Non-local gravity wave turbulence in presence of condensate

Abstract

The $k^{-23/6}$ wave action spectrum with an inverse cascade is one of the fundamental Kolmogorov–Zakharov solutions for gravity wave turbulence, which is part of the citation for the Dirac Medal in 2003. Instead of confirming this solution, however, several existing simulations and experiments suggest a spectrum of $k^{-3}$ in set-ups corresponding to the inverse cascade. We provide a theoretical explanation for the latter, considering the condensate that naturally forms in finite domains of experiments/simulations. Our new theory hinges on: (1) derivation of a spectral diffusion equation when non-local interactions with the condensate become dominant, for the first time systematically formulated for quartet-interaction systems; and (2) careful analysis of the asymptotics of interaction coefficient with a remarkable cancellation of all leading-order terms.

1. Introduction

Wave turbulence theory (WTT) is a statistical closure that provides a coarse-grained description for evolution of random weak wave fields (Zakharov, Lvov & Falkovich 1992; Nazarenko 2011; Galtier 2022). It has been applied successfully to very different physical cases, from quantum (Proment, Nazarenko & Onorato 2012; Kolmakov, McClintock & Nazarenko 2014) to classical (Zakharov & Filonenko 1967; Zakharov 1999; Connaughton, Nazarenko & Quinn 2015) to cosmological (Galtier & Nazarenko 2017) wave systems. One of the most important applications of WTT is the water surface gravity waves, understanding which is important for sea navigation and industrial offshore activities. A review of important achievements in this area can be found in Nazarenko & Lukaschuk (2016). For a kinetic description of weakly nonlinear gravity waves, there are two quadratic (in the wave amplitude) invariants: the energy and the wave action. This system is characterized by a dual cascade behaviour, for which there exist two local Kolmogorov–Zakharov (KZ) spectra, one in which the energy is cascading downscale (Zakharov & Filonenko 1967), and the other one where the wave action is cascading upscale (Zakharov & Zaslavskii 1982). Realizability of the KZ spectra requires locality of interactions in the scale space, which in turn is possible only if there are sufficiently wide inertial ranges of scales without local energy pile-up in the spectral space. However, often there is no efficient dissipation mechanism at the large scales, which leads to a spectrum pile-up (or condensation) at the scales of the order of the largest available scale in the system – the size of the containing basin. We emphasize that, in contrast with a uniform condensate in an infinite system, our condensate is weakly non-uniform, i.e. it contains small but non-zero wavenumbers. In laboratory experiments and numerical simulations, the largest available scale is limited by the size of the flume or the computational domain, whereas in open seas, this scale may be set by a swell or the water depth. Such condensation can lead to breakdown of locality of interactions.

In the present paper, we will develop a WTT description of gravity waves in the presence of a strong large-scale condensate formed due to inverse cascade. This will allow us to propose an explanation for deviations from the inverse-cascade KZ spectrum that were recently observed numerically in Korotkevich (2023) and earlier reported experimentally in Nazarenko & Lukaschuk (2016) and Deike, Laroche & Falcon (2011).

A standard approach in describing the gravity waves (including that in Korotkevich 2023) is to consider a two-dimensional surface over a three-dimensional potential flow of an ideal incompressible infinitely deep fluid with velocity $\boldsymbol{v} = \boldsymbol{\nabla} \varPhi (x,y,z;t)$, where $\varPhi (x,y,z;t)$ is the velocity potential. Capillary effects are neglected, assuming that they are small with respect to gravity. The Hamiltonian variables for this system are elevation of the surface $\eta (x,y;t)$ and velocity potential on the surface $\psi (x,y;t)=\varPhi (x,y,z;t)|_{z=\eta }$ (Zakharov 1967). It is convenient to introduce normal canonical variables $a_{\boldsymbol{k}}$, corresponding to expansion in Fourier harmonics as follows:

(1.1)\begin{equation} a_{\boldsymbol{k}} = \sqrt \frac{\omega_k}{2k}\,\eta_{\boldsymbol{k}} + \mathrm i\,\sqrt \frac{k}{2\omega_k}\,\psi_{\boldsymbol{k}},\quad \mbox{where } \omega_k = \sqrt {g k}. \end{equation}

In terms of these variables, the surface dynamics is Hamiltonian,

(1.2)\begin{equation} \dot a_{\boldsymbol{k}} =-\mathrm i\,\frac{\delta H}{\delta a_{\boldsymbol{k}}^{*}}, \end{equation}

with the Hamiltonian $H$ being the total physical energy of the wave system (not shown).

For statistical description of a stochastic wave field one can use a pair correlation function called the wave action spectrum,

(1.3)\begin{equation} \langle a_{\boldsymbol{k}} a_{\boldsymbol{k}'}^*\rangle = n_{\boldsymbol{k}} \delta (\boldsymbol{k} - \boldsymbol{k}'), \end{equation}

where the angular bracket denotes averaging over the ensemble of initial conditions (which have random phases), and $\delta$ denotes the Dirac delta function. The spectrum $n_{\boldsymbol{k}}$ is a measurable quantity, directly related to observable correlation functions; e.g. from the definition of $a_{\boldsymbol{k}}$, one can get

(1.4)\begin{equation} \langle \eta_{\boldsymbol{k}} \eta_{\boldsymbol{k}'}^*\rangle = I_{\boldsymbol{k}} \delta (\boldsymbol{k} - \boldsymbol{k}') \quad \longrightarrow \quad I_{\boldsymbol{k}} = \frac{1}{2}\,\frac{\omega_k}{g}\,(n_{\boldsymbol{k}} + n_{-\boldsymbol{k}}). \end{equation}

Under the WTT assumptions, i.e. small wave amplitudes, random phases and large-basin limit, spectrum $n_k$ obeys the Hasselmann wave kinetic equation (WKE) (Nordheim 1928; Peierls 1929; Hasselmann 1962):

(1.5)\begin{equation} \frac{\partial n_{\boldsymbol{k}}}{\partial t} = I_{St} + f_p (k) - f_d (k). \end{equation}

Here, $f_p$ and $f_d$ are some pumping and damping terms respectively, and $I_{St}$ is a so-called collision integral:

(1.6)$$\begin{gather} I_{St}=4{\rm \pi} \int \left| T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}_2,\boldsymbol{k}_3}\right|^2 n_{\boldsymbol{k}} n_{\boldsymbol{k}_1} n_{\boldsymbol{k}_2} n_{\boldsymbol{k}_3}\left(\frac{1}{n_{\boldsymbol{k}}} + \frac{1}{n_{\boldsymbol{k}_1}} - \frac{1}{n_{\boldsymbol{k}_2}} - \frac{1}{n_{\boldsymbol{k}_3}}\right)\delta (\boldsymbol{k} + \boldsymbol{k}_1- \boldsymbol{k}_2 - \boldsymbol{k}_3)\nonumber\\ {}\times \delta (\omega_k + \omega_{k_1}- \omega_{k_2} - \omega_{k_3}) \,\mathrm{d} \boldsymbol{k}_1\,\mathrm{d} \boldsymbol{k}_2\,\mathrm{d} \boldsymbol{k}_3. \end{gather}$$

The interaction coefficients $T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}_2,\boldsymbol{k}_3}$ can be found in e.g. Pushkarev, Resio & Zakharov (2003) and in Appendix B. The kinetic equation and its modifications are the basis for all wave forecasting models (Cavaleri et al. 2007).

One can consider stationary solutions of the WKE in the so-called inertial interval – range of scales far from pumping and damping regions. Such solutions have to obey the $I_{St}=0$ equation. Beyond obvious Rayleigh–Jeans thermodynamic equilibrium (zero flux) solutions $n_{\boldsymbol{k}} \sim 1/(\omega _k +{\rm const}$), there are dynamic equilibrium solutions, corresponding to finite fluxes of conserved quantities, the so-called KZ spectra (Zakharov et al. 1992; Nazarenko 2011). Equations (1.5)–(1.6) describe a four-wave process of scattering of two waves into two waves. This means that in addition to the total energy $E=\int \omega _k n_{\boldsymbol{k}} \,\mathrm {d}\boldsymbol{k}$, there is a conservation of the total wave action (‘number of waves’) $N=\int n_{\boldsymbol{k}} \,\mathrm {d}\boldsymbol{k}$. Thus there are two KZ spectra: one describing a local downscale (with respect to the forcing scale) energy cascade,

(1.7)\begin{equation} n_k^{DC} = C_P P^{1/3} k^{-4}, \end{equation}

and the other one a local upscale (towards smaller $\boldsymbol{k}$) wave action cascade (Zakharov & Zaslavskii 1982),

(1.8)\begin{equation} n_k^{IC} = C_Q Q^{1/3} k^{-23/6}\sim k^{-3.83}. \end{equation}

Here, $P$ and $Q$ are the fluxes of energy and wave action, respectively, $C_P$ and $C_Q$ are some constants, and $k=|\boldsymbol{k}|$.

In a recent paper Korotkevich (2023), the primordial dynamical equations were simulated in a region $L_x = L_y = 2{\rm \pi}$ with double periodic boundary conditions, which means that components of $\boldsymbol{k}$'s were integer numbers. Grid resolution was $N_x=N_y=512$. Pumping was isotropic with respect to angle, and concentrated in a ring $k \in (60,64)$ with random phase of every pumped harmonic. Damping started at $k_d=128$. As a result, in the range of wavenumbers smaller than $k=60$, one would expect to find a spectrum similar to (1.8). Nevertheless, it was reported that the observed inverse-cascade spectrum had a different slope, close to $n_k \sim k^{-3.07}$, which was indistinguishable for significantly different levels of nonlinearity in the system. This numerically observed spectrum is rather close to two previous independent experimental results. One of the experiments (Nazarenko & Lukaschuk 2016), conducted in a tank of size $42\,{\rm cm}\times 42\,{\rm cm}$, showed $n_k \sim k^{-2.5}$ or $n_k \sim k^{-3}$ depending on the forcing frequency (see figure 12 in Nazarenko & Lukaschuk (2016) for the corresponding energy spectra $E_k$). The other experiment (Deike et al. 2011), conducted earlier in a $20\,{\rm cm}\times 20\,{\rm cm}$ tank filled with mercury, showed that the spectral slope varies from $n_k \sim k^{-3}$ to $n_k \sim k^{-3.65}$ (see figure 2 in Deike et al. (2011) for the corresponding frequency spectrum $E_\omega$), from low to high nonlinearity levels. In particular, the $k^{-3}$ spectrum at low nonlinearity is associated with a clear condensate at large scales, whereas the $k^{-3.65}$ spectrum (closer to the KZ solution) is present without a clear condensate, perhaps due to some effective large-scale dissipation mechanism at high nonlinearity in their experiment, e.g. bottom or/and wall friction.

In previous works (Korotkevich 2008, 2012), it was shown that condensate plays an important role in the nonlinear interaction processes for such systems. Specifically, the measured dispersion relation in Korotkevich (2013) demonstrated that the influence of the condensate on the harmonics in the region of inverse cascade cannot be neglected. There are several approaches that allow one to take the condensate influence into account. Probably the most obvious one is based on Bogolyubov's transformation technique, as in the case of dilute Bose gas, similar to the recent work in Griffin et al. (2022) – see sections about degenerated almost ideal Bose gas in Abrikosov, Gorkov & Dzyaloshinskii (1962) or Lifshitz & Pitaevskii (1978). This approach requires the condensate to be a coherent object, which is usually achieved due to the fact that it is located at a single harmonic, $k=0$. In the case of numerical simulations in Korotkevich (2023), the condensate was roughly a ring in the $k$-space, whose further cascade to low wavenumbers is prohibited by the finite size effect of the domain. The pictures of the $|a_{\boldsymbol{k}}|^2$ distributions are shown in figure 1.

Figure 1. Surfaces of $|a_{\boldsymbol{k}}|^2\times 10^{8}$ in the condensate region for four simulations of Korotkevich (2023) corresponding to four different mean wave slopes $\mu$, for (a) $\mu = 0.054$, (b) $\mu = 0.067$, (c) $\mu = 0.093$, (d) $\mu = 0.135$.

One can see that the wave amplitudes are randomized and, in most cases, are significantly different from each other in amplitude even for harmonics close in ${\bf k}$. So the Bogolyubov approach is clearly not applicable. There are more than a hundred harmonics in the condensate ring. Since averaging over an ensemble of realizations is computationally unfeasible, we average the $|a_{\boldsymbol{k}}|^2$ function over the azimuthal angle, and obtain an object $\langle |a_{\boldsymbol{k}}|^2\rangle$, which can be considered as an approximation for $n_k$. Thus one can try to modify the WKE (1.5) to describe the statistics of the wave field on the background of condensate.

In the remainder of the paper, we will describe our new theoretical development following the above idea that allows an explanation of the $k^{-3}$ spectrum observed in experiments and simulations. There are two major elements of our development. First, there is the derivation of a spectral diffusion equation for surface gravity waves in the presence of condensate. In addition to the derivation from the WKE that we have in the main paper, we also put an alternative derivation in Appendix C, which relies on a Wentzel–Kramers–Brillouin (WKB) approximation of the Zakharov equation. The two approaches yield exactly the same result, including the detailed formulation of the diffusion coefficient. We note that the spectral diffusion equation has been developed previously for triad-resonant systems, such as internal gravity waves (McComas & Bretherton 1977), Rossby waves (Connaughton et al. 2015) and magnetohydrodynamic (MHD) turbulence (Nazarenko, Newell & Galtier 2001). Our current work provides the first systematic formulation for a quartet-resonant system, which involves much more complexity in both approaches of derivation. The second element is a careful analysis on the asymptotics of the interaction coefficient at the limit of non-local interactions. We will show that this calculation yields a remarkable cancellation of all leading-order terms, which is crucial in obtaining the correct scaling of the diffusion coefficient. Finally, it is the combination of these two elements that allows us to achieve the correct stationary solution with the $k^{-3}$ spectrum.

2. New theory on non-local gravity wave turbulence

2.1. Derivation of the diffusion equation in the presence of condensate

Assume that the condensate modes are much stronger than any other harmonics, so that for any four waves in the resonant quartet, the dominant contribution comes from the situation that two waves $\boldsymbol{k}_1$ and $\boldsymbol{k}_3$ are in the condensate (see figure 2), satisfying

(2.1a,b)\begin{equation} \boldsymbol{k} + \boldsymbol{k}_1 = \boldsymbol{k}_2 + \boldsymbol{k}_3,\quad \omega_k + \omega_{k_1} = \omega_{k_2} + \omega_{k_3}. \end{equation}

We consider the case $k\gg k_{1,3}$ and strong condensate $n_{\boldsymbol{k}_{1,3}}\gg n_{\boldsymbol{k}}, n_{\boldsymbol{k}_2}$. The case when both condensate waves are at one side of resonant conditions (2.1a,b) cannot be realized. The situation is isotropic with respect to azimuthal angle. For simplicity, let us consider the case where the condensate is supported in a ring as in figure 2. Writing $\boldsymbol{q} = \boldsymbol{k}_1 - \boldsymbol{k}_3$ for the difference of wave vectors, i.e. $\boldsymbol{k}_2 = \boldsymbol{k} + \boldsymbol{q}$, and neglecting $1/n_{\boldsymbol{k}_{1,3}}$ terms in (1.6), one gets

(2.2)\begin{equation} \frac{\partial n_{\boldsymbol{k}}}{\partial t} \approx 8{\rm \pi} \int \left\{\left| T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}_2,\boldsymbol{k}_3}\right|^2 n_{\boldsymbol{k}_1} n_{\boldsymbol{k}_3}( n_{\boldsymbol{k}_2} - n_{\boldsymbol{k}}) \displaystyle \delta (\varOmega)\right\}_{\boldsymbol{k}_2 = \boldsymbol{k}+\boldsymbol{q}} \,\mathrm{d} \boldsymbol{k}_1\,\mathrm{d} \boldsymbol{k}_3, \end{equation}

where $\varOmega = \omega _k + \omega _{k_1}- \omega _{k_2} - \omega _{k_3}$. The additional factor of two relative to (1.6) is due to the fact that the condensate is supported on either $\boldsymbol{k}_2$ or $\boldsymbol{k}_3$.

Figure 2. Scheme of considered wave vectors with respect to position of condensate ring.

Next, we exploit Taylor expansion in $\boldsymbol{q}$ of the expression

(2.3)\begin{equation} \left|T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}_2,\boldsymbol{k}_3}\right|^2( n_{\boldsymbol{k}_2} - n_{\boldsymbol{k}}), \end{equation}

where $\boldsymbol{k}_2= \boldsymbol{k} + \boldsymbol{q}$ to second order (see Appendix A):

(2.4)\begin{align} \frac{\partial n_{\boldsymbol{k}}}{\partial t} &\approx 8{\rm \pi} \int n_{\boldsymbol{k}_1} n_{\boldsymbol{k}_3} \left|T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}, \boldsymbol{k}_3}\right|^2\boldsymbol{q}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\boldsymbol{k}}n_{\boldsymbol{k}} \left.\delta (\varOmega)\right|_{\boldsymbol{k}_2 = \boldsymbol{k}+\boldsymbol{q}} \,\mathrm{d} \boldsymbol{k}_1\,\mathrm{d} \boldsymbol{k}_3\nonumber\\ &\quad +4{\rm \pi}\int n_{\boldsymbol{k}_1} n_{\boldsymbol{k}_3} \left\{ 2\left.\left(\boldsymbol{q}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\boldsymbol{k}_2}\left|T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}_2, \boldsymbol{k}_3}\right|^2\right)\right|_{\boldsymbol{k}_2 =\boldsymbol{k}}\boldsymbol{q}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\boldsymbol{k}}n_{\boldsymbol{k}} \right.\nonumber\\ &\quad+ \left.\left|T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}, \boldsymbol{k}_3}\right|^2\boldsymbol{q}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\boldsymbol{k}}\left(\boldsymbol{q}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\boldsymbol{k}}n_{\boldsymbol{k}}\right) \right\}\left.\delta (\varOmega)\right|_{\boldsymbol{k}_2 = \boldsymbol{k}+\boldsymbol{q}} \,\mathrm{d} \boldsymbol{k}_1\,\mathrm{d} \boldsymbol{k}_3. \end{align}

To simplify (2.4), we consider the expansion of $\varOmega$ in $\boldsymbol{q}$:

(2.5)\begin{equation} \varOmega = \omega_{k_1} -\omega_{k_3} -\boldsymbol{\nabla}_{\boldsymbol{k}}\omega_k\boldsymbol{\cdot} \boldsymbol{q} + O(|\boldsymbol{q}|^2). \end{equation}

Using the equation $\omega _k= \sqrt {gk}$, we obtain

(2.6)\begin{align} \varOmega &\approx \omega_{k_1}-\omega_{k_3} -\frac{\sqrt{g}}{2}\,\frac{\boldsymbol{k}}{k^{3/2}}\boldsymbol{\cdot}(\boldsymbol{k}_1- \boldsymbol{k}_3)\nonumber\\ &= \omega_{k_1}-\omega_{k_3} -\frac{\sqrt{g}}{2}\,\frac{k_1\cos\alpha - k_3\cos\beta}{k^{1/2}}, \end{align}

where $\alpha$ and $\beta$ are as in figure 2. Here, we keep the next order term, which becomes part of the leading one if $k_1\approx k_3$. It should be noted that $\delta (\varOmega )$ (here and below we use the last expression for $\varOmega$) is invariant with respect to exchange $\boldsymbol{k}_1\leftrightarrow \boldsymbol{k}_3$. Using this symmetry as well as $T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}_2, \boldsymbol{k}_3}=T_{\boldsymbol{k}_2,\boldsymbol{k}_3}^{\boldsymbol{k}, \boldsymbol{k}_1}$, and taking into account the fact that $\boldsymbol{k}_1$ and $\boldsymbol{k}_3$ are dummy variables so that value of an integral must not change after exchange $\boldsymbol{k}_1 \leftrightarrow \boldsymbol{k}_3$, the first term of (2.4) is integrated to zero, due to change of sign of $\boldsymbol{q}$ after the mentioned exchange of integration variables. For evaluation of the second term in (2.4) at $\boldsymbol{k}_2=\boldsymbol{k}$, one can use

(2.7)\begin{equation} \left.\boldsymbol{\nabla}_{\boldsymbol{k}_2}\left|T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}_2, \boldsymbol{k}_3}\right|^2\right|_{\boldsymbol{k}_2 =\boldsymbol{k}} = \left.\boldsymbol{\nabla}_{\boldsymbol{k}}\left|T_{\boldsymbol{k}_2,\boldsymbol{k}_1}^{\boldsymbol{k}, \boldsymbol{k}_3}\right|^2\right|_{\boldsymbol{k}_2 =\boldsymbol{k}} = \frac{1}{2}\,\boldsymbol{\nabla}_{\boldsymbol{k}}\left|T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}, \boldsymbol{k}_3}\right|^2, \end{equation}

where the second equality should be understood in the context of integration over $\boldsymbol{k}_1$ and $\boldsymbol{k}_3$ as in (2.4). As a result, (2.4) can be transformed (after combining the second and third terms) into the expression

(2.8)\begin{equation} \frac{\partial n_{\boldsymbol{k}}}{\partial t} \approx 4{\rm \pi}\int n_{\boldsymbol{k}_1} n_{\boldsymbol{k}_3} \boldsymbol{\nabla}_{\boldsymbol{k}}\boldsymbol{\cdot} \left(\boldsymbol{q} \left|T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k}, \boldsymbol{k}_3}\right|^2\boldsymbol{q}\boldsymbol{\cdot} \boldsymbol{\nabla}_{\boldsymbol{k}}n_{\boldsymbol{k}}\right) \displaystyle \delta (\varOmega) \,\mathrm{d} \boldsymbol{k}_1\,\mathrm{d} \boldsymbol{k}_3 . \end{equation}

This is a continuity equation for the wave action $n_{\boldsymbol{k}}$,

(2.9)\begin{equation} \frac{\partial n_{\boldsymbol{k}}}{\partial t} + \boldsymbol{\nabla}_{\boldsymbol{k}} \boldsymbol{\cdot} \boldsymbol{Q}_{\boldsymbol{k}}=0 \end{equation}

with action flux $\boldsymbol{Q}_{\boldsymbol{k}}$ for an isotropic spectrum, where

(2.10)\begin{equation} \boldsymbol{Q}_{\boldsymbol{k}} =- 4{\rm \pi} \int n_{\boldsymbol{k}_1}n_{\boldsymbol{k}_3} \boldsymbol{q} \left| T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k},\boldsymbol{k}_3}\right|^2 \frac{\boldsymbol{q}\boldsymbol{\cdot}\boldsymbol{k}}{k}\,\frac{\partial n_{k}}{\partial k}\,\delta (\varOmega) \,\mathrm{d} \boldsymbol{k}_1\,\mathrm{d} \boldsymbol{k}_3. \end{equation}

Under the same isotropic consideration, we have

(2.11)$$\begin{gather} N=\int n_{\boldsymbol{k}}\,\mathrm{d}\boldsymbol{k} =\int n_{k} 2{\rm \pi} k\,\mathrm{d} k, \end{gather}$$

(2.12)$$\begin{gather}\frac{\partial (2{\rm \pi} k n_k)}{\partial t} +\frac{\partial Q_{ k}}{\partial k}=0. \end{gather}$$

Here, we introduced the isotropic wave action flux

(2.13)\begin{align} Q_k &=2{\rm \pi}\boldsymbol{k} \boldsymbol{\cdot}\boldsymbol{Q}_{\boldsymbol{k}}=-\left[\frac {8{\rm \pi}^2} k \int n_{\boldsymbol{k}_1}n_{\boldsymbol{k}_3}\, \delta (\varOmega) \right.\nonumber\\ &\quad \left.{}\times\left(\left| T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k},\boldsymbol{k}_3}\right|^2 (\boldsymbol{q} \boldsymbol{\cdot}\boldsymbol{k})^2 \right) \,\mathrm{d} \boldsymbol{k}_1\,\mathrm{d} \boldsymbol{k}_3\right]\frac{\partial n_{k}}{\partial k}. \end{align}

One can integrate out $k_3$ using $\delta (\varOmega )$. In order to do this, we need to set $\varOmega =0$ in (2.6). Solving this quadratic equation for $\sqrt {k_3}$ and taking into account condition $k_1\ll k$, we get

(2.14)\begin{equation} k_3 \approx k_1 -k_1\sqrt{\frac{k_1}{k}}\,(\cos\alpha-\cos\beta). \end{equation}

The integration in (2.13) is over $0\le \alpha,\beta <2{\rm \pi}$ and $k_1$. If the relative width of the condensate ring $\varepsilon /k_0$ is small, then from (2.14), $k_3$ will be outside the ring for various values of $\alpha, \beta \in (0,2{\rm \pi} )$; see figure 2. However, if the condensate ring is wide ($\varepsilon /k_0 \gg 2 (k_0/k)^{1/2}$), then we can always find $k_3 \approx k_1$ for all $0\le \alpha,\beta <2{\rm \pi}$. In other words, in the non-local resonant quartets, the lengths of vectors $\boldsymbol{k}_1$ and $\boldsymbol{k}_3$ are nearly equal, whereas their respective directions are allowed to be arbitrary (details can be found in Appendix D). As a result of the above strong inequality, we can take $n_{k_3}=n_{k_1}$, thus we have a diffusion equation

(2.15)\begin{equation} 2{\rm \pi} k\,\frac{\partial n_{k}}{\partial t} = \frac{\partial}{\partial k} \left(D_k\,\frac{\partial n_{k}}{\partial k} \right), \end{equation}

with diffusion coefficient

(2.16)\begin{equation} D_k = 8{\rm \pi}^2 k \int_{k_1 \ll k}n_{k_1}^2\,\frac{k_1^4}{\left|\dfrac{{\rm d} \omega_{k_1}}{{\rm d} k_1}\right|}\left[\int_{0}^{2{\rm \pi}} \int_{0}^{2{\rm \pi}} \left| T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k},\boldsymbol{k}_3}\right|^2 (\cos\beta-\cos\alpha)^2 \,\mathrm{d}\alpha\,\mathrm{d} \beta\right]\mathrm{d} k_1. \end{equation}

2.2. Asymptotics of the interaction coefficient and stationary solution

In Appendix B, it is shown that the first non-vanishing term in the expansion of $T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k},\boldsymbol{k}_3}$ in $k_{1,3}/k$ (after cancellation of all leading-order terms of order $k^2k_1$) is

(2.17)\begin{equation} T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k},\boldsymbol{k}_3}=-\frac{(k k_1)^{3/2}}{16{\rm \pi}^2}\,(\cos\alpha -\cos\beta)^2. \end{equation}

To find the diffusion coefficient (2.16), one needs to compute the following integral using (2.17):

(2.18)\begin{equation} \int_{0}^{2{\rm \pi}}\int_{0}^{2{\rm \pi}} \left| T_{\boldsymbol{k},\boldsymbol{k}_1}^{\boldsymbol{k},\boldsymbol{k}_3}\right|^2 (\cos\beta-\cos\alpha)^2 \,\mathrm{d}\alpha\,\mathrm{d} \beta = \frac{25(k k_1)^{3}}{256{\rm \pi}^2}, \end{equation}

which results in

(2.19)\begin{equation} D_k =\lambda k^4, \quad \lambda = \frac{25{\rm \pi}}{16} \int_{k_1 \ll k}n_{k_1}^2 k_1^{15/2} \,\mathrm{d} k_1 = \text{const}. \end{equation}

Note that we lose the universal (i.e. independent of the spectrum shape) scaling $D_k \sim k^4$ if the width of the condensate is $\varepsilon /k_0 \sim (k_0/k)^{1/2}$.

Note also that the expression for the diffusion coefficient (2.19) is different from the one previously obtained in Zakharov (2010), which is a consequence of an error in obtaining the asymptotics of the interaction coefficient made in the latter paper. The calculation of the latter paper is for a different purpose of obtaining the locality window of power-law spectra of gravity waves. As a result, we can now correct the upper boundary of locality for the power-law spectra $n_k \sim k^{-x}$, which follows from the convergence condition in the integral of (2.19), giving $x<17/4$ (cf. the respective condition $x<19/4$ written in Zakharov 2010). This correction makes the locality window narrower than previously thought, but does not affect the forward and inverse-cascade KZ spectra as local solutions (just by the margin of $1/4$ in the forward cascade!). The correction in (2.19) also likely affects the lower boundary of the locality window according to the procedure outlined in Zakharov (2010). However, the calculation for the UV boundary is much more involved, and the procedure needs to be put under close scrutiny before a definitive answer can be given. We leave this task to our future work.

From $Q=-D_k\,\partial n_{k}/\partial k = {\rm const}$, we immediately get

(2.20)\begin{equation} n_{k} = \frac{Q}{3\lambda}\,k^{-3}, \end{equation}

which is a constant-flux power-law solution. Note that since both $n_{k}$ and $\lambda$ are positive, the wave action flux $Q$ is positive too, i.e. it is towards high $k$ and has the opposite direction with respect to the wave action flux on the local KZ solution with $Q={\rm const}$. But by the standard Fjortoft argument, the wave action cannot be continuously transferred to high $k'$ values – otherwise, the amount of energy transferred to these wavenumbers would be greater than the energy produced by the forcing, which is impossible. Therefore, the non-local cascade must drain energy from the condensate. Indeed, for the energy of the out-of-condensate modes, we have from (2.15) that

(2.21)\begin{equation} \dot E= \int \omega_k (D_k n_k')' \,{\rm d}k = \frac{7\lambda}{4} \int k^{5/2} n_k \,{\rm d}k >0. \end{equation}

Thus for any shape of $n_k$, the energy of the out-of-condensate modes is growing, and since the total energy must be conserved, in the absence of forcing and dissipation, we conclude that the condensate energy must be decreasing at the same rate. We also conclude that for a steady state to exist, there should exist a mechanism of supplying the energy from the forcing region to the condensate that is not the diffusive mechanism arising from the WKE in the non-local regime (since the latter transfers energy in the opposite direction).

3. Conclusions and discussion

In this paper, we considered turbulence of water surface gravity waves and developed a WTT for non-local wave action spectrum evolution in the presence of a strong large-scale condensate. We derived a singular inhomogeneous spectral diffusion equation governing such an evolution. We found a stationary solution corresponding to a constant wave action cascade, $n_{k}\sim k^{-3}$, and proposed it as an explanation of the spectrum observed numerically in Korotkevich (2023) and experimentally in recent wave tank experiments (Nazarenko & Lukaschuk 2016; Deike et al. 2011).

Regarding the mechanism of supply of energy and wave action to the condensate, one can envisage two scenarios. First, the condensate could be built up via a local inverse cascade at the initial stages, followed by the non-local regime switching the cascade direction. Clearly, in this case the non-local regime would not be able to sustain itself due to the condensate drainage (the existence of a stationary solution in the absence of a large-scale forcing is prohibited by the wave action conservation), and the system would return to the local regime with a possible periodic repetition of the local–non-local cycle. We rule out this possibility because no such oscillations are observed in the numerical simulations. The second possibility is excitation of the condensate modes directly from a non-resonant three-wave process following modulational instability of the forcing modes. Such a mechanism requires that the spectrum of the waves in the forcing range is sufficiently strong – clearly, a feature seen in the numerical results of Korotkevich (2023). Moreover, the latter process could remain active simultaneously with the non-local direct cascade and thereby contribute to formation of a stationary spectrum. However, this scenario, proposing an implicit forcing mechanism, which could possibly explain the existence of the steady non-local cascade in the numerics of Korotkevich (2023), remains speculative and requires further validation.

In this paper, we mostly considered the effect of the resonant wave quartets, which implies that all the scales, including the condensate, are populated by weakly nonlinear waves. However, the WKB description (C20) derived in Appendix C does include the non-resonant non-local interactions; it is valid for the cases when the condensate scales are strongly nonlinear. Making analytical predictions about the spectra in this case is difficult as it implies knowledge of the low-$k$ dynamics, which is not a part of the WKB description itself. This can be done in future by assuming a simplified model for the low-k dynamics or by using direct numerical simulations for resolving the low wavenumber part, and using a ‘particle-in-cell method’ for the high-frequency WKB equations, as done previously for acoustic waves in Nazarenko, Zabusky & Scheidegger (1995).