Three-dimensional Periodic Fully Nonlinear Potential Waves

An exact numerical scheme for a long-term simulation of three-dimensional potential fully-nonlinear periodic gravity waves is suggested. The scheme is based on a surfacefollowing non-orthogonal curvilinear coordinate system and does not use the technique based on expansion of the velocity potential. The Poisson equation for the velocity potential is solved iteratively. The Fourier transform method, the secondorder accuracy approximation of the vertical derivatives on a stretched vertical grid and the fourth-order Runge-Kutta time stepping are used. The scheme is validated by simulation of steep Stokes waves. The model requires considerable computer resources, but the one-processor version of the model for PC allows us to simulate an evolution of a wave field with thousands degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of the nonlinear two-dimensional surface waves, for generation of extreme waves as well as for the direct calculations of a nonlinear interaction rate. After implementation of the wave breaking parameterization and wind input, the model can be used for the direct simulation of a two-dimensional wave field evolution under the action of wind, nonlinear wavewave interactions and dissipation. The model can be used for verification of different types of simplified models. INTRODUCTION The majority of the models designed for investigation of the three-dimensional wave dynamics are based on the simplified equations (see review [1]). Much closer to reality is an approach based on Zakharov’s equations [2]. He suggested using the boundary condition for the velocity potential written on an interface. The Laplace equation remains written in Cartesian coordinates according to this approach. For utilization of the boundary condition for the Laplace equation a procedure of interpolation of the potential from a free surface to a fixed level was suggested [3,4]. Later this method was titled as HOS (High Order Scheme) and used in many works (see, for example, [5,6,7]). An advantage of this approach is that the method is simple, computationally effective and robust. However, applicability of this method for simulation of highly nonlinear waves and for such delicate problem as the nonlinear wave-wave interaction remains unclear (see [8]]. The most developed methods are based on the full three dimensional equations and the surface integral formulation [9,10,11]. The method can be applied to the periodic and non-periodic flows. The main advantage of the method is exactness. The method does not impose any restriction on the wave steepness, so it can be used for simulation of waves approaching the breaking. However, the method seems quite complicated, so it is unlikely to be applied to the large-scale modeling of a long-term evolution of the real sea waves. Another method for the 3-D waves includes an elliptic boundary layer problem solved by the finitedifference methods. Such approaches to simulation of the unsteady free surface flows based on full equations have been developed at least over three decades (see, for example [12.]). The related applications were later described in [13,14,15]. The main advantage of this method is that it is based on the initial equations transformed into the surface-following coordinate system. The Laplace-type equation obtained by transformation into the sigma-coordinate system, was solved in [16] by the iterative conjugate gradient method using the three dimensional finite element discretization. The finitedifference multi-grid model for 3-D flow was developed in [17]. All papers of this group were mostly devoted to the technical application of the water wave theory, i.e., for the calculations of the dynamic load on the submerged bodies, or for simulation of wave dynamics in a domain with a complicated shape. A long-term evolution of a multi-mode wave field was not reproduced; this is why the exact conservation of energy was not the priority target of such models. In this work we suggest a new approach specifically targeted at simulation of a long-term multi-mode periodic wave field evolution in the deep ocean. It is well known that the nonlinear transformation and growth of waves occur over hundreds and thousands of wave periods. This property imposes tough restrictions on the model because such modifications of waves should not be obscured with the numerical errors. It means the model should be exact enough to reproduce a relatively slow spectrum evolution. This condition is well satisfied in the 1-D model in conformal coordinates. The 3-D waves represent a far more difficult object, since it is probably impossible to reduce the problem to the surface problem (in fact, the surface integral method cannot be referred to as the surface method, since it uses the Green function); this is why a velocity potential should be calculated from the elliptic equation. Such model is described in this paper. In section 1 the primary equations, as well as transformation of coordinates and the numerical scheme in the non-orthogonal and nonstationary curvilinear coordinates are given. The results of validation of the approach and the codes of the model are discussed in Section 2. The results of the long-term simulations of a multi-mode three-dimensional wave field are described and Section 3. The main results and prospects of the investigation are discussed in Conclusions. 1. 3-D DEEP WATER WAVE MODEL Let us introduce the non-stationary surfacefollowing non-orthogonal coordinate system: ( ) , , , , , x y z t ξ θ ζ η ξ θ τ τ = = = − = (1) where ) , , ( ) , , ( τ θ ξ η η = t y x is the shape of the wave surface, which maps the original domain ( ) , , , , x y z x y t η −∞ < < ∞ − ∞ < < ∞ − ∞ < ≤ (2) which maps the original domain (4) onto the layer , , 0 ξ θ ζ −∞ < < ∞ − ∞ < < ∞ − ∞ < ≤ (3) with the periodicity conditions over ‘horizontal’ coordinates ξ and θ : The coordinates (1) is constructed for the deep water case. As seen, the vertical fluctuation of ‘horizontal’ coordinates ξ andθ does not attenuate with depth. the lower boundary condition is applied at the variable level H ζ η = + . Such fluctuations do not create any problems with approximation since all variables in the wave motion attenuate wth depth exponentially. For depth H η >> the difference between the fixed and fluctuating levels becomes negligible. The model formulated above, is integrated in the domain { } 0 2 , 0 2 , 0 H ξ π θ π ζ < ≤ < < < ≤ . This corresponds to a moving periodic wave surface given by Fourier series ( ) ( ) , , , , , M k M M h k l k l η ξ θ τ τ ≤ ≤ = Θ ∑ − (4)