The numerical modeling of two-dimensional surface wave development under the action of wind is performed. The model is based on three-dimensional equations of potential motion with a free surface written in a surface-following nonorthogonal curvilinear coordinate system in which depth is counted from a moving surface. A three-dimensional Poisson equation for the velocity potential is solved iteratively. A Fourier transform method, a second-order accuracy approximation of vertical derivatives on a stretched vertical grid and fourth-order Runge–Kutta time stepping are used. Both the input energy to waves and dissipation of wave energy are calculated on the basis of earlier developed and validated algorithms. A one-processor version of the model for PC allows us to simulate an evolution of the wave field with thousands of degrees of freedom over thousands of wave periods. A long-time evolution of a two-dimensional wave structure is illustrated by the spectra of wave surface and the input and output of energy.

The phase-resolving modeling of sea waves is the mathematical modeling of surface waves including explicit simulations of surface elevation and a velocity field evolution. As compared with spectral wave modeling, phase-resolving modeling is more general since it reproduces a real visible physical process and is based on well-formulated full equations. Phase-resolving models usually operate with a large number of degrees of freedom. In general, this method is more complicated and requires more computational resources. The simplest way to model like this is to calculate wave field evolution based on linear equations. Such an approach allows the reproduction of the main effects of the linear wave transformation due to the superposition of wave modes, reflections, refractions, etc. This approach is useful for many technical applications but it cannot reproduce a nonlinear nature of waves and the transformation of wave field due to the nonlinearity. Another example of a relatively simple object is a case of the shallow-water waves. The nonlinearity can be taken into account in the more sophisticated models derived from the fundamental fluid mechanics equations with some simplifications. The most popular approach is based on a nonlinear Schrödinger equation of different orders (see Dysthe, 1979) obtained by expansion of the surface wave displacement. This approach is also used for solving the problem of freak waves. The main advantage of a simplified approach is that it allows the reduction of a three-dimensional (3-D) problem to a two-dimensional one (or 2-D problem to 1-D problem). However, it is not always clear which of the nonrealistic effects is eliminated or included in the model after simplifications. This is why the most general approach being developed over the past years is based on the initial two-dimensional or three-dimensional equations (still potential). All the tasks based on these equations can be divided into two groups: the periodic and nonperiodic problems. An assumption of periodicity considerably simplifies construction of the numerical models, though such formulation can be applied to the cases when the condition of periodicity is acceptable, for example, when domain is considered to be a small part of a large uniform area. For the limited domains with no periodicity the problem becomes more complicated since the Fourier presentation cannot be used directly.

From the point of view of physics, the problem of phase-resolving modeling can be divided into two groups: the adiabatic and nonadiabatic modeling. A simple adiabatic model assumes that the process develops with no input or output of energy. Being not completely free of limitations, such a formulation allows the investigation of the wave motion on the basis of true initial equations. Including the effects of input energy and its dissipation is always connected with the assumptions that generally contradict the assumption of potentiality, i.e., the new terms added to the equations should be referred to as pure phenomenological. This is why the treatment of a nonadiabatic approach is often based on quite different constructions.

All of the phase-resolving models use the methods of computational mathematics and inherit all their advantages and disadvantages; i.e., on the one side, there is the possibility of a detailed description of the processes, and on the other side, there are a bunch of the specific problems connected with the computational stability, space and time resolution. The mathematical modeling produces tremendous volumes of information, the processing of which can be more complicated than the modeling itself.

The phase-resolving wave modeling takes a lot of computer time since it normally uses a surface-following coordinate system, which considerably complicates the equations. The most time-consuming part of the model is an elliptic equation for the velocity potential usually solved with iterations. Luckily, for a two-dimensional problem this trouble is completely eliminated by use of the conformal coordinates, reducing the problem to a one-dimensional system of equations which can be solved with high accuracy (Chalikov and Sheinin, 1998). For a three-dimensional problem, the reduction to a two-dimensional form is evidently impossible; hence, the solution of a 3-D elliptical equation for the velocity potential becomes an essential part of the entire problem. This equation is quite similar to the equation for pressure in a nonpotential problem. It follows that the 3-D Euler equations, being more complicated, can still be solved over the acceptable computer time.

There is a large volume of papers devoted to the numerical methods developed for the investigation of wave processes over the past decades. It includes a finite-difference method (Engsig-Karup et al., 2009, 2012), a finite-volume method (Causon et al., 2010), a finite-element method (Ma and Yan, 2010; Greaves, 2010), a boundary (integral) element method (Grue and Fructus, 2010), and spectral methods (Ducroset et al., 2007, 2012, 2016; Touboul and Kharif, 2010; Bonnefoy et al., 2010). These include a smoothed-particle hydrodynamics method (Dalrymple et al., 2010), a large-eddy simulation (LES) method (Issa et al., 2010; Lubin and Caltagirone, 2010), a moving particle semi-implicit method (Kim et al., 2014), a constrained interpolation profile method (Zhao, 2016), a method of fundamental solutions (Young et al., 2010) and a meshless local Petrov–Galerkin method (Ma and Yan, 2010). A fully nonlinear model should be applied to many problems. Most of the models were designed for engineering applications such as overturning waves, broken waves, waves generated by landslides, freak waves, solitary waves, tsunamis, violent sloshing waves, an interaction of extreme waves with beaches and an interaction of steep waves with the fixed structures or with different floating structures. The references given above make up less than 1 % of the publications on those topics.

A two-dimensional approach (like a conformal method) considers a strongly idealized wave field since even monochromatic waves in the presence of lateral disturbances quickly obtain a two-dimensional structure. The difficulty arising is not a direct result of the increase in the dimension. The fundamental complication is that the problem cannot be reduced to a two-dimensional problem, and even for the case of a double-periodic wave field, the problem of solution of a Laplace-like equation for the velocity potential arises. The majority of the models designed for investigation of the three-dimensional wave dynamics are based on simplified equations such as the second-order perturbation methods in which the higher-order terms are ignored. Overall, it is unclear which effects are missing in such simplified models.

The most sophisticated method is based on the full three-dimensional equations and surface integral formulations (Beale, 2001; Xue et al., 2001; Grilli et al., 2001; Clamond and Grue, 2001; Clamond et al., 2005, 2006; Fructus et al., 2005; Guyenne et al., 2006; Fochesato et al., 2006). A fully nonlinear model of three-dimensional water waves, which extends an approach suggested by Craig and Sulem (1993), was originally given in a two-dimensional setting. The model is based upon the Hamiltonian formulation (Zakharov, 1968), which allows the reduction of the problem of surface variable computation by introducing a Dirichlet–Neumann operator, which is expressed in terms of its Taylor series expansion in homogeneous powers of surface elevation. Each term in this Taylor series can be obtained from the recursion formula and efficiently computed using a fast Fourier transform.

The main advantage of the boundary integral equation methods (BIEMs) is that they are accurate and can describe highly nonlinear waves. A method of solution of the Laplace equation is based on the use of Green's function, which allows us to reduce a 3-D water wave problem to a 2-D boundary integral problem. The surface integral method is well suited for simulation of the wave effects connected with very large steepness, specifically, for investigation of the freak wave generation. These methods can be applied both to the periodic and nonperiodic flows. The methods do not impose any limitations on the wave steepness; thus they can be used for simulation of the waves that even approach breaking (Grilli et al., 2001) when the surface obtains a non-single value shape. The method allows us to take into account the bottom topography (Grue and Fructus, 2010) and investigate an interaction of waves with the fixed structures or with the freely responding floating structures (Liu et al., 2016; Gou et al., 2016).

However, the BIEM seems to be quite complicated and time consuming when applied to the long-term evolution of a multimode wave field in large domains. The simulation of the relatively simple wave fields illustrates an application of the method, and it is unlikely that the method can be applied to the simulation of the long-term evolution of a large-scale multimode wave field with a broad spectrum. An implementation of a multipole technique for a general problem of the sea wave simulation (Fochesato et al., 2006) can solve the problem but obviously leads to considerable algorithmic difficulties.

Currently, the most popular approach
in the oceanography approach is a HOS (high-order scheme) model developed by
Dommermuth and Yue (1987) and West et al. (1987). The HOS model is based on a paper by
Zakharov (1968) in which a convenient form of the dynamic and kinematic
surface conditions was suggested. The equations used by Zakharov were not
intended for modeling, but rather for investigation of stability of the
finite amplitude waves. In fact, a system of coordinates in which depth is
counted from the surface was used, but the Laplace equation for the velocity
potential was taken in its traditional form. However, the Zakharov's
followers have accepted this idea literally. They used the two coordinate
systems: a curvilinear surface-fitting system for the surface conditions and
the Cartesian system for calculation of the surface vertical velocity. An
analytical solution for the velocity potential in the Cartesian coordinate
system is known. It is based on the Fourier coefficients on a fixed level,
while the true variables are the Fourier coefficients for the potential on a
free surface. Here a problem of transition from one coordinate system to
another arises. This problem is solved by expansion of the surface potential
into the Taylor series in the vicinity of the surface. The accuracy of this
method depends on that of the representation of the exponential function with
a finite number of the Taylor series. For the small-amplitude waves and for a
narrow wave spectrum, such accuracy is evidently satisfactory. However, for
the case of a broad wave spectrum that contains many wave modes, the order of
the Taylor series should be high. The problem is now that the waves with high
wave numbers are superposed over the surface of larger waves. Since the
amplitudes of a surface potential attenuate exponentially, the amplitude of a
small wave at a positive elevation increases, and conversely, it can approach
zero at negative elevations. It is clear that such a setting of the HOS model
cannot reproduce high-frequency waves, which actually reduces the
nonlinearity of the model. This is why such a model can be integrated for
long periods using no high-frequency smoothing. In addition, an accuracy of
the calculation of a vertical velocity on the surface depends on full
elevation at each point. Hence, the accuracy is not uniform along the wave
profile. A substantial extension of the Taylor series can definitely result
in numerical instability due to the occasional amplification of modes with
high wave numbers. The authors of a surface integral method share a similar
point of view (Clamond et al., 2005). We should note, however, that the
comparison of the HOS method based on the West et al. (1987) approach using a
method of the surface integral for an idealized wave field (Clamond et al.,
2006) shows quite acceptable results. It was shown in the previous paper that
a method suggested by Dommermuth et al. (1987) demonstrates poorer divergence
of the expansion for the vertical velocity than the method by West et
al. (1987). The HOS model has been widely used (for example, Tanaka, 2001;
Toffoli et al., 2010; Touboul and Kharif, 2010) and it has shown its ability
to efficiently simulate the wave evolution (propagation, nonlinear wave–wave
interactions, etc.) in a large-scale domain (Ducrozet et al., 2007, 2012). It
is obvious that the HOS model can be used for many practical purposes.
Recently, Ecole Centrale Nantes, LHEEA Laboratory (CNRÑ) announced that
the nonlinear wave models based on HOS are published as an open source
(

Opposite to the HOS method based on the analytical solution of the Laplace equation in Cartesian coordinates, a group of models is based on a direct solution of the equation for the velocity potential in the curvilinear coordinates (Engsig-Karup et al., 2009, 2012; Chalikov et al., 2014). The main advantage of a surface-following coordinate system is that a variable surface is mapped onto the fixed plane. Since the wave motion is very conservative, the highly accurate numerical schemes should be used for a good description of the nonlinearity and spectrum transformation. This most universal approach is being developed at the Technical University of Denmark (TUD) (see Engsig-Karup, 2009). Actually, the models ModelWave3D developed at TUD are targeted at the solution of a variety of problems, including such problems as the modeling of wave interaction with submerged objects as well as the simulation of wave regime in basins with a real shape and topography.

The model is based on the equations of a potential flow with a free surface.
An effect of variable bathymetry is taken into account by using the
so-called

A comparison of a ModelWave3D with a HOS model was presented by Ducrozet et al. (2012). It was shown that both models demonstrate high accuracy, while the HOS model shows a better performance. Note that the comparison of the speed of the models in this case is irrelevant since the ModelWave3D was designed for investigation of complicated processes, taking into account the real shape of a basin, variable depth and even the presence of engineering constructions. All these features are obviously not included in the HOS model.

The development of waves under the action of wind is a process that is difficult to simulate since surface waves are very conservative and change their energy for hundreds and thousands of periods. This is why the most popular method is spectral modeling. Waves as physical objects in this approach are actually absent since an evolution of the spectral distribution of wave energy is simulated. The description of input and dissipation in this approach is not directly connected with the formulation of the problem, but rather it is adopted from other branches of wave theory in which waves are the objects of investigation. However, the spectral approach was found to be the only method capable of describing the space and time evolution of wave field in the ocean. The phase-resolving models (or “direct” models) designed for reproducing the waves themselves cannot compete with the spectral models since the typical size of the domain in such models does not exceed several kilometers. Such a domain includes just several thousands of large waves. Nevertheless, the direct wave modeling plays an ever-increasing role in geophysical fluid dynamics because it gives the possibility of investigating the processes which cannot be reproduced with spectral models. One such problem is that of an extreme wave generation (Chalikov, 2009; Chalikov and Babanin, 2016a). Direct modeling is also a perfect instrument for the development of parameterization of physical processes for spectral wave models. In addition, such models can be used for direct simulation of wave regimes of small water basins, for example, port harbors. Other approaches of direct modeling are discussed in Chalikov et al. (2014) and Chalikov (2016).

Until recently the direct modeling was used for reproduction of a quasi-stationary wave regime when the wave spectrum did not change significantly. A unique example of the direct numerical modeling of a surface wave evolution is given in Chalikov and Babanin (2014), in which the development of a wave field was calculated with the use of a two-dimensional model based on the full potential equations written in the conformal coordinates. The 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 an evolution of wave spectrum under the action of wind. However, the strictly one-dimensional (unidirected) waves are not realistic; hence, a full problem of wave evolution should be formulated on the basis of the three-dimensional equations. An example of such modeling is given in the current paper.

Let us introduce a nonstationary surface-following nonorthogonal coordinate system:

The 3-D equations of potential waves in the system of coordinates (1) at

It is suggested in Chalikov et al. (2014) that it is convenient to represent
the velocity potential

The analytical component

The nonlinear component satisfies an equation:

Equation (12) is solved with the boundary conditions

The derivatives of a linear component

Equations (4)–(6) are written in a nondimensional form by using the
following scales: length

The energy input to waves is described by a pressure term

According to the linear theory (Miles, 1957), the Fourier components of
surface pressure

It is a traditional suggestion that both coefficients are the functions of
the virtual nondimensional frequency

The most reliable data on

It was indicated above that an initial wave field is assigned as a
superposition of linear modes whose amplitudes
are calculated with a JONSWAP
spectrum with an initial peak wave number

Real (dashed curve) and imaginary (solid curve) parts of the

A nonlinear flux of energy directed to the small wave numbers produces the
downshifting of the spectrum, while an opposite flux forms the shape of the
spectral tail. The second process can produce the accumulation of energy
near a “cut” wave number. Both processes become more intensive with an increase
in the energy input. The growth of amplitudes at high wave numbers is
followed by that of the local steepness and numerical instability. This
well-known phenomenon in the numerical fluid mechanics is eliminated by use
of a highly selective filter simulating the nonlinear viscosity. To support
stability, additional terms are included in the right-hand sides of
Eqs. (4) and (5):

The main process of wave dissipation is wave breaking. This process is taken into account in all the spectral wave forecasting models similar to WAVEWATCH (see Tolman and Chalikov, 1996). Since there are no waves in the spectral models, no local criteria of wave breaking can be formulated. This is why the breaking dissipation is represented in spectral models in a distorted form. A real breaking occurs in relatively narrow areas of the physical space; however, a spectral image of such breaking is stretched over the entire wave spectrum, while in reality the breaking decreases height and energy of dominant waves. This contradiction occurs because the waves in spectral models are assumed to be the linear ones, while in fact the breaking occurs in the physical space with a nonlinear sharp wave, usually composed of several modes. However, progress has been gradually made in spectral wave modeling over the past decade. It became clear that state-of-the-art wave models should account for the threshold behavior of the dominant wave breaking, i.e., waves will not break unless their steepness exceeds the threshold (Alves and Banner, 2003; Babanin et al., 2010).

The mechanics of wave breaking at a developed wave spectrum differs from that in a wave field represented by the few modes normally considered in many theoretical and laboratory investigations (e.g., Alberello et al., 2018). Since the breaking in laboratory conditions is initiated by special assignment of amplitudes and phases, it cannot be similar to the breaking in natural conditions. To some degree the wave breaking is similar to the development of an extreme wave that appears suddenly with no pronounced prehistory (Chalikov and Babanin, 2016a, b). There are no signs of modulational instability in both phenomena, which suggests a process of energy consumption from other modes. The evolution leading to the breaking or “freaking” seems just opposite: the full energy of a main wave remains nearly constant while the columnar energy is focused around the crest of this wave, which becomes sharper and unstable. Probably even more frequent cases of wave breaking and extreme wave appearance can be explained by a local superposition of several modes.

The instability of interface leading to the breaking is an important and poorly developed problem of fluid mechanics. In general, this essentially nonlinear process should be investigated for a two-phase flow. Such an approach was demonstrated, for example, by Iafrati (2009). However, progress in solving this highly complicated problem is slow.

A problem of the breaking parameterization includes two points: (1) establishing of a criterion of the breaking onset and (2) development of an algorithm of the breaking parameterization. The problem of breaking is discussed in detail in Babanin (2011). Chalikov and Babanin (2012) performed a numerical investigation of the processes leading to the breaking. It was found that a clear predictor of the breaking formulated in dynamical and geometrical terms probably does not exist. The most evident criterion of the breaking is the breaking itself, i.e., the process when some part of the upper portion of a sharp wave crest is falling down. This process is usually followed by separation of the detached volume of liquid into the water and air phases. Unfortunately, there is no possibility of describing this process within the scope of the potential theory.

Some investigators suggest using a physical velocity approaching the rate of surface movement in the same direction as a criterion of the breaking onset. This is incorrect since a kinematic boundary condition suggests that these quantities are exactly equal to each other. It is quite clear that the onset of breaking can be characterized by the appearance of a non-single-value piece of surface. This stage can be investigated with a two-dimensional model, which due to a high flexibility of the conformal coordinates allows us to reproduce a surface with an inclination in the Cartesian coordinates exceeding 90 degrees. (In the conformal coordinates the dependence of elevation on a curvilinear coordinate is always a single value). The duration of this stage is extremely short, the calculations being always interrupted by the numerical instability with sharp violation of the conservation laws (constant integral invariants, i.e., full energy and volume) and strong distortion of the local structure of flow. The numerous numerical experiments with a conformal model showed that after the appearance of a non-single value the model never returns to stability. However, the introduction of a non-single surface as a criterion of the breaking instability even in a conformal model is impossible since a behavior of the model at a critical point is unpredictable, and the run is most likely to be terminated, no matter what kind of parameterization of breaking is introduced. It means that even in a precise conformal model the stabilization of the solution should be initiated prior to the breaking.

A consideration of an exact criterion for the breaking onset for the models
using transformation of the coordinate type of point (1) is useless since the
numerical instability in such models arises not because of the approach of
breaking but because of the appearance of large local steepness. The multiple
experiments with a direct 3-D wave model show that the appearance of the local
steepness

It is seen that the probability of large negative values of the curvilinearity is calculated over an ensemble of linear modes by orders larger than the probability with the spectra generated by a nonlinear model.

The curvilinearity turned out to be very sensitive to the shape of surface.
This is why it was chosen as a criterion of the breaking approach.
Coefficients

Probability of the curvilinearity

We do not think that the suggested breaking parameterization is a final solution to the problem. Other schemes will be tested in the next version of the model. However, the results presented below show that the scheme is reliable and provides a realistic energy dissipation rate.

The elevation and surface velocity potential fields are approximated in the
current calculations by

The parameters chosen were used for solution of a problem of a wave field
evolution over the acceptable time (of the order of 10 days). The initial
conditions were assigned on the basis of the empirical spectrum JONSWAP
(Hasselmann et al., 1973) with a maximum placed at the wave number

The total energy of a wave motion

An evolution of the characteristics calculated by Eq. (30) is shown in
Fig. 3. The sharp variations in all the characteristics at

The data on the evolution of the wave spectrum are shown in Fig. 4. A 2-D wave
spectrum

An angle

Evolution of integral characteristics of the solution, a rate of
evolution of the integral energy multiplied by

The wave spectra

As seen, each spectrum consists of separated peaks and holes

The wave spectrum looks more like the Sagrada Família (Gaudí) in Barcelona than the St Mary Axe (“The Gherkin”) in London.

. This phenomenon was first observed and discussed by Chalikov et al. (2014). The repeated calculations with different resolution showed that such a structure of the 2-D spectrum is typical. It cannot be explained by a fixed combination of interacting modes since in different runs (with the same initial conditions but a different set of phases for the modes) the peaks are located at different locations in a Fourier space.Another presentation is given in Fig. 6 in which the

Sequence of 3-D images of

Sequence of 2-D images of

Evolution of the integrated-over-angles

For calculation of

Evolution of the input spectra (Fig. 7) is in general similar to that of the wave spectra shown in Fig. 4. Note that the maximum of the spectra is located at the maximum of the wave spectra since the input depends mainly on the spectral density, while the dependence on frequency is less important.

An algorithm (Eq. 30) was applied for calculation of the dissipation spectra due to dumping of a high-wave-number part of spectrum (tail dissipation) and for calculation of the spectrum of the breaking dissipation. In the first case, the fictitious time step was made taking into account the terms described by Eqs. (19)–(23), while in the second case the time step was made using the terms described by Eqs. (24)–(27).

The spectra of the tail dissipation calculated similarly to the spectra

The 2-D Fourier spectral “tail” looks like a peacock tail.

. This is why such dissipation, averaged over angles, seems to affect the middle part of a 1-D spectrum. The tail dissipation effectively stabilizes the solution.The spectrum of energy input

The tail dissipation spectra

The breaking dissipation averaged over angles is presented in Fig. 8. As seen, the breaking dissipation has a maximum at the spectral peak. This does not mean that in the vicinity of the wave peak the probability of large curvilinearity is quite high. A high rate of the breaking dissipation can be explained by high wave energy in the vicinity of the wave peak. The energy lost through the breaking, described by the diffusion mechanism, correlates with the energy of breaking waves. Opposite to the high-wave-number dissipation which regulates the shape of the spectral tail, the breaking dissipation forms the main energy-containing part of the spectrum.

The diffusion mechanism suggested in Eqs. (24)–(27) modifies an elevation and surface stream function in close vicinity of the breaking point. The amplitudes of side perturbation are small and decrease very quickly throughout the distance from the breaking point.

An example of the profile of the energy input due to the breaking

The breaking dissipation spectra

In general, for the specific conditions considered in this paper, the
breaking is an occasional process taking place in a small part of the domain.
The kurtosis of the input energy due to the breaking

The number of breaking points in terms of percentage of the total number of
points is given in Fig. 11. As seen, the number of breaking events decreases to

An integral term describing a nonlinear interaction

Example of the energy input due to the breaking

As can be seen, the nonlinearity is quite an important property of surface waves.
The contribution of nonlinearity can be estimated, for example, by
comparison of the kinetic energy of a linear component

The time evolution of the integral spectral characteristics is presented in
Fig. 14. Curve 1 corresponds to the weighted frequency

Evolution of a number of the wave breaking events

The value of fetch in a periodic problem can be calculated by integration of
a peak phase velocity

The JONSWAP dependencies for the frequency of a spectral peak

Sequence of wave spectra

Time evolution of the ratio

Time evolution of weighted frequency

A model based on the full three-dimensional equations of a potential motion with a free surface was used for simulation of development of wave fields. The model is written in a surface-following nonstationary nonorthogonal coordinate system. The details of a numerical scheme and the results of the model validation were described in Chalikov et al. (2014). The main difference between the given model and HOS model (Ducroset et al., 2016) is that our model is based on a direct solution of the 3-D equations for the velocity potential. This approach is similar to that developed at the Technical University of Denmark (TUD; see Engsig-Karup et al., 2009). Actually, the models developed at TUD are targeted at the solution of a variety of problems including such problems as the modeling of wave interaction with submerged objects and the simulation of a wave regime in basins with real shape and topography.

In the current paper a three-dimensional model was used for simulation of the
development of a wave field under the action of wind and dissipation. The
input energy is described by a single term, i.e., surface pressure

We are not familiar with any observational data that can be used for the
formulation of a statistically provided scheme for calculation of the input
energy to waves. The only method that can give more or less reliable results
is the mathematical modeling of the statistical structure of a turbulent
boundary layer above a curvilinear moving surface whose characteristics
satisfy the kinematic conditions. The method described above is based on
several millions of values of the pressure referred strictly to the surface.
As a whole, the problem of a boundary layer seems even more complicated than
the wave problem itself. Some early attempts to solve this problem were made
on the basis of a finite-difference two-dimensional model of a boundary layer
written in a simple surface following the coordinate (see review Chalikov, 1986).
Waves were assigned as a superposition of linear modes with random phases,
corresponding to the empirical wave spectrum. This approach was not quite
accurate since it did not take into account the nonlinear properties of
surface (for example, the sharpness of real waves and the absence of a
dispersive relation for the waves of medium and high
frequencies). The next step was the
formulation of coupled models for a boundary layer and potential waves, both
written in the conformal coordinates (Chalikov and Rainchik,
2010). The calculations showed that the pressure
field consists mostly of random fluctuations not directly connected with the
waves. A small part of these fluctuations are in phase with the surface
disturbances. The calculated values of

The wave breaking is obviously even more complicated than the input energy. Nevertheless, this problem can be simplified, if the common ideas used in the numerical fluid mechanics are accepted. For example, in the LES modeling a more or less artificial viscosity is introduced to prevent too large local velocity gradients. In fact the numerical instability terminating computations precedes the wave breaking. Hence, the scheme should prevent the breaking approach to preserve stability of the numerical scheme. Hence, a wave model should contain the algorithms preventing the appearance of too large slopes. A criterion of breaking is introduced not for recognizing the breaking itself, but for the choice of places where it might happen (or, unfortunately, might not happen). Finally, the algorithm should produce the local smoothing of elevation (and the surface potential). The algorithm should be highly selective so that the “breaking” could occur within narrow intervals and not affect the entire area. The exact criteria of the breaking events (most evident of them is the breaking itself) cannot be used for parameterization of breaking since in a coordinate system (1) the numerical instability occurs long before the breaking. In our opinion, the most sensitive parameter indicating potential instability is the curvilinearity (second derivative) of elevation.

In the current work, the breaking is parameterized by a diffusion algorithm with a nonlinear coefficient of diffusion providing high selectivity of the smoothing. We admit that such an approach can be realized in many different forms. The same situation is observed in a problem of the turbulence modeling for parameterization of subgrid scales. Note that the breaking dissipation in phase-resolving models is included in a more realistic manner than in spectral models. For example, the breaking is simulated in a physical space, which allows us to reduce the height and energy of the nonlinear waves composed of several modes. In spectral models the dissipation is distributed more or less arbitrarily over the entire spectrum. The spectral models sometimes include additional dissipation of short waves due to their modulation by long waves (Young and Babanin, 2006; Babanin et al., 2010). In the phase-resolving models this process has been included explicitly.

We can finally conclude that the physics included in wave models still rests on shaky ground. Nevertheless, the result of the calculations looks quite realistic, which convinces us that the approach deserves further development.

The numerical models of waves similar to that considered in this paper have a lot of important applications. First, they are a perfect tool for the development of physical parameterization schemes in spectral wave models. Second, a direct model can be used in future for the numerical simulation of wave processes in the basins of small and medium size. These investigations can be based on the HOS model (Ducrozet et al., 2016) or the model used in the current paper. However, the most universal approach seems to be developed at the Technical University of Denmark (see Engsig-Karup, 2009). Any model used for the long-term simulation of wave field evolution should include the algorithms describing transformation of energy similar to those considered in the current paper.

The underlying data (150 Gb) are not publicly accessible. Any number of them can be shared upon request.

The authors declare that they have no conflict of interest.

The authors thank Olga Chalikova for her assistance in preparation of the paper as well as the anonymous reviewers for their constructive comments. This investigation was supported by Russian Science Foundation, project 16-17-00124. Edited by: Neil Wells Reviewed by: two anonymous referees