**3. Energy input**

The detailed description of the algorithm for calculation of energy input is given in Ref. [26]. The energy and momentum are transferred from air to water by the surface pressure field and tangent stress. According to the most reasonable theory

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

[30], the Fourier components of surface pressure *p* are connected with those of the surface elevation through the following expression:

$$\mathbf{i}\_{k,l} + \mathbf{i}\mathbf{j}\_{-k,-l} = \frac{\rho\_a}{\rho\_w} \left(\beta\_{k,l} + \mathbf{i}\beta\_{-k,-l}\right) (h\_{k,l} + \mathbf{i}h\_{-k,-l}),\tag{14}$$

where *hk*,*l*, *h*�*k*,�*l*, *βk*,*l*, *β*�*k*,�*<sup>l</sup>* are real and imaginary parts of elevation *η*, and the so-called *β*-function, *ρa=ρw*, is the ratio of air and water densities. Both *β* coefficients are the functions of the nondimensional frequency

$$
\boldsymbol{\Omega} = \boldsymbol{\alpha} \, \mathbf{U}/\mathbf{g},\tag{15}
$$

that characterizes the ratio of wind velocity to phase velocity of *ck*:

$$
\Omega = U / c\_k \tag{16}
$$

Since the supplying of wave with the energy and momentum occurs in a layer whose height is proportional to the wave length, it is reasonable to suggest that the reference height for the wind velocity should be different for a different virtual wave length (distance *λk=* cos *θ<sup>i</sup>* between the wave peaks in wind direction; the index *i* denotes a direction of mode). The wind velocity can be found by interpolation or extrapolation to the level:

$$z\_{i,k} = \mathbf{0}.\mathbf{5}\lambda\_k / \cos\theta\_i\tag{17}$$

The definition of Ω*<sup>к</sup>* should take into account the angle *θ<sup>i</sup>* between the vector *U* and the direction of wave mode. Finally, the virtual nondimensional frequency takes the form:

$$\Omega\_{i,k} = a\flat\_k \cos\theta\_i U(\mathbf{z}\_k) / \mathbf{g} = \cos\theta\_i U(\mathbf{z}\_k) / c\_k \tag{18}$$

where *ck* ¼ *g=ω<sup>k</sup>* is the phase velocity of *k*th mode.

For experimental derivation of the shape of *β*-function, it is necessary to simultaneously measure the wave surface elevation and nonstatic pressure on the surface [31–35]. The data obtained in this way allowed constructing an imaginary part of *β*function used in some versions of the wave forecasting models [36]. The data on experimental *β*-function are compared in Ref. [4]. The values of *β* within the interval 0ð Þ < Ω <10 differ by decimal orders. Hence, the question arises: in what way, using such a different input, the spectral models provide a reasonable agreement with the observations. The answer is very simple: the researchers have the possibility to modify the parameterization of dissipation. Despite the hundreds of papers, the knowledge on dissipation is even poorer than the knowledge on the energy input. Finally, only the sum of those source terms regulates the growth of total wave energy. Such situation is far from being perfect since the energy input and dissipation have totally different spectral properties.

The second way of the *β*-function evaluation is based on the results of numerical investigations of the statistical structure of the boundary layer above waves with the use of Reynolds equations and an appropriate closure scheme. In general, this method works so well that many problems in the technical fluid mechanics are often solved not experimentally but by using the numerical models [37, 38]. This method was being developed beginning from Refs. [39, 40] and followed by Refs. [41–43]. The results were implemented in the WAVEWATCH model, i.e., the third-generation wave forecast model [44], and thoroughly validated against the experimental data in the course of developing WAVEWATCH-III [45]. Most of the schemes for the calculations of *β*-function consider a relatively narrow interval of the nondimensional frequencies Ω. In the current work, the range of frequencies covers the interval 0ð Þ < Ω <10 , and occasionally, the values of Ω >10 can appear.

The most reliable data on *β*-function are concentrated in the interval �10 < Ω <10 (the negative values of Ω correspond to the wave modes running against wind). In the current calculations, the modes running against wind are absent. The function *β* can be approximated by the formulas:

$$\beta\_i = \begin{cases} \beta\_0 + a\_0(\boldsymbol{\Omega} - \boldsymbol{\Omega}\_0) + a\_1(\boldsymbol{\Omega} - \boldsymbol{\Omega}\_0)^2 & \boldsymbol{\Omega} > \boldsymbol{\Omega}\_0 \\ \beta\_0 - a\_0(\boldsymbol{\Omega} - \boldsymbol{\Omega}\_0) + a\_1(\boldsymbol{\Omega} - \boldsymbol{\Omega}\_0)^2 & \boldsymbol{\Omega} < \boldsymbol{\Omega}\_0 \end{cases},\tag{19}$$

$$\beta\_r = \begin{cases} \beta\_1 + a\_3(\boldsymbol{\Omega} - \boldsymbol{\Omega}\_2) & \boldsymbol{\Omega} < \boldsymbol{\Omega}\_2 \\\ a\_2(\boldsymbol{\Omega} - \boldsymbol{\Omega}\_1)^2 & \boldsymbol{\Omega}\_2 < \boldsymbol{\Omega} < \boldsymbol{\Omega}\_3 \\\ \beta\_1 - a\_3(\boldsymbol{\Omega} - \boldsymbol{\Omega}\_3) & \boldsymbol{\Omega} > \boldsymbol{\Omega}\_3 \end{cases} \tag{20}$$

where Ω<sup>0</sup> ¼ 0*:*355, Ω<sup>1</sup> ¼ 1*:*20, Ω<sup>2</sup> ¼ �18*:*8, Ω<sup>3</sup> ¼ 21*:*2, *a*<sup>0</sup> ¼ 0*:*0228, *a*<sup>1</sup> ¼ 0*:*0948, *a*<sup>2</sup> ¼ �0*:*372, *a*<sup>3</sup> ¼ 14*:*8, *β*<sup>0</sup> ¼ �0*:*02, *β*<sup>1</sup> ¼ �148*:*0.

The wind velocity remains constant throughout the integration. The values of Ω for other wave numbers are calculated by assuming that the wind profile is logarithmic.

Note that the formulation of wind and waves interaction can be significantly improved by coupling the wave model with the 1-D Wave Boundary Layer model [4]. The next step can be the coupling of wave model with the 3-D model of WBL based on the closure schemes or LES model (see Ref. [46]).
