**1. Introduction**

The development of waves under the action of wind is a process that is difficult to simulate since surface waves are very conservative and their energy changes over hundreds and thousands of periods. This is why the most popular method is spectral modeling based on the averaged over phase equations for spectral energy. In this approach, the waves as physical objects are actually absent since the evolution of spectral distribution of the wave energy is simulated. The description of input and dissipation in this approach is not directly connected with the formulation of the problem, but it is rather adopted from other branches of the wave theory where waves are the objects of investigation. However, the spectral approach was found to be the only method capable to describe the space and time evolution of wave field in the ocean. The phase resolving models (or 'direct' models) designed for reproducing waves themselves cannot compete with spectral models since such models typically can reproduce the evolution of just several thousands of large waves. Nevertheless, the direct wave modeling plays an ever-increasing role in the geophysical fluid dynamics because it gives the possibility to investigate the processes that cannot be reproduced by spectral models.

The spectral model assumes that wave field consists of a superposition of linear waves with random phases and arbitrary angle distribution. Being converted to a physical wave field, it looks unreal because real waves usually have prolonged smooth troughs and sharp peaks. Such shape suggests that the waves are similar to Stokes waves. For any given wave spectrum, the wave field can be represented as a superposition of linear Fourier modes with random phases [1]. It can be represented also as a superposition of Stokes modes. The calculations of statistical characteristics for both wave fields show that they are nearly identical. However, such conclusion could be made with no calculations because typical steepness of sea waves at reasonable spectral resolution is of the order of 0.01–0.05, so all the amplitudes of Stokes modes starting from the second one are small. It follows that the specific shape of sea waves is a dynamic property.

The breaking is usually initiated in the vicinity of wave peaks, so the breaking parameterization describes the processes, which are, in principle, impossible in a linear wave field. The breaking is concentrated in the separated narrow intervals; an instantaneous spectrum is discrete and is shifted to high frequencies. The spectral description of the isolated extreme waves is also impossible. The Fourier transform of a large-scale wave field including separated large waves does not provide any indications of their appearance.

The input energy to waves is based on the assumption that each mode induces a pressure mode with a certain amplitude and phase. In reality, the pressure field is quite complicated and not directly connected with the surface elevation because of the systematic separation of air flow behind wave crests (which was shown experimentally in Refs. [2, 3] with the coupled wind-wave model). There are many other complications in the coupled wind-wave dynamics [4] including the processes connected with sprays, bubbles, foam, and structure of the high-frequency wave spectrum.

Most (but not all) of the processes mentioned above can be investigated using the numerical modeling that is a perfect instrument for development of parameterization of physical processes for spectral wave models.

The phase-resolving models can completely replace the spectral models for direct simulation of wave regimes of small water basins, for example, port harbors (see Refs. [5, 6]). Other approaches of direct modeling are discussed in Refs. [7, 8].

Over the past decades, a big volume of papers devoted to the numerical methods developed for investigation of wave processes has been published. The most advanced among them are Finite Difference Method [5, 6], Finite Volume Method [9], Finite Element Method [10, 11], Boundary (Integral) Element Method [12], Spectral HOS Methods [13–17], the Smoothed Particle Hydrodynamics Method [18], Large Eddy Simulation Method (LES) [19, 20], Moving Particle Semi-Implicit Method [21], Constrained Interpolation Profile Method [22], and Method of Fundamental Solutions [23]. Most of the models were designed for engineering application handling such processes as overturning waves, broken waves, waves generated by landslides, freak waves, solitary waves, tsunamis, violent sloshing waves, interaction of extreme waves with beaches, as well as interaction of steep waves with fixed and different floating structures. The wave models designed for engineering applications seem to be more advanced than the models for pure

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

geophysical research. However, as a rule, the engineering models pay little attention to description of physical processes that are responsible for a long-term evolution of wave spectrum. A more detailed review of direct numerical models is given in Ref. [8].

Until recently, the direct modeling was used for reproduction of a quasistationary wave regime when wave spectrum does not change significantly. An example of direct numerical modeling of surface wave evolution is given in Ref. [24] where the development of wave field was calculated by using of a twodimensional model based on full potential equations written in the conformal coordinates. A model included the algorithms for parameterization of the input and dissipation of energy (a description of similar algorithms is given below). The model successfully reproduced the evolution of wave spectrum under the action of wind. That model was a prototype of 3-D model, because being very fast it was convenient for development of the physical process parameterization. However, the strictly one-dimensional (unidirected) waves are not quite realistic since the unidirected waves in the presence of the small-amplitude perturbations relatively quickly turn into the two-dimensional wave field [25]. Hence, the full problem of wave evolution should be formulated on the basis of three-dimensional equations. Such 3-D calculations were done by Chalikov [26]. The model included parameterization of the main physical processes: input energy, different types of dissipation, and transformation of spectrum due to the nonlinear interaction. The last process does not require any parameterization because the nonlinearity is described with equations. The model used a relatively poor resolution (1024 � 512 nodes in *x* and *y* directions). However, the calculations reproduced the evolution of wave spectrum and the spectra of the main physical processes such as input and dissipation of energy and nonlinear interactions. As long as we know, it was the first attempt to reproduce the development of waves based on full three-dimensional equations with a direct solution of 3-D equation for the velocity potential. The current paper is devoted to development of the method, including the tuning and modifications of the algorithm, the increase of resolution, and the integration for longer periods. The most important but pure technical modifications were introduced in the numerical scheme for Poisson equation for the velocity potential. The algorithms for calculation of the input and dissipation remain nearly the same, but the numerical parameters in those schemes were changed to achieve a better agreement with the rate of spectrum evolution given by JONSWAP approximation.
