**6. Statistical properties of wave field**

The phase-resolving modeling requires a higher computer capacity for calculations of any statistical characteristics of sea waves. In the course of simulations, 1.200 two-dimensional fields of the elevation and surface potential with the size 2048 � 1024 points were recorded. Following the solution of the 3-D equation for the velocity potential, these data allow us to reproduce any kinematic and dynamic characteristics of the three-dimensional structure of waves.

The most important statistical characteristics of wave field are mean *Hs*, variance *V*, skewness *Sk,* and kurtosis *Ku* calculated by the averaging over each of 1200 fields:

$$\mathcal{V} = \overline{\left(\eta - \overline{\eta}\right)^2}, \quad \text{Sk} = \overline{\left(\eta - \overline{\eta}\right)^3} V^{-3/2}, \quad \text{K} \\ \mu = \overline{\left(\eta - \overline{\eta}\right)^4} V^{-2} - \text{3.} \tag{38}$$

The evolution of these characteristics in time is shown in **Figure 9**.

The volume of the domain characterized by *η* is preserved with the accuracy of the order of 10�8. The variance *V* is the potential energy that is growing up to the saturation. When the wave field is a superposition of a large number of linear waves, both the skewness and kurtosis are equal to zero. The skewness *S* characterizes asymmetry of the probability distribution indicating that the positive values of *η* are larger than the negative ones, then *S*> 0. The kurtosis *Ku* is positive if the crests are sharper and the troughs are smoother than in the case of linear waves.

The probabilities of the geometrical characteristics (elevation, first and second derivatives over *x*) are shown in **Figure 10**. The elevation *Z* (normalized by the significant wave height) is characterized by asymmetry: the heights of waves are significantly larger than the depths of troughs, that is, the wave field is closer to the superposition of Stokes waves than to that of the harmonic modes. The distribution of slopes exhibits horizontal asymmetry: the negative slopes are larger than the

**Figure 9.**

*Evolution of the statistical characteristics of elevation: (1) mean value η, (2) variance V, (3) skewness* Sk, *and (4) kurtosis* Ku*.*

*High-Resolution Numerical Simulation of Surface Wave Development under the Action of Wind DOI: http://dx.doi.org/10.5772/intechopen.92262*

positive ones, that is, the waves, on the average, are inclined in the direction of movements. The second derivative (curvilinearity) has the most striking tendency for asymmetry: the negative values corresponding to the sharpness of crests are much larger by absolute value than the positive values corresponding to the curvilinearity of troughs. This property of curvilinearity was used for the parameterizing of breaking. The limit value *Zxx* ¼ �50 was used as a criterion for the initiating of breaking (see Eq. (26)).

The probability for three components of the surface velocity is given in **Figure 11**. The distributions of the vertical and transverse components of velocity are symmetrical. For a horizontal component, the values of positive fluctuations are considerably larger than the negative fluctuations. This effect cannot be explained by the influence of Stokes drift, which value for those specific conditions does not exceed 10�<sup>3</sup> *:* The asymmetry of the probability distribution for the *u*-components is definitely connected with the asymmetry of the probability distribution for inclinations of surface (**Figure 10**, panel 2).

The number of extreme waves with a high crest *Zc=Hs* >1*:*2 is shown in **Figure 12**. Because such wave is not presented in each of the wave fields, the picture looks as discrete bars of different heights. The total number of values *Z=Hs* >1*:*2 is 17.214. The formally calculated probability of the values equals 0*:*<sup>67</sup> � <sup>10</sup>�<sup>5</sup> . Note that the data on the probability of wave height contain uncertainty because it is not always clear which event should be considered as a single freak wave. The straightforward way suggests calculation of a portion of all the records including freak waves, out of the total volume of the data.

However, some of the records can belong to the single moving freak waves. The cause of this uncertainty is the absence of a strict definition of freak wave being either a case or a process. The number of extreme waves grows with development of wave field.

The integral probability of the total wave height *Ztc=Hs*, the wave height above mean level *Zc=Hs*, and the depth of trough *Zt=Hs* are shown in **Figure 13**.

Thin lines show that *Ztc=Hs* ¼ 2 correspond approximately to *Zc=Hs* ¼ 1*:*2 and �*Zt=Hs* ¼ �*:*86. It is worth to remind that here the nondimensional 'extreme' waves are considered. The true extreme waves are the product of the real wave field. The probability of real extreme waves can be estimated by multiplying the probability of the nondimensional wave by the probability of significant wave height.

The statistical connection between the total wave heights, crest heights, and trough depths is shown in **Figure 14**.

### **Figure 10.**

*Geometric characteristics of elevation: (1) probability of elevation P Z*ð Þ*; (2) probability of slopes P Z*ð Þ*<sup>x</sup> ; and (3) probability of curvilinearity P Z*ð Þ *xx .*

### **Figure 11.**

*Probability of the longitudinal u, transverse v, and vertical w components of the surface velocity calculated for the last of nine periods corresponding approximately to the quasi-stationary regime.*

### **Figure 12.**

*The number of points where the nondimensional height Zc=Hs exceeds 1.2. The total number of values is* <sup>2</sup>*:*<sup>57</sup> 109*.*

### **Figure 13.**

*The cumulative probability of crest-to-trough wave height Ztc=Hs (curve 3); crest height Zc=Hs (curve 2); and trough depth Zt=Hs (curve 3). The number of points in each filed is equal to* 2048 1024*. The number of fields is 1200.*

*High-Resolution Numerical Simulation of Surface Wave Development under the Action of Wind DOI: http://dx.doi.org/10.5772/intechopen.92262*

The dependences between these characteristics can be approximated by the formulas:

$$\begin{aligned} \tilde{Z}\_{\text{c}} &= \text{-0.105} + \text{0.626} \tilde{Z}\_{\text{tc}} + \text{0.015} \tilde{Z}\_{\text{tc}}^2 \\ \tilde{Z}\_{\text{t}} &= \text{-0.105-0.374} \tilde{Z}\_{\text{tc}} + \text{0.015} \tilde{Z}\_{\text{tc}}^2 \end{aligned} \tag{39}$$

where the tilde denotes the normalizing by significant wave height *Hs*. Note that the first and third coefficients in (39) turned out to be a match. The correlation coefficient between *<sup>Z</sup>*~*<sup>t</sup>* and *<sup>Z</sup>*~*<sup>c</sup>* is �0*:*354, while between *<sup>Z</sup>*~*<sup>t</sup>* and <sup>Z</sup>~*tc*, it is �0.721, and between *Z*~*<sup>c</sup>* and Z~*tc*, it is 0.903, that is, the correlation between the full wave height *Z*~*tc* and the wave height above mean level *Z*~*<sup>c</sup>* is so high that *Z*~*<sup>c</sup>* can be used for identification of extreme waves.

The last characteristics that we consider here is the angle distribution of the spectral density. This characteristic can be described by the function ϒ *ω=ω<sup>p</sup>* (see Ref. [60]).

**Figure 14.** *Dependence of crest height Zc=Hs (top section) and depth trough Zt=Hs(bottom section) on the total wave height Ztc=Hs.*

$$\Upsilon = \frac{\int \mathcal{S}(\boldsymbol{\alpha}, \boldsymbol{\psi}) | \boldsymbol{\theta} | d \boldsymbol{\alpha} d \boldsymbol{\psi}}{\int \mathcal{S}(\boldsymbol{\alpha}, \boldsymbol{\psi}) d \boldsymbol{\alpha} d \boldsymbol{\psi}} \tag{40}$$

where the integrals are taken over the domain 0ð Þ <*ω*< *ω<sup>c</sup>* f g ,ð Þ �*π=*2 <*ψ* < *π=*2 . The value ϒ is weighted by the absolute spectrum value of wave direction. The wave spectra as the functions of frequency *ω* normalized by peak frequency *ω<sup>p</sup>* for the first seven periods are shown in the upper panel of **Figure 15**.

The function ϒ *ω=ω<sup>p</sup>* � � calculated for the same spectra is given in the bottom panel. As seen, the ϒ curves corresponding to different wave ages are close to each other. All of them have a sharp maximum at the frequencies below the spectral peak, a well-pronounced minimum in the spectral peak, and a relatively slow

### **Figure 15.**

*The shape of wave spectrum as a function of the nondimensional frequency ω=ω<sup>p</sup> (top panel) and a function* ϒ *(Eq. (40); ω<sup>p</sup> is the frequency in the spectral peak.*

*High-Resolution Numerical Simulation of Surface Wave Development under the Action of Wind DOI: http://dx.doi.org/10.5772/intechopen.92262*

growth above the spectral peak. The decrease of ϒ at high frequencies is probably caused by the high-frequency dumping. The angle distribution was investigated in Refs. [56–60]. The approximations of ϒ *ω=ω<sup>p</sup>* from the different sources collected in Ref. [61] show considerable scatter, but the general features are quite similar to those calculated in the current work. Note that the spectrum has undergone a long development; hence, the characteristics presented in **Figure 14** were produced by the numerical model itself.
