#  An effective semilocal model for wave turbulence in 2D nonlinear optics

Jonathan Skipp<EMAIL_ADDRESS>Department of Mathematics, College of
Engineering and Physical Sciences, Aston University, Aston Triangle,
Birmingham, B4 7ET, UK  Jason Laurie<EMAIL_ADDRESS>Department of
Mathematics, College of Engineering and Physical Sciences, Aston University,
Aston Triangle, Birmingham, B4 7ET, UK  Sergey Nazarenko
<EMAIL_ADDRESS>Université Côte d’Azur, CNRS-Institut de
Physique de Nice, 17 Rue Julien Lauprêtre, 06200, Nice, France

###### Abstract

The statistical evolution of ensembles of random, weakly-interacting waves is
governed by wave kinetic equations. To simplify the analysis, one frequently
works with reduced differential models of the wave kinetics. However, the
conditions for deriving such reduced models are seldom justified self-
consistently. Here, we derive a reduced model for the wave kinetics of the
Schrödinger-Helmholtz equations in two spatial dimensions, which constitute a
model for the dynamics of light in a spatially-nonlocal, nonlinear optical
medium. This model has the property of sharply localising the frequencies of
the interacting waves into two pairs, allowing for a rigorous and self-
consistent derivation of what we term the semilocal approximation model (SLAM)
of the wave kinetic equation. Using the SLAM, we study the stationary spectra
of Schrödinger-Helmholtz wave turbulence, and characterise the spectra that
carry energy downscale, and waveaction upscale, in a forced-dissipated setup.
The latter involves a nonlocal transfer of waveaction, in which waves at the
forcing scale mediate the interactions of waves at every larger scale. This is
in contrast to the energy cascade, which involves local scale-by-scale
interactions, familiar from other wave turbulent systems and from classical
hydrodynamical turbulence.

##  I Introduction

Wave turbulence is the statistical theory of large ensembles of random, weakly
nonlinear, dispersive waves [1, 2]. Accordingly, when developing the wave
turbulence description of a physical system, one is most frequently concerned
with the mean square of the wave intensity: the waveaction spectrum. The
equation of motion for the spectrum is known as the wave kinetic equation
(WKE), and describes the irreversible evolution of the spectrum over nonlinear
timescales (which are long compared to the linear wave period), due to
resonant $M$-wave interactions. For a system in $d$ spatial dimensions, the
WKE involves an integration over $\mathbb{R}^{(M-1)d}$, constrained to the
resonant manifold of interacting waves. The complexity of this so-called
collision integral makes solving the WKE a challenging task in general.
Nonetheless, certain analytic techniques exist, in particular the Zakharov-
Kraichnan transform that allows one to find the Kolmogorov-Zakharov (KZ)
cascade spectra [3]. These are stationary solutions of the WKE on which, for
many systems, the dynamical invariants cascade with constant flux through
spatial scales, via a self-similar, spectrally local (scale-by-scale)
transfer, analogous to the Kolmogorov energy spectrum in classical
hydrodynamics. The stationary spectrum of thermodynamic equilibrium—the
Rayleigh-Jeans (RJ) spectrum—can also be derived trivially as the spectrum on
which the collision integral has an integrand that vanishes pointwise.

In a seminal paper, Dyachenko et. al. [4] demonstrated that the collision
integral can be greatly simplified if one makes the _ad-hoc_ assumption that
the wave interaction coefficient is sharply peaked, so that all $M$ waves
taking part in interactions have approximately the same frequency. This
assumption, which we will refer to as superlocality, allows one to reduce the
collision integral to a differential operator. The resulting equation—the
differential approximation model (DAM)—preserves a great deal of the structure
of the original WKE, namely its conserved quantities, the degree of
nonlinearity with respect to the spectrum, its scaling with frequency, and, as
a result of the latter, the stationary RJ and KZ solutions. DAMs are the wave
turbulence equivalent of the Leith model of classical hydrodynamics [5, 6].

Being differential equations, DAMs are much easier to work with than the
collision integrals from which they are derived. They have been used in a wide
variety of physical systems to examine topics such as the stationary RJ and KZ
wave turbulence spectra [4, 7, 8, 9, 10, 11, 12], thermalisation at the end of
a cascade spectrum [13], the crossover from strong to weak turbulence [14],
and the nature of transient spectra before the KZ spectra are established,
including the anomalous scaling of spectral fronts [15, 16, 17, 18].

The reduction of a WKE to a DAM is predicated on the assumption of
superlocality. However, this assumption is rarely justified in the cases where
DAMs are applied, indeed Dyachenko et. al. [4] introduced the DAM in the
context of the cubic nonlinear Schrödinger equation, whose interaction
coefficient is a constant across Fourier space. Furthermore, DAMs are often
constructed heuristically, based on the scaling properties of the interaction
coefficient, with the desired stationary solutions and degree of nonlinearity
built in. To our knowledge, there has been no rigorous derivation of the DAM
for any system whose interaction coefficient has the required properties to
justify any locality assumption.

In this paper, we derive such a reduced model for the wave kinetics of the
Schrödinger-Helmholtz equations (SHE). We introduce the SHE in Sec. II, along
with their physical context, the dynamical invariants that they conserve
(namely energy and waveaction), their WKE, and the directions in Fourier space
that their invariants flow during the wave kinetic evolution. The SHE are of
interest to us because they comprise the first system studied in the wave
turbulence context in which the spectral locality of interactions arises
naturally from the functional form of the interaction coefficient. In fact,
the locality manifested by the SHE is one in which distinct pairs of
interacting waves are localised in frequency space. We refer to the latter as
a semilocal, as opposed to a superlocal, limit. We exploit this property in
Sec. III to reduce the kinetic equation of the SHE to a simpler model, in the
same spirit as the derivation of the DAM in Ref. [4]. It transpires that the
semilocality property allows the collision integral to be reduced to an
integro-differential operator, rather than a purely differential one. The
resulting reduction of the WKE we term the semilocal approximation model
(SLAM).

Analysis of the SLAM allows us to extract the stationary spectra of the WKE,
including prospective candidates for the KZ cascade spectra, in Sec. IV.
However, we demonstrate that the KZ waveaction cascade spectrum is
pathological, as it leads to a divergence of the SLAM at high frequency.
Furthermore, in Sec. V we show how the KZ spectra lead to flux directions that
are inconsistent with the more general argument we present in Sec. II.3.2,
requiring us to reconsider the spectra that establish the turbulent transport
of dynamical invariants across scales of the system. We proceed to find the
true waveaction flux spectrum in Sec. VI, and conclude that both stationary
solutions that describe the flux of energy on the one hand, and waveaction on
the other, are very closely related to the RJ equilibrium spectrum. The
waveaction flux spectrum, which carries waveaction to large scales, is
dominated by nonlocal interactions, with waves at the forcing scale mediating
wave interactions at all larger scales. By contrast, the energy flux spectrum,
which carries energy to small scales, has local scale-by-scale interactions.
We start by introducing the SHE in the next section.

##  II Schrödinger-Helmholtz equation

The SHE consist of a nonlinear Schrödinger equation for the dynamical variable
$\psi({\bf x},t)\in\mathbb{C}$,

$\displaystyle i\frac{\partial\psi}{\partial t}+\nabla^{2}\psi-V(\psi)\psi$
$\displaystyle=0,$ (1a) coupled, via the potential $V(\psi)\in\mathbb{R}$, to
the Helmholtz equation, $\displaystyle\nabla^{2}V-\Lambda V$
$\displaystyle=\alpha|\psi|^{2}.$ (1b)

We take the spatial domain to be a $d$-dimensional periodic box of side length
$L$. The SHE (1) are a spatially nonlocal111 The spatial nonlocality
originates from the inversion of the Helmholtz operator in Eq. (1b). The term
“local” is used here in a different sense to the locality of interactions in
frequency space, to which the semilocal approximation refers.  version of the
familiar cubic nonlinear Schrödinger equation (NLSE, also known as the Gross-
Pitaevskii equation),

$\displaystyle i\frac{\partial\psi}{\partial
t}+\nabla^{2}\psi\pm|\psi|^{2}\psi=0.$ (2)

The NLSE is obtained from Eqs. (1) by sending $\alpha,\Lambda\to\infty$ in
such a way that $\alpha/\Lambda$ remains constant, and renormalising $\psi$.

The physical applications of the SHE were discussed in Ref. [12]; in brief,
for $d=3$ they describe so-called Fuzzy Dark Matter [19, 20, 21] in a universe
with cosmological constant $\Lambda$. In $d=2$, Eqs. (1) describe the
perpendicular dynamics of laser light in a thermo-optic or elasto-optic
nonlinear medium [22, 23, 24]. In the optics case, $\Lambda$ is the normalised
Kerr coefficient of spatially-local interactions. The dynamical variable
$\psi({\bf x},t)$ represents, respectively, the wavefunction of the putative
dark matter boson, or the envelope of the electric field. Here we restrict
ourselves to the $d=2$ case.

Also closely associated with the SHE are the Schrödinger-Newton equations
(SNE) [25, 26, 27],

$\displaystyle i\frac{\partial\psi}{\partial t}+\nabla^{2}\psi-V(\psi)\psi=0,$
(3a) $\displaystyle\nabla^{2}V=\alpha|\psi|^{2},$ (3b)

which are obtained by formally setting $\Lambda=0$ in the SHE. However, as
discussed in Ref. [12], the SNE are ill-posed in periodic settings, or when
one wants to describe fluctuations over an infinite, static background,
because that background does not solve the Poisson equation (3b), c.f. the
“Jeans swindle” [28]. Non-trivial dynamics are recovered by introducing a
spatially local term to the left-hand side of Eq. (3b), i.e. moving to the SHE
[29]. This fact is reflected in the divergence of the SLAM when we set
$\Lambda=0$, see Sec. VII.1.

###  II.1 Hamiltonian and invariants of the SHE

Most commonly in wave turbulence, the equation of motion under study can be
derived via Hamilton’s equation $i\partial_{t}\psi=\delta H/\delta\psi^{*}$.
Equations (1) are no exception, with the Hamiltonian functional being

$\displaystyle H=\int|\nabla\psi|^{2}\,d{\bf
x}+\int\frac{\alpha}{2}\left[(\nabla^{2}-\Lambda)^{-1/2}|\psi|^{2}\right]^{2}\,d{\bf
x}.$ (4)

The first term on the right-hand side of Eq. (4) is the quadratic energy,
associated with the free propagation of dispersive waves. The second term is
the energy contribution due to the nonlinear interaction of waves via the
spatially-nonlocal potential
$V(\psi)=\alpha(\nabla^{2}-\Lambda)^{-1}|\psi|^{2}$ that solves Eq. (1b). The
operator $(\nabla^{2}-\Lambda)^{-q}$ with $q$ rational and positive, is to be
understood as a formal power series, and is made concrete in its Fourier-space
representation (the latter is found in Ref. [12]).

In the absence of forcing and dissipation (see below), the Hamiltonian $H$ is
conserved under the evolution via the SHE, and is strictly positive. The other
positive invariant is the waveaction (a.k.a. number of particles, in reference
to the application to Bosonic systems),

$\displaystyle N=\int|\psi|^{2}\,d{\bf x}.$ (5)

The momentum ${\bf P}=(i/2)\int(\psi\nabla\psi^{*}-\psi^{*}\nabla\psi)\,d{\bf
x}$ is yet another conserved quantity. However, not being sign-definite, it
plays no role in the argument regarding the invariant cascade directions, see
Sec. II.3.2. The momentum will not feature in the work we carry out in this
paper.

###  II.2 SHE in Fourier space

In Fourier space Eqs. (1) become an equation of motion for $\psi_{\bf
k}(t)=(1/L)^{d}\int\psi({\bf x},t)\exp(-i{\bf k}\cdot{\bf x})d{\bf x}$, the
Fourier series coefficient of $\psi({\bf x},t)$ for the wave mode with
wavevector ${\bf k}$:

$i\frac{\partial\psi_{\bf k}}{\partial t}-k^{2}\psi_{\bf
k}-\\!\\!\\!\sum_{{\bf k}_{1},{\bf k}_{2},{\bf k}_{3}}\\!\\!\\!W^{12}_{3{\bf
k}}\psi_{{\bf k}_{1}}\psi_{{\bf k}_{2}}\psi_{{\bf
k}_{3}}^{*}\delta^{12}_{3{\bf k}}=0.$ (6)

The quantity

$\displaystyle W^{12}_{3{\bf k}}=-\frac{\alpha}{2}\left[\frac{1}{|{\bf
k}_{1}-{\bf k}|^{2}+\Lambda}+\frac{1}{|{\bf k}_{2}-{\bf
k}|^{2}+\Lambda}\right]$ (7)

is the interaction coefficient for the SHE. It gives the strength of nonlinear
interactions between tetrads of waves that satisfy ${\bf k}_{1}+{\bf
k}_{2}-{\bf k}_{3}-{\bf k}=0$, to which the sum in Eq. (6) is constrained via
the Kronecker delta $\delta^{12}_{3{\bf k}}\coloneqq\delta^{{\bf k}_{1},{\bf
k}_{2}}_{{\bf k}_{3},{\bf k}}$. The functional dependence of $W^{12}_{3{\bf
k}}$ on the wavevectors is indicated in the super- and subscript indices, and
its functional form is obtained by writing Eqs. (1a) and (1b) in Fourier space
and combining. Using the Kronecker delta $\delta^{12}_{3{\bf k}}$, we
establish the symmetries $W^{12}_{3{\bf k}}=W^{21}_{3{\bf k}}=W^{12}_{{\bf
k}3}=\left(W_{12}^{3{\bf k}}\right)^{*}$, which are required by the fact that
the Hamiltonian (4) is real.

####  II.2.1 Freely-evolving vs. forced-dissipated systems

In a closed system, namely one with no sources or sinks of dynamical
invariants, Eq. (6) (equivalently Eq. (1)) describes the conservative dynamics
of the field $\psi_{\bf k}(t)$ (equivalently $\psi({\bf x},t)$) evolving
freely from an initial condition. Alternatively, one can consider a system in
which invariants are injected into the system by some forcing mechanism, and
removed from it by dissipation. These mechanisms manifest as extra terms
$F_{\bf k}$ and $D_{\bf k}$, respectively, on the right-hand side of Eq. (6)
(and their Fourier inverses on the right-hand side of Eq. (1a)). Analogous
terms appear on the right-hand side of the WKE (8) in the forced-dissipated
case.

In turbulence theory, one usually considers forcing to be isotropic (depending
only on $k\coloneqq|{\bf k}|$), statistically time-independent, and to act in
a narrow band of lengthscales around a characteristic wavenumber $k_{f}$.
$F_{\bf k}$ is then a stochastic term supported within the forcing range and
negligible elsewhere. Dissipation is usually significant at the largest and/or
smallest lengthscales of the system, and negligible elsewhere. In this paper,
we assume both large-scale and small-scale isotropic dissipation, with
respective characteristic wavenumbers $k_{d-}$ and $k_{d+}$.

Furthermore, we take the forcing and dissipation scales to be widely
separated: $k_{d-}\ll k_{f}\ll k_{d+}$. Lengthscales in between, but far from,
the forcing and dissipation scales are known as the inertial ranges. Inside
the inertial ranges the dynamics are conservative, described by Eq. (6)
(equivalently (1a), and when we consider wave turbulence, the WKE (8)) with
zero right-hand side. The role of forcing and dissipation is to provide a
source and sink of invariants on either side of the inertial ranges, setting
up a flux of invariants through them. We discuss the directions of these
fluxes in section II.3.2.

###  II.3 Wave kinetic equation

Wave turbulence is primararily concerned with the waveaction spectrum $n_{\bf
k}(t)=(L/2\pi)^{d}\langle|\psi_{\bf k}(t)|^{2}\rangle$. The operator
$\langle\cdot\rangle$ denotes an average over a statistical ensemble of
realisations, starting from independent initial conditions $\psi_{\bf k}(0)$,
with phases uniformly distributed in $[0,2\pi)$, and identically distributed
amplitudes. Taking the domain size $L\to\infty$, and then assuming weak
nonlinearity, one can derive the following WKE describing the evolution of the
spectrum at intermediate times due to the nonlinear 4-wave interactions of the
$2\leftrightarrow 2$ type [2]:

$\displaystyle\frac{\partial n_{\bf k}}{\partial t}=4\pi\int|W^{12}_{3{\bf
k}}|^{2}\delta^{12}_{3{\bf k}}\delta(\omega^{12}_{3{\bf
k}})n_{1}n_{2}n_{3}n_{\bf k}\left[\frac{1}{n_{\bf
k}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right]d{\bf k}_{1}\,d{\bf
k}_{2}\,d{\bf k}_{3}.$ (8)

The right-hand side of Eq. (8) is the collision integral, and is taken across
the joint ${\bf k}$-space
$\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathbb{R}^{2}$. Here,
$\delta^{12}_{3{\bf k}}\coloneqq\delta({\bf k}_{1}+{\bf k}_{2}-{\bf
k}_{3}-{\bf k})$ and $\delta(\omega^{12}_{3{\bf k}})$ are Dirac delta
functions that constrain interacting wave tetrads to the resonant manifold
defined by

$\displaystyle{\bf k}_{1}+{\bf k}_{2}-{\bf k}_{3}-{\bf k}=0,$ (9a)
$\displaystyle\omega^{12}_{3{\bf
k}}\coloneqq\omega_{1}+\omega_{2}-\omega_{3}-\omega_{\bf k}=0.$ (9b)

Here $\omega_{\bf k}=k^{2}$ is the linear dispersion relation. We have also
used the shorthand notation $n_{i}=n_{{\bf k}_{i}},\omega_{i}=\omega_{{\bf
k}_{i}}$ etc. for $i=1,2,3$.

Inspecting Eq. (7), we see manifestly that the interaction coefficient
$W^{12}_{3{\bf k}}$ decays rapidly when all wavevectors are very different.
The first term in $W^{12}_{3{\bf k}}$ becomes dominant when ${\bf
k}_{1}\to{\bf k}$. By Eq. (9a) we then have ${\bf k}_{2}\to{\bf k}_{3}$. If we
also have $\Lambda\ll k^{2}$, then the interaction coefficient becomes sharply
peaked in the joint ${\bf k}$-space where ${\bf k}_{1}\approx{\bf k}$ and
${\bf k}_{3}\approx{\bf k}_{2}$, with the latter following from the above
symmetries. Likewise, if the second term in $W^{12}_{3{\bf k}}$ is dominant
then it becomes peaked over ${\bf k}_{1}\approx{\bf k}_{3}$ and ${\bf
k}_{2}\approx{\bf k}$. These pairings are equivalent to the first pairings by
exchange of dummy variables, as $W^{12}_{3{\bf k}}$ always appears under a sum
or integral.

Thus, the 4-wave interactions responsible for evolution of the system under
the SHE (1), and therefore the corresponding WKE (8), are dominated by
interactions in which ${\bf k}_{1}\approx{\bf k}$ and ${\bf k}_{3}\approx{\bf
k}_{2}$. This property of the interaction coefficient, of picking out dominant
interactions when pairs of wavevectors become equal, we refer to as
semilocality. It is this property that will allow the collision integral to be
reduced to a simpler operator. We will retain the possibility that ${\bf
k}_{1}\not\approx{\bf k}_{2}$, so the reduction will be to an integro-
differential, rather than a purely differential, operator.

At this point we note that taking the NLSE limit $\alpha,\Lambda\to\infty$
with $\alpha/\Lambda\to\text{const.}$ sends $W^{12}_{3{\bf k}}\to\pm 1$. In
this case there is no natural pairing of $({\bf k}_{1},{\bf k})$ and $({\bf
k}_{2},{\bf k})$, and we lose the semilocality property. We return to this
point in Sec. VII.1.

####  II.3.1 Invariants of the wave kinetic equation

In general, WKEs of the $M/2\leftrightarrow M/2$ type ($M$ being an even
integer denoting the order of the resonant wave interaction), such as Eq. (8),
conserve the two quadratic invariants

$\displaystyle E=\int\omega_{\bf k}n_{\bf k}\,d{\bf k}\quad\text{and}\quad
N=\int n_{\bf k}\,d{\bf k}.$ (10)

Comparing with Eq. (5) we see that $N$ is the total waveaction, expressed in
Eq. (10) as an integral over Fourier space. Both the original SHE and the WKE
derived from them conserve the waveaction exactly. By contrast, $E$ is the
Fourier-space representation of the quadratic energy (first term on the right-
hand side of Eq. (4)). Recall that the WKE is derived under the assumption of
weak nonlinearity. Under this condition, $E$ will be the leading contribution
to the total Hamiltonian $H$, i.e. the original equations of motion conserve
$H$ exactly, while their WKE conserves $H$ asymptotically, and $E$ exactly.
Here, we are interested in the quantities that are conserved by the WKE and
its reduced model, the SLAM. Therefore, we will simply refer to $E$ as the
energy hereafter.

The interaction coefficient $W^{12}_{3{\bf k}}$ is unchanged under global
rotations. We further assume that when the system is forced and dissipated, it
is done so in a spatially homogeneous and isotropic manner. Therefore, we
expect that the spectrum $n_{\bf k}$ will be isotropic, depending only on
$|{\bf k}|$, or equivalently on frequency. Accordingly, we can consider the
spectrum as a function of either ${\bf k}$, or frequency $\omega$ at that
value of ${\bf k}$, via the dispersion relation $\omega=k^{2}$. Namely, we
adopt the notation $n_{\omega_{i}}\coloneqq n({\bf k}_{i}(\omega_{i}))=n_{{\bf
k}_{i}}\eqqcolon n_{i}$. Converting the ${\bf k}$-space integrals in Eqs. (10)
into integrals over $\omega$, the invariants of the WKE become, for a 2D
isotropic spectrum,

$\displaystyle E=\pi\int\omega n_{\omega}\,d\omega\quad\text{and}\quad
N=\pi\int n_{\omega}\,d\omega.$ (11)

####  II.3.2 Flux directions - the Fjørtoft argument

The action of the WKE is to redistribute the spectral density of $E$ and $N$
across ${\bf k}$-space. The qualitative manner of this redistribution is
predicted by the argument of Fjørtoft [30]. This argument is recapitulated in
many places in the wave turbulence literature, see for example Refs. [11, 12]
for its application to the forced-dissipated SHE, and Ref. [2] for a version
of the argument in freely-evolving systems.

We restrict our discussion to the isotropic case, which allows us to elide
from ${\bf k}$-space to $\omega$-space via the dispersion relation, and speak
of scales when referring to frequencies. The conclusion of the Fjørtoft
argument is that the presence of each invariant constrains how the
$\omega$-space distribution of the other invariant can evolve, so that the
bulk of each invariant moves to the sector of $\omega$-space where its
spectral density dominates. For the SHE this means that the majority of the
energy $E$, which has a spectral density of $\omega=k^{2}$, moves towards high
$\omega$, whereas most of the waveaction $N$, having a spectral density of
$1$, moves towards low $\omega$.

More specifically, for a freely-evolving system the Fjørtoft argument predicts
that the invariant densities are redistributed by the WKE so that the centroid
of $E$, defined as $(\pi/E)\int\\!\omega^{2}n_{\omega}\,d\omega$, moves
towards small scales, while the $N$-centroid, $(\pi/N)\int\\!\omega
n_{\omega}\,d\omega$, moves towards the largest scale in the system [2]. The
total $E$ and $N$ of course remain constant during the evolution.

In a forced-dissipated system, $E$ and $N$ are injected at the intermediate
forcing scale $\omega_{f}$, and transported by the WKE until they reach the
dissipation scales at $\omega_{d-}$ (large scale), and $\omega_{d+}$ (small
scale), where they are removed. We assume a wide scale separation
$\omega_{d-}\ll\omega_{f}\ll\omega_{d+}$. We further assume that the system
has reached a non-equilibrium stationary state in which forcing and
dissipation are continuous, and the rate of dissipation has adjusted to match
the rate of forcing, so that the total $E$ and $N$ remain constant. We
conjecture that this steady state condition is universal, independent of the
detailed manner of forcing and dissipation, and will be attained from a wide
range of initial conditions after a transient phase. In these circumstances,
the Fjørtoft argument predicts that most ofthe energy injected at $\omega_{f}$
will be transferred with constant positive energy flux $P$ through the direct
inertial range (namely, scales $\omega$ such that
$\omega_{f}\ll\omega\ll\omega_{d+}$), to be dissipated at small scales around
$\omega_{d+}$. Likewise, most of the waveaction injected at $\omega_{f}$ will
be transferred with constant negative waveaction flux $Q$ through the inverse
inertial range ($\omega_{d-}\ll\omega\ll\omega_{f}$), until it is dissipated
at large scales near $\omega_{d-}$. This scenario, of one invariant moving to
small scales and the other moving to large scales, is common to all wave
turbulence systems with two quadratic invariants, and also 2D hydrodynamic
turbulence [2]. It is referred to in the literature as the dual cascade,
although the term “cascade” usually implies a scale-by-scale flux of
invariants in which all the waves taking part in the transfer are localised in
$\omega$-space. We will see that for the SLAM, the direct transfer of energy
involves entirely local interactions, whereas the inverse waveaction transfer
is nonlocal, involving waves at $\omega_{f}$ participating in every tetrad of
interacting waves throughout the inertial range. In this paper we will still
speak of the direct cascade of energy, and inverse cascade of waveaction,
which together make up the dual cascade, having made this caveat about the
(non)locality of interactions in these cascades.

The Fjørtoft argument is premised only on having positive-definite integral
invariants, which are quadratic in wave amplitude, but which have different
spectral densities, and on having widely-separated forcing and dissipation
scales. Having such parsimonious assumptions, the predictions of the Fjørtoft
argument are robust, and must be recovered by any subtler manipulation of the
WKE. More concretely, once we derive the SLAM, we can look for its stationary
solutions that realise the dual cascade, but these solutions must have fluxes
$P>0$ and $Q<0$ in their respective inertial ranges, to correspond to the
predictions of the Fjørtoft argument. On the other hand, the argument makes no
assumptions about the locality of interactions in $\omega$-space. This must be
determined by examining individual candidate solutions, which is particularly
straightforward in the SLAM.

##  III Derivation of the semilocal approximation model

To derive the SLAM, we follow the initial strategy set out in Ref. [4] for
deriving the DAM. First, we multiply Eq. (8) by an arbitrary test function
$\varphi_{\bf k}=\varphi({\bf k})$, integrate with respect to ${\bf k}$, and
use the resulting symmetries of the integrand to split it into four pieces:

$\displaystyle\int\varphi_{\bf k}\frac{\partial n_{\bf k}}{\partial t}\,d{\bf
k}$ $\displaystyle=4\pi\int\varphi_{\bf k}|W^{12}_{3{\bf
k}}|^{2}\delta^{12}_{3{\bf k}}\delta(\omega^{12}_{3{\bf
k}})n_{1}n_{2}n_{3}n_{\bf k}\left[\frac{1}{n_{\bf
k}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right]d{\bf k}_{1}\,d{\bf
k}_{2}\,d{\bf k}_{3}\,d{\bf k}$ $\displaystyle=\pi\int\left[\varphi_{\bf
k}+\varphi_{3}-\varphi_{1}-\varphi_{2}\right]|W^{12}_{3{\bf
k}}|^{2}\delta^{12}_{3{\bf k}}\delta(\omega^{12}_{3{\bf
k}})n_{1}n_{2}n_{3}n_{\bf k}\left[\frac{1}{n_{\bf
k}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right]d{\bf k}_{1}\,d{\bf
k}_{2}\,d{\bf k}_{3}\,d{\bf k}.$ (12)

Next we assume that the spectra $n_{\bf k}$ and test functions $\varphi_{\bf
k}$ are isotropic, and consider both as functions of frequency (see Sec.
II.3.1).

At this point, following the discussion after Eq. (9b), we make the
semilocality assumption that ${\bf k}_{1}\approx{\bf k}$, and hence ${\bf
k}_{3}\approx{\bf k}_{2}$, but retain the possibility of ${\bf k}_{1}$ and
${\bf k}_{2}$ being distinct. This is in contrast to the procedure of
Dyachenko et al. [4], who assume that _all_ interactions are superlocal in
frequency space.

Taylor expanding the terms in square brackets in Eq. (12) up to first order in
frequency, and using Eq. (9b), we have

$\displaystyle\left[\frac{1}{n_{\bf
k}}+\frac{1}{n_{3}}-\frac{1}{n_{1}}-\frac{1}{n_{2}}\right]$
$\displaystyle\approx\partial_{\omega}{n_{\omega}}^{-1}(\omega-\omega_{1})-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}(\omega_{2}-\omega_{3})$
(13)
$\displaystyle=\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)(\omega-\omega_{1})$
$\displaystyle=\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)({\bf
k}-{\bf k}_{1})\cdot({\bf k}+{\bf k}_{1})$
$\displaystyle\approx\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)(-{\bf
p}_{1})\cdot 2{\bf k},$

where the difference vectors ${\bf p}_{i}\coloneqq{\bf k}_{i}-{\bf k}$, shown
in green in Fig. 1(a). Similarly,

$\displaystyle\left[{\varphi_{\bf
k}}+{\varphi_{3}}-{\varphi_{1}}-{\varphi_{2}}\right]\approx\left(\partial_{\omega}\varphi_{\omega}-\partial_{\omega_{2}}\varphi_{\omega_{2}}\right)(-{\bf
p}_{1})\cdot 2{\bf k}.$

Therefore, Eq. (12) simplifies to

$\displaystyle\int\varphi_{\bf k}\frac{\partial n_{\bf k}}{\partial t}\,d{\bf
k}$
$\displaystyle=\pi\int\frac{\alpha^{2}}{\left(\Lambda+p_{1}^{2}\right)^{2}}\delta(-2{\bf
p}_{1}\cdot{\bf p}_{2})n_{2}^{2}n_{k}^{2}({\bf p}_{1}\cdot 2{\bf
k})^{2}\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)\left(\partial_{\omega}\varphi_{\omega}-\partial_{\omega_{2}}\varphi_{\omega_{2}}\right)d{\bf
k}_{1}\,d{\bf k}_{2}\,d{\bf k},$

where we have also used Eq. (37) in the argument of the frequency delta
function, and exhausted the delta function of wavevectors by integrating out
${\bf k}_{3}$.

Figure 1:  Geometry of wavevectors and definition of angles involved in
deriving the SLAM. (a) A tetrad of wavevectors $\\{{\bf k}_{1},{\bf
k}_{2},{\bf k}_{3},{\bf k}\\}$ that satisfy the resonant conditions Eqs. (9)
are shown in blue. Note the semilocal approximation ${\bf k}_{1}\approx{\bf
k}$ and ${\bf k}_{3}\approx{\bf k}_{2}$. The difference vectors $\\{{\bf
p}_{i}\coloneqq{\bf k}_{i}-{\bf k}\\}$ are shown in green. (b) Triangle formed
by $\\{{\bf p}_{i}\\}$. Note that due to the frequency resonance condition
$\delta(\omega^{12}_{3{\bf k}})$ in Eq. (8), only wavevectors with difference
vectors that form a right-angle triangle, i.e. $\theta=\pi/2$, will contribute
to the collision integral, see Appendix A.

To constrain the integral to the resonant manifold, we fix ${\bf k}$ and ${\bf
k}_{2}$ and change variables from ${\bf k}_{1}$ to $(p_{1},\theta)$, where
$\theta$ is the angle between ${\bf p}_{1}$ and ${\bf p}_{2}$, see Fig. 1(b).
The volume element transforms as $d{\bf k}_{1}=p_{1}dp_{1}\,d\theta$, and we
perform the $\theta$ integral as follows:

$\displaystyle\int(\ldots)\delta(-2{\bf p}_{1}\cdot{\bf
p}_{2})p_{1}\,d\theta\,dp_{1}=\int(\ldots)\frac{1}{p_{2}}\,dp_{1},$

where we have taken into account the fact that ${\bf p}_{1}$ and ${\bf p}_{2}$
are orthogonal, see Appendix A. Then, using the properties of the scalar
triple product, we can write ${\bf k}\cdot{\bf p}_{1}=\pm{\bf k}\cdot({\bf
e}_{z}\times{\bf p}_{2})p_{1}/p_{2}=\pm{\bf e}_{z}\cdot({\bf p}_{2}\times{\bf
k})p_{1}/p_{2}$. Thus, we have

$\displaystyle\int\varphi_{\bf k}\frac{\partial n_{\bf k}}{\partial t}\,d{\bf
k}=4\pi\int\frac{\alpha^{2}p_{1}^{2}}{\left(\Lambda+p_{1}^{2}\right)^{2}}\frac{|{\bf
p}_{2}\times{\bf
k}|^{2}}{p_{2}^{3}}n_{2}^{2}n_{k}^{2}\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)\left(\partial_{\omega}\varphi_{\omega}-\partial_{\omega_{2}}\varphi_{\omega_{2}}\right)dp_{1}\,d{\bf
k}_{2}\,d{\bf k}.$ (14)

The dependence on $p_{1}$ can be factored into the reduced interaction
coefficient, which can be calculated exactly,

$\displaystyle
S_{\Lambda}=4\pi\int_{0}^{\infty}\frac{\alpha^{2}p_{1}^{2}}{\left(\Lambda+p_{1}^{2}\right)^{2}}\,dp_{1}=\frac{\pi^{2}\alpha^{2}}{\sqrt{\Lambda}}.$
(15)

This last step highlights the important feature of the SHE in this analysis:
it is in this step that we have used the peaked nature of the interaction
coefficient to reduce it to the coefficient $S_{\Lambda}$ analytically. To our
knowledge, the SHE constitute the first system analysed in wave turbulence
theory whose interaction coefficient can be reduced in this way.

To express the ${\bf p}_{2}$ dependence of (14) in terms of ${\bf k}_{2}$ and
${\bf k}$, we denote the angle between vectors ${\bf k}$ and ${\bf k}_{2}$ by
$\phi$ (see Fig. 1(a)), note that $|{\bf p}_{2}\times{\bf k}|^{2}=|{\bf
k}_{2}\times{\bf k}|^{2}=k_{2}^{2}k^{2}\sin^{2}(\phi)$, and use basic
trigonometry to re-express $p_{2}$, giving

$\displaystyle\int\varphi_{\bf k}\frac{\partial n_{\bf k}}{\partial t}\,d{\bf
k}=S_{\Lambda}\int\frac{k_{2}^{2}k^{2}\sin^{2}(\phi)}{(k_{2}^{2}-2k_{2}k\cos(\phi)+k^{2})^{3/2}}n_{2}^{2}n_{k}^{2}\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)\left(\partial_{\omega}\varphi_{\omega}-\partial_{\omega_{2}}\varphi_{\omega_{2}}\right)d{\bf
k}_{2}\,d{\bf k}.$

The ${\bf k}\leftrightarrow{\bf k}_{2}$ symmetry of the integrand allows us to
replace
$(\ldots)\left(\partial_{\omega}\varphi_{\omega}-\partial_{\omega_{2}}\varphi_{\omega_{2}}\right)\to
2(\ldots)\left(\partial_{\omega}\varphi_{\omega}\right)$.

The next step is to move to frequency space by writing the integrations over
${\bf k}$ and ${\bf k}_{2}$ in polar form, so that $d{\bf
k}=(1/2)d\chi\,d\omega$ and $d{\bf k}_{2}=(1/2)d\phi\,d\omega_{2}$. Here
$\chi$ is the polar angle of wavevector ${\bf k}$ (see Fig. 1(a)), which we
integrate out immediately and cancel from both sides. We obtain

$\displaystyle\int\varphi_{\omega}\frac{\partial n_{\omega}}{\partial
t}\,d\omega$
$\displaystyle=S_{\Lambda}\int\frac{\omega_{2}\omega\sin^{2}(\phi)}{(\omega_{2}-2\sqrt{\omega_{2}\omega}\cos(\phi)+\omega)^{3/2}}n_{\omega_{2}}^{2}n_{\omega}^{2}\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)\left(\partial_{\omega}\varphi_{\omega}\right)d\phi\,d\omega_{2}\,d\omega.$
(16)

Finally, we integrate by parts with respect to $\omega$ to isolate the test
function $\varphi_{\omega}$ on both sides, and use the fact that
$\varphi_{\omega}$ is arbitrary, to obtain the SLAM for the SHE:

$\displaystyle\frac{\partial n_{\omega}}{\partial
t}=-\frac{1}{\pi}\frac{\partial Q}{\partial\omega},$ (17a) where
$\displaystyle Q=\pi S_{\Lambda}\int
f\\!\left(\\!\sqrt{\frac{\omega_{2}}{\omega}}\right)\frac{\omega_{2}}{\sqrt{\omega}}\,n_{\omega_{2}}^{2}n_{\omega}^{2}\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)d\omega_{2}$
(17b) is the waveaction flux flowing through $\omega$ in frequency space (or
circle of radius $k=\sqrt{\omega}$ in wavevector space; the factor of $\pi$
arises from the transformation between the two spaces). In Eq. (17b) the
function $f(s)$ is defined as $\displaystyle
f(s)=\int_{0}^{2\pi}\frac{\sin^{2}(\phi)}{(1-2s\cos(\phi)+s^{2})^{3/2}}\,d\phi.$
(17c)

In Appendix B we note some properties of $f(s)$.

The SLAM, defined by Eqs. (17), is the main result of the present paper. The
SLAM must be supplemented with initial and boundary conditions to produce a
well-posed problem. To retain consistency with the semilocal approximation,
the initial condition can be any arbitrary function supported on
$\omega\gg\Lambda$, and whose characteristic scale of variation
$\partial\omega/\partial(\ln n_{\omega})\gg\Lambda$. Respectively, these
conditions ensure that $W^{12}_{3{\bf k}}$ decays sufficiently rapidly that
superlocality is satisfied, and that the Taylor expansion in Eq. (13) can be
truncated. As mentioned in section II.3.2, we conjecture that the steady-state
dual cascade spectra in the forced-dissipated problem will be independent of
initial conditions.

As for the boundary conditions, if one were concerned with a solution on the
whole real line then physicality requires that the spectrum and fluxes vanish
as $\omega\to\infty$. The situation as $\omega\to 0$ is quite involved, as one
expects solutions that exhibit finite-time blowup there, associated with
condensation [31]. At this point not only does the support of the spectrum
violate the $\omega\gg\Lambda$ condition, but the system becomes strongly
nonlinear, meaning that the WKE (and hence the SLAM) no longer describes the
dynamics. However, when examining the dual cascade spectra, we assume that
dissipation forces the spectrum to vanish at $\omega_{d+}$ and $\omega_{d-}$.
We take these as boundary conditions of the solution in sections V and VI when
we characterise these spectra.

###  III.1 Conservation of invariants in the SLAM

To show that the original invariants of the WKE continue to be conserved in
the SLAM, we first note that (17a) is a continuity equation for waveaction,
and so $N$ is manifestly conserved. Secondly, we note that the energy density
is $\omega n_{\omega}$, and so the continuity equation for energy is

$\displaystyle\frac{\partial(\omega n_{\omega})}{\partial
t}=-\frac{1}{\pi}\frac{\partial P}{\partial\omega}$ (18)

where $P$ is the energy flux. Together with Eq. (17a), this gives
$\partial_{\omega}P=\omega\partial_{\omega}Q$. Integrating from $0$ to
$\omega$, we obtain

$\displaystyle P(\omega)=\omega
Q(\omega)-\int_{0}^{\omega}Q(\tilde{\omega})\,d\tilde{\omega}.$ (19)

Integrating Eq. (18) over all $\omega$ gives
$\partial_{t}E=-[P(\infty)-P(0)]/\pi$. Using Eq. (19), and assuming that the
particle flux decays fast enough at large and small $\omega$, so that $\omega
Q(\omega)|_{\infty}=\omega Q(\omega)|_{0}=0$, we obtain

$\displaystyle\frac{\partial E}{\partial
t}=S_{\Lambda}\int_{0}^{\infty}\\!\int_{0}^{\infty}f\\!\left(\\!\sqrt{\frac{\omega_{2}}{\tilde{\omega}}}\right)\frac{\omega_{2}}{\sqrt{\tilde{\omega}}}\,n_{\omega_{2}}^{2}n_{\tilde{\omega}}^{2}\left(\partial_{\tilde{\omega}}n_{\tilde{\omega}}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)d\omega_{2}d\tilde{\omega}.$
(20)

Now we observe that by Eq. (41), the factor
$(\omega_{2}/\\!\sqrt{\tilde{\omega}})\,f(\\!\sqrt{\omega_{2}/\tilde{\omega}})$
is symmetric under $\tilde{\omega}\leftrightarrow\omega_{2}$. This leaves the
integrand on the right-hand side of Eq. (20) antisymmetric under exchange of
the integration variables, and so we must have $\partial_{t}E=0$.

Therefore, in a closed system the SLAM preserves the same quadratic invariants
as the WKE from which it is derived. The rest of this paper is devoted to
obtaining solutions of the SLAM, particularly the solutions that realise the
dual cascade of invariants that is predicted by the Fjørtoft argument in a
forced-dissipated system.

##  IV Stationary solutions of the SLAM

In this section we show that the usual stationary solutions of the WKE—the
equilibrium RJ spectrum, and the KZ cascade spectra—are stationary solutions
of the SLAM (17). We are particularly interested in spectra that are self-
similar, i.e. of power-law form $n_{\omega}=C\omega^{-x}$, where $C$ is a
constant that is positive for physical spectra.

###  IV.1 Thermodynamic equilibrium (RJ) spectrum

The RJ spectrum describes the state of thermodynamic equilibrium where a
linear combination of the integral invariants is partitioned equally over
${\bf k}$-space:

$\displaystyle n_{\omega}=\frac{T}{\mu+\omega}\quad\text{(equipartition of
$\mu N+E$)},$ (21)

where the thermodynamic potentials are the temperature $T$ and chemical
potential $\mu$ (both constants). This spectrum is a stationary solution of
the SLAM because the bracket
$\left(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1}\right)$
in Eq. (17b) vanishes when Eq. (21) is substituted.

The RJ spectrum has the asymptotic limits

$\displaystyle\begin{aligned}
n_{\omega}&\propto\omega^{0}\quad\text{(equipartition of $N$)},\\\
n_{\omega}&\propto\omega^{-1}\quad\text{(equipartition of $E$)},\end{aligned}$
(22)

which are self-similar spectra with spectral indices $x=0$ and $x=1$,
respectively.

###  IV.2 Stationary nonequilibrium cascade (KZ) spectra

As mentioned in Sec. I, in many systems one can find stationary solutions of
the WKE that are of power-law form, and which describe the constant flux of
invariants via a self-similar, scale-by-scale cascade. These are the KZ
cascade spectra, and they are the first candidate for the spectra that realise
the dual cascade predicted by the Fjørtoft argument for forced-dissipated
systems. When the KZ spectra are physically relevant, the flux of each
dynamical invariant will be described by its own KZ spectrum, and on that
spectrum the flux of all other dynamical invariants will be zero.

To find the KZ spectra, we first substitute $n_{\omega}=C\omega^{-x}$ into Eq.
(17b), giving for the waveaction flux

$\displaystyle Q=\pi S_{\Lambda}C^{3}x\int
f\\!\left(\\!\sqrt{\frac{\omega_{2}}{\omega}}\right)\frac{\omega_{2}}{\sqrt{\omega}}\,\omega_{2}^{-2x}\omega^{-2x}(\omega^{x-1}-\omega_{2}^{x-1})\,d\omega_{2}.$
(23)

####  IV.2.1 Energy cascade spectrum

In the wave turbulence literature, KZ spectra are frequently found by making
non-identity transformations of the collision integral that allow one to read
off spectral indices $x$ that make the integrand of the transformed collision
integral vanish. This technique is known as the Zakharov-Kraichnan transform
[3].

We now adapt this method to the SLAM in order to find the KZ energy cascade
spectrum. We split the right-hand side of Eq. (23) into two halves. In the
second half, we substitute $\omega_{2}=\omega^{2}/\tilde{\omega}_{2}$. Using
Eq. (41), and dropping tildes immediately, we obtain

$\displaystyle Q=\frac{\pi
S_{\Lambda}C^{3}}{2}x\int\left[1-\left(\frac{\omega_{2}}{\omega}\right)^{y}\right]f\\!\left(\\!\sqrt{\frac{\omega_{2}}{\omega}}\right)\frac{\omega_{2}}{\sqrt{\omega}}\,\omega_{2}^{-2x}\omega^{-2x}(\omega^{x-1}-\omega_{2}^{x-1})\,d\omega_{2},$

with $y=3x-3/2$. Choosing the spectral index $x=1/2$ leads to a vanishing
waveaction flux $Q$, suggesting that this represents the KZ energy cascade
spectrum.

To see that this is indeed the case, we extract the overall $\omega$
dependence in Eq. (23), leaving a reduced, dimensionless collision integral
$I(x)$, as follows:

$\displaystyle Q(\omega)=2\pi
S_{\Lambda}C^{3}\omega^{(1-6x)/2}I(x),\qquad\text{where}\qquad
I(x)=x\int_{0}^{\infty}f(s)s^{3-4x}(1-s^{2x-2})\,ds,$ (24)

and $s=\sqrt{\omega_{2}/\omega}$. Substituting Eq. (24) into Eq. (19) gives
for the energy flux

$\displaystyle P(\omega)=2\pi
S_{\Lambda}C^{3}\frac{1-6x}{3-6x}\omega^{(3-6x)/2}I(x).$ (25)

Setting $x=1/2$ in Eq. (24) reproduces the result that $Q(\omega)=0$, because
$I(1/2)\propto\int_{0}^{\infty}\\!sf(s)\,ds-\int_{0}^{\infty}\\!f(s)\,ds$,
which vanishes by the $s\to 1/s$ symmetry in Eq. (41) (note that the
transformation $s\to 1/s$ is exactly equivalent to making the Zakharov-
Kraichnan transform).

When $x=1/2$, Eq. (25) gives $P(\omega)=0/0$. To resolve this indeterminacy we
use L’Hôpital’s rule, obtaining

$\displaystyle P(\omega)=\frac{2\pi
S_{\Lambda}C^{3}}{3}I^{\prime}(1/2)=\frac{2\pi
S_{\Lambda}C^{3}}{3}\left(3\int_{0}^{\infty}f(s)\ln(s)\,ds\right)=-4.85432\left(\pi
S_{\Lambda}C^{3}\right),$

where the prime denotes differentiation with respect to $x$, and we have again
used Eq. (41).

Thus, when $x=1/2$ the energy flux $P$ is a constant, independent of $\omega$,
while the waveaction flux $Q$ vanishes, indicating that this is indeed the KZ
energy cascade spectrum. However, we note that on this spectrum the sign of
$P$ is negative, which is opposite to the sign predicted by the Fjørtoft
argument. We elaborate on this in Sec. V.

####  IV.2.2 Waveaction cascade spectrum

To determine the KZ spectrum for the cascade of waveaction, we put $x=1/6$ in
Eq. (24), obtaining $Q(\omega)=2\pi S_{\Lambda}C^{3}I(1/6)$, which is
independent of $\omega$. Likewise, Eq. (25) gives $P(\omega)=0\times I(1/6)$.
This would satisfy the requirements to be the KZ waveaction cascade spectrum
if $I(1/6)$ converged. However, from the second equation in (24) we see that

$\displaystyle I(1/6)=\frac{1}{6}\int_{0}^{\infty}f(s)(s^{7/3}-s^{2/3})\,ds.$

Noting the asymptotic behaviour of $f(s)$ from Eq. (40c), we see that $I(1/6)$
diverges as $s\to\infty$.

Therefore, even though the power-law spectrum with $x=1/6$ superficially gives
the correct properties for the KZ waveaction cascade spectrum, we must rule it
out because the collision integral is divergent on that spectrum.

####  IV.2.3 Summary of KZ spectra

To summarise, the formal KZ cascade spectra that are our first candidates for
realising the dual cascade are

$\displaystyle\begin{aligned} n_{\omega}&\propto\omega^{-1/2}\quad\text{(KZ
spectrum: cascade of $E$)},\\\ n_{\omega}&\propto\omega^{-1/6}\quad\text{(KZ
spectrum: cascade of $N$)}.\end{aligned}$ (26)

However, both spectra suffer pathologies: on the first spectrum the flux of
energy is in the wrong direction, and the second spectrum causes the collision
integral to diverge. We must therefore rule them out, and seek other spectra
on which the dual cascade can be supported.

These pathologies notwithstanding, it is still worth noting that the original
interaction coefficient $W^{12}_{3{\bf k}}$ is not a homogeneous function of
the four wavevectors, i.e. it possesses no obvious properties that would lead
to a self-similar scaling behaviour. Nevertheless, the semilocality property
of $W^{12}_{3{\bf k}}$ allows us to integrate out its non-homogeneous part,
giving the constant coefficient $S_{\Lambda}$. The resulting equation, the
SLAM, is self-similar. However, unlike KZ spectra, the relevant solutions of
the SLAM that manifest the dual cascade do not turn out to be self-similar
themselves, as we demonstrate in the following sections.

###  IV.3 Interpretation of divergent spectra

As mentioned above, the divergence of the collision integral at a certain
scale causes us to rule out a prospective KZ spectrum. However, we expect the
true solution to retain some characteristics indicated by this divergence.
Namely, waves of a scale that approaches the divergent scale will be
increasingly dominant in every tetrad of interacting waves in which they
participate. In other words, wave interactions at every scale will be mediated
by waves whose scale approaches the divergent scale. In this situation, the
true solution is termed a nonlocal flux spectrum, as opposed to the
spectrally-local cascades that are described by physically-realisable KZ
solutions.

In the specific case here, the divergence of the KZ waveaction cascade
spectrum as $s\to\infty$ implies that the true waveaction flux spectrum is
nonlocal, dominated by interactions at $\omega_{2}\gg\omega$. By contrast, the
fact that the KZ energy cascade spectrum gives convergence of the collision
integral signals that the true cascade solution has local interactions. We
need only resolve the matter of the cascade direction, which we do in Sec. V.

In Appendix. C we present a full convergence study of the collision integral
on general power-law spectra, allowing us to see the KZ spectra in their full
context. The results of this convergence study are shown in Fig. 2(b).

##  V Flux directions on power-law spectra

For the sake of completeness, we present in this section a general
diagrammatic argument [2] that determines the directions of both the energy
flux $P$ and the waveaction flux $Q$, on all power-law spectra
$n_{\omega}=C\omega^{-x}$ (with $C>0$). In order to present the argument, we
neglect for a moment the divergence of the collision integral on the KZ
waveaction cascade spectrum. Recall that if the sign of a flux is positive
(negative), the invariant flows towards large (small) $\omega$.

It is natural to assume that for very sharply peaked spectra, the resulting
fluxes will flatten the spectra out. Thus, for $x\to\infty$ (spectrum sharply
peaked around $\omega\approx 0$), we expect the associated fluxes to be
strongly positive. Likewise for $x\to-\infty$ (spectrum sharply rising), the
fluxes will be strongly negative.

In between these two, the fluxes will both be zero on each of the
thermodynamic spectra $x=0,1$. As for the KZ spectra, by construction the KZ
energy cascade spectrum is for the _pure_ flux of energy, with no waveaction
flux. Likewise, the energy flux is zero on the KZ spectrum for a pure
waveaction cascade. In our case we respectively have $Q=0$ for $x=1/2$, and
$P=0$ for $x=1/6$ (were the latter to give a convergent collision integral).
Assuming that the fluxes vary continuously with spectral index $x$ forces them
to behave qualitatively as shown in Fig. 2(a).

Figure 2:  (a) Sketch of the energy flux $P$ and waveaction flux $Q$
dependence on spectral index $x$, where $n_{\omega}=C\omega^{-x}$. The signs
of the fluxes are determined by the relative ordering of the RJ and KZ spectra
(Eqs. (22) and (26) respectively), where one or both of the fluxes is zero,
and the behaviour at large and small $x$. The qualitative behaviour of the
fluxes in between the zeros follows by continuity, see Sec. V. (b) Convergence
(green) or divergence (red) of Eq. (17b) with respect to spectral index $x$.
Above the $x$ axis refers to $\omega_{2}$ in the ultraviolet range, and below
the $x$ axis refers to $\omega_{2}$ in the infrared range, see Appendix C.
(Convergence is unconditional exactly on the thermodynamic spectra $x=0,1$.
This is indicated by the narrow green strips around these two spectra.)

We see that the ordering of the zero crossings forces $P$ to be negative on
the KZ energy cascade spectrum, as found in Sec. IV.2.1, and also forces $Q$
to be positive on the KZ waveaction cascade spectrum. These are both in direct
contradiction to the conclusion of the Fjørtoft argument, which is that $P$
must be positive and $Q$ negative on the stationary spectra that realise the
direct and inverse portions of the dual cascade, see Sec. II.3.2. If the
respective flux-carrying solutions are to be realised by the KZ spectra, the
only way to reconcile the two arguments is for the KZ spectra to have non-
positive (i.e. negative or even complex) prefactor constants $C$, which is
clearly unphysical. Therefore, we conclude once more that the KZ spectra found
in Sec. IV cannot realise the dual cascade in any physically-relevant
scenario.

As the KZ spectrum must be ruled out, the true solution to realise a steady-
state cascade must be related to the other stationary solution: the RJ
spectrum [4]. Indeed, experience with other wave turbulence systems suggests
that the true cascade solution is an RJ spectrum with small deviations that
are nonetheless responsible for carrying the entire flux, see Eq. (27). Such
solutions are termed warm cascade spectra [4, 10, 12].

We therefore hypothesise that the flux-carrying spectra that realise the dual
cascade are warm spectra in both the direct and inverse inertial ranges.
However, anticipating the results of Sec. VI, we will conclude that the
inverse cascade of $N$ is not only nonlocal in character, but is also realised
by a warm spectrum with negative thermodynamic potentials $T$ and $\mu$.

By contrast, the convergence of the KZ energy cascade spectrum found in Sec.
IV.2.1, and the discussion of Sec. IV.3, indicate that the true direct cascade
of $E$ is local. We therefore expect the direct cascade to be warm, with
positive $T$ and $\mu$, and spectrum

$n_{\omega}^{\text{dir}}=\frac{T}{\mu+\omega+\Delta(\omega)}.$ (27)

Here, $\Delta(\omega)$ is the deviation from the RJ spectrum, which remains
small in the inertial range, far from the forcing and dissipation scales. At
the end of the inertial range $\Delta(\omega)$ becomes large, until the
spectrum terminates at the dissipation scale $\omega_{d+}$. We sketch the warm
direct cascade $n_{\omega}^{\text{dir}}$ qualitatively in red in Fig. 3.

It is a prediction from the superlocal DAM that the warm spectrum terminates
in a logarithmic compact front, and that the temperature of the cascade
spectrum $T$ is determined by the energy flux $P$, and small-scale dissipation
range $\omega_{d+}$ [12]. We leave it to future work, reinforced by numerical
simulations, to examine these relations for the direct warm cascade realised
by the SLAM.

Figure 3:  Qualitative sketch of the steady-state dual cascade predicted by
the SLAM, which realises the prediction of the Fjørtoft argument. Forcing at
$\omega_{f}$ injects waveaction and energy into the system. The negative
waveaction flux $Q$ is realised by the nonlocal inverse cascade spectrum
$n_{\omega}^{\text{inv}}$ (blue), which terminates at the scale $\omega_{d-}$
where the majority of the waveaction is dissipated. The positive energy flux
$P$ is realised by the local direct cascade spectrum $n_{\omega}^{\text{dir}}$
(red), which terminates at $\omega_{d+}$ where most of the energy is
dissipated. The asymptotic solutions $n_{\omega}^{\ll}$ and $n_{\omega}^{\gg}$
are overlaid in white dashes. (Note that the inverse and direct cascade
spectra meet at $\omega_{f}$, which is strictly less than, but of the same
order as, $\omega_{*}$. In a realistic system, the break in gradient at
$\omega_{f}$ will be regularised by the specific forcing protocol, which we do
not attempt to show here.)

##  VI Nonlocal inverse cascade solution

In this section, we seek the stationary solution of the SLAM that realises a
constant inverse flux of waveaction, and that is nonlocal in the sense
suggested by the divergence of the corresponding KZ spectrum, see Sec. IV.3.
We also seek to parameterise the solution in terms of quantities that we can
control externally, for example in simulations. These will turn out to be the
flux $Q$, the forcing and dissipation scales $\omega_{f}$ and $\omega_{d-}$,
and the temperature of the inverse warm cascade $T$.

We set $Q$ to be negative in Eq. (17b) to specify an inverse flux, and
substitute $f(s)=\pi/s^{3}$, its $\omega_{2}\gg\omega$ limit. Equation (17b)
becomes

$\displaystyle\frac{\partial
n_{\omega}}{\partial\omega}=\frac{\hat{Q}}{A\omega}+\frac{B}{A}n_{\omega}^{2}$
(28)

where $\hat{Q}=-Q/\pi^{2}S_{\Lambda}>0$ and the integrals over $\omega_{2}$
are absorbed into the constants

$\displaystyle
A=\int\frac{n_{\omega_{2}}^{2}}{\sqrt{\omega_{2}}}\,d\omega_{2}\quad\text{and}\quad
B=\int\frac{1}{\sqrt{\omega_{2}}}\frac{\partial
n_{\omega_{2}}}{\partial\omega_{2}}\,d\omega_{2}.$ (29)

Manifestly $A>0$. Self-consistency of the asymptotic solutions of Eq. (28)
demands that $B>0$ also (see discussion after Eq. (31) below).

###  VI.1 Nonlocal inverse cascade: asymptotics

First, we examine the asymptotics of Eq. (28) to extract key characteristics
of the full solution. We denote the frequency at which the two terms on the
right-hand side are equal by $\omega_{s}$.

For $\omega\ll\omega_{s}$ the first term on the right-hand side of Eq. (28)
dominates. The solution to the resulting asymptotic equation is

$\displaystyle
n_{\omega}^{\ll}=\frac{\hat{Q}}{A}\ln\left(\frac{\omega}{\omega_{d-}}\right).$
(30)

Here we have written the constant of integration as the frequency at which the
solution $n_{\omega}^{\ll}$ vanishes, and interpreted it as the dissipation
scale $\omega_{d-}$. Note that the solution finds a vanishing point naturally,
without specifying a dissipation mechanism. This is in common with warm
solutions of superlocal DAMs that contain compact fronts at which the solution
vanishes logarithmically, see e.g. [10, 12]. (We expect that if dissipation is
not provided, so that the flux is drained from the system by the time it
reaches $\omega_{d-}$, the spectrum would grow in this vicinity so that the
situation would not be time-independent. Eventually the nonlinearity would
become strong here, so that the wave turbulence assumptions would become
violated.)

For $\omega\gg\omega_{s}$ the second term on the right-hand side of Eq. (28)
is dominant, and we have the asymptotic solution

$\displaystyle n_{\omega}^{\gg}=\frac{A/B}{\omega_{*}-\omega}.$ (31)

Here the constant of integration appears as $\omega_{*}$, the frequency at
which the solution becomes singular. By hypothesis, $\omega_{*}$ is greater
than any $\omega$ in the inverse cascade range. The integral of
$n_{\omega}^{\gg}$ is weakly (logarithmically) divergent as
$\omega\to\omega_{*}$, which is consistent with the assumption of a nonlocal
solution that is dominated by interactions with $\omega_{2}\gg\omega$.
Obviously, in any realistic scenario the solution cannot continue up to
$\omega_{*}$. We therefore cut the solution off at $\omega_{f}$ where
$\omega_{d-}\ll\omega_{f}<\omega_{*}$. This cutoff represents the end of the
inverse cascade inertial range; in a forced-dissipated setup this is none
other than the forcing scale. By choosing $\omega_{f}$ in the vicinity of
$\omega_{*}$, so that $\omega_{2}$ can approach the singularity frequency
$\omega_{*}$, we keep consistency with the nonlocality assumption
$\omega_{2}\gg\omega$.

If we define the temperature $T\coloneqq-A/B$ and chemical potential
$\mu\coloneqq-\omega_{*}$, we also see that $n_{\omega}^{\gg}$ is actually a
thermodynamic spectrum (21) with negative $T$ and $\mu$. The interpretation of
RJ equilibria with negative thermodynamic potentials was given in Ref. [32]
for the case of three sign-definite invariants. For the present case with two
invariants, these are exactly equilibria with spectra diverging at some
nonzero $\mu$ (see appendix of Ref. [32]).

Note that, had we chosen $B<0$, we would have obtained the asymptotic solution
$n_{\omega}^{B<0}=A/|B|(\omega+\omega_{*})$. For $\omega_{*}<0$ this is
negative in $0<\omega<|\omega_{*}|$, which is unphysical. For $\omega_{*}\geq
0$, if we substitute $n_{\omega}^{B<0}$ back into Eq. (28), the first term on
the right-hand side dominates for all $\omega\geq 0$, which is inconsistent
with the assumptions for deriving $n_{\omega}^{B<0}$. We therefore rule out
the $B<0$ case. Had we chosen $B=0$, the full solution $n_{\omega}^{\ll}$ is
the only solution, but then the second integral in (29) gives $B\neq 0$.
Hence, we rule out $B=0$ as well, and therefore we must have $B>0$.

Thus, the full solution of Eq. (28) resembles an RJ spectrum for
$\omega\gg\omega_{d-}$, Eq. (31), but has a deviation that grows towards the
infrared, and that terminates at $\omega=\omega_{d-}$ with a logarithmic
compact front, Eq. (30). This is exactly to say that it is a warm cascade
spectrum, but with negative thermodynamic potentials.

####  VI.1.1 Determination of constants $A$ and $B$

The integrals in Eqs. (29) must be taken over the whole inverse cascade range,
from $\omega_{d-}$ up to $\omega_{f}$. Since the inverse cascade spectrum is
nonlocal, the dominant contributions to the integrals occurs at large
$\omega$. Using the asymptotic spectrum $n_{\omega}^{\gg}$, and evaluating
Eqs. (29) at the upper limit $\omega_{f}$, we obtain, to leading order,
$B^{2}/A=1/\sqrt{\omega_{f}}(\omega_{*}-\omega_{f})$. In terms of the
temperature $T$ this gives

$\displaystyle
A=\frac{T^{2}}{\sqrt{\omega_{f}}(\omega_{*}-\omega_{f})}\quad\text{and}\quad
B=-\frac{T}{\sqrt{\omega_{f}}(\omega_{*}-\omega_{f})},$ (32)

i.e. we have expressed $A$ and $B$ in terms of $T,\omega_{f}$ and
$\omega_{*}$.

###  VI.2 Nonlocal inverse cascade: full solution

Equation (28) can be solved analytically by noting that it is a Ricatti
equation. Using standard techniques [33], its solution is found to be

$\displaystyle
n_{\omega}^{\text{inv}}=-\sqrt{\frac{\hat{Q}}{B\omega}}\left(\frac{Y_{0}(K)+J_{0}(K)c}{Y_{1}(K)+J_{1}(K)c}\right),\quad\text{with}\quad
K=\frac{2\sqrt{B\hat{Q}\omega}}{A},$ (33)

where $J_{n}(K),Y_{n}(K)$ are $n$-th order Bessel functions of the first and
second kinds respectively, and $c$ is the constant of integration.

We can relate $c$ to the integration constants of the asymptotic solutions by
noting that $\omega_{d-}$ corresponds to the first zero of the right-hand side
of Eq. (33). This will be at the first root of the numerator
$Y_{0}(K)+J_{0}(K)c$. Using the asymptotics of the Bessel functions for $K\ll
1$ gives, to leading order,

$\displaystyle\omega_{d-}=\frac{A^{2}}{B\hat{Q}}e^{-2\gamma-\pi c},$ (34)

where $\gamma\approx 0.5772$ is the Euler-Mascheroni constant. Likewise,
$\omega_{*}$ corresponds to the first root of $Y_{1}(K)+J_{1}(K)c$, the
denominator of Eq. (33). To leading order this gives

$\displaystyle\frac{A^{2}}{B\hat{Q}\omega_{*}}+\ln\left(\frac{A^{2}}{B\hat{Q}\omega_{*}}\right)=\pi
c+2\gamma-1.$ (35)

This has solution $\omega_{*}=A^{2}/[B\hat{Q}\,W(e^{\pi c+2\gamma-1})]$, where
$W(x)$ is the Lambert-W function.

Firstly, we note that the first term on the right-hand side of Eq. (35) is
dominant as we send $c\to\infty$, while the left-hand side is greater than
$A^{2}/(B\hat{Q}\omega_{*})$. This gives $\omega_{*}^{-1}(c)=\mathcal{O}(c)$,
whereas from Eq. (34), $\omega_{d-}\to 0$ exponentially as $c\to\infty$. Thus,
the ratio $\omega_{d-}/\omega_{*}\to 0$ as $c\to\infty$, and so by adjusting
$A,B,$ and $\hat{Q}$ to set the overall scaling, we can make the inverse
inertial range arbitrarily wide.

Furthermore, we can eliminate $c$ between Eqs. (34) and (35), and eliminate
$A$ and $B$ using Eq. (32), obtaining

$\displaystyle\ln\left(\frac{\omega_{*}}{\omega_{d-}}\right)=1+\frac{T^{3}}{\hat{Q}\sqrt{\omega_{f}}(\omega_{*}-\omega_{f})\omega_{*}}.$
(36)

Equation (36) implicitly expresses $\omega_{*}$ in terms of the control
parameters $(\hat{Q},T,\omega_{f},\omega_{d-})$. Solving for $\omega_{*}$ (for
example numerically, or to any desired accuracy by iteration), and
substituting into Eq. (32) allows $A$ and $B$ to be written in terms of the
same set of parameters. We can likewise express $c$ via Eq. (34), and finally
obtain the solution $n_{\omega}$ via Eq. (33), in terms of the control
parameters $(\hat{Q},T,\omega_{f},\omega_{d-})$.

Note that, unlike the case for superlocal DAMs, we cannot close the set of
control parameters by writing $T$ as a function of the flux and the forcing
and dissipation scales. This is reminiscent of two-free-parameter stationary
solutions of the Leith model [6]. (Closures could be provided by specific
assumptions about the forcing or dissipation, for example that the forcing
starts with a given flux and temperature, but these assumptions would not be
universal.)

In Fig. 3 we sketch the qualitative behaviour of the nonlocal inverse
waveaction cascade spectrum $n_{\omega}^{\text{inv}}$ in blue. We also show
the asymptotic solutions $n_{\omega}^{\ll}$ and $n_{\omega}^{\gg}$ in white
dashes, and the frequency $\omega_{*}$ where the inverse cascade spectrum
becomes singular.

##  VII Discussion and conclusion

###  VII.1 Comparison with the NLSE and SNE limits

Before concluding, we make some remarks about the two limits of the SHE that
were mentioned in Sec. II. The first is the NLSE limit, where we send
$\alpha,\Lambda\to\infty$, while $\alpha/\Lambda\to\text{const.}$ After
rescaling $\psi$ we obtain Eq. (2). The second is the SNE limit, where we set
$\Lambda=0$ to obtain Eqs. (3).

In both these limits, the interaction coefficient becomes a homogeneous
function, in the sense that $W^{\mu{\bf k}_{1},\mu{\bf k}_{2}}_{\mu{\bf
k}_{3},\mu{\bf k}}=\mu^{\beta}W^{{\bf k}_{1},{\bf k}_{2}}_{{\bf k}_{3},{\bf
k}}$, with $\beta=0$ in the NLSE limit and $\beta=-2$ in the SNE limit. This
observation allowed us to heuristically construct superlocal DAMs of the NLSE
and SNE in Ref. [12]. There, we used the DAMs to examine the respective KZ
spectra, and found that in both cases the flux directions contradicted the
Fjørtoft argument. We therefore proposed that the flux-carrying spectra were
warm spectra in both the NLSE and SNE. We are now in a position to revisit
this work, in light of the rigorously-derived SLAM.

The first thing to note is that in the NLSE limit, the interaction coefficient
becomes a constant across all wavevectors. In particular, this no longer
respects the semilocality property: pairs of wavevectors are no longer picked
out by the sharp decay of the interaction coefficient. We therefore cannot
approximate the full WKE by the SLAM—to do so would neglect the majority of
wave interactions, all of which are important in evolving the spectrum.
However, we can still use the DAM for qualitative understanding, e.g. the
argument about flux directions and the prediction of warm cascades [4, 12].

By contrast, the SNE are a singular limit of the SHE. We noted in Sec. II that
the SNE are ill-posed, and that their regularisation requires restoring
$\Lambda\neq 0$, i.e. moving to the SHE. To elaborate: in order to develop the
wave turbulence theory and derive the WKE, one starts with a periodic system
[2], but in a periodic system, Eq. (3b) has no non-trivial solutions. Once we
derive the SLAM, this ill-posedness is revealed in Eq. (15): setting
$\Lambda=0$ sends $S_{\Lambda}\to\infty$, i.e. the SLAM diverges for every
spectrum $n_{\omega}$. This indicates that one can formally write down the
kinetic equation of the SNE, but the collision integral becomes infinite when
any two wavevectors become equal. Likewise, one can obtain KZ spectra for the
SNE based on dimensional arguments, but these spectra will be invalid because
the collision integral will be divergent on these spectra. Moreover, the KZ
spectra will change discontinuously when we regularise the kinetic equation by
setting $\Lambda\neq 0$. This is indeed what we find when we compare the KZ
spectra found in Ref. [12] (namely $\omega^{0}$ for the KZ waveaction cascade
spectrum, and $\omega^{-1/3}$ for the KZ energy cascade spectrum, for the 2D
case) to Eqs. (26).

Thus, we see that retaining $\Lambda\neq 0$ in the SHE is necessary in order
to regularise the singular SNE limit. This is a salutary lesson as it
highlights the hidden pitfalls of such heuristic derivations of DAMs: their
predictions are misleading if, as in our case, they do not respect essential
properties of the original interaction coefficient. We speculate that a
similar derivation of a semilocal model might be applied to other examples in
the literature, e.g. in the theory of gravitational waves in Einstein’s vacuum
field model [34].

###  VII.2 Conclusion

Starting from the wave kinetic equation of the SHE, we have rigorously derived
a reduced kinetic equation, the SLAM, by exploiting the natural locality
properties of the interaction coefficient. We believe this to be the first
such derivation of a reduced kinetic equation in which the locality assumption
can be justified self-consistently.

Having derived the SLAM, we use it to obtain the stationary spectra that are
responsible for realising the dual cascade of energy and waveaction that is
predicted by the Fjørtoft argument. After deriving the formal KZ cascade
spectra, and examining their flux directions and locality, we conclude that
neither the direct cascade of energy nor inverse cascade of waveaction are
realised by the respective KZ spectra.

Instead, we predict that the dual cascade is carried by warm spectra. This
concurs with our examination of the limits of the SHE in Ref. [12], even
though some of that was carried out in the SNE limit, which is, in fact,
singular. Here, though, the SLAM allows us to refine our prediction about the
character of the warm spectra. We predict that the direct energy cascade
spectrum will have positive thermodynamic parameters, and that interactions
will be between waves that are local in frequency. By contrast, the inverse
cascade of waveaction will be carried by a nonlocal spectrum, with
interactions at every frequency $\omega$ being dominated by the spectrum near
the forcing scale $\omega_{f}$. Accordingly, we derive a nonlocal, warm,
inverse cascade spectrum, that is parameterised by a negative temperature and
chemical potential.

Our results on the dual cascade were derived for the forced-dissipated 2D SHE,
which is the setup that leads to the clearest manifestation of the cascades.
Results on the inverse cascade may also apply to the case of turbulence that
evolves freely from an initial condition in a closed system, due to the
inverse cascade having finite capacity (the integral of the inverse cascade
spectrum converges when we send $\omega_{d-}\to 0$). Experience with finite
capacity KZ spectra shows that an initial condition fills out its respective
inertial range in finite time, with the KZ spectrum establishing after an
initial transient [35, 36, 8]. It remains to be tested whether this
phenomenology carries over to the inverse cascade spectrum of the SHE. By
contrast, the direct cascade has infinite capacity for energy (the integral
defining energy diverges as $\omega_{d+}\to\infty$), and so it can absorb an
arbitrary amount of energy that is sent into it, unless there is some small-
scale cutoff that arrests the direct cascade e.g. the finite numerical
resolution of a simulation. For such systems, the cascade spectrum typically
does not form behind the front that propagates from an initial condition,
unless continuously forced.

The finite-capacity inverse cascade cannot absorb an arbitrary amount of
waveaction. If no large-scale dissipation is provided, waveaction will arrive
at the end of the inertial range and start to accumulate into coherent large-
scale structures: condensates and solitons. The dynamics of these structures
are attracting much interest, particularly in astrophysics, where they could
represent galactic dark matter halos [21, 37, 38, 39], or their 1D and 2D
analogues in nonlinear optics [40, 24, 41, 42]. The dual cascade process is a
universal mechanism whereby such large-scale structures emerge due to the
interaction of weakly turbulent small-scale waves, at least in the initial
transient phase where weak waves exist without any coherent structures
present.

Our results are therefore directly applicable to the 2D case, which may be
accessible in optical experiments. Indeed, experiments modelled by the 2D SHE
have already been conducted, using thermo-optic crystals as the nonlinear
medium, particularly in the context of tabletop analogues of dark matter
haloes or boson stars [40, 43, 44]. 1D liquid crystal experiments were carried
out specifically looking at the wave turbulence of the SHE [11], and it would
be feasible to extend this to a 2D experiment.

The next step in this work is an extensive numerical comparison of the SHE,
its WKE, and the SLAM, in both the freely-evolving and forced-dissipated case.
This is currently in progress by the authors and will be published separately
at a later date. Such a comparison was recently carried out for the NLSE and
its WKE [45, 46]. Furthermore, we envisage that one could derive a similar
SLAM for the 3D SHE, which will be applicable to the turbulent formation of
galactic dark matter halos, and the 1D SHE, relevant to optical experiments
such as those carried out in [47]. The same methodology should carry over to
those cases, with the technical subtlety in the 1D case, the leading-order
wave process is 6-wave, rather than the 4-wave case in 2D and 3D.

##  VIII Acknowledgements

This work was supported by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
823937 for the RISE project HALT, and by the Simons Foundation Collaboration
grant Wave Turbulence (Award ID 651471). J.L. and J.S. are supported by the
Leverhulme Trust Project Grant RPG-2021-014.

##  Appendix A Wavevectors relative to ${\bf k}$ form a right-angle triangle

Waves on the resonant manifold are constrained to have a particular geometric
relation. The frequency and wavevector resonance conditions (9) give

$\displaystyle\omega^{12}_{3{\bf k}}$ $\displaystyle=({\bf k}_{1}-{\bf
k})\cdot({\bf k}_{1}+{\bf k})+({\bf k}_{2}-{\bf k}_{3})\cdot({\bf k}_{2}+{\bf
k}_{3})$ $\displaystyle=2({\bf k}_{1}-{\bf k})\cdot({\bf k}-{\bf k}_{2})=0.$
(37)

Recalling the definition ${\bf p}_{i}\coloneqq{\bf k}_{i}-{\bf k}$, we thus
have that ${\bf p}_{1}$ is orthogonal to ${\bf p}_{2}$. A similar calculation
gives the Pythagorean relation $p_{1}^{2}+p_{2}^{2}=p_{3}^{2}$. We conclude
that on the resonant manifold, ${\bf p}_{1},{\bf p}_{2},{\bf p}_{3}$ form a
right-angle triangle, see Fig. 1(b).

##  Appendix B Properties of $f(s)$

In this appendix we examine the function

$\displaystyle
f(s)=\int_{0}^{2\pi}\frac{\sin^{2}(\phi)}{(1-2s\cos(\phi)+s^{2})^{3/2}}\,d\phi,$
(38)

where in our application to the SLAM (17), $s=\sqrt{\omega_{2}/\omega}$ and so
$0\leq s<\infty$.

###  B.1 Writing $f(s)$ in terms of complete elliptic integrals

Writing Eq. (38) as a derivative, integrating by parts, using symmetry under
$\phi\to 2\pi-\phi$, and double-angle formulae, we obtain

$\displaystyle f(s)$
$\displaystyle=-\int_{0}^{2\pi}\frac{\sin(\phi)}{s}\frac{\partial}{\partial\phi}\left(\frac{1}{\sqrt{1-2s\cos(\phi)+s^{2}}}\right)d\phi$
$\displaystyle=\frac{2(1+s)}{s^{2}}\left[\frac{1+s^{2}}{(1+s)^{2}}\,K\\!\left(\frac{4s}{(1+s)^{2}}\right)-E\\!\left(\frac{4s}{(1+s)^{2}}\right)\right],$
(39)

where

$\displaystyle
K(z)=\int_{0}^{\pi/2}\frac{1}{\sqrt{1-z\sin^{2}(\sigma)}}\,d\sigma\quad\text{and}\quad
E(z)=\int_{0}^{\pi/2}\sqrt{1-z\sin^{2}(\sigma)}\,d\sigma$

are the complete elliptic integrals of the first and second kind,
respectively, and $\sigma=\phi/2$.

###  B.2 Asymptotics of $f(s)$

From Eq. (39), and using the asymptotics of $K(z)$ and $E(z)$ around
$s=0,1,\infty$, we obtain

$\displaystyle f(s)=\pi+\mathcal{O}(s^{2})$
$\displaystyle\qquad\text{as}\qquad s\to 0,$ (40a) $\displaystyle
f(s)=-2\ln|s-1|+\mathcal{O}(1)$ $\displaystyle\qquad\text{as}\qquad s\to 1,$
(40b) $\displaystyle
f(s)=\frac{\pi}{s^{3}}+\mathcal{O}\left(\frac{1}{s^{5}}\right)$
$\displaystyle\qquad\text{as}\qquad s\to\infty.\ $ (40c)

The integrand in Eq. (38) is undefined at $s=1$ when $\phi=0$. This behaviour
is resolved by Eq. (40b): we see that $f(s)$ has a logarithmic singularity as
$s\to 1$. This singularity is integrable in Eq. (17b) as long as the rest of
the integrand is regular as $\omega_{2}\to\omega$. This regularity holds for
all cases presented in this paper.

###  B.3 $f(s)$ with reciprocal argument

Note that from Eqs. (38) and (39), transforming $s\to 1/s$ gives

$\displaystyle f(1/s)=s^{3}f(s).$ (41)

##  Appendix C Locality of power-law spectra

In this appendix we carry out an analysis of the convergence of the collision
integral for general power-law spectra $n_{\omega}=C\omega^{-x}$. We do this
for completeness, and as a demonstration of the ease of analysis that the SLAM
permits.

The integral in Eq. (23) could diverge as $\omega_{2}\to 0$ or
$\omega_{2}\to\infty$. The behaviour of $f(s)$ in these ranges is noted in
Appendix B.2.

For $\omega_{2}\to\infty$, i.e. $s\to\infty$, we have that $f(s)\to\pi/s^{3}$,
therefore $Q\propto\int^{\infty}\omega_{2}^{1-3/2-2x}(1-\omega_{2}^{x-1})\
d\omega_{2}$, which is convergent for $x>\max\\{1/4,-1/2\\}=1/4$.

For $\omega_{2}\to 0$, i.e. $s\to 0$, we have $f(s)\to\pi$, and so
$Q\propto\int_{0}\omega_{2}^{1-2x}(1-\omega_{2}^{x-1})\ d\omega_{2}$, which is
convergent for $x\leq 1$.

For either choice of $\omega_{2}$, the waveaction and energy equipartition
spectra lead to convergence of the collision integral, since the factor
$(\partial_{\omega}n_{\omega}^{-1}-\partial_{\omega_{2}}n_{\omega_{2}}^{-1})$
in Eq. (17b) vanishes exactly for any RJ spectrum (21).

Thus, for $\omega_{2}\to\infty$, the SLAM converges for power-law spectra with
spectral index $x\in\\{(1/4,\infty)\cup 0\\}$, and for $\omega_{2}\to 0$ it
converges for $x\in(-\infty,1]$. Otherwise, the SLAM is divergent. These
convergence (green) and divergence (red) zones are indicated in Fig. 2(b), for
the two choices of $\omega_{2}\to\infty$ or $0$, above and below the $x$ axis
respectively. The thin green (convergence) strips around the thermodynamic
spectra are indicative only, and in reality shrink to the single points
$x=0,1$.

We see that the KZ energy cascade spectrum, with $x=1/2$, gives convergence of
the SLAM, whereas the KZ waveaction cascade spectrum, with $x=1/6$, gives
divergence as $\omega_{2}\to\infty$ (These results were found in Sec. IV.2,
but now we see them in their full context of convergence or divergence on
general power-law spectra).

Note that analysing the locality of general power-law spectra is made possible
in the SLAM precisely because of the _semilocality_ manifested by the
interaction coefficient of the SHE. This analysis is not possible when working
with a DAM, because in order to construct a DAM one assumes (without proof)
from the outset that interacting waves are superlocal.

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See pages 1-last of paper.pdf