)

$\hat{\bm{w}}_{k}=\hat{\psi}_{k}(k^{2}\bm{e}_{\parallel}-k_{\parallel}\bm{k})+ik^{2}\hat{\phi}_{k}\bm{k}\times\bm{e}_{\parallel}.$
(4)

It is straightforward to show in Fourier space that the linear contribution of
equation (1) leads, after projection, to

$\displaystyle\frac{\partial\hat{\phi}_{k}}{\partial t}$ $\displaystyle=$
$\displaystyle 2i\Omega_{0}\frac{k_{\parallel}}{k^{2}}\hat{\psi}_{k},$ (5a)
$\displaystyle\frac{\partial\hat{\psi}_{k}}{\partial t}$ $\displaystyle=$
$\displaystyle 2i\Omega_{0}k_{\parallel}\hat{\phi}_{k}.$ (5b)

The linear solutions are the well-known (helical) inertial waves with the
angular frequency ($\partial^{2}_{t}=-\omega^{2}_{k}$ can be used)

$\omega^{2}_{k}=4\Omega^{2}_{0}\frac{k_{\parallel}^{2}}{k^{2}}.$ (6)

From this property, we introduce the canonical variables

$A^{s}_{k}\equiv A^{s}(\bm{k})=k^{2}\hat{\phi}_{k}-sk\hat{\psi}_{k},$ (7)

with $s=\pm$ the directional polarity. With such a choice of canonical
variables, we have

$|A^{+}_{k}|^{2}+|A^{-}_{k}|^{2}=2|\hat{\bf u}_{k}|^{2}$ (8)

and at the linear level

$\frac{\partial A^{s}_{k}}{\partial t}+is\omega_{k}A^{s}_{k}=0.$ (9)

###  II.2 Resonance condition

The resonance condition for three-wave interactions can be written (Galtier,
2023a)

$\displaystyle s\omega_{k}+s_{p}\omega_{p}+s_{q}\omega_{q}$ $\displaystyle=$
$\displaystyle 0,$ (10a) $\displaystyle\bm{k}+\bm{p}+\bm{q}$ $\displaystyle=$
$\displaystyle 0.$ (10b)

In the case of inertial waves, these relations are equivalent to the
conditions

$\frac{s_{q}q-s_{p}p}{s\omega_{k}}=\frac{sk-
s_{q}q}{s_{p}\omega_{p}}=\frac{s_{p}p-sk}{s_{q}\omega_{q}}.$ (11)

Assuming that the system is initially excited at large scale in a narrow
isotropic domain in Fourier space, a situation often considered in DNS, the
dynamics will initially be dominated by local interactions such that $k\simeq
p\simeq q$. As the locality of the interactions is a property of turbulence
that is generally verified, we can extend its use beyond the initial instant.
We obtain

$\frac{s_{q}-s_{p}}{sk_{\parallel}}\simeq\frac{s-s_{q}}{s_{p}p_{\parallel}}\simeq\frac{s_{p}-s}{s_{q}q_{\parallel}}.$
(12)

From this expression, we can show that the associated cascade is necessarily
anisotropic. Indeed, if $k_{\parallel}$ is non-zero, the left-hand term will
only give a non-negligible contribution when $s_{p}=-s_{q}$. The immediate
consequence is that either the middle or the right-hand term has its numerator
which cancels (to leading order), which implies that the associated
denominator must also cancel (to leading order) to satisfy the equality: for
example, if $s=s_{p}$ then $q_{\parallel}\simeq 0$. This condition means that
the transfer in the parallel direction is negligible: indeed, the integration
of the wave amplitude equation in the parallel direction (see below) is then
reduced to a few modes (since $p_{\parallel}\simeq k_{\parallel}$) which
strongly limits the transfer between parallel modes. The cascade in the
parallel direction is thus possible but relatively weak compared to that in
the perpendicular direction. In the following, we will take advantage of this
property and consider the anisotropic limit $k_{\perp}\gg k_{\parallel}$ to
simplify the derivation. Note that once turbulence is anisotropic, we can
still use the locality condition with $k\sim k_{\perp}$; we then obtain
$k_{\perp}\sim p_{\perp}\sim q_{\perp}$, whereas the parallel wavenumbers are
limited to a narrow domain.

###  II.3 Wave amplitude equation

In the derivation of the wave amplitude equation, we will 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 Benney and Saffman
(1966). Note that the anisotropic limit ($k_{\perp}\gg k_{\parallel}$) will
also be taken before the (asymptotic) long time limit.

The first non-linear term of equation (1) writes

$\displaystyle\widehat{({\bm{w}}\cdot{\boldsymbol{\nabla}}){\bf u}}_{k}$
$\displaystyle=$ $\displaystyle i\int(\hat{\bm{w}}_{p}\cdot\bm{q})\hat{\bf
u}_{q}\delta_{k,pq}d\bm{p}d\bm{q}$ (13) $\displaystyle=$ $\displaystyle
i\int\left[i\hat{\phi}_{p}\hat{\phi}_{q}p^{2}\left(\bm{q}\cdot(\bm{p}\times\bm{e}_{\parallel})\right)(q^{2}\bm{e}_{\parallel}-q_{\parallel}\bm{q})-\hat{\phi}_{p}\hat{\psi}_{q}p^{2}\left(\bm{q}\cdot(\bm{p}\times\bm{e}_{\parallel})\right)(\bm{q}\times\bm{e}_{\parallel})\right.$
$\displaystyle\left.\mbox{}-\hat{\psi}_{p}\hat{\phi}_{q}\left(p_{\parallel}\bm{p}\cdot\bm{q}-p^{2}q_{\parallel}\right)(q^{2}\bm{e}_{\parallel}-q_{\parallel}\bm{q})-i\hat{\psi}_{p}\hat{\psi}_{q}\left(p_{\parallel}\bm{p}\cdot\bm{q}-p^{2}q_{\parallel}\right)(\bm{q}\times\bm{e}_{\parallel})\right]$
$\displaystyle\mbox{}\times\delta_{k,pq}d\bm{p}d\bm{q},$

with $\delta_{k,pq}\equiv\delta(\bm{k}-\bm{p}-\bm{q})$. In the anisotropic
limit ($k_{\perp}\gg k_{\parallel}$), a first simplification arises

$\displaystyle\widehat{({\bm{w}}\cdot{\boldsymbol{\nabla}}){\bf u}}_{k}$
$\displaystyle=$ $\displaystyle
i\int\left[i\hat{\phi}_{p}\hat{\phi}_{q}p_{\perp}^{2}q_{\perp}^{2}\left(\bm{e}_{\parallel}\cdot(\bm{q_{\perp}}\times\bm{p_{\perp}})\right)\bm{e}_{\parallel}-\hat{\phi}_{p}\hat{\psi}_{q}p_{\perp}^{2}\left(\bm{e}_{\parallel}\cdot(\bm{q_{\perp}}\times\bm{p_{\perp}})\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})\right.$
(14)
$\displaystyle\left.\mbox{}-\hat{\psi}_{p}\hat{\phi}_{q}q_{\perp}^{2}\left(p_{\parallel}\bm{p_{\perp}}\cdot\bm{q_{\perp}}-p_{\perp}^{2}q_{\parallel}\right)\bm{e}_{\parallel}-i\hat{\psi}_{p}\hat{\psi}_{q}\left(p_{\parallel}\bm{p_{\perp}}\cdot\bm{q_{\perp}}-p_{\perp}^{2}q_{\parallel}\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})\right]$
$\displaystyle\mbox{}\times\delta_{k,pq}d\bm{p}d\bm{q}.$

The second non-linear term of equation (1) reads

$\displaystyle\widehat{({\bf u}\cdot{\boldsymbol{\nabla}}){\bm{w}}}_{k}$
$\displaystyle=$ $\displaystyle i\int(\hat{\bf
u}_{p}\cdot\bm{q})\hat{\bm{w}}_{q}\delta_{k,pq}d\bm{p}d\bm{q}$ (15)
$\displaystyle=$ $\displaystyle
i\int\left[i\hat{\phi}_{p}\hat{\phi}_{q}\left(p^{2}q_{\parallel}-p_{\parallel}\bm{p}\cdot\bm{q}\right)q^{2}(\bm{q}\times\bm{e}_{\parallel})+\hat{\phi}_{p}\hat{\psi}_{q}\left(p^{2}q_{\parallel}-p_{\parallel}\bm{p}\cdot\bm{q}\right)(q^{2}\bm{e}_{\parallel}-q_{\parallel}\bm{q})\right.$
$\displaystyle\left.\mbox{}-\hat{\psi}_{p}\hat{\phi}_{q}\left(\bm{q}\cdot(\bm{p}\times\bm{e}_{\parallel})\right)q^{2}(\bm{q}\times\bm{e}_{\parallel})+i\hat{\psi}_{p}\hat{\psi}_{q}\left(\bm{q}\cdot(\bm{p}\times\bm{e}_{\parallel})\right)(q^{2}\bm{e}_{\parallel}-q_{\parallel}\bm{q})\right]$
$\displaystyle\mbox{}\times\delta_{k,pq}d\bm{p}d\bm{q},$

which simplifies in the anisotropic limit to

$\displaystyle\widehat{({\bf u}\cdot{\boldsymbol{\nabla}}){\bm{w}}}_{k}$
$\displaystyle=$ $\displaystyle i\int
q_{\perp}^{2}\left[i\hat{\phi}_{p}\hat{\phi}_{q}\left(p_{\perp}^{2}q_{\parallel}-p_{\parallel}\bm{p_{\perp}}\cdot\bm{q_{\perp}}\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})+\hat{\phi}_{p}\hat{\psi}_{q}\left(p_{\perp}^{2}q_{\parallel}-p_{\parallel}\bm{p_{\perp}}\cdot\bm{q_{\perp}}\right)\bm{e}_{\parallel}\right.$
(16)
$\displaystyle\left.\mbox{}-\hat{\psi}_{p}\hat{\phi}_{q}\left(\bm{e}_{\parallel}\cdot(\bm{q_{\perp}}\times\bm{p_{\perp}})\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})+i\hat{\psi}_{p}\hat{\psi}_{q}\left(\bm{e}_{\parallel}\cdot(\bm{q_{\perp}}\times\bm{p_{\perp}})\right)\bm{e}_{\parallel}\right]$
$\displaystyle\mbox{}\times\delta_{k,pq}d\bm{p}d\bm{q}.$

The addition of these two non-linear contributions leads to the simplified
expression

$\displaystyle\widehat{NL}(\bm{k})$ $\displaystyle=$
$\displaystyle\widehat{({\bm{w}}\cdot{\boldsymbol{\nabla}}){\bf
u}}_{k}-\widehat{({\bf u}\cdot{\boldsymbol{\nabla}}){\bm{w}}}_{k}$ (17)
$\displaystyle=$
$\displaystyle\int\hat{\phi}_{p}\hat{\phi}_{q}p_{\perp}^{2}q_{\perp}^{2}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)\bm{e}_{\parallel}\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}+i\int\hat{\phi}_{p}\hat{\psi}_{q}p_{\perp}^{2}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}-i\int\hat{\psi}_{p}\hat{\phi}_{q}q_{\perp}^{2}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}-\int\hat{\psi}_{p}\hat{\psi}_{q}q_{\perp}^{2}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)\bm{e}_{\parallel}\delta_{k,pq}d\bm{p}d\bm{q}.$

The introduction of the canonical variables

$\displaystyle\hat{\psi}_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{2k_{\perp}}\sum_{s}sA_{k}^{s},$ (18a)
$\displaystyle\hat{\phi}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2k_{\perp}^{2}}\sum_{s}A_{k}^{s},$ (18b)

gives

$\displaystyle\widehat{NL}(\bm{k})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)\bm{e}_{\parallel}\delta_{k,pq}d\bm{p}d\bm{q}$
(19) $\displaystyle\mbox{}-\frac{i}{4}\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}\frac{s_{q}}{q_{\perp}}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}+\frac{i}{4}\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}\frac{s_{p}}{p_{\perp}}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)(\bm{q_{\perp}}\times\bm{e}_{\parallel})\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}-\frac{1}{4}\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}s_{p}s_{q}\frac{q_{\perp}}{p_{\perp}}\left(\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})\right)\bm{e}_{\parallel}\delta_{k,pq}d\bm{p}d\bm{q}.$

The dummy variables $\bm{p}$, $\bm{q}$ and $s_{p}$, $s_{q}$, can be exchanged
to symmetrise the equation; we find

$\displaystyle\widehat{NL}(\bm{k})$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}\frac{\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})}{p_{\perp}q_{\perp}}(p_{\perp}^{2}-q_{\perp}^{2})s_{p}s_{q}\bm{e}_{\parallel}\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}+\frac{i}{8}\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}\frac{\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})}{p_{\perp}q_{\perp}}\left(s_{p}q_{\perp}-s_{q}p_{\perp}\right)(\bm{k_{\perp}}\times\bm{e}_{\parallel})\delta_{k,pq}d\bm{p}d\bm{q}.$

Coming back to the wave amplitude equation, we can write

$\displaystyle\left(\frac{\partial\hat{\psi}_{k}}{\partial
t}-2i{\bm{\Omega}}_{0}k_{\parallel}\hat{\phi}_{k}\right)k_{\perp}^{2}\bm{e}_{\parallel}+\left(ik_{\perp}^{2}\frac{\partial\hat{\phi}_{k}}{\partial
t}+2{\bm{\Omega}}_{0}k_{\parallel}\hat{\psi}_{k}\right)\bm{k}\times\bm{e}_{\parallel}=\widehat{NL}(\bm{k}),$
(21)

therefore, after projection and use of the dispersion relation, we obtain

$\displaystyle\frac{\partial\hat{\psi}_{k}}{\partial
t}-i\omega_{k}k_{\perp}\hat{\phi}_{k}$ $\displaystyle=$
$\displaystyle\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}\frac{\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})}{8k_{\perp}^{2}p_{\perp}q_{\perp}}(p_{\perp}^{2}-q_{\perp}^{2})s_{p}s_{q}\delta_{k,pq}d\bm{p}d\bm{q},\quad$
(22a) $\displaystyle\frac{\partial\hat{\phi}_{k}}{\partial
t}-i\omega_{k}\frac{\hat{\psi}_{k}}{k_{\perp}}$ $\displaystyle=$
$\displaystyle\sum_{s_{p}s_{q}}\int
A_{p}^{s_{p}}A_{q}^{s_{q}}\frac{\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})}{8k_{\perp}^{2}p_{\perp}q_{\perp}}\left(s_{p}q_{\perp}-s_{q}p_{\perp}\right)\delta_{k,pq}d\bm{p}d\bm{q}.\quad$
(22b)

With the introduction of the canonical variables (7), the weighted addition of
the previous expressions gives

$\displaystyle\frac{\partial A^{s}_{k}}{\partial t}+is\omega_{k}A^{s}_{k}$
$\displaystyle=$ $\displaystyle\sum_{s_{p}s_{q}}\int
ss_{p}s_{q}\frac{\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})}{8k_{\perp}p_{\perp}q_{\perp}}\left(q_{\perp}^{2}-p_{\perp}^{2}+sk_{\perp}\left(s_{q}q_{\perp}-s_{p}p_{\perp}\right)\right)$
(23) $\displaystyle\mbox{}\times
A_{p}^{s_{p}}A_{q}^{s_{q}}\delta_{k,pq}d\bm{p}d\bm{q}.$

Remaking that

$q_{\perp}^{2}-p_{\perp}^{2}=(s_{q}q_{\perp}-s_{p}p_{\perp})(s_{p}p_{\perp}+s_{q}q_{\perp}),$
(24)

we can rearrange the expression in the following manner

$\displaystyle\frac{\partial A^{s}_{k}}{\partial t}+is\omega_{k}A^{s}_{k}$
$\displaystyle=$ $\displaystyle\sum_{s_{p}s_{q}}\int
ss_{p}s_{q}\frac{\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})}{8k_{\perp}p_{\perp}q_{\perp}}(s_{q}q_{\perp}-s_{p}p_{\perp})(sk_{\perp}+s_{p}p_{\perp}+s_{q}q_{\perp})$
(25) $\displaystyle\mbox{}\times
A_{p}^{s_{p}}A_{q}^{s_{q}}\delta_{k,pq}d\bm{p}d\bm{q}.$

We introduce the interaction representation for waves of weak amplitude
($0<\epsilon\ll 1$)

$A_{k}^{s}=\epsilon a_{k}^{s}e^{-is\omega_{k}t},$ (26)

and get eventually the wave amplitude equation after a few last manipulations

$\frac{\partial a^{s}_{k}}{\partial t}=\epsilon\sum_{s_{p}s_{q}}\int
L_{kpq}^{ss_{p}s_{q}}a_{p}^{s_{p}}a_{q}^{s_{q}}e^{i\Omega_{k,pq}t}\delta_{k,pq}d\bm{p}d\bm{q},$
(27)

with $\Omega_{k,pq}\equiv s\omega_{k}-s_{p}\omega_{p}-s_{q}\omega_{q}$ and

$L_{kpq}^{ss_{p}s_{q}}\equiv\omega_{k}\frac{\bm{e}_{\parallel}\cdot(\bm{p_{\perp}}\times\bm{q_{\perp}})}{8k_{\perp}p_{\perp}q_{\perp}}s_{p}s_{q}\left(\frac{s_{q}q_{\perp}-s_{p}p_{\perp}}{s\omega_{k}}\right)(sk_{\perp}+s_{p}p_{\perp}+s_{q}q_{\perp}).$
(28)

Expression (28) satisfies the following properties (relation (11) is used)

$\displaystyle L_{0pq}^{ss_{p}s_{q}}$ $\displaystyle=$ $\displaystyle 0,$
(29a) $\displaystyle L_{kqp}^{ss_{q}s_{p}}$ $\displaystyle=$ $\displaystyle
L_{kpq}^{ss_{p}s_{q}},$ (29b) $\displaystyle L_{pkq}^{s_{p}ss_{q}}$
$\displaystyle=$
$\displaystyle-\frac{s_{p}\omega_{p}}{s\omega_{k}}L_{kpq}^{ss_{p}s_{q}},$
(29c) $\displaystyle L_{kpq}^{-s-s_{p}-s_{q}}$ $\displaystyle=$
$\displaystyle-L_{kpq}^{ss_{p}s_{q}},$ (29d) $\displaystyle
L_{-kpq}^{ss_{p}s_{q}}$ $\displaystyle=$ $\displaystyle
L_{kpq}^{ss_{p}s_{q}},$ (29e) $\displaystyle L_{-k-p-q}^{ss_{p}s_{q}}$
$\displaystyle=$ $\displaystyle L_{kpq}^{ss_{p}s_{q}}.$ (29f)

The wave amplitude equation (27) governs the slow evolution of inertial waves
of weak amplitude in the anisotropic limit. It is a quadratic non-linear
equation which corresponds to the interactions between waves propagating along
$\bm{p}$ and $\bm{q}$, in the positive ($s_{p},s_{q}>0$) or negative
($s_{p},s_{q}<0$) direction. The multiple time scale method introduced in the
next section is based on this expression. The symmetries listed above will
also be used to simplify the derivation of the kinetic equation. Unlike the
original derivation by Galtier (2003), expression (27) has been derived
without going through a complex helicity basis. The wave amplitude equation
tells us that the non-linear coupling between the states associated with the
wavevectors $\bm{p_{\perp}}$ and $\bm{q_{\perp}}$ vanishes when these
wavevectors are collinear. Moreover, we note that the non-linear coupling
disappears when the wavenumbers $p_{\perp}$ and $q_{\perp}$ are equal if their
associated directional polarities, $s_{p}$ and $s_{q}$, are also equal. These
are general properties for helical waves (Kraichnan, 1973; Waleffe, 1992;
Turner, 2000; Galtier, 2003, 2006, 2014).

##  III Asymptotic sequential closures

The method outlined here was first proposed by Benney and Saffman (1966) for
three-wave interactions, but to our knowledge it has never been explicitly
applied to a physical system. In fact, originally the main motivation for such
a development was four-wave interactions that describe gravity waves
(Hasselmann, 1962; Newell, 1968), and for which the main prediction of wave
turbulence (Kolmogorov-Zakharov spectrum) is now well observed (Lenain and
Melville, 2017).

Unlike the classical perturbation analysis, the multiple time scale method is
based on the existence of a sequence of time scales, $T_{0}$, $T_{1}$,
$T_{2}$, …, with by definition (Nayfeh, 2004)

$T_{0}\equiv t,\quad T_{1}\equiv\epsilon t,\quad
T_{2}\equiv\epsilon^{2}t,...\,.$ (30)

Because of the weak dependence in $t$ of $T_{1}$, $T_{2}$, …, all these
variables will be treated (it is an approximation) as independent. And the
smaller $\epsilon$ is, the better the approximation. The variation of the wave
amplitude with $T_{1}$ and $T_{2}$ represents the slow variation that we wish
to extract. (In practice, our analysis will be limited to
$\mathcal{O}(\epsilon^{2})$.) Using the chain rule, we obtain ($T_{0}$ being
replaced by $t$)

$\displaystyle\left({\partial\over\partial t}+\epsilon{\partial\over\partial
T_{1}}+\epsilon^{2}{\partial\over\partial
T_{2}}+...\right)a^{s}_{k}=\epsilon\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}a^{s_{p}}_{p}a^{s_{q}}_{q}e^{i\Omega_{k,pq}t}\delta_{k,pq}d\bm{p}d\bm{q}.$
(31)

The variable $a^{s}_{k}$ must also be expanded to the power of $\epsilon$ and
make the various scales appear in time

$a^{s}_{k}=\sum_{n=0}^{+\infty}\epsilon^{n}a^{s}_{k,n}(t,T_{1},T_{2},...)=a^{s}_{k,0}+\epsilon
a^{s}_{k,1}+\epsilon^{2}a^{s}_{k,2}+...\,.$ (32)

Expression (32) is then introduced into the fundamental equation (31); we
obtain for the first three terms

$\displaystyle{\partial a^{s}_{k,0}\over\partial t}$ $\displaystyle=0,$ (33a)
$\displaystyle{\partial a^{s}_{k,1}\over\partial t}$ $\displaystyle=-{\partial
a^{s}_{k,0}\over\partial T_{1}}+\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}a_{p,0}^{s_{p}}a_{q,0}^{s_{q}}e^{i\Omega_{k,pq}t}\delta_{k,pq}d\bm{p}d\bm{q},$
(33b) $\displaystyle{\partial a^{s}_{k,2}\over\partial t}$
$\displaystyle=-{\partial a^{s}_{k,1}\over\partial T_{1}}-{\partial
a^{s}_{k,0}\over\partial T_{2}}+\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}\left[a_{p,1}^{s_{p}}a_{q,0}^{s_{q}}+a_{p,0}^{s_{p}}a_{q,1}^{s_{q}}\right]e^{i\Omega_{k,pq}t}\delta_{k,pq}d\bm{p}d\bm{q}.$
(33c)

To lighten the writing, the time dependency of the variables has been omitted.
After integration on $t$, one finds

$\displaystyle a^{s}_{k,0}$ $\displaystyle=a^{s}_{k,0}(T_{1},T_{2},...),$
(34a) $\displaystyle a^{s}_{k,1}$ $\displaystyle=-t{\partial
a^{s}_{k,0}\over\partial T_{1}}+b^{s}_{k,1},$ (34b) $\displaystyle
a^{s}_{k,2}$
$\displaystyle=\frac{t^{2}}{2}{\partial^{2}a^{s}_{k,0}\over\partial
T_{1}^{2}}-t{\partial a^{s}_{k,0}\over\partial T_{2}}-\int_{0}^{t}{\partial
b^{s}_{k,1}\over\partial T_{1}}dt+\tilde{b}^{s}_{k,2},$ (34c)

with by definition

$b^{s}_{k,1}\equiv\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}a_{p,0}^{s_{p}}a_{q,0}^{s_{q}}\Delta(\Omega_{k,pq})\delta_{k,pq}d\bm{p}d\bm{q},$
(35) $\displaystyle\Delta(X)\equiv\int_{0}^{t}e^{iXt}dt=\frac{e^{iXt}-1}{iX},$
(36)

and

$\tilde{b}^{s}_{k,2}\equiv\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}\int_{0}^{t}\left[a_{p,1}^{s_{p}}a_{q,0}^{s_{q}}+a_{p,0}^{s_{p}}a_{q,1}^{s_{q}}\right]e^{i\Omega_{k,pq}t}dt\delta_{k,pq}d\bm{p}d\bm{q}.$
(37)

The previous equation is modified when expression (34b) is introduced; one
finds

$\displaystyle\tilde{b}^{s}_{k,2}$ $\displaystyle=$
$\displaystyle-\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}\frac{\partial(a^{s_{p}}_{p,0}a_{q,0}^{s_{q}})}{\partial
T_{1}}\left(\int_{0}^{t}te^{i\Omega_{k,pq}t}dt\right)\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}+\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}\left(\int_{0}^{t}(b^{s_{p}}_{p,1}a_{q,0}^{s_{q}}+b^{s_{q}}_{q,1}a_{p,0}^{s_{p}})e^{i\Omega_{k,pq}t}dt\right)\delta_{k,pq}d\bm{p}d\bm{q}.$

Expression (34c) becomes

$a^{s}_{k,2}=\frac{t^{2}}{2}{\partial^{2}a^{s}_{k,0}\over\partial
T_{1}^{2}}-t{\partial a^{s}_{k,0}\over\partial T_{2}}+b^{s}_{k,2},$ (39)

with

$\displaystyle b^{s}_{k,2}$ $\displaystyle=$
$\displaystyle-\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}\frac{\partial(a^{s_{p}}_{p,0}a_{q,0}^{s_{q}})}{\partial
T_{1}}\left(\int_{0}^{t}\left[\Delta(\Omega_{k,pq})+te^{i\Omega_{k,pq}t}\right]dt\right)\delta_{k,pq}d\bm{p}d\bm{q}$
(40) $\displaystyle\mbox{}+\sum_{s_{p}s_{q}s_{p^{\prime}}s_{q^{\prime}}}\int
2L^{ss_{p}s_{q}}_{kpq}L^{s_{p}s_{p^{\prime}}s_{q^{\prime}}}_{pp^{\prime}q^{\prime}}a_{p^{\prime},0}^{s_{p^{\prime}}}a_{q^{\prime},0}^{s_{q^{\prime}}}a_{q,0}^{s_{q}}\left(\int_{0}^{t}\Delta(\Omega_{p,p^{\prime}q^{\prime}})e^{i\Omega_{k,pq}t}dt\right)$
$\displaystyle\mbox{}\times\delta_{k,pq}\delta_{p,p^{\prime}q^{\prime}}d\bm{p}d\bm{q}d\bm{p}^{\prime}d\bm{q}^{\prime}.$

The time integrals give the relations

$\displaystyle\int_{0}^{t}\left[\Delta(\Omega_{k,pq})+te^{i\Omega_{k,pq}t}\right]dt$
$\displaystyle=$ $\displaystyle t\Delta(\Omega_{k,pq}),$ (41)
$\displaystyle\int_{0}^{t}\Delta(\Omega_{p,p^{\prime}q^{\prime}})e^{i\Omega_{k,pq}t}dt$
$\displaystyle=$
$\displaystyle\frac{\Delta(\Omega_{k,p^{\prime}q^{\prime}q})-\Delta(\Omega_{k,pq})}{i(\Omega_{k,p^{\prime}q^{\prime}q}-\Omega_{k,pq})},$
(42)

that will be used below in the long time limit.

###  III.1 First asymptotic closure at time $T_{1}$

With the previous definitions, the perturbative expansion of the second-order
moment writes

$\displaystyle\langle a^{s}_{k}a^{s^{\prime}}_{k^{\prime}}\rangle$
$\displaystyle=$ $\displaystyle\langle(a^{s}_{k,0}+\epsilon
a^{s}_{k,1}+\epsilon^{2}a^{s}_{k,2}+...)(a^{s^{\prime}}_{k^{\prime},0}+\epsilon
a^{s^{\prime}}_{k^{\prime},1}+\epsilon^{2}a^{s^{\prime}}_{k^{\prime},2}+...)\rangle$
$\displaystyle=$ $\displaystyle\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},0}\rangle+\epsilon\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},1}+a^{s}_{k,1}a^{s^{\prime}}_{k^{\prime},0}\rangle+\epsilon^{2}\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},2}+a^{s}_{k,1}a^{s^{\prime}}_{k^{\prime},1}+a^{s}_{k,2}a^{s^{\prime}}_{k^{\prime},0}\rangle+...,$

where $\langle\rangle$ denotes the ensemble average. 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,
$q^{ss^{\prime}}(\bm{k},\bm{k}^{\prime})\equiv q_{k}^{ss^{\prime}}$, such that

$\langle
a^{s}_{k}a^{s^{\prime}}_{k^{\prime}}\rangle=q_{k}^{ss^{\prime}}\delta(\bm{k}+\bm{k}^{\prime}),$
(44)

where the presence of $\delta(\bm{k}+\bm{k}^{\prime})$ is the consequence of
the statistical homogeneity (Galtier, 2023a). We have to adapt the original
formalism developed for isotropic problems (Benney and Saffman, 1966) to this
anisotropic case where the dispersion relation depends not only on the
wavenumber $k$ but also on the component $k_{\parallel}$. In this case, a
contribution from $q^{ss^{\prime}}$ is only relevant if $s=s^{\prime}$,
whereas it is for $s=-s^{\prime}$ in the case of an isotropic problem. (With
such conditions $q^{ss^{\prime}}_{k}$ is real.) We assume – and this is the
basic idea of the method – that the second-order moment (in fact, the
coefficient $q^{ss^{\prime}}_{k}$ in front of the delta function) in the left
hand side of expression (III.1) remains bounded at all time (Benney and
Saffman, 1966). 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 at each order in $\epsilon$. We will see that
secular terms can appear at different orders in $\epsilon$; this leads to
certain conditions to cancel their contributions 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. The main problem is
therefore to account for all the secular contributions.

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

At order $\mathcal{O}(\epsilon^{1})$, we have the contribution

$\displaystyle\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},1}+a^{s}_{k,1}a^{s^{\prime}}_{k^{\prime},0}\rangle$
$\displaystyle=$ $\displaystyle\left\langle a^{s}_{k,0}\left(-t{\partial
a^{s^{\prime}}_{k^{\prime},0}\over\partial
T_{1}}+b^{s^{\prime}}_{k^{\prime},1}\right)+\left(-t{\partial
a^{s}_{k,0}\over\partial
T_{1}}+b^{s}_{k,1}\right)a^{s^{\prime}}_{k^{\prime},0}\right\rangle$ (45)
$\displaystyle=$ $\displaystyle-t\frac{\partial}{\partial T_{1}}\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},0}\rangle+\langle
a^{s}_{k,0}b^{s^{\prime}}_{k^{\prime},1}\rangle+\langle
b^{s}_{k,1}a^{s^{\prime}}_{k^{\prime},0}\rangle.$

The first term on the right hand side gives a secular contribution
proportional to $t$. For the second term, we have

$\sum_{s_{p}s_{q}}\int L^{ss_{p}s_{q}}_{kpq}\langle
a^{s}_{k,0}a_{p,0}^{s_{p}}a_{q,0}^{s_{q}}\rangle\Delta(\Omega_{k,pq})\delta_{k,pq}d\bm{p}d\bm{q}.$
(46)

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 generalised
functions)

$\displaystyle\Delta(X)$ $\displaystyle=$
$\displaystyle\frac{e^{iXt}-1}{iX}\xrightarrow{\text{t
$\to+\infty$}}\pi\delta(X)+i{\cal P}\left(\frac{1}{X}\right),$ (47)

where ${\cal P}$ is the Cauchy principal value of the integral. Therefore, the
long time limit of expression (46) gives no secular contribution. The same
conclusion is obtained for the third term of expression (45). Also, the
condition to cancel the unique secular term is

$\frac{\partial\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},0}\rangle}{\partial T_{1}}=0,$ (48)

which means that the second-order moment does not evolve over a time scale
$T_{1}$. As will be seen later, a turbulent cascade in inertial wave
turbulence is only expected on a time scale $T_{2}$. (Here we have a point of
disagreement with expression (2.43) in Benney and Saffman (1966): it is not
correct, but this has no impact on the rest of the paper.)

###  III.2 Second asymptotic closure at time $T_{2}$

The analysis continues at order $\mathcal{O}(\epsilon^{2})$. With expression
(39), the next contribution reads

$\displaystyle\left\langle
a^{s}_{k,1}a^{s^{\prime}}_{k^{\prime},1}+a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},2}+a^{s}_{k,2}a^{s^{\prime}}_{k^{\prime},0}\right\rangle=\left\langle\left(-t{\partial
a^{s}_{k,0}\over\partial T_{1}}+b^{s}_{k,1}\right)\left(-t{\partial
a^{s^{\prime}}_{k^{\prime},0}\over\partial
T_{1}}+b^{s^{\prime}}_{k^{\prime},1}\right)\right\rangle$ (49)
$\displaystyle\mbox{}+\left\langle
a^{s}_{k,0}\left(\frac{t^{2}}{2}{\partial^{2}a^{s^{\prime}}_{k^{\prime},0}\over\partial
T_{1}^{2}}-t{\partial a^{s^{\prime}}_{k^{\prime},0}\over\partial
T_{2}}+b^{s^{\prime}}_{k^{\prime},2}\right)+a^{s^{\prime}}_{k^{\prime},0}\left(\frac{t^{2}}{2}{\partial^{2}a^{s}_{k,0}\over\partial
T_{1}^{2}}-t{\partial a^{s}_{k,0}\over\partial
T_{2}}+b^{s}_{k,2}\right)\right\rangle,$

which gives after development and simplifications

$\displaystyle\left\langle
a^{s}_{k,1}a^{s^{\prime}}_{k^{\prime},1}+a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},2}+a^{s}_{k,2}a^{s^{\prime}}_{k^{\prime},0}\right\rangle$
$\displaystyle=$ $\displaystyle\frac{t^{2}}{2}\frac{\partial^{2}\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},0}\rangle}{\partial
T_{1}^{2}}-t\frac{\partial\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},0}\rangle}{\partial T_{2}}+\left\langle
b^{s}_{k,1}b^{s^{\prime}}_{k^{\prime},1}\right\rangle$ $\displaystyle-$
$\displaystyle\mbox{}t\left\langle{\partial a^{s}_{k,0}\over\partial
T_{1}}b^{s^{\prime}}_{k^{\prime},1}+b^{s}_{k,1}{\partial
a^{s^{\prime}}_{k^{\prime},0}\over\partial T_{1}}\right\rangle+\left\langle
a^{s}_{k,0}b^{s^{\prime}}_{k^{\prime},2}+a^{s^{\prime}}_{k^{\prime},0}b^{s}_{k,2}\right\rangle.$

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

$\displaystyle\left\langle
b^{s}_{k,1}b^{s^{\prime}}_{k^{\prime},1}\right\rangle$ $\displaystyle=$
$\displaystyle\sum_{s_{p}s_{q}s_{p^{\prime}}s_{q^{\prime}}}\int
L^{ss_{p}s_{q}}_{kpq}L^{s^{\prime}s_{p^{\prime}}s_{q^{\prime}}}_{k^{\prime}p^{\prime}q^{\prime}}\langle
a_{p,0}^{s_{p}}a_{q,0}^{s_{q}}a_{p^{\prime},0}^{s_{p^{\prime}}}a_{q^{\prime},0}^{s_{q^{\prime}}}\rangle\Delta(\Omega_{k,pq})\Delta(\Omega_{k^{\prime},p^{\prime}q^{\prime}})$
(51)
$\displaystyle\mbox{}\times\delta_{k,pq}\delta_{k^{\prime},p^{\prime}q^{\prime}}d\bm{p}d\bm{q}d\bm{p}^{\prime}d\bm{q}^{\prime}.$

Here again, the theory of generalised functions gives us the long time
behaviour of this oscillating integral (with the Poincaré-Bertrand formula –
see e.g. Benney and Newell (1969))

$\displaystyle\Delta(X)\Delta(-X)\xrightarrow{\text{t $\to+\infty$}}2\pi
t\delta(X)+2{\cal P}\left(\frac{1}{X}\right)\frac{\partial}{\partial X}.$ (52)

Therefore, a secular contribution (proportional to $t$) involving a Dirac
function is possible. The fourth-order moment in expression (51) can be
decomposed into products of second-order cumulant plus a fourth-order cumulant
such that (the statistical homogeneity is used as well as $\langle
a_{k,0}^{s}\rangle=0$)

$\displaystyle\langle
a_{p,0}^{s_{p}}a_{q,0}^{s_{q}}a_{p^{\prime},0}^{s_{p^{\prime}}}a_{q^{\prime},0}^{s_{q^{\prime}}}\rangle=q^{s_{p}s_{q}s_{p^{\prime}}s_{q^{\prime}}}_{pqp^{\prime},0}\delta(\bm{p}+\bm{q}+\bm{p}^{\prime}+\bm{q}^{\prime})+q^{s_{p}s_{q}}_{p,0}q^{s_{p^{\prime}}s_{q^{\prime}}}_{p^{\prime},0}\delta(\bm{p}+\bm{q})\delta(\bm{p}^{\prime}+\bm{q}^{\prime})$
$\displaystyle\mbox{}+q^{s_{p}s_{p^{\prime}}}_{p,0}q^{s_{q}s_{q^{\prime}}}_{q,0}\delta(\bm{p}+\bm{p}^{\prime})\delta(\bm{q}+\bm{q}^{\prime})+q^{s_{p}s_{q^{\prime}}}_{p,0}q^{s_{q}s_{p^{\prime}}}_{q,0}\delta(\bm{p}+\bm{q}^{\prime})\delta(\bm{q}+\bm{p}^{\prime}).$
(53)

Note that according to expression (III.1) by homogeneity we also have the
relation $\bm{k}=-\bm{k}^{\prime}$. We obtain

$\displaystyle\left\langle
b^{s}_{k,1}b^{s_{k^{\prime}}}_{k^{\prime},1}\right\rangle=\sum_{s_{p}s_{q}s_{p^{\prime}}s_{q^{\prime}}}\int
L^{ss_{p}s_{q}}_{kpq}L^{s^{\prime}s_{p^{\prime}}s_{q^{\prime}}}_{k^{\prime}p^{\prime}q^{\prime}}\left[q^{s_{p}s_{q}s_{p^{\prime}}s_{q^{\prime}}}_{pqp^{\prime},0}\delta(\bm{p}+\bm{q}+\bm{p}^{\prime}+\bm{q}^{\prime})\right.$
(54)
$\displaystyle\mbox{}+q^{s_{p}s_{q}}_{p,0}q^{s_{p^{\prime}}s_{q^{\prime}}}_{p^{\prime},0}\delta(\bm{p}+\bm{q})\delta(\bm{p}^{\prime}+\bm{q}^{\prime})+q^{s_{p}s_{p^{\prime}}}_{p,0}q^{s_{q}s_{q^{\prime}}}_{q,0}\delta(\bm{p}+\bm{p}^{\prime})\delta(\bm{q}+\bm{q}^{\prime})$
$\displaystyle\left.\mbox{}+q^{s_{p}s_{q^{\prime}}}_{p,0}q^{s_{q}s_{p^{\prime}}}_{q,0}\delta(\bm{p}+\bm{q}^{\prime})\delta(\bm{q}+\bm{p}^{\prime})\right]\Delta(\Omega_{k,pq})\Delta(\Omega_{k^{\prime},p^{\prime}q^{\prime}})\delta_{k,pq}\delta_{k^{\prime},p^{\prime}q^{\prime}}d\bm{p}d\bm{q}d\bm{p}^{\prime}d\bm{q}^{\prime}.$

We are looking for secular contributions. In the second line, the first term
does not contribute since it imposes $\bm{k}={\bf 0}$ which cancels
$L^{ss_{p}s_{q}}_{kpq}$, but the second term can contribute (in this
anisotropic problem) when the conditions $s_{p}=s_{p^{\prime}}$ and
$s_{q}=s_{q^{\prime}}$ are satisfies. Likewise, in the third line a
contribution is possible when $s_{p}=s_{q^{\prime}}$ and
$s_{q}=s_{p^{\prime}}$. 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). In summary, in the long time limit,
the secular contribution, written ${\cal C}_{t}\left\langle
b^{s}_{k,1}b^{s^{\prime}}_{k^{\prime},1}\right\rangle$, is

$\displaystyle{\cal C}_{t}\left\langle
b^{s}_{k,1}b^{s^{\prime}}_{k^{\prime},1}\right\rangle$ $\displaystyle=$
$\displaystyle 4\pi t\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}L^{ss_{p}s_{q}}_{k-p-q}q^{s_{p}s_{p}}_{p,0}q^{s_{q}s_{q}}_{q,0}\delta(\Omega_{k,pq})\delta_{k,pq}\delta_{kk^{\prime}}d\bm{p}d\bm{q}$
(55) $\displaystyle=$ $\displaystyle 4\pi
t\sum_{s_{p}s_{q}}\int|L^{ss_{p}s_{q}}_{kpq}|^{2}q^{s_{p}s_{p}}_{p,0}q^{s_{q}s_{q}}_{q,0}\delta(\Omega_{k,pq})\delta_{k,pq}\delta_{kk^{\prime}}d\bm{p}d\bm{q}.$

The fourth term on the right hand side of equation (III.2) 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

$\displaystyle\langle
a_{k}^{s}a_{k^{\prime}}^{s^{\prime}}a_{k^{\prime\prime}}^{s^{\prime\prime}}...\rangle$
$\displaystyle=$ $\displaystyle\langle
a_{k,0}^{s}a_{k^{\prime},0}^{s^{\prime}}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...\rangle$
$\displaystyle\mbox{}+\epsilon\langle
a_{k,1}^{s}a_{k^{\prime},0}^{s^{\prime}}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...+a_{k,0}^{s}a_{k^{\prime},1}^{s^{\prime}}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...+a_{k,0}^{s}a_{k^{\prime},0}^{s^{\prime}}a_{k^{\prime\prime},1}^{s^{\prime\prime}}....+....\rangle$
$\displaystyle\mbox{}+\epsilon^{2}\langle....\rangle+...$

As for the second-order moment, we demand that the moments (ie. the
coefficients in front of the delta functions) of order n are bounded. At order
$\mathcal{O}(\epsilon)$, we obtain the relation

$\displaystyle\langle
a_{k,1}^{s}a_{k^{\prime},0}^{s^{\prime}}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...+a_{k,0}^{s}a_{k^{\prime},1}^{s^{\prime}}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...+a_{k,0}^{s}a_{k^{\prime},0}^{s^{\prime}}a_{k^{\prime\prime},1}^{s^{\prime\prime}}...+...\rangle$
(57) $\displaystyle=\left\langle\left(-t{\partial a^{s}_{k,0}\over\partial
T_{1}}+b^{s}_{k,1}\right)a_{k^{\prime},0}^{s^{\prime}}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...+a_{k,0}^{s}\left(-t{\partial
a^{s^{\prime}}_{k^{\prime},0}\over\partial
T_{1}}+b^{s^{\prime}}_{k^{\prime},1}\right)a_{k^{\prime\prime},0}^{s^{\prime\prime}}...+...\right\rangle$
$\displaystyle=-t{\partial\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},0}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...\rangle\over\partial
T_{1}}+\langle
b^{s}_{k,1}a_{k^{\prime},0}^{s^{\prime}}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...\rangle+\langle
a^{s}_{k,0}b^{s^{\prime}}_{k^{\prime},1}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...\rangle+...$

Only the first term of the last line gives a secular contribution over the
long time, which means that we have to impose the asymptotic condition

${\partial\langle
a^{s}_{k,0}a^{s^{\prime}}_{k^{\prime},0}a_{k^{\prime\prime},0}^{s^{\prime\prime}}...\rangle\over\partial
T_{1}}=0,$ (58)

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 (III.2) does not contribute in the long time.

The last term of equation (III.2) writes

$\displaystyle\left\langle
a^{s}_{k,0}b^{s^{\prime}}_{k^{\prime},2}+a^{s^{\prime}}_{k^{\prime},0}b^{s}_{k,2}\right\rangle=\sum_{s_{p}s_{q}s_{p^{\prime}}s_{q^{\prime}}}\int
2L^{s^{\prime}s_{p}s_{q}}_{k^{\prime}pq}L^{s_{p}s_{p^{\prime}}s_{q^{\prime}}}_{pp^{\prime}q^{\prime}}\left\langle
a^{s}_{k,0}a_{p^{\prime},0}^{s_{p^{\prime}}}a_{q^{\prime},0}^{s_{q^{\prime}}}a_{q,0}^{s_{q}}\right\rangle$
$\displaystyle\mbox{}\times\left(\frac{\Delta(\Omega_{k^{\prime},p^{\prime}q^{\prime}q})-\Delta(\Omega_{k^{\prime},pq})}{i(\Omega_{k^{\prime},p^{\prime}q^{\prime}q}-\Omega_{k^{\prime},pq})}\right)\delta_{k^{\prime},pq}\delta_{p,p^{\prime}q^{\prime}}d\bm{p}d\bm{q}d\bm{p}^{\prime}d\bm{q}^{\prime}$
$\displaystyle+\sum_{s_{p}s_{q}s_{p^{\prime}}s_{q^{\prime}}}\int
2L^{ss_{p}s_{q}}_{kpq}L^{s_{p}s_{p^{\prime}}s_{q^{\prime}}}_{pp^{\prime}q^{\prime}}\left\langle
a^{s^{\prime}}_{k^{\prime},0}a_{p^{\prime},0}^{s_{p^{\prime}}}a_{q^{\prime},0}^{s_{q^{\prime}}}a_{q,0}^{s_{q}}\right\rangle\left(\frac{\Delta(\Omega_{k,p^{\prime}q^{\prime}q})-\Delta(\Omega_{k,pq})}{i(\Omega_{k,p^{\prime}q^{\prime}q}-\Omega_{k,pq})}\right)$
$\displaystyle\mbox{}\times\delta_{k,pq}\delta_{p,p^{\prime}q^{\prime}}d\bm{p}d\bm{q}d\bm{p}^{\prime}d\bm{q}^{\prime}.$
(59)

The secular contributions will be given by the theory of generalised functions
with the following relation

$\frac{\Delta(X)-\Delta(0)}{iX}\xrightarrow{\text{t $\to+\infty$}}\pi
t\delta(X)+it{\cal P}\left(\frac{1}{X}\right).$ (60)

We also need to use the following development (and its symmetric in
$\bm{k}^{\prime}$)

$\displaystyle\langle
a_{k,0}^{s}a_{p^{\prime},0}^{s_{p^{\prime}}}a_{q^{\prime},0}^{s_{q^{\prime}}}a_{q,0}^{s_{q}}\rangle=q^{ss_{p^{\prime}}s_{q^{\prime}}s_{q}}_{kp^{\prime}q^{\prime},0}\delta(\bm{k}+\bm{p}^{\prime}+\bm{q}^{\prime}+\bm{q})+q^{ss_{q}}_{k,0}q^{s_{p^{\prime}}s_{q^{\prime}}}_{p^{\prime},0}\delta(\bm{k}+\bm{q})\delta(\bm{p}^{\prime}+\bm{q}^{\prime})$
$\displaystyle\mbox{}+q^{ss_{q^{\prime}}}_{k,0}q^{s_{p^{\prime}}s_{q}}_{p^{\prime},0}\delta(\bm{k}+\bm{q}^{\prime})\delta(\bm{p}^{\prime}+\bm{q})+q^{ss_{p^{\prime}}}_{k,0}q^{s_{q^{\prime}}s_{q}}_{q^{\prime},0}\delta(\bm{k}+\bm{p}^{\prime})\delta(\bm{q}^{\prime}+\bm{q}).$
(61)

In the right hand side of expression (III.2), 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 $s=s_{q^{\prime}}$,
$s_{p^{\prime}}=s_{q}$ and $s=s_{p^{\prime}}$, $s_{q^{\prime}}=s_{q}$,
respectively. After substitution and simplification, we obtain the secular
contributions in the long time limit

$\displaystyle{\cal C}_{t}\left\langle
a^{s}_{k,0}b^{s^{\prime}}_{k^{\prime},2}+a^{s^{\prime}}_{k^{\prime},0}b^{s}_{k,2}\right\rangle=$
(62) $\displaystyle\mbox{}+4t\sum_{s_{p}s_{q}}\int
L^{s^{\prime}s_{p}s_{q}}_{k^{\prime}pq}L^{s_{p}s_{q}s^{\prime}}_{p-qk^{\prime}}q^{s^{\prime}s^{\prime}}_{k^{\prime},0}q^{s_{q}s_{q}}_{q,0}\left(\pi\delta(\Omega_{k^{\prime},pq})+i{\cal
P}\left(\frac{1}{\Omega_{k^{\prime},pq}}\right)\right)\delta_{k^{\prime},pq}\delta_{kk^{\prime}}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}+4t\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}L^{s_{p}s_{q}s}_{p-qk}q^{ss}_{k,0}q^{s_{q}s_{q}}_{q,0}\left(\pi\delta(\Omega_{k,pq})+i{\cal
P}\left(\frac{1}{\Omega_{k,pq}}\right)\right)\delta_{k,pq}\delta_{kk^{\prime}}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}=8\pi t\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}L^{s_{p}s_{q}s}_{p-qk}q^{ss}_{k,0}q^{s_{q}s_{q}}_{q,0}\delta(\Omega_{k,pq})\delta_{k,pq}\delta_{kk^{\prime}}d\bm{p}d\bm{q}.$

The last writing is obtained by using the general property
$L^{ss_{p}s_{q}}_{-k-p-q}=L^{ss_{p}s_{q}}_{kpq}$ and the symmetry in
$\bm{p}\to-\bm{p}$ and $\bm{q}\to-\bm{q}$.

If in expression (III.2) we impose the nullity of the sum of the different
secular contributions, we find the asymptotic condition (after integration
over $k^{\prime}$)

$\displaystyle\frac{\partial q^{ss^{\prime}}_{k,0}}{\partial T_{2}}$
$\displaystyle=$ $\displaystyle
4\pi\sum_{s_{p}s_{q}}\int|L^{ss_{p}s_{q}}_{kpq}|^{2}q^{s_{p}s_{p}}_{p,0}q^{s_{q}s_{q}}_{q,0}\delta(\Omega_{k,pq})\delta_{k,pq}d\bm{p}d\bm{q}$
$\displaystyle\mbox{}+8\pi\sum_{s_{p}s_{q}}\int
L^{ss_{p}s_{q}}_{kpq}L^{s_{p}s_{q}s}_{p-qk}q^{ss}_{k,0}q^{s_{q}s_{q}}_{q,0}\delta(\Omega_{k,pq})\delta_{k,pq}d\bm{p}d\bm{q}.$

Introducing $e^{s}_{k}\equiv q^{ss}_{k,0}$, we end up with (some simple
manipulations are also used to symmetrise the equation)

$\displaystyle\frac{\partial e_{k}^{s}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{\pi\epsilon^{2}}{16s\omega_{k}}\sum_{s_{p}s_{q}}\int\left(\frac{\sin\theta_{k}}{k_{\perp}}\right)^{2}(s_{p}p_{\perp}-s_{q}q_{\perp})^{2}(sk_{\perp}+s_{p}p_{\perp}+s_{q}q_{\perp})^{2}$
$\displaystyle\mbox{}\times\left[s\omega_{k}e_{p}^{s_{p}}e_{q}^{s_{q}}+s_{p}\omega_{p}e_{k}^{s}e_{q}^{s_{q}}+s_{q}\omega_{q}e_{k}^{s}e_{p}^{s_{p}}\right]\delta(\Omega_{kpq})\delta_{kpq}d\bm{p}d\bm{q},$

with $\theta_{k}$ the opposite angle to $\bm{k_{\perp}}$ in the triangle
$\bm{k_{\perp}}=\bm{p_{\perp}}+\bm{q_{\perp}}$. Expression (III.2) is the
kinetic equation for inertial wave turbulence in the anisotropic limit (see
equation 7 in Galtier (2003); the small difference (sign and numerical factor)
depends only on the normalisation of the canonical variables (7)). This result
proves that the anisotropic and asymptotic limits commute. Note that the
kinetic equation for inertial wave turbulence does not describe the slow mode
($k_{\parallel}=0$) which involves strong turbulence.

##  IV Discussion and conclusion

Our study completes the original derivation of Galtier (2003) where the
delicate issue of the asymptotic limit was mentioned but not explicitly used.
We obtained the kinetic equation using a multiple time scale method that leads
to a sequential asymptotic closures at times $T_{1}$ and $T_{2}$. This method
consists of imposing the nullity of secular terms (proportional to $t$) which
emerge over asymptotically long times in order to guarantee a bounded value
for the associated moments and thus keep the development uniform in time.
These secular terms do not involve fourth-order cumulants but only products of
second-order cumulants. Consequently, the derivation of the kinetic equation
(III.2) is performed in a systematic and consistent manner, and does not
require any closure assumptions such as quasi-Gaussianity. In addition, the
random phase approximation is natural as it arises dynamically from the
separation of time scales. The latter property was first mentioned by Benney
and Saffman (1966), but has not always been recognized as an inherent property
of wave turbulence (Zakharov et al., 1992; Nazarenko, 2011). Generally, the
random phase approximation is introduced (initially or subsequently) to fully
justify the closure. This fundamental difference with the multiple time scale
method suggests that the latter is the most natural method for deriving the
kinetic equation of wave turbulence. 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. The proof of this
conjecture remains a mathematical challenge (Deng and Zaher, 2021).

The domain of validity of wave turbulence has already been discussed in
Galtier (2003). The conclusion is that we can always find a finite domain
where inertial wave turbulence exists. To show this, we introduce the linear
time $\tau_{W}\sim 1/\omega_{k}\sim k/(k_{\parallel}\Omega_{0})$ and the non-
linear time $\tau_{NL}\sim 1/(ku)$. In the anisotropic limit, we have $k\sim
k_{\perp}$, and we obtain the time ratio

$\chi=\frac{\tau_{W}}{\tau_{NL}}\sim\frac{k_{\perp}^{2}u}{k_{\parallel}\Omega_{0}},$
(64)

which depends on the scale. If the system under study is initially excited
locally at large scale, isotropically, and with a Rossby number much smaller
than $1$, then we can obtain $\chi\ll 1$ at large scale, which is the
condition for having weak turbulence. As we have explained, such turbulence
becomes anisotropic, with the energy cascading towards a region of Fourier
space where $k_{\perp}\gg k_{\parallel}$. This leads to an increase of $\chi$
(the dependence of $u$ on wavenumbers does not alter this trend). Clearly, at
sufficiently large values of $k_{\perp}$, we can find $\chi\sim 1$, which is
synonymous with strong turbulence (and the critical balance regime). There is
therefore a domain in Fourier space where the condition for the validity of
wave turbulence can be satisfied even in the anisotropic limit.

In summary, it can be said that inertial wave turbulence for three-wave
interactions is characterised by a dynamics on two time scales. On short time
scales, of the order of the wave period, there is phase mixing which leads,
due to the dispersive nature of the waves, to the decoupling of the
correlations if there are initially present and to a statistics close to
Gaussianity, as expected from the central limit theorem. This happens with a
decay in $1/t$. On a longer time scale, the non-linear coupling – weak at
short times – becomes non-negligible due to 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.

The second novelty of our study is the demonstration that anisotropy and
asymptotic closure commute. (Note that this property was also found in Alfvén
wave turbulence (Galtier et al., 2002).) Therefore, to obtain the main
properties of inertial wave turbulence (exact power law solution, direction of
the cascade, Kolmogorov constant, existence of an inertial range), the limit
$k_{\perp}\gg k_{\parallel}$ can be taken before any statistical development.
The study also reveals that it is not necessary to use an helicity basis,
which simplifies the treatment. The third novelty is that the multiple time
scale method has been generalized to an anisotropic problem involving
different types of correlation in terms of directional polarity.

Note that in the present derivation, the system under study is assumed to be
of infinite size and can therefore be treated as continuous. The numerical
simulation with its grid of points escapes this description. Effects (freezing
of the cascade) linked to the discretisation of the Fourier space can appear
because the resonance conditions are a priori more difficult to satisfy (see
for example Connaughton et al. (2001) for capillary waves). In theory, the
weaker the non-linearities, the more important these effects are. In inertial
wave turbulence, Bourouiba (2008) has shown that discretisation effects become
non-negligible 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, with the resonances,
contribute to the energy transfer.

###### Acknowledgements.

I would like to thank and pay tribute to Professor Vladimir Zakharov, whose
contribution to wave turbulence has been profound and inspiring for me.

## References

  * Galtier (2003) S. Galtier, Phys. Rev. E 68, 015301 (2003).
  * Galtier (2023a) S. Galtier, _Physics of Wave Turbulence_ (Cambridge University Press, 2023a).
  * Galtier and Nazarenko (2017) S. Galtier and S. Nazarenko, Phys. Rev. Lett. 119, 221101 (2017).
  * Hassaini and Mordant (2017) R. Hassaini and N. Mordant, Phys. Rev. Fluids 2, 094803 (2017).
  * Hassaini et al. (2019) R. Hassaini, N. Mordant, B. Miquel, G. Krstulovic, and G. Düring, Phys. Rev. E 99, 033002 (2019).
  * Savaro et al. (2020) C. Savaro, A. Campagne, M. Linares, P. Augier, J. Sommeria, T. Valran, S. Viboud, and N. Mordant, Phys. Rev. Fluids 5, 073801 (2020).
  * Galtier and Nazarenko (2021) S. Galtier and S. Nazarenko, Phys. Rev. Lett. 127, 131101 (2021).
  * Ricard and Falcon (2021) G. Ricard and E. Falcon, EPL (Europhysics Letters) 135, 64001 (2021).
  * David and Galtier (2022) V. David and S. Galtier, J. Plasma Phys. 88, 905880509 (2022).
  * Falcon and Mordant (2022) E. Falcon and N. Mordant, Ann. Rev. Fluid Mech. 54, 1 (2022).
  * Griffin et al. (2022) A. Griffin, G. Krstulovic, V. L’vov, and S. Nazarenko, Phys. Rev. Lett. 128, 224501 (2022).
  * Hrabski and Pan (2022) A. Hrabski and Y. Pan, J. Fluid Mech. 936, A47 (2022).
  * Kochurin and Kuznetsov (2022) E. A. Kochurin and E. A. Kuznetsov, Sov. J. Exp. Theor. Phys. Lett. 116, 863 (2022).
  * Onorato et al. (2022) M. Onorato, G. Dematteis, D. Proment, A. Pezzi, M. Ballarin, and L. Rondoni, Phys. Rev. E 105, 014206 (2022).
  * Rodda et al. (2022) C. Rodda, C. Savaro, G. Davis, J. Reneuve, P. Augier, J. Sommeria, T. Valran, S. Viboud, and N. Mordant, Phys. Rev. Fluids 7, 094802 (2022).
  * Zhang and Pan (2022) Z. Zhang and Y. Pan, J. Fluid Mech. 933, A58 (2022).
  * Dematteis and Lvov (2023) G. Dematteis and Y. Lvov, J. Fluid Mech. 954, A30 (2023).
  * Galtier (2023b) S. Galtier, J. Fluid Mech. 89, 905890205 (2023b).
  * Lanchon et al. (2023) N. Lanchon, D. Mora, E. Monsalve, and P.-P. Cortet, Phys. Rev. Fluids 8, 054802 (2023).
  * Novkoski et al. (2023) F. Novkoski, C.-T. Pham, and E. Falcon, Phys. Rev. E 107, 045101 (2023).
  * Zhu et al. (2023) Y. Zhu, B. Semisalov, G. Krstulovic, and S. Nazarenko, Phys. Rev. Lett. 130, 133001 (2023).
  * Hopfinger et al. (1982) E. Hopfinger, Y. Gagne, and F. Browand, J. Fluid Mech. 125, 505 (1982).
  * Jacquin et al. (1990) L. Jacquin, O. Leuchter, C. Cambon, and J. Mathieu, J. Fluid Mech. 220, 1 (1990).
  * Morize et al. (2005) C. Morize, F. Moisy, and M. Rabaud, Phys. Fluids 17, 095105 (2005).
  * Hossain (1994) M. Hossain, Phys. Fluids 6, 1077 (1994).
  * Zeman (1994) O. Zeman, Phys. Fluids 6, 3221 (1994).
  * Zhou (1995) Y. Zhou, Phys. Fluids 7, 2092 (1995).
  * Cambon et al. (1997) C. Cambon, N. Mansour, and F. Godeferd, J. Fluid Mech. 337, 303 (1997).
  * Smith and Waleffe (1999) L. Smith and F. Waleffe, Phys. Fluids 11, 1608 (1999).
  * Baroud et al. (2002) C. Baroud, B. Plapp, Z. She, and H. Swinney, Phys. Rev. Lett. 88, 114501 (2002).
  * Godeferd and Moisy (2015) F. Godeferd and F. Moisy, Appl. Mech. Rev. 67, 030802 (2015).
  * van Bokhoven et al. (2009) L. van Bokhoven, H. Clercx, G. van Heijst, and R. Trieling, Phys. Fluids 21, 096601 (2009).
  * Yarom and Sharon (2014) E. Yarom and E. Sharon, Nature Physics 10, 510 (2014).
  * Monsalve et al. (2020) E. Monsalve, M. Brunet, B. Gallet, and P.-P. Cortet, Phys. Rev. Lett. 125, 254502 (2020).
  * Bellet et al. (2006) F. Bellet, F. Godeferd, J. Scott, and C. Cambon, J. Fluid Mech. 562, 83 (2006).
  * Scott (2014) J. Scott, J. Fluid Mech. 741, 316 (2014).
  * Clark di Leoni and Mininni (2016) P. Clark di Leoni and P. Mininni, J. Fluid Mech. 809, 821 (2016).
  * Le Reun et al. (2017) T. Le Reun, B. Favier, A. Barker, and M. Le Bars, Phys. Rev. Lett. 119, 034502 (2017).
  * Sharma et al. (2018) M. Sharma, M. Verma, and S. Chakraborty, Phys. Fluids 30, 115102 (2018).
  * Galtier and David (2020) S. Galtier and V. David, Phys. Rev. Fluids 5, 044603 (2020).
  * Le Reun et al. (2020) T. Le Reun, B. Favier, and M. Le Bars, EPL (Europhysics Letters) 132, 64002 (2020).
  * Yokoyama and Takaoka (2021) N. Yokoyama and M. Takaoka, J. Fluid Mech. 908, A17 (2021).
  * David and Galtier (2023) V. David and S. Galtier, J. Fluid Mech. Rapids 955, R2 (2023).
  * Gelash et al. (2017) A. Gelash, V. L’vov, and V. Zakharov, J. Fluid Mech. 831, 128 (2017).
  * Benney and Saffman (1966) D. Benney and P. Saffman, Proc. R. Soc. Lond. A 289, 301 (1966).
  * Kraichnan (1973) R. H. Kraichnan, J. Fluid Mech. 59, 745 (1973).
  * Waleffe (1992) F. Waleffe, Phys. Fluids A 4, 350 (1992).
  * Turner (2000) L. Turner, J. Fluid Mech. 408, 205 (2000).
  * Galtier (2006) S. Galtier, J. Plasma Phys. 72, 721 (2006).
  * Galtier (2014) S. Galtier, J. Fluid Mech. 757, 114 (2014).
  * Hasselmann (1962) K. Hasselmann, J. Fluid Mech. 12, 481 (1962).
  * Newell (1968) A. Newell, Rev. Geophys. Space Physics 6, 1 (1968).
  * Lenain and Melville (2017) L. Lenain and W. Melville, J. Phys. Ocean. 47, 2123 (2017).
  * Nayfeh (2004) A. Nayfeh, _Perturbation Methods_ (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2004).
  * Benney and Newell (1969) D. Benney and A. Newell, Stud. App. Maths. 48, 29 (1969).
  * Zakharov et al. (1992) V. Zakharov, V. L’Vov, and G. Falkovich, _Kolmogorov spectra of turbulence I: Wave turbulence_ (Springer Series in Nonlinear Dynamics, Berlin: Springer, 1992, 1992).
  * Nazarenko (2011) S. Nazarenko, _Wave Turbulence_ , vol. 825 of _Lecture Notes in Physics_ (Berlin Springer Verlag, 2011).
  * Deng and Zaher (2021) Y. Deng and H. Zaher, Forum Math. Pi 9, 1 (2021).
  * Galtier et al. (2002) S. Galtier, S. V. Nazarenko, A. C. Newell, and A. Pouquet, Astrophys. J. 564, L49 (2002).
  * Connaughton et al. (2001) C. Connaughton, S. Nazarenko, and A. Pushkarev, Phys. Rev. E 63, 046306 (2001).
  * Bourouiba (2008) L. Bourouiba, Phys. Rev. E 78, 056309 (2008).