### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The 2-D Model
- 3. The 3-D Model
- 4. Comparison With Classical Nonlinear Theories
- 5. Comparison With Experimental Data
- 6. Conclusion
- Acknowledgments
- References

[1] We investigate the statistical properties of a three-dimensional simple and versatile model for weakly nonlinear gravity waves in infinite depth, referred to as the “choppy wave model” (CWM). This model is analytically tractable, numerically efficient, and robust to the inclusion of high frequencies. It is based on horizontal rather than vertical local displacement of a linear surface and is a priori not restricted to large wavelengths. Under the assumption of space and time stationarity, we establish the complete first- and second-order statistical properties of surface random elevations and slopes for long-crested as well as fully two-dimensional surfaces, and we provide some characteristics of the surface variation rate and frequency spectrum. We establish a relationship between the so-called “dressed spectrum,” which is the enriched wave number spectrum of the nonlinear surface, and the “undressed” one, which is the spectrum of the underlying linear surface. The obtained results compare favorably with other classical analytical nonlinear theories. The slope statistics are further found to exhibit non-Gaussian peakedness characteristics. Compared to observations, the measured non-Gaussian omnidirectional slope statistics can only be explained by non-Gaussian effects and are consistently approached by the CWM.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The 2-D Model
- 3. The 3-D Model
- 4. Comparison With Classical Nonlinear Theories
- 5. Comparison With Experimental Data
- 6. Conclusion
- Acknowledgments
- References

[2] The development of fully consistent inversions of sea surface short wave characteristics via the ever increasing capabilities (radiometric precision, spatial resolution) of remote sensing measurements has considerably advanced. Yet, difficulties remain, mostly associated to stringent requirements to have adequate understandings and means to describe very precisely the sea surface statistical properties in relation to surface wave dynamics. The simplest linear superposition and Gaussian models remain in common use. Such models provide insight and are often accurate enough for many practical purposes. Yet, common visual inspections of natural ocean surface waves often reveal geometrical asymmetries. Namely, when the steepness of a wave locally increases, its crest becomes sharper and its trough flatter. Harmonic phase couplings occur, and an ocean surface wave field can become rapidly a non-Gaussian random process. For remote sensing applications and model developments, the statistical description of random nonlinear gravity waves is then certainly not straightforward, but must be taken into account to improve uses and interpretation of measurements, e.g., to correct for the sea state bias in altimetry, to explain the upwind/downwind asymmetry of the radar cross section or to interpret the role of fast scatterer in Doppler spectra.

[3] As usually described, nonlinear surface gravity waves are generally prescribed in the context of the potential flow of an ideal fluid. For small wave steepness, the resulting nonlinear evolution equations can first been solved by means of a perturbation expansion [*Tick*, 1959]. This approach consists in finding iteratively a perturbative solution of the equations of motion for both the surface elevation and the velocity potential, by matching the boundary conditions at the bottom and at the free surface [*Hasselmann*, 1962; *Longuet-Higgins*, 1963; *Weber and Barrick*, 1977]. Following an other approach, *Zakharov* [1968] showed that the wave height and velocity potential evaluated on the free surface are canonically conjugate variables. This helps to uniquely formulate the water wave equations as a Hamiltonian system. For water waves, the Hamiltonian is the total energy *E* of the fluid. The Hamiltonian approach is based on operators expansions technique [*Zakharov*, 1968; *Creamer et al.*, 1989; *Watson and West*, 1975; *West et al.*, 1987; *Fructus et al.*, 2005], albeit using truncated Hamiltonian. We refer to *Elfouhaily* [2000] for a comparison and discussion between the two approaches. For two-dimensional water waves, where the free surface evolves as a function of one variable in space, effective methods have been improved and include conformal mapping variables [*Zakharov et al.*, 2002; *Ruban*, 2005; *Chalikov and Sheinin*, 2005]. A recent review on numerical methods for irrotational waves can be found in the paper by *Dias and Bridges* [2006]. For the three-dimensional problem, one loses the possibility to employ complex analysis, except to still consider a quasi-planar approximation, i.e., very long crested waves. Consequently, for the general problem, the perturbative technique has the advantage of simplicity, but remains essentially a low-frequency expansion and produces some nonphysical effects at higher frequencies, such as the divergence of the second-order spectrum. The Hamiltonian approach will be capable of handling stronger nonlinearities but is more tedious, remains essentially numerical and does not provide explicit statistical formulas. Finally, a Lagrangian description of surface wave motion may be more appropriate to describe steep waves [*Chalikov and Sheinin*, 2005]. In such a context, the Gerstner wave [*Gerstner*, 1809] is a first well-known exact solution for rotational waves in deep water, and *Stokes* [1847] derived a second-order Lagrangian approximation for irrotational waves leading to a well-known and observed net mass transport, the Stokes drift phenomenon, in the direction of the wave propagation.

[4] The aim of this paper is to build on this latter simplified phase perturbation methodology to propose a simple, versatile model, that can reproduce the lowest-order nonlinearity of the perturbative expansion but does not suffer from its related shortcomings. This analytical model is certainly not properly new, as it is widely used by the computer graphics community [*Fournier and Reeves*, 1986; J. Tessendorf, unpublished data, 2004] to produce real-time realistic looking sea surfaces. The terminology choppy wave model (henceforth abbreviated to “CWM”) originates from the visual effect imposed by the transformation compared to linear waves. In addition to gravity waves nonlinear interaction, the model can incorporate further physical features such as the horizontal skewness induced by wind action over the waves, an effect that we will not consider in this paper and which will be left for subsequent work.

[5] On the mathematical level, the model identifies completely with the perturbative expansion in Lagrangian coordinates as proposed four decades ago by *Pierson* [1962, 1961]. In the case of a single wave, it coincides with the Gerstner solution and is consistent with the Stokes expansion [*Stokes*, 1880] at third order in slope. Our present contribution is to provide a complete, nontrivial statistical study of this model and a comparison with the classical approaches. As understood, the CWM does not claim to compete with Hamiltonian-based methods and is in fact limited to the lowest-order nonlinearity. Its main strength is to provide a good compromise between simplicity, stability and accuracy. More precisely, it is (1) numerically efficient, as time evolving sample surfaces can be generated by FFT; (2) analytically tractable, as it provides explicit formulas for the first- and second-order point statistics; and (3) robust to the frequency regime, as it is found to be equivalent to the canonical approach [*Creamer et al.*, 1989] at low frequencies while remaining stable at higher frequencies.

[6] In the following we have studied the two- and three-dimensional case pertaining to long-crested or truly two-dimensional sea surfaces, which from now on we will refer to as the 2-D and 3-D case. Since the methodology remains the same in both instances, we have chosen to give a complete exposure of the technique in the 2-D case which is considerably simpler. All the analytical results of the 2-D case (section 2) have their counterpart in the 3-D case (section 3). In the subsequent study, the emphasis will be put on the spatial properties of a “frozen” surface, even through some temporal properties will also be discussed. Using a phase perturbation in the Fourier domain, the nonlinear local transformation simply consists in shifting the horizontal surface coordinates. Starting with a linear, reference surface, assumed to be a second-order Gaussian stationary process in space and time with given power spectrum, the complete first- and second-order properties of the resulting, non-Gaussian, random process is derived and related to the statistics of the reference surface. In particular, the resulting spectrum, which we refer to as dressed, has been related to the reference spectrum, termed undressed, in a way which is found to be very similar to *Weber and Barrick*'s [1977] and *Creamer et al.*'s [1989], but corrects the former and extends the latter to the 3-D case. As well, the sea surface slope statistical description is modified to exhibit a non-Gaussian behavior with a measurable peakedness effect, i.e., an excess of zero and steep slopes. A comparison with recent airborne laser measurements which allows to discriminate the slope statistics of gravity waves from smaller, short gravity, and capillary waves, is presented in section 4. As found, the CWM brings the excess kurtosis of omnidirectional slopes significantly closer to the data.

### 3. The 3-D Model

- Top of page
- Abstract
- 1. Introduction
- 2. The 2-D Model
- 3. The 3-D Model
- 4. Comparison With Classical Nonlinear Theories
- 5. Comparison With Experimental Data
- 6. Conclusion
- Acknowledgments
- References

[24] In the 3-D case, *Pierson* [1961] has provided the solution of the linearized equations of motion for an inviscid irrotational fluid in Lagrangian coordinates. In deep water, the particle positions at the free surface have following parameterization:

where _{j} is a two-dimensional vector, _{j} = **k**_{j}/∣**k**_{j}∣ and **r**_{0} = (*x*_{0}, *y*_{0}) labels the particles at rest on the flat surface. Similarly to the 2-D case, the corresponding surface can be realized through horizontal displacements of a reference, linear, surface:

where **r** = (*x*, *y*) is the horizontal coordinate. The function

is the so-called Riesz transform of the function *h*, and

is its two-dimensional spatial Fourier transform.

#### 3.1. First-Order Properties of the Space Process

[25] The calculations are similar to the 2-D case, although more involved.

[26] Let us introduce the partial and total absolute moments of the spectrum:

[27] Standard calculations lead to the following expression for the characteristic function (43) of elevations:

with Σ_{1} = *σ*_{111}^{4} − *σ*_{201}^{2}*σ*_{021}^{2}. An example of deviation from the normal distribution is shown on Figure 5 for an input linear surface with directional Elfouhaily spectrum [*Elfouhaily et al.*, 1997] at 10 m s^{−1} wind speed.

[28] From the characteristic function, the following moments are easily obtained:

as well as the pdf of elevations:

where as before *P*_{0} is the Gaussian pdf of the linear surface.

[29] We can also derive the skewness (_{3}) and the kurtosis (_{4}) of elevation. The respective values for isotropic spectra are given in parenthesis:

[30] Again, there is a negative skewness and a positive excess of kurtosis, and the msh is diminished by a negligible amount.

[31] We have not been able to calculate explicitly the pdf of slopes _{2}(**z**). However, we could establish the following integral representation, which can be estimated numerically:

with

where ∣*M*∣ denote the determinant of the matrix *M*. Figure 6 displays the pdf of slopes in the upwind and crosswind direction for a directional Elfouhaily spectrum [*Elfouhaily et al.*, 1997]. A comparison is given with the associated Gaussian distribution. The tail of the distribution decreases slower for the CWM and is significantly higher than the Gaussian tail for slope magnitudes beyond 0.5. Again, the slopes larger than some threshold (about 0.7) are not physical and the distribution must be truncated beyond this value.

#### 3.2. Second-Order Properties of the Space Process

[32] As in the 2-D case, we can derive the two-dimensional Fourier transform of the two-point characteristic function on the diagonal, namely:

Operating the change of variable **r** ↦ **r** + **D**(**r**) we obtain:

[33] Here, *J* is the Jacobian matrix:

Discarding the quadratic terms in the Jacobian,

and using standard properties of Gaussian processes [e.g., *Papoulis*, 1965] we obtain after tedious but straightforward calculations the following expression for the functional Ψ:

Here ∇*C* and Δ*C* are the gradient and the Laplacian of the correlation function, respectively. The dependence in the space variable is implicit. The auxiliary functions Φ_{u} and *S*_{1} are defined by:

[34] Using the same technique as in 2-D we obtain for the dressed spectrum:

The calculation of the low-frequency expansion of the dressed spectrum is similar to the 2-D case, leading to:

with **k**″ = **k** − **k**′.

#### 3.3. Undressing the Spectrum

[35] As can be seen on Figure 3, the dressed spectrum has an enhanced curvature with respect to the undressed one. This is natural since the inclusion of bound waves enriches the high-frequency content of the spectrum. In the 3-D case, we might also expect a enhancement of the spreading function at high frequencies through the nonlinear interaction of strongly directive long waves and weakly directive short waves. Now it is the dressed spectrum which is measured experimentally. To generate a nonlinear surface with a preassigned spectrum, it is thus necessary to go through an undressing procedure of the latter. The CWM transformation of the linear, fictitious, surface with undressed spectrum will eventually produce a nonlinear surface with suitable dressed spectrum.

[36] *Soriano et al.* [2006] introduced a simple undressing method assuming a power law form of the high-frequency part of the undressed spectrum. The parameters were fitted in such a way that the dressed spectrum leads to the correct values of the mean square height and slope after the nonlinear transformation proposed by *Creamer et al.* [1989]. *Elfouhaily et al.* [1999] used another method to retrieve the lowest-order cumulants of the nonlinear surface.

[37] The equation (55) can be incorporated in a simple iterative procedure to undress a spectrum with prescribed curvature and spreading function _{target}, _{target}. Assuming a second harmonic azimuthal expansion of the dressed and undressed spectra:

the iterative procedure to find undressed curvature (*B*) and spreading (Δ) functions runs as follows:

with *B*^{(0)} = _{target} and Δ^{(0)} = _{target}. As an example, Figure 7 shows the first few iterates for a fully developed Elfouhaily dressed spectrum by a *U*_{10} = 11 m s^{−1} wind.

#### 3.4. Numerical Surface Generation

##### 3.4.1. Frozen Surface

[38] Sample nonlinear surfaces at a given time can be generated efficiently at the cost of three successive two-dimensional fast Fourier transforms: one for the spectral representation of the linear surface *h*(**r**) and the other two for its Riesz transform **D**(**r**):

where Γ is the prescribed spectrum and ϕ_{ij} are random uniform and independent phases on [0, 2*π*]. The nonlinear surface is parameterized by the points (**r**_{mn} + **D**(**r**_{mn}), *h*(**r**_{mn})). An example is given on Figures 8 and 9with an Elfouhaily directional spectrum at wind *U*_{10} = 15 m s^{−1}. It is a 3 m × 3 m patch of a total 50 m × 50 m sea surface generated with 8192 × 8192 points. The spectrum has been truncated at *k*_{max} = 500 rad m^{−1}, corresponding to a minimal surface wavelength of 1 cm, and the surface is sampled regularly at the Shannon frequency 2*k*_{max}. Since the operation is based on abscissa displacements, the resulting surface is given on a non regular grid. As can be seen on the encircled region of the plot, the crests of the CWM are sharpened while those of the linear surface are smoother.

##### 3.4.2. Time Evolution

[39] The time evolution of nonlinear surface is a challenging issue, essentially because of the absence of a simple and well-defined dispersion relation. However, CWM surface at a given time is obtained by the same transformation (1) of a time-dependent linear surface. Therefore, it suffices to let the reference linear surface evolve and to perform the local transformation at the current time. If we, in addition, assume a fully developed time-independent spectrum, the evolution of the linear surface is simply obtained by use of the gravity-waves dispersion relation *ω*^{2} = *g*∣**k**∣, and amounts to change the original phases ϕ_{ij} by an additional factor −*ω*_{ij}*t* = ±*t* in the FFT (60), depending on the travel direction of the waves.

[40] Figure 10 exemplifies the time evolution of a 2-D linear surface with one-sided time spectrum and the corresponding nonlinear surface. The undressed spectrum was chosen to be the fully developed Elfouhaily spectrum by a wind *U*_{10} = 3 m s^{−1}. The sample surface was taken to be 64 m long, with extreme frequencies *k*_{min} = 1.10^{−3} rad m^{−1} and *k*_{max} = 100 rad m^{−1} and a sampling of 4096 points. The evolution of 4 m patch is represented for both the linear and CWM surface, with a time step Δ*t* = 0.1 s.

### 4. Comparison With Classical Nonlinear Theories

- Top of page
- Abstract
- 1. Introduction
- 2. The 2-D Model
- 3. The 3-D Model
- 4. Comparison With Classical Nonlinear Theories
- 5. Comparison With Experimental Data
- 6. Conclusion
- Acknowledgments
- References

[41] As mentioned in the Introduction, a certain number of fully nonlinear and numerically efficient solutions of potential flows have been developed in recent years. This makes it, in principle, possible to validate approximate theories. In practice, comparing the latter with various exact numerical solutions raises some difficulties, such as the lack of control of the final spectrum in an evolving nonlinear solution, the sensibility to the initial state or the relevance of sample surfaces comparisons. However, fast numerical schemes now allow the derivation of statistical properties of the surface through the use of extensive Monte-Carlo computations, especially for one-dimensional surfaces [e.g., *Chalikov*, 2005; *Toffoli et al.*, 2008]. Nevertheless, going through a validation procedure by systematic comparisons of relevant statistical quantities is an important work which goes far beyond the scope of this paper and is left for further investigation.

#### 4.1. Stokes Expansion

[42] A perturbative expansion of the implicit function can be obtained in the case of small displacements. We will here limit the discussion to the 2-D case. Supposed that the profile is obtained by dilation of a single dimensionless template *h*_{0}:

where *a* and *s* = *Ka* are height and slope parameters, respectively. Then easy algebra leads to the following expansion, correct at second order in slope

[43] In the case of a single wave *h*(*x*) = −*a*cos(*Kx*), this perturbative series can be compared with a Stokes expansion. After rearrangement of the different terms in (62) we obtain:

which coincides with a Stokes expansion at third order in slope. Note, however, that the CWM is more general than a mere superposition of Stokes waves, as frequency and phase coupling between the different modes comes in play through the nonlinear terms of the spatial expansion.

#### 4.2. Longuet-Higgins Theory

[44] The classical approach [*Hasselmann*, 1962; *Longuet-Higgins*, 1963] to the nonlinear theory of gravity waves is to seek both the elevation *h* and velocity potential Φ in a perturbation series,

where the first terms are given by the linear spectral representation of a Gaussian process,

and the following terms in the expansion involve nth-order multiplicative combinations of these linear spectral components. The perturbative expansions of elevation and velocity potential are identified simultaneously by injecting the successive Fourier expansions in the equations of motion. The leading, quadratic, nonlinear term for elevation was provided by *Longuet-Higgins* [1963] in the form (the factor 1/2 in the kernels *K*_{ij} and *K*′_{ij} is missing in the original paper by *Longuet-Higgins* [1963], as was later acknowledged by the author himself [see *Srokosz and Longuet-Higgins*, 1986]:

where

To simplify the comparison we will again concentrate on the 2-D case. For long-crested waves we may operate the substitution **k**_{i} · **k**_{j} sign(*k*_{i}*k*_{j})∣*k*_{i}*k*_{j}∣, leading to simplified expressions of the kernels:

Now, in the perturbative expansion (62) after the CWM we have:

Since the former process (68) is centered while the latter (72) is not, we must rather compare with a recentered right-hand side:

where *δ*_{ij} is the Kronecker symbol. This last expression is resembling but not identical to the second-order correction (68) of Longuet-Higgins. Note, however, that the respective kernels coincide on the diagonal.

[45] This makes the CWM consistent with Longuet-Higgins theory, at least for narrow spectra. Passing to the limit of infinitely many spectral components, *Longuet-Higgins* [1963] could also derive general formula for the first few cumulants of the second-order nonlinear surface. The mean and RMS of elevation at second-order are found to be identical to those of linear process and the third cumulant turns out to be nonvanishing (in the paper by *Longuet-Higgins* [1963], the following expression is given for one-sided spectrum only):

where as usual Γ is the spectrum of the linear process *h*^{(1)}. The corresponding skewness,

has opposite sign with respect to the skewness (14) derived in the framework of the CWM. However, the absolute values of these quantities are too small for their sign to be meaningful. A quick estimation can be performed with a power law Phillips omnidirectional spectrum, Γ(*k*) = 0.0025 × ∣*k*∣^{−3}, for ∣*k*∣ > *k*_{peak}, in which case we the skewness predicted by the two models are found quasi-independent of the peak wave number, *λ*_{3} ≃ 0.015 for the Longuet-Higgins theory and *λ*_{3} ≃ −3.10^{−6} for the CWM. Note that some recent numerical experiments for one-dimensional surfaces after the so-called ChSh method [*Chalikov*, 2005] show a unambiguously positive skewness, so that the precision of the CWM might no be sufficient to capture the latter correctly.

#### 4.3. Weber and Barrick Theory

[46] In their 1977 companion papers *Weber and Barrick* [1977] and *Barrick and Weber* [1977] revisited the nonlinear theory for random seas with continuous spectra. The adopted methodology is essentially the same as *Longuet-Higgins* [1963] but the perturbative expansion is operated on the continuous Fourier components of the surface. The time-evolving surface elevation is sought in the form:

and a perturbative expansion is operated on the Fourier components:

The first-order term correspond to free waves propagating with the gravity wave dispersion relation *ω* = (the “linear” term),

while the second-order term is found to be:

Here the kernel *A* is given by:

where Ω_{12}^{+} is given by (70), and *A* = 0 whenever **k**_{2} = −**k**_{1} and *ω*_{2} = −*ω*_{1}. Even through this is not obvious at first sight, this kernel is consistent with Longuet-Higgins perturbative theory since:

To make a comparison with the CWM, we will consider the surface frozen at a given time, say *t* = 0, in which case the spatial process *h*(**r**) = *h*(**r**, 0) at first- and second-order can be written:

with

and

Denoting as usual Γ and the first- and higher-order wave number spectra,

we can easily establish the following relationship:

with

In the 2-D case, the kernel reduces to:

and thus

which is the first integrand appearing in the low-frequency expansion after the CWM (29). As discussed later by *Creamer et al.* [1989], retaining this sole term leads to a divergence of the second-order correction at higher wave numbers. This is explained by the fact that the second-order spectrum (fourth order in surface amplitude) is not complete, since it misses the contribution of the *h*_{1} × *h*_{3} term.

#### 4.4. Creamer Theory

[47] In order to generate nonlinear sea surfaces *Creamer et al.* [1989] uses a canonical transformation of physical variables (surface elevation and potential) in order to improve the accuracy of the Hamiltonian expansion. This transformation has the same domain of validity of the CWM in a sense that it can be used for surface gravity waves and reproduces the effects of the lowest-order nonlinearities for the first-order development of the transformation. The 3-D formulation remains, however, quite involved and its numerical implementation require further approximations [*Soriano et al.*, 2006]. In the 2-D Creamer model, the nonlinear process is given by:

where the corrective term *δh* is expressed by its Fourier transform

and *D* is the Hilbert transform of *h*. This expression is unpractical for further analytical investigation. However, at low frequencies (*kD* ≪ 1) the exponential may be expanded,

leading to the lowest-order approximation for the dressed spectrum [*Creamer et al.*, 1989, equation 6.11]:

This expression is similar to the first integral in the low-frequency expansion (29). Figure 11 displays a comparison of *Creamer et al.*'s [1989], *Weber and Barrick*'s [1977] and CWM low-frequency expansion, for an omnidirectional *k*^{−3} spectrum with exponential cutoff at peak frequency *kp* = 0.7 rad m^{−1} (corresponding to a wind of 3 m s^{−1}) and upper limit *k*_{u} = 120 rad m^{−1}. The undressed (linear) spectrum is shown together with the corrections brought by the dressed spectrum. *Creamer et al.*'s [1989] and CWM expansions are extremely close at low frequency but CWM eventually diverges at higher frequency (*k* > 100 rad m^{−1}). *Weber and Barrick* [1977] diverge very early (*k* > 3 rad m^{−1}) and is slightly higher than CWM and *Creamer et al.*'s [1989] corrections.

### 5. Comparison With Experimental Data

- Top of page
- Abstract
- 1. Introduction
- 2. The 2-D Model
- 3. The 3-D Model
- 4. Comparison With Classical Nonlinear Theories
- 5. Comparison With Experimental Data
- 6. Conclusion
- Acknowledgments
- References

[48] The reference data basis for the sea wave slope distribution is the optically derived measurement of *Cox and Munk* [1954], which has been used to calibrate many models of the literature. Since the CWM in its current state is restricted to gravity waves only, it cannot describe the scales smaller than, say 5 cm, and the related slopes, making the comparison with Cox and Munk data irrelevant. Instead, we will resort to a recent airborne campaign [*Vandemark et al.*, 2004], which has provided laser measurements of the omnidirectional slope statistics of long gravity waves. This amounts to filter out in the slope statistics the contribution of wavelengths smaller than about 2 m and renders the comparison with the CWM possible. The main outcome of this study was an elevated kurtosis for the omnidirectional slope, a result that can be put on the account of either the strong directionality of the wavefield or its non-Gaussian character. We will investigate the respective contributions of these two effects in the framework of the CWM. Denote *P*_{2-omni}(*S*) the omnidirectional slope distribution, that is the distribution of absolute magnitude of slope *S* = ∣∇**h**∣. For an isotropic Gaussian distribution with variance *σ*_{2}^{2}, this is a Rayleigh distribution with parameter *σ*_{2},

whose kurtosis is *λ*′_{4} = 3.245. For a directional Gaussian slope distribution with upwind/crosswind mean square slope (mss) ratio *ρ*^{2} = *σ*_{200}^{2}/*σ*_{020}^{2} and total mss *σ*_{2}^{2}, the nth moments *M*_{n} = 〈*S*^{n}*P*_{omni}(*S*)〉 of the omnidirectional slope distributions are found to be:

where *R*_{n} is the nth moment of the normalized Rayleigh distribution (*S* exp(−*S*^{2}/2)). The variation of kurtosis with the directionality parameter *ρ* can be estimated numerically. A maximum value *λ*′_{4} = 4.166 is reached at *ρ* = 3, while the minimum kurtosis is obtained at *ρ* = 1 for the Rayleigh distribution (*λ*_{4} = 3.245), It follows that the elevated values of kurtosis reported by *Vandemark et al.* [2004], which ranges from 4.5 to 6, cannot be explained by mere directional effects of the slope distribution.

[49] The kurtosis has been computed as a function of wind speed for both linear and CWM surfaces generated with a directional Elfouhaily (undressed) spectrum. The fourth moment of the theoretical CWM slope distribution is in principle infinite, but the corresponding integral can be shown to have a slow, logarithmic divergence. Therefore, the slope distribution has been truncated to a maximum value of 1.7, corresponding to a steep wave of about 60 degree. For small and moderate winds (*U*_{10} ≤ 12 m s^{−1}), the resulting fourth moment is quite insensitive to the chosen threshold. Furthermore, we have checked that the lack of normalization of the slope distribution after truncation has a negligible impact on the computation of the first cumulants. At higher winds, the slope kurtosis is found to increase slightly with the slope threshold. However, we do not expect the CWM to remain meaningful for steep waves. The simulated excess kurtosis is shown on Figure 12 and compared with recorded data. To reproduce the filtering of small waves slopes realized in the paper by *Vandemark et al.* [2004], the Elfouhaily spectrum has been truncated to a maximum wave number of *k*_{u} = 6 rad m^{−1}. The corresponding surfaces are referred to as “long gravity waves.” The comparison is given with the untruncated gravity waves Elfouhaily spectrum (*k*_{u} = 200 rad m^{−1}). The horizontal line at *γ* = 0.245 is the excess kurtosis of the Rayleigh distribution, obtained for Gaussian isotropic slope distribution. The line at *γ* = 0.7 is the estimation of *Cox and Munk* [1954], which is insensitive to wind and identical for slick and clean surfaces. The inclusion of nonlinearities through the CWM drastically increases the excess kurtosis and brings it to values intermediate between *Vandemark et al.*'s [2004] data and *Cox and Munk*'s [1954] data, while the linear model remains closer to the Rayleigh distribution.

### 6. Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. The 2-D Model
- 3. The 3-D Model
- 4. Comparison With Classical Nonlinear Theories
- 5. Comparison With Experimental Data
- 6. Conclusion
- Acknowledgments
- References

[50] As reported, CWM provides an analytically tractable, numerically efficient solution to approach the geometrical description of nonlinear surface waves. CWM is also robust to the inclusion of high frequencies. CWM explicitly builds on a phase perturbation method to modify the surface coordinates, and statistical properties can be derived. We establish the complete first- and second-order statistical properties of surface elevations and slopes for long-crested as well as fully two-dimensional surfaces. As compared to standard approximation, the CWM is shown to be a reasonably accurate model for weak nonlinear gravity-wave interactions. It is based on the local deformation of a reference Gaussian process and the first few cumulants up to fourth order can be expressed in terms of the underlying Gaussian statistics. Relations between dressed and undressed spectra have been established and found to favorably extend the classical low-frequency formulations of *Weber and Barrick* [1977] and *Creamer et al.* [1989].

[51] As already pointed out [*Elfouhaily et al.*, 1999], it can be crucial to determined the required input undressed spectrum for which the simulated moments remain consistent with a measured spectrum. The CWM can then be used to define an inversion scheme to consistently evaluate the first-order cumulants (elevation skewness, the elevation and slope cross skewness) to evaluate the predicted long wave geometrical contribution to altimeter sea state bias [*Elfouhaily et al.*, 2000; *Vandemark et al.*, 2005].

[52] Moreover, the nonlinear surface wave geometry with shallow troughs and enhanced crests, implies an excess of both zero and steep slope occurrences. As numerically derived, CWM predictions unambiguously confirm that bound harmonics associated to the simplified surface coordinate changes will indeed lead to non negligible surface slope kurtosis. Compared to measurements, CWM is found to help to bridge the differences between a linear Gaussian model and reported large slope kurtosis.

[53] For short gravity waves, the CWM can also be used to heuristically introduce the skewness of individual slopes. These effects can indeed be subsequently incorporated in the model through a generalization of the horizontal displacement **D**(**r**, *t*) to steepen slightly the forward face of individual waves, especially when the local steepness exceeds a threshold value.