Dimension-Reduced Model for Deep-Water Waves

Starting from the 2D Euler equations for an incompressible potential flow, a dimension-reduced model describing deep-water surface waves is derived. Similar to the Shallow-Water case, the z-dependence of the dependent variables is found explicitly from the Laplace equation and a set of two one-dimensional equations in x for the surface velocity and the surface elevation remains. The model is nonlocal and can be formulated in conservative form, describing waves over an infinitely deep layer. Finally, numerical solutions are presented for several initial conditions. The side-band instability of Stokes waves and stable envelope solitons are obtained in agreement with other work. The conservation of the total energy is checked.


Introduction
The procedure for deriving reduced equations by expressing the dependence of the dependent variables on a certain, geometrically distinguished spatial coordinate analytically is widespread in nonlinear pattern formation and wave dynamics. As a first attempt, one can consider the Shallow-Water or Saint-Venant equations [1], which are systematically derived from the hydrodynamic basic equations for an inviscid potential flow by solving iteratively the Laplace equation for the velocity potential function. Thereby, all expansions are controlled by a geometry parameter 1 d δ = (1) where only the horizontal coordinates occur. Standard equations in normalized form like the Allen-Cahn, the (complex) Ginzburg-Landau or the Swift-Hohenberg equations can be derived, partially by heuristic arguments concerning at least their nonlinear parts [5].
Stokes [6] initiated the theory of nonlinear deep-water waves. Three branches of development for strongly nonlinear theory have grown from seminal papers. 1) The highest Stokes wave [7]. 2) The instability of Stokes waves [8]. 3) Breaking of deep-water waves [9].
The early nonlinear theories in 2D have now grown into full 3D theories.
However, a long-standing problem is that of dimensional reduction of the nonlinear 3D problem to a 2D analysis, as it can be done straightforwardly for the case of shallow water. For deep water waves, the Hamiltonian formulation [10] [11] provides a nonlinear system for two two-dimensional field variables, namely surface elevation and potential at the surface. To compute the velocity normal to the surface, the solution of the underlying 3D-Laplace equation is achieved by an expansion with respect to powers of the surface elevation into spectral function. These algorithms have been named 'higher order spectral methods' (HOSM) and go back to the work of West et al. [12] and Dommermuth and Yue [13].
Another attempt is the systematic derivation of an equation describing the slowly varying envelope of a given monochromatic wave train. This equation has the form of a nonlinear cubic Schrödinger equation [10] [14] and reflects the side-band instability of Stokes waves as well as envelope solitons as localized stable solutions.
Contrary to the Shallow-Water or Thin-Film equations, no intrinsic length scale exists for the deep-water case. The only available scales are given by the solutions sought for, namely the wave length λ (if the waves are nearly coherent) and the amplitude A of those waves. In deep-water waves, the ratio of amplitude to wave length is normally rather small. Thus, using the wave steepness for expanding the nonlinearities seems to be quite natural.
The present paper offers a consistent scheme of reducing the spatial dimen- The conservation of the total energy can be considered as a quality test for both the derived system and the numerical method applied for its solutions. We show that the energy is conserved within a few percents which give us confidence in our derivation and results.
The model can be extended to the case of a 3D potential flow, resulting in a set of three two-dimensional evolution equations.

Potential Flow
For the sake of simplicity we restrict the treatment first to two dimensions, . We are looking for solutions of the non-dimensional Euler eqs. for an incompressible fluid in the half space Figure 1: where time is scaled by g τ = (6) and lengths with an arbitrary which will be identified later (sect. 3.2) with the characteristic wave length of the initial condition. We assume that the flow field can be derived from a potential = ∇Φ v (7) which then, due to (5) must be a solution of the Laplace eq.
be the horizontal velocity component at the free surface located at From (3) we derive Equation (11) serves as a boundary condition (b.c.) for (9) at The surface function h is determined by the kinematic b.c.
For the following treatment, it is of advantage to formulate (12) also in con- For an infinitely deep layer, the asymptotic b.c. (15) must hold. Hence the general solution of (9) reads ( )e e . .
with c.c. as the complex conjugate.

Nonlocal Expansion
Considering (11) and (13) as evolution Equations, the original 3D (here 2D) problem is reduced to a 2D (here 1D) system in the horizontal coordinates x,y (here x) only. However, to evaluate the flux (14) one needs to know ( ) , , x v x z t . Taking (16), Equation (14) reads To determine the amplitudes k u from h u , we evaluate (16) at z h = and expand the exponential function up to a given order N h : Next we introduce the fractional differential operator of order N h : for every function ( ) f x via its Fourier transform k f . We note that (20) is one of the definitions of the fractal Laplacian, here in 1D [15]. Now Equation (18) can be written as (in the 3D case, xx ∂ must be replaced by the horizontal Laplacian xx yy ∂ + ∂ ).
Using the same technique, S from (17) can be expressed as Thus, the equations (11) and (13) are closed by the relation (23). Note that is a nonlocal operator only for odd n. For n even it can be expressed by spatial derivatives of order n in position space. Moreover, assuming periodic lateral boundary conditions, it is self-adjoint for all n.

Second Order Model
In the following we shall restrict ourselves to 1 N = , resulting in a model of second order in the dependent variables ( ) ( ) its inverse 1 11 , and the self-adjoint operator ( ) where only terms up to the second order in , h u h are included.
In (11), the pressure gradient evaluated at z h = turns into where h P is the pressure along the free surface and assumed to be constant (external pressure). To find z P ∂ we evaluate (4) at the surface and include only linear terms in v , leading to where again only linear terms are considered. Finally, we have ( ) 1 1 , where the first term on the right-hand side corresponds to the hydrostatic pressure, the second one to the dynamic part. Using (33), Equation (11) can be cast into the form ( )

Model
After inversion of the operator on the left-hand side of (34), the complete system up to the second order reads ( )

Linear Waves
Taking only linear terms into account one finds from ((35), (36)) or, eliminating h u , the linear wave equation where we have used the identity . Equation (38) possesses the well-known short-wave dispersion relation

Two Horizontal Dimensions
If we add the second horizontal dimension we are able to describe a 3D flow which, however, is still potential. Extending the surface velocity to a 2D vector the derivation of the reduced system is straightforward and it reads

Similarities and Differences with HOSM
Contrary to the higher-order spectral methods (HOSM) presented in [12] [13], we use the surface elevation and the surface velocity instead of the surface potential. With the kinematic b.c.

Numerical Method
We concentrate on the 2 nd order model ( (35) with the Green's function To evaluate S according to (28) it is therefore necessary to compute ˆh Du and

Benjamin-Feir Instability of Slightly Disturbed Coherent Wave Trains
The initial conditions are based on surface elevations in form of a traveling wave with given amplitude and wave number 0 k . If for the spatial scaling introduced in (6) the wave length is chosen, we can put 0 2π k = . The x-dimension is discretized with 2048 N = mesh points, the step sizes used are 60 0.029 As first shown in [8], coherent Stokes waves with a single frequency and wave length are unstable with respect to the side-band instability. The time series in Figure 2 presents the evolution of a 2 nd -order Stokes wave [19] traveling to the right side as initial condition:  The absolute values of the Fourier amplitudes over time for the carrier wave ( 0 k ) as well as for its five neighbors ( 0 Figure 3. In the beginning, the side bands grow exponentially, taking the energy from the carrier wave that decreases.

Envelope Solitons
If a coherent wave is enveloped by a slowly varying function ( ) Zakharov [10] showed that for small but finite amplitudes an equation for the complex valued The nonlinear coefficient ν depends on the ratio of the water depth to the wave length of the carrier wave [11] and reads for infinite depth To see how envelope solitons behave in our model, we take as initial condition Figure 4 shows snapshots from the evolution for and 60 Γ = . The envelope soliton travels thereby with the group speed ω′ as given in (57)  Finally we show in Figure 6 the interaction of two colliding envelope solitons, generated by the initial condition ( )

Conservation of Energy
It is interesting to check the temporal variation of the energy during the runs.
The conservation of (63) can serve as a quality feature for the numerical solutions as well as for the accuracy of the derived system itself.   Figure 2 and Figure 6, respectively. In Figure 8, potential as well as kinetic energy varies strongly during the collision phases but the sum (63) remains fairly constant. Table 1 shows the standard deviations normalized with the square mean defined as Figure 7. Energies plotted over time of the evolution shown in Figure 2. solid: total, dashed: kinetic, dashed-dotted: potential. Figure 8. Energies of the evolution shown in Figure 6. solid: total, dashed: kinetic, dashed-dotted: potential. Table 1. Standard deviations of the energies shown in Figure 7, Figure 8.  Note that all computations are only up to second order which then applies also for the energy conservation so that where ξ stands for h, h u and combinations.

High Waves Caused by Dispersive Focusing of an Unstable Stokes
Wave Waves on deep water are generated by wind and have a broad spectrum of wave lengths. Since their phase and group speed is proportional to 1 ω − , longer waves are faster than shorter ones and, if they propagate in the same direction, may catch up and enhance the total amplitude. This is already a linear effect but can be amplified by nonlinearities what then leads to even higher wave heights . In this way, very large heights can be obtained, which is one of the mechanisms responsible for the occurrence of freak or rogue waves [14] and named "dispersive focusing" or "dispersive enhancement".
To classify waves with respect to their height, the "abnormality index" AI Then, a freak wave can be defined by AI > 2. The probability for freak waves depends strongly on the initial wave pattern and on the dimensionality of the system. In 2D, only two propagation directions are possible and freak waves should occur quite often. Figure 9 shows the evolution of an unstable Stokes wave as (52), but for a larger aspect ratio and a higher grid resolution A freak wave shows up with AI ≈ 2.8. In Figure 10 we plot AI over time, the circle marks the situation shown in Figure 9. Figure 11 shows the the absolute values of the Fourier amplitudes of ( ) h x depicted in Figure 9. A scaling law according to 2 15 α ≈ can be seen over wide regions, in agreement with the findings of Zakharov et al. [21].

Conclusions
We presented the derivation and numerical solutions of dimension-reduced deep-water equations. The system can be considered as a systematic expansion  as well as to three-dimensional potential flows, resulting in a reduced 2D set.
Thus, the numerical examination of 2D envelope solitons occurring on the surface of a 3D potential flow should become feasible. This work is currently in progress and will be presented in a following paper.