Nonstationary distributions of wave intensities in Wave Turbulence

TL;DR

This paper derives a general solution for wave intensity distributions in non-stationary Wave Turbulence, revealing how wave statistics evolve over time and under different conditions, including steady states and decay.

Contribution

It provides a comprehensive analytical framework linking wave intensity distributions to the evolving wave action spectrum in non-stationary turbulence.

Findings

Wave statistics tend to Gaussian in steady-state forced systems without wave breaking.

Initial Gaussian statistics remain Gaussian over time.

Non-Gaussian initial conditions can persist indefinitely in decaying turbulence.

Abstract

We obtain a general solution for the probability density function of wave intensities in non-stationary Wave Turbulence. The solution is expressed in terms of the wave action spectrum evolving according the the wave-kinetic equation. We establish that, in absence of wave breaking, the wave statistics asymptotes to a Gaussian distribution in forced-dissipated wave systems that approach a steady state. Also, in non-stationary systems, if the statistics is Gaussian initially, it will remain Gaussian at any time. Generally, the statistics that is not Gaussian initially will remain non-Gaussian over the characteristic nonlinear time of the wave spectrum. In freely decaying wave turbulence, substantial deviations from Gaussianity may persist infinitely long.

Equations

$$i\dot{a_{\mathbf{k}}}=\frac{\partial \mathcal{H}}{\partial a_{\mathbf{k}}^{\ast }},\mathcal{H}=\sum _{\mathbf{k}}{\omega }_{\mathbf{k}}|a_{\mathbf{k}}{|}^2+\frac{1}{2}\sum _{{\mathbf{k}}_1,{\mathbf{k}}_2,{\mathbf{k}}_3,{\mathbf{k}}_4}{W}_{{\mathbf{k}}_3,{\mathbf{k}}_4}^{{\mathbf{k}}_1,{\mathbf{k}}_2}{a}_{{\mathbf{k}}_1}^{\ast }{a}_{{\mathbf{k}}_2}^{\ast }{a}_{{\mathbf{k}}_3}{a}_{{\mathbf{k}}_4},$$

$$\mathcal{P}(t,s_{\mathbf{k}})=⟨\delta (s_{\mathbf{k}}-J_{\mathbf{k}})⟩,$$

$$\frac{\partial \mathcal{P}(t,s_{\mathbf{k}})}{\partial t}+\frac{\partial F}{\partial s_{\mathbf{k}}}=0,$$

$$F=-s_{\bf k}\Big{(}\gamma_{\bf k}{\mathcal{P}}+\eta_{\bf k}\frac{\partial{\mathcal{P}}}{\partial s_{\bf k}}\Big{)}$$

$${\eta }_{\mathbf{k}}(t)$$

$${\gamma }_{\mathbf{k}}(t)$$

$$\mathcal{Z}(t,{\lambda }_{\mathbf{k}})=⟨e^{-{\lambda }_{\mathbf{k}}|a_{\mathbf{k}}(t){|}^2}⟩={\int }_0^{\infty }\mathcal{P}({\lambda }_{\mathbf{k}},t)e^{-{\lambda }_{\mathbf{k}}{s}_{\mathbf{k}}}d{s}_{\mathbf{k}}$$

$$\mathcal{P}(t,s)=\frac{1}{2\pi i}\underset{T\to \infty }{\lim }{\int }_{T-i\infty }^{T+i\infty }\mathcal{Z}(\lambda ,t)e^{s\lambda }d\lambda .$$

$$M_{\mathbf{k}}^{(p)}\equiv ⟨|a_{\mathbf{k}}{|}^{2p}⟩=(-1{)}^p{\mathcal{Z}}_{\lambda \cdots \lambda }{|}_{\lambda =0}=(-1{)}^p⟨|a{|}^{2p}{e}^{\lambda |a{|}^2}⟩{|}_{\lambda =0},$$

$$n_{\mathbf{k}}=-{\mathcal{Z}}_{\lambda }{|}_{\lambda =0}.$$

$$\dot{\mathcal{Z}}=-\lambda \eta \mathcal{Z}-({\lambda }^2\eta +\lambda \gamma ){\mathcal{Z}}_{\lambda }.$$

$$\mathcal{Z}=\frac{1}{1+{\lambda }_{\mathbf{k}}{n}_{\mathbf{k}}}.$$

$$\frac{d\lambda}{dt}=\Big{(}{\gamma}+\lambda\eta\Big{)}\lambda,\quad\frac{d\mathcal{Z}}{dt}=-\lambda\eta\mathcal{Z}.$$

$$\frac{d\mu (t)}{dt}=\eta \mu \lambda =\eta {\mu }^2{e}^{{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }},$$

$$-\frac{1}{\mu (t)}+\frac{1}{{\mu }_0}={\int }_0^t\eta (t^{\prime })e^{{\int }_0^{t^{\prime }}\gamma (t^{\prime \prime })d{t}^{\prime \prime }}d{t}^{\prime },$$

$$\lambda (t)=\frac{{\lambda }_0{e}^{{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}}{1-{\lambda }_0{\int }_0^t\eta (t^{\prime })e^{{\int }_0^{t^{\prime }}\gamma (t^{\prime \prime })d{t}^{\prime \prime }}d{t}^{\prime }}.$$

$$\dot{n}=\eta -\gamma n,$$

$$n(t)=n(0)e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}+e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}{\int }_0^t\eta (t^{\prime })e^{{\int }_0^{t^{\prime }}\gamma (t^{\prime \prime })d{t}^{\prime \prime }}d{t}^{\prime }.$$

$$\lambda (t)=\frac{{\lambda }_0}{e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}-{\lambda }_0\left(n(t)-n(0)e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}\right)}.$$

$${\lambda }_0=\frac{\lambda e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}}{1+\lambda \left(n(t)-n(0)e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}\right)}.$$

$$\mathcal{Z}(t,\lambda )={\mathcal{Z}}_0\frac{{\mu }_0}{\mu }=\frac{{\mathcal{Z}}_0{\lambda }_0}{\lambda }{e}^{{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}=\frac{{\mathcal{Z}}_0}{1+\lambda \left(n(t)-n(0)e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}\right)}.$$

$$\underset{t\to \infty }{\lim }\frac{n(0)e^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}}{n(t)}=0.$$

$$\underset{t\to \infty }{\lim }{\lambda }_0(t,\lambda )=0,\underset{t\to \infty }{\lim }{\mathcal{Z}}_0=\mathcal{Z}(0,0)=1and\underset{t\to \infty }{\lim }\mathcal{Z}(t,\lambda )=\frac{1}{1+\lambda n}.$$

$$\mathcal{P}(t,s)={\int }_0^{\infty }\mathcal{P}(0,J){\mathcal{P}}_J(t,s)dJ.$$

$$\begin{gathered} {\mathcal{P}}_J(t,s)=\frac{1}{2\pi i}\underset{T\to +\infty }{\lim }{\int }_{T-i\infty }^{T+i\infty }{e}^{\lambda s}\mathcal{Z}(\lambda )d\lambda \\ =\frac{1}{2\pi i}\underset{T\to +\infty }{\lim }{\int }_{T-i\infty }^{T+i\infty }\frac{e^{\lambda s-{\lambda }_{0}J}}{1+\lambda \tilde{n}}d\lambda . \end{gathered}$$

$$\tilde{n}=n(t)-J{e}^{-{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }}$$

$${\mathcal{P}}_J(t,s)=\frac{e^{-\frac{s}{\tilde{n}}-a\tilde{n}}}{2\pi i\tilde{n}}\underset{T\to +\infty }{\lim }{\int }_{T-i\infty }^{T+i\infty }\frac{e^{s\rho +\frac{a}{\rho }}}{\rho }d\rho =\frac{1}{\tilde{n}}{e}^{-\frac{s}{\tilde{n}}-a\tilde{n}}{I}_0(2\sqrt{as}).$$

$${\mathcal{P}}_J(s,t)\to \frac{{\mathcal{P}}_G}{(2\pi {)}^{1/2}(as{)}^{1/4}}{e}^{2\sqrt{as}-as}\ll {\mathcal{P}}_{G}foras\gg 1,{\int }_0^t\gamma (t^{\prime })d{t}^{\prime }\gg 1.$$

Full text

I Introduction.

Wave Turbulence is a theory that describes random weakly nonlinear wave systems with broadband spectra (see e.g. ref. naza11 ). The main object in this theory is a wave action spectrum which is the second-order moment of the wave amplitude and which evolves according to the so-called wave-kinetic equation. Special attention in past literature was given to studies of stationary scaling solutions of this equation which are analogous to the Kolmogorov spectrum of hydrodynamic turbulence, the so-called Kolmogorov-Zakharov spectra. However, as it was shown in refs. lvov_2004 ; choi_2004 ; choi_2005 ; choi_2005b ; naza11 , Wave Turbulence approach can also be extended to describing the higher-order moments and even to the entire probability density function (PDF) of the wave amplitude. A formal justification of such an extension based on a rigorous statistical formulation was later presented in ref. Eyink . An introduction to Wave Turbulence as well as a summary recent developments in this area can be found in book naza11 and in an older text ZLF .

It has been widely believed that the statistics of random weakly nonlinear wave systems is close to being Gaussian. Derivation of the evolution equation for the PDF of the wave intensities presented in ref. choi_2005 has made it possible to examine this belief. It was shown in ref. choi_2005 indeed has a stationary solution corresponding to the Gaussian state, but it was also noted that the typical evolution time of the PDF is the same as the one for the spectrum. Thus, for non-stationary wave systems one can expect significant deviations from the Gaussianity if the initial wave distribution is non-Gaussian. Note that non-Gaussian (typically deterministic) initial conditions for the wave intensity are typical in numerical simulations in Wave Turbulence. Also, there is no reason to believe that initial waves excited in natural conditions, e.g. sea waves excited by wind, should be Gaussian. Therefore, study of evolution of the wave statistics is important for both understanding of fundamental nonlinear processes as well as for the practical predictions such as e.g. wave weather forecast.

In the present paper we will present the full general solution for the PDF equation derived in ref. choi_2005 . Based on that solution we will formulate the condition under which the wave statistics relaxes to the Gaussian state.

II Evolution equations for the wave amplitude and for the PDF

Consider a weakly nonlinear wave system dominated by the four-wave interactions bounded by an $L$-periodic cube in the $d$ dimensional physical space. (Four-wave systems are considered here as an illustrative example only. All results of this paper hold for the $N$-wave systems with any $N$. The only difference will be in the expressions for $\gamma_{\bf k}$ and $\eta_{\bf k}$ below; see ref. naza11 .) We have the Hamiltoninan equations for the Fourier coefficients as follows,

[TABLE]

where ${\bf k},{\bf k}_{1},{\bf k}_{2},{\bf k}_{3},{\bf k}_{4}\in\frac{L}{2\pi}\mathbb{Z}^{d}$ are the wave vectors, $a_{\bf k}\in\mathbb{C}$ is the wave action variable, $W^{{\bf k}_{1},{\bf k}_{2}}_{{\bf k}_{3},{\bf k}_{4}}\in\mathbb{R}$ is an interaction coefficient which is a model-specific function of ${{\bf k}_{1},{\bf k}_{2},{\bf k}_{3},{\bf k}_{4}}$ (e.g. $W^{{\bf k}_{1},{\bf k}_{2}}_{{\bf k}_{3},{\bf k}_{4}}=1$ for the Gross-Pitaevskii equation) and $\omega_{\bf k}\in\mathbb{R}$ is the frequency of mode $\bf k$.

Let us consider the PDF ${\mathcal{P}}(t,s_{\bf k})$ of the wave intensity $J_{\bf k}=|a_{\bf k}|^{2}$ defined in a standard way as so that the probability for $J_{\bf k}$ to be in the range from $s_{\bf k}$ to $s_{\bf k}+ds_{\bf k}$ is ${\mathcal{P}}(t,s_{\bf k})ds_{\bf k}$. In symbolic form,

[TABLE]

Suppose that the waves are weakly nonlinear, so that the quadric part of the Hamiltonian is much less than its quadratic part. Suppose also that the complex wave amplitudes $a_{\bf k}$ are independent random variables for each $\bf k$ and that the initial phases of $a_{\bf k}$ are random and equally probable in the range from [math] to $2\pi$. These are the main assumptions of the weak Wave Turbulence theory (see ref. naza11 ), leading, upon taking the infinite-box limit $L\to\infty$, to the following evolution equation for ${\mathcal{P}}(t,s_{\bf k})$, as derived in ref. choi_2005 :

[TABLE]

where

[TABLE]

and, for the four-wave systems,

[TABLE]

where $n_{\bf k}=\langle J_{\bf k}\rangle$ is the wave action spectrum. The infinite-box limit resulted to passing continuous wave number description; each wave number integration in the above equations is over $\mathbb{R}^{d}$.

In this paper, we will find the time-dependent solution of the PDF equation (3).

III Generating function

Let us introduce the generating function

[TABLE]

where $\lambda_{\bf k}$ is a real parameter. Note that this definition is different from the one used in Ref.choi_2005 by the sign of the exponent. Here, we have changed the sign in order to comply with the standard relation between $\mathcal{P}$ and $\mathcal{Z}$ via the Laplace transform, as expressed in eqn. (7).

In what follows we will drop subscripts $k$ for brevity whenever it does not cause ambiguity.

The inverse Laplace transformation of $\mathcal{Z}(t,\lambda)$ gives:

[TABLE]

Given ${\mathcal{Z}}$, one can easily calculate the moments of the wave intensity,

[TABLE]

where $p\in\mathbb{N}$ is the order of the moments and subscript $\lambda$ means taking derivative with respect to $\lambda$. In particular, for the waveaction spectrum we have

[TABLE]

The evolution equation for $\mathcal{Z}$ can be obtained by Laplace transforming eqn. (7), which gives

[TABLE]

Note that the sign differences in this equation with respect to the corresponding equation in Ref.choi_2005 is due to the sign difference in our definition of $\mathcal{Z}$.

Previously in Ref.choi_2005 , the general steady state solution eqn.(10) was presented:

[TABLE]

This solution corresponds to gaussian statistics of the wave field (Rayleigh distribution for the wave intensity respectively).

Below, we will concentrate the fully time-evolving problem, in which the parameters $\eta,\gamma$ have time-dependency. The goal of this paper is first to find the solution of the eqn. (10) and then obtain the respective time-dependent PDF.

IV Solution for $\mathcal{Z}$ by the method of characteristics

We can find the solution for the fully time-evolving case of the eqn. (10) by using the method of characteristics. Rewriting this equation in the characteristic form we have:

[TABLE]

Changing variable to $\mu=\lambda e^{-\int_{0}^{t}{\gamma(t^{\prime})}dt^{\prime}}$ in the first of these equations, we transform it into

[TABLE]

solving which we have:

[TABLE]

where $\mu_{0}=\mu(0)=\lambda_{0}$. This gives for $\lambda(t)$:

[TABLE]

This relation has a simpler form in terms of $n$ rather than $\eta$. Indeed, $n$ satisfies the following (kinetic) equation,

[TABLE]

integrating which we have

[TABLE]

Using this identity, we have:

[TABLE]

Conversely, we have:

[TABLE]

Now, from the second of the eqns. (12) and from the first equality in eqn. (13) we see that the log derivative of $\mathcal{Z}$ is equal to the negative log derivative of $\mu$. Thus,

[TABLE]

where $\mathcal{Z}_{0}=\mathcal{Z}(0,\lambda_{0})$ and $\lambda_{0}$ must be substituted in terms of $\lambda$ solving for it from eqn. (19).

Eqns. (20) and (19) allow us to prove the following theorem.

Theorem**.**

Wave fields which are Gaussian initially will remain Gaussian for all time. 2. 2.

Wave turbulence asymptotically becomes Gaussian if

[TABLE]

To prove the first part we simply substitute $\mathcal{Z}_{0}=1/(1+\lambda_{0}n_{0})$ into eqn. (20) and, after using (19), obtain $\mathcal{Z}=1/(1+\lambda n)$, which corresponds to the Gaussian statistics.

To prove the second part we notice that if condition (21) is satisfied then

[TABLE]

Remarks:

Condition (21) is satisfied for the inertial range modes in forced-dissipated systems which tend to a steady state. Indeed, in this case $\gamma\to\eta/n$ which is a positive constant (at fixed mode $\bf k$), so the time integral of this quantity diverges as $t\to\infty$. 2. 2.

In absence of forcing and dissipation, spectrum $n_{\bf k}$ decays to zero at any mode $\bf k$ as $t\to\infty$, and so does $\gamma_{\bf k}$. Thus the integral of $\gamma_{\bf k}(t)$ may converge as $t\to\infty$, which means that non-Gaussianity of some (or all) wave modes may persist as $t\to\infty$. 3. 3.

In general, function $\gamma_{\bf k}(t)$ is not sign definite, and there may be transient time periods where $\gamma_{\bf k}(t)<0$. The deviation from Gaussianity of some (or all) wave modes may increase during these periods.

V Evolution of the PDF

Now let us analyse the PDF of transient states. Let us think of a simple case with a deterministic initial wave intensity, $P(0,s)=\delta(s-J)$. We will call such a solution $\mathcal{P}_{J}(s,t)$. Then $\mathcal{Z}(0,\lambda)=e^{-\lambda J}$. In fact, since the inverse Laplace transform is a linear operation, the considered solution is nothing but Green’s function for the general problem with an arbitrary initial condition $P(0,s)$:

[TABLE]

Let us take the inverse Laplace transform of $\mathcal{Z}(t,\lambda)$ given by eqn. (20) to obtain the $\mathcal{P}_{\delta}(s,t)$ at $t>0$:

[TABLE]

where

[TABLE]

(note that $n(0)=J$). Substituting $\lambda_{0}$ from (18) and changing the integration variable as $\rho=\lambda+1/\tilde{n}$, we have:

[TABLE]

where $a=\frac{J}{{\tilde{n}}^{2}}e^{-\int^{t}_{0}\gamma(t^{\prime})dt^{\prime}}$ and $I_{0}(x)$ is the zeroth modified Bessel function of the first kind. Note that $I_{0}(0)=1$, so we recover $\mathcal{P}_{\delta}\to\mathcal{P}_{G}=\frac{1}{n}e^{-s/n}$ as $t\to\infty$ if condition (21) is satisfied provided that $s$ is not too large, $as\ll 1$.

Now let us suppose that condition (21) is satisfied and let consider the asymptotic behaviour of the probability distribution at large $s$ and large $t$, and $as\gg 1$ (i.e. $s$ is much larger than $1/a$ which is itself large). Taking into account that $I_{0}(x)\xrightarrow{x\rightarrow\infty}\frac{e^{x}}{\sqrt{2\pi x}}$, we have:

[TABLE]

Thus, we see a front at $s\sim s^{*}(t)=1/a$ moving toward large $s$ as $t\to\infty$. The PDF ahead of this front is depleted with respect to the Gaussian distribution, whereas behind the front it asymptotes to Gaussian. Obviously, the same kind of behaviour will be realised for any solution (23) arising from initial data having a finite support in $s$.

VI Conclusions and Discussion

In this paper we have obtained the general solutions for the generating function and for the PDF of wave intensities in Wave Turbulence, equations (20) and (19), and equation (26) respectively. This allowed us to prove a theorem stating that wave fields which are Gaussian initially will remain Gaussian for all time and that Wave Turbulence asymptotically becomes Gaussian if condition (21) is satisfied. We have also found (when condition (21) is satisfied) an asymptotic solution for the PDF (27) where the Gaussian distribution forms behind a front propagating toward large wave intensities.

Condition (21) is satisfied for the inertial range modes in forced-dissipated systems approaching a steady state. Thus, the Gaussian statistics will form at large time for such modes in these systems. An interesting subclass of solutions in forced-dissipated systems is when the spectrum is in a steady state from the initial moment of time (i.e. it is a stationary solution of the wave-kinetic equation), while the PDF is not Rayleigh initially (i.e. the initial wave field is not Gaussian). For example, the initial wave intensities can be deterministic, i.e. their PDFs are delta-functions, as it is often taken in numerical simulations of Wave Turbulence. In this case, equation (26) looks the simplest, with $\int^{t}_{0}\gamma(t^{\prime})dt^{\prime}=\gamma t$.

Since the characteristic evolution times are the same for the spectrum $n_{\bf k}$ and the PDF, the latter will remain non-Gaussian over a substantial time in the initial field is non-Gaussian. Such situations should be considered typical rather than exception in natural conditions (where initial waves arise, e.g., from an instability which does not necessarily produce Gaussian waves) and in numerical simulations (where typically the wave intensities are taken to be deterministic).

Moreover, in absence of forcing and dissipation, spectrum $n_{\bf k}$ decays to zero at any mode $\bf k$ as $t\to\infty$, and so does $\gamma_{\bf k}$. Thus the integral $\int^{t}_{0}\gamma_{\bf k}(t^{\prime})dt^{\prime}$ may converge as $t\to\infty$, which means that non-Gaussianity of some (or all) wave modes may persist as $t\to\infty$. Furthermore, since $\gamma_{\bf k}(t)$ is not sign definite, there may be transient time periods where $\gamma_{\bf k}(t)<0$. The deviation from Gaussianity of some (or all) wave modes may increase during these periods.

The present paper considers the four-wave systems as an illustrative example, but it is clear that the obtained solutions are more general and apply to the wave systems with resonances of any order (one would simply have to use different expressions for the integrals $\gamma_{\bf k}(t)$ and $\eta_{\bf k}(t)$ corresponding to resonance of the considered order; see e.g. book naza11 ). Note that our solution for the PDF (26) is expressed in terms of the spectrum $n_{\bf k}(t)$ (recall that $\gamma_{\bf k}(t)$ depends on $n_{\bf k}(t)$ via eqn. (6)). On the other hand, $n_{\bf k}(t)$ obeys the wave-kinetic equation which is not easy to solve for non-stationary systems. However, it is quite straightforward to solve the wave-kinetic equation numerically, after which the resulting $n_{\bf k}(t)$ can be used in the analytical formula for the PDF (26). Since the latter formula is very simple, we believe that it can be very effective in practical calculations especially in the situations where non-Gaussianity is important, e.g., in wave weather forecasts including prediction of anomalously strong waves – the so-called freak waves.

Acknowledgements.

Yeontaek Choi’s work is supported by National Institute for Mathematical Sciences(NIMS) and Korean Union of Public sector Research and Professional workers(KUPRP). The work of Young-Sam Kwon is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2013R1A1A2057662).