General theory

Our starting point is the Zakharov equation (, ), which is a deterministic nonlinear evolution equation for surface gravity waves in deep water. Let's consider the potential flow of an ideal fluid of infinite depth. Coordinates are chosen in such a way that the undisturbed surface of the fluid coincides with the $x-y$ plane. The $z$-axis is pointed upward, and the acceleration of gravity $g$ is pointed in the negative $z$-direction. The surface elevation $eta$ may be written in terms of a Fourier expansion as

$$\eta = \int_{-\infty}^{\infty}{\rm d}\vec{k} \left[ a(\vec{k})+a^*(-\vec{k}) \right]e^{i\vec{k}.\vec{x}},$$ (1)

where $a(vec{k},t)=sqrt{(omega/2g)}B(vec{k},t)$ and $B(vec{k},t)$ is the normal variable. Here, $vec{k}$ is the wave number vector, $k$ is its absolute value, and $omega = sqrt{gk}$ denotes the dispersion relation of deep-water, gravity waves. Alternatively one may write for $eta$

$$\eta = \int_{-\infty}^{\infty}{\rm d}\vec{k}\; a(\vec{k}) e^{i \vec{k}.\vec{x}} +c.c,$$ (2)

where c.c denotes the complex conjugate.

() obtained from the Hamilton equations an approximate evolution equation for the amplitude of the free surface gravity waves, that contained the third order non-resonant and resonant four-wave interactions. In order to eliminate the effects of bound waves he applied on $B$ a canonical transformation of the type

$$B = B(b,b^*),$$ (3)

where $b$ is the normal variable of the free gravity waves. The evolution equation for $b$, called the Zakharov equation, becomes

$$\frac{\partial b_1}{\partial t}+i\omega_1 b_1 = -i\int{\rm d}\vec{k}_{2,3,4} T_{1,2,3,4}b^*_2 b_3 b_4 \delta_{1+2-3-4},$$ (4)

where, for brevity we have introduced the notation $b_1=b(vec{k}_1)$, etc, and the nonlinear transfer function $T_{1,2,3,4}$ as found by () enjoys a number of symmetries which guarantee that the Zakharov equation is Hamiltonian and conserves wave energy. In () it was shown that in the context of the deep-water version of the Zakharov equation extreme surface gravity waves are generated by nonlinear focusing in a random wave field. This process also causes the Benjamin-Feir instability of a uniform wave train. As a consequence, for deep-water waves a considerable enhancement of the probability for extreme waves is found. However, when going to shallow waters, the effect of nonlinear focusing is greatly reduced. In fact for a narrow band wave train (, ) it can be shown that the nonlinear transfer coefficient $T_{1,2,3,4}$ vanishes when the dimensionless water depth $kh=1.36$. For smaller dimensionless depth nonlinear effects will lead to defocusing and, consequently a considerable reduction of extreme events. In this paper we only concentrate on the deep-water case by restricting ourselves to $kh>2$.

Thus, the nonlinear term in Eqn.() will generate deviations from the normal, Gaussian probability distribution function (PDF) for the surface elevation. It is of interest to determine these deviations because it gives us information on the occurrence of extreme sea states. However, because the nonlinearity is of third order in amplitude, the skewness of the free waves vanishes, while the fourth cumulant is finite. On the other hand, the bound waves contribute to both the skewness and the kurtosis of surface elevation PDF. In order to determine the effects of the bound waves on the PDF of a random, nonlinear wave field one needs to utilize the canonical transformation Eq.(). This results in complicated expressions for skewness and kurtosis because one has to go to third order in amplitude. The problem simplifies considerably when the narrow-band approximation is adopted, because than one can simply use the Stokes solution for the surface elevation. For example, in the narrow-band approximation it is easy to show (see $ 2.2) that for a `typical' steepness of the order $0.1$ bound waves contribute less than $2\%$ to the value of the kurtosis, so we can safely ignore their contribution.

Here, we are interested in obtaining the high-order moments of the surface elevation for a homogeneous random sea. The homogeneity and stationarity conditions imply

$$\langle b_1 b^*_2 \rangle = N_1 \delta(\vec{k}_1-\vec{k}_2),\;{\rm and}\; \langle b_1 b_2 \rangle =0,$$ (5)

where we have introduced the usual action density $N(vec{k})$ and the angle brackets denote an ensemble average. As for narrow-band spectra bound waves only have a small effect on the statistics of the surface gravity waves we have to good approximation

$$\langle a_1 a^*_2 \rangle \simeq \frac{\omega}{2g} N_1 \delta(\vec{k}_1-\vec{k}_2)\;{\rm and}\;\langle a_1 a_2 \rangle =0.$$ (6)

Envelope $A$ and phase $phi$ are now defined as

$$\frac{1}{2} A e^{i\phi} = \int_{-\infty}^{\infty}{\rm d}\vec{k}\; a e^{i\vec{k}.\vec{x}},$$ (7)

hence,

$$\eta = A\cos \phi$$ (8)

where for a narrow-band wave train $A$ and $phi$ are slowly varying functions in time and space. Now introduce the auxiliary variable $zeta$ in such a way that the random variables $eta$ and $zeta$ are not correlated in the linear wave field, $langle eta \; zeta rangle=0$. Thus,

$$\zeta = A\sin \phi,$$ (9)

or,

$$\zeta=-i \left( \int_{-\infty}^{\infty}{\rm d}\vec{k}\; a e^{i\vec{k}.\vec{x}} -c.c. \right)$$ (10)

Assuming zero mean for $eta$, the second order moment $mu_2$ is given by

$$\mu_2 = \langle \eta^2 \rangle = m_0$$ (11)

According to Eq.(29) of Janssen (2003), the fourth moment $langle eta^4 rangle$ and kurtosis $mu_4$ can be obtained in terms of the action density $N$ and of the nonlinear transfer function $T_{1,2,3,4}$. The result is

$$\kappa_{40} = \frac{\langle \eta^4 \rangle}{m_0^2} - 3$$ (12)

$$= \mu_4 - 3$$ (13)

$$= \frac{12}{g^2m_0^2} \int \mbox{\rm d}\ensuremath{\vec{k}}_{1,2,3,... ...a_1\omega_2\omega_3\omega_4} \delta_{1+2-3-4} R_r(\Delta\omega,t) N_1 N_2 N_3$$ (14)

where $kappa_{40}$ is the fourth order cumulant of the surface elevation $eta$ and is equivalent to $mu_4-3$, where $mu_4$ is the normalized fourth order moment, kurtosis of the surface elevation. The transfer function $R_r=(1-cos(Deltaomega t))/Delta omega rightarrow mathcal{P}/Delta omega$ for large time t, where $Delta omega=omega_1+omega_2-omega_3-omega_4$ and $mathcal{P}$ denotes the principle value of the integral to avoid singularity in the integral.

For the cross-correlation between $eta$ and $zeta$, using Eq.() it immediately follows that for homogeneous ocean waves indeed there is no correlation between $eta$ and $zeta$. Then,

$$\langle \eta \; \zeta \rangle = -i \left \langle\int_{-\inft... ...ght) \left( a_2e^{i\vec{k}_2.\vec{x}}-c.c\right) \right\rangle$$ (15)

Using the second equation of () this becomes

$$\langle \eta \; \zeta \rangle = -i \int_{-\infty}^{\infty} {\rm d... ...{x}}+ \langle a_2a^*_1 \rangle e^{i(\vec{k}_2-\vec{k}_1).\vec{x}} \right] = 0 !$$ (16)

which vanishes because of the first equation of (). The second cumulant we need is termed $kappa_{22}$ and is defined as

$$\kappa_{22}=\frac{\langle \eta^2 \zeta^2 \rangle}{m_0^2}-1$$ (17)

where it is noted that $langle zeta^2 rangle =langle eta^2 rangle$. Evaluating the RHS of () using the definitions of $eta$ and $zeta$ one finds

$$\kappa_{22} = \frac{1}{3} \kappa_{40},$$ (18)

hence $kappa_{22}$ is, as expected, precisely three times smaller as $kappa_{40}$.