One-mode statistics

We have established above that the one-point statistics is at the heart of the WT theory. All one-point statistical objects can be derived from the one-point amplitude generating function,

$$Z_a (\lambda_j) = \left< e^{\lambda_j A_j^2} \right>$$

which can be obtained from the $N$-point $Z$ by taking all $\mu$'s and all $\lambda$'s, except for $\lambda_j$, equal to zero. Substituting such values to (64) (for the three-wave case) or to (72) (four-wave) we get the following equation for $Z_a$,

$$\frac{\partial Z_a}{\partial t} = \lambda_j \eta_j Z_a +(\lambda_j^2 \eta_j - \lambda_j \gamma_j) \frac{\partial Z_a}{\partial \lambda_j},$$ (74)

where for the three-wave case we have:

$$\eta_j = 4 \pi \epsilon^2 \int \left(\vert V^j_{lm}\vert^2 \delta... ...\delta^m_{jl} \delta(\omega^m_{jl} ) \right) n_{l} n_{m} \, d { k_l} d { k_m} ,$$ (75)

$$\gamma_j = 8 \pi \epsilon^2 \int \left( \vert V^j_{lm}\vert^2 \de... ...elta^m_{jl} \delta(\omega^m_{jl}) (n_{l}- n_{m}) \right) \, d { k_l} d { k_m} .$$ (76)

and in the four-wave case:

$$\eta_j = 4 \pi \epsilon^2 \int \vert W^{jl}_{nm}\vert^2 \delta^{jl}_{nm} \delta(\omega^{jl}_{nm}) n_l n_m n_n \, d { k_l} d { k_m} d { k_n,}$$ (77)

$$\gamma_j = 4 \pi \epsilon^2 \int \vert W^{jl}_{nm}\vert^2 \delta^{jl}_{nm} \... ...^{jl}_{nm}) \Big[ n_l (n_m + n_n) - n_m n_n\Big] \, d { k_l} d { k_m} d { k_n.}$$ (78)

Here we introduced the wave-action spectrum,

$$n_j = \langle A_j^2 \rangle.$$ (79)

Differentiating (80) with respect to $\lambda$ we get an equation for the moments $M^{(p)}_j = \langle A_j^{2p} \rangle$:

$$\dot M^{(p)}_j = -p \gamma_j M^{(p)}_j + p^2 \eta_j M^{(p-1)}_j.$$ (80)

which, for $p=1$ gives the standard kinetic equation,

$$\dot n_j = - \gamma_j n_j + \eta_j .$$ (81)

First-order PDE (80) can be easily solved by the method of characteristics. Its steady state solution is

$$Z={1 \over 1 -\lambda n_k}$$ (82)

which corresponds to the Gaussian values of momenta

$$M^{(p)} = p! n_k^p.$$ (83)

However, these solutions are invalid at small $\lambda$ and high $p$'s because large amplitudes $s=\vert a\vert^2$, for which nonlinearity is not weak, strongly contribute in these cases. Because of the integral nature of definitions of $M^{(p)}$ and $Z$ with respect to the $s=\vert a\vert^2$, the ranges of amplitudes where WT is applicable are mixed with, and contaminated by, the regions where WT fails. Thus, to clearly separate these regions it is better to work with quantities which are local in $s=\vert a\vert^2$, in particular the one-mode probability distribution $P_a$. Equation for the one-mode PDF can be obtained by applying the inverse Laplace transform to (80). This gives:

$${\partial P_a \over \partial t}+ {\partial F \over \partial s_j} =0,$$ (84)

with $F$ is a probability flux in the s-space,

$$F=-s_j (\gamma P_a +\eta_j {\delta P_a \over \delta s_j}).$$ (85)

Let us consider the steady state solutions, $\dot P_{a} =0$ so that

$$-s(\gamma P+\eta\partial_{s}P) = F= \hbox{const}.$$ (86)

Note that in the steady state $\gamma /\eta = n$ which follows from kinetic equation (87). The general solution to (92) is

$$P=P_{hom} + P_{part}$$ (87)

where

$$P_{hom} = \hbox{const} \, \exp{(-s/n)}$$ (88)

is the general solution to the homogeneous equation (corresponding to $F=0$) and $P_{part}$ is a particular solution,

$$P_{part} = -({F}/{\eta}) Ei({s}/{n}) \exp{(-s/n)},$$ (89)

where $Ei(x)$ is the integral exponential function.

At the tail of the PDF, $s \gg n_k$, the solution can be represented as series in $1/s$,

$$P_{part} = - \frac{F}{s\gamma}-\frac{\eta F}{(\gamma s)^2}+ \cdots.$$ (90)

Thus, the leading order asymptotics of the finite-flux solution is $1/s$ which decays much slower than the exponential (Rayleigh) part $P_{hom}$ and, therefore, describes strong intermittency.

Note that if the weakly nonlinearity assumption was valid uniformly to $s=\infty$ then we had to put $F=0$ to ensure positivity of $P$ and the convergence of its normalisation, $\int P \, ds =1$. In this case $P=P_{hom} = n \, \exp{(-s/n)}$ which is a pure Rayleigh distribution corresponding to the Gaussian wave field. However, WT approach fails for for the amplitudes $s \ge s_{nl}$ for which the nonlinear time is of the same order or less than the linear wave period and, therefore, we can expect a cut-off of $P(s)$ at $s = s_{nl}$. Estimate for the value of $s_{nl}$ can be obtained from the dynamical equation (14) by balancing the linear and nonlinear terms and assuming that if the wave amplitude at some $k$ happened to be of the critical value $s_{nl}$ then it will also be of similar value for a range of $k$'s of width $k$ (i.e. the $k$-modes are strongly correlated when the amplitude is close to critical). This gives:

$$s_{nl}= \omega/\epsilon W k^2$$ (91)

This cutoff can be viewed as a wavebreaking process which does not allow wave amplitudes to exceed their critical value, $P(s) =0$ for $s > s_{nl}$. Now the normalisation condition can be satisfied for the finite-flux solutions. Note that in order for our analysis to give the correct description of the PDF tail, the nonlinearity must remain weak for the tail, which means that the breakdown happens far from the PDF core $s \ll s_{nl}$. When $s_{nl} \sim n$ one has a strong breakdown predicted in [32] which is hard to describe rigorously due to strong nonlinearity.

Depending on the position in the wavenumber space, the flux $F$ can be either positive or negative. As we discussed above, $F<0$ results in an enhanced probability of large wave amplitudes with respect to the Gaussian fields. Positive $F$ mean depleted probability and correspond to the wavebreaking value $s_{nl}$ which is closer to the PDF core. When $s_{nl}$ gets into the core, $s_{nl} \sim n$ one reaches the wavenumbers at which the breakdown is strong, i.e. of the kind considered in [32]. Consider for example the water surface gravity waves. Analysis of [32] predicts strong breakdown in the high-$k$ part of the energy cascade range. According to our picture, these high wavenumbers correspond to the highest positive values of $F$ and, therefore, the most depleted PDF tails with respect to the Gaussian distribution. When one moves away from this region toward lower $k$'s, the value of $F$ gets smaller and, eventually, changes the sign leading to enhanced PDF tails at low $k$'s. This picture is confirmed by the direct numerical simulations of the water surface equations reported in [23] the results of which are shown in figure 1.

Figure: One-mode amplitude PDF's at two wavenumbers: upper curve at $k_1$ and the lower curve at and $k_2$ such that $k_1>k_2$. Dashed line corresponds to the Rayleigh distribution.

[width=.5]1.eps

This figure shows that at a high $k$ the PDF tail is depleted with respect to the Rayleigh distribution, whereas at a lower $k$ it is enhanced which corresponds to intermittency at this scale. Similar conclusion that the gravity wave turbulence is intermittent at low rather than high wavenumbers was reached on the basis of numerical simulations in [33]. To understand the flux reversal leading to intermittency appears in the one-mode statistics, one has to consider fluxes in the multi-mode phase space which will be done in the next section.