1. Introduction

As it is well known, there have been several attempts at numerical calculation of the nonlinear kinetic integral (for references see Hasselmann and Hasselmann 1985)since the pioneering paper by Hasselmann (1962). Herewith, the first reliable results of such calculations have been obtained nearly 20 years later (Webb 1978; Masuda1980; Hasselmann and Hasselmann 1981), but these studies were not detailed. They encompassed only the conceptual investigation of the peculiarities of nonlinear interactions between waves.

The most detailed investigation of the problem, for the case of deep water, was carried out by the author recently (Polnikov 1989, 1990). In terms of physics, four distinct properties of the nonlinear energy transferas a function of two-dimensional spectrum shape parameters were found and formulated (Polnikov 1989).In Polnikov (1990), the peculiarities of a long-scale temporal evolution of the two-dimensional frequency-angular wave spectrum S(ω, θ), provided by the nonlinearity of waves only, were studied numerically. Manynew and interesting results regarding the role of nonlinearity of waves were obtained.

This paper is a natural continuation of the previous studies, focusing on the case of finite depth. The necessity of such an investigation is based on the following reasons: First, it is induced by the lack of a detailed description of nonlinear energy transfer for the case of finite depth at present. The only two papers dealing with the point (Herterich and Hasselmann 1980; Hasselmann andHasselmann 1985) are insufficient for the full investigation of the problem. Second, the numerical technique of computation in this case is more complicated than for the deep water case, and it should be described in detail.

The initial point of investigation is the well-known nonlinear kinetic equation of the kind

View Expanded

where n_i ≡ n(k_i) is the wave action spectrum in k space related to the energy spectrum S(k) by the formula

View Expanded

where M_1,2,3,4 ≡ M_k1k2k3k4 are the matrix elements of the four-wave interactions and ω_i ≡ ω(k_i) and k_i are the frequency and the wave vector related to each other by the dispersion relationship of the kind

ω²kgkkh(3)

The explicit expressions for the elements M_1,2,3,4 for the case of finite depth are presented, for example, in the paper by Krasitskii (1994).

There are several techniques of integration of the kinetic integral (1) (Webb 1978; Masuda 1980; Hasselmann and Hasselmann 1981). For these calculations a modified method of Masuda, derived by the author (Polnikov 1989), was used. In this method, for the deep water case, integration of the threefold δ-function in (1)is carried out analytically in an explicit form. Unexcludable singularities under integral (1), connected with the zero values of the Jacobian denominator arising due to the integration of the complicated δ function, are integrated analytically in the vicinity of singular points as well. In this way, the mathematical difficulties of numerical integration are totally overcome. Details of the algorithm are given in Polnikov (1989).

However, for the case of finite depth, the direct utilization of relationship (3) in the algorithm stated above is impossible due to the irrational equation for the frequency resulting from the function δ(ω₁ + ω₂ − ω₃ −ω₄). Thus, the problem requires adaptation of the previous algorithm for the case of finite depth. This problem is solved and described below.

The main aim of this paper is to find and describe the principle peculiarities of the nonlinear energy transfer (NET) through the spectrum of surface gravity waves for the finite-depth case. It will be shown that these peculiarities are not confined only by a trivial increase of the NET intensity with decreasing depth h, but they are more numerous and complicated.

In section 2 the modification of the deep water algorithm of the kinetic integral calculation is described. Methods and results of the calculations are presented insections 3 and 4, respectively. Analysis and analytical parameterization of the NET properties for the case of finite depth are given in section 5. Finally, a discussion of the results is carried out in section 6.

2. Modification of calculation algorithm

According to Polnikov (1989), the algorithm of kinetic integral calculation comprises the following items:

1. analytical integration of the δ(k) function with respect to k₂;

2. transformation to polar coordinates (ω_i, θ_i) and analytical integration of δ(ω₁ + ω₂ − ω₃ − ω₄) with respect to θ₁;

3. determination of singular points of the Jacobian denominator (by the analysis of its explicit expression) and the limits of integration with respect to ω₁;

4. analytical integration over the vicinity of singular points and further numerical integration over the domain outside these points.

The procedure stated above determines the needed modification of the algorithm provided by the dependence of the dispersion relationship (3) on depth.

By introducing the definitions

View Expanded

where vector k_a in polar coordinates is defined by the couple (k_a, θ_a) (see appendix A), one can derive the following formula for the δ function with respect to frequencies (taking into account the prior integration with respect to k₂):

View Expanded

From (6) it is easy to find an analytical solution for zero points of argument of function δ(ω) with respect to θ₁,

View Expanded

Here k₁ and k_;at2 are expressed as functions of ω₁ and ω_;at2 ≡ ω_a − ω₁ by Eq. (4), respectively. The Jacobian of the δ-function integration (at points θ₁ = θ^±₁) is as follows:

View Expanded

Hereafter the tilde over subindex 2 is omitted everywhere. Consider now the denominator of the Jacobian more closely.

In contrast to the case of deep water, the exact expressions for lower and upper limits of ω₁, denoted further by ω_1mi and ω_1ma, cannot be calculated analytically in the case of finite depth. Because of this, for each set {ω₃, θ₃, ω₄, θ₄}, determining a respective set {k_a, θ_a,ω_a}, the solutions of Eqs. (9a,b) must be obtained numerically. For this purpose, Eqs. (9a,b) are rewritten in an explicit form using an approximation (4) for variablesk₁(ω₁) and k₂(ω_a − ω₁) (see appendix A).

As a result, after transformation to polar coordinates(ω_i, θ_i) and nondimensional variables, with normalization of frequency by the peak value ω_p and the spectrumS(ω, θ) by S_p ≡ S(ω_p, θ_p), one can obtain the final expression for ∂S/∂t ≡ Ṡ of the kind

View Expanded

Here C = πg⁻⁴ω¹¹_pS³_p/8 is the dimensional constant, and the expression in brackets under the integral is the regular part of integrand presented in appendix A.

Further transformation of the algorithm for the kinetic integral calculation includes estimation of the type of singularity and analytical integration of it over the vicinity of the singular points ω₁ = ω_1mi and ω₁ = ω_1ma. A direct check shows that both singular points have a square root integrable type of singularity (see appendixA).

While integrating the kinetic integral, it is useful to consider Longuet–Higgins’s diagrams for four-wave interactions (Hasselmann and Hasselmann 1981). For the case of finite depth, these diagrams have the same form of so-called figure-of-eight curves for some nondimensional interaction parameter Δ. The parameter Δ, separating the coupled (Δ > 0) and uncoupled (Δ < 0)isolines of Δ, is defined in our case by the formula

View Expanded

Use of relationship (11) permits estimation of the singular point contribution by analogy with previous calculations (Polnikov 1989). Thus, the technique of calculations, described previously, is transferred to the case of finite depth. A few final expressions are given in appendix A.

3. Method of calculations

The most important divergence between our calculations and those known from the literature (Herterichand Hasselmann 1980; Hasselmann and Hasselmann1985) consists of the large variety of two-dimensional spectral shapes considered here. The representative series of shapes S(ω, θ), presented in Table 1, covers a wide range of spectral shape parameters, defined as follows: the frequency width δ:

View Expanded

angular narrowness A(ω):

View Expanded

This choice ensures a reasonably thorough investigation of the problem.

Note that the normalizing coefficient C(ω) for the angular spreading function Ψ(θ) of spectrum S(ω, θ) ≡S(ω);asΨ(θ) [in that case C(ω) = A(ω)/Ψ(θ_p)] is not accounted for here because, in our choice of Ψ(θ), C(ω)= const for each variant, and it simply changes the magnitude of S_p but does not influence the shape of the two-dimensional function of NET T(ω, θ) ≡ ∂S(ω, θ)/∂t.

As in previous studies, the spectral shape S(ω) is taken in the form typical for a deep water case. For this reason, in the present work we investigate only the influence of finite depth on the NET due to the matrix elements. The impact of shoaling on changing the spectral shape with the depth (as pointed out by Kitaigorodskii et al. 1975) should be considered separately.

With the goal of finding objective relationships of the two-dimensional NET function T(ω, θ) with the parameters of spectral shape, we introduce the following informative parameters of function T(ω, θ):

1. T⁺ the absolute positive extremum (maximum) ofT(ω, θ)

2. T⁻ the absolute negative extremum (minimum) ofT(ω, θ)

3. ω₁ the frequency coordinate of T⁺

4. ω₂ the frequency coordinate of T⁻

5. ω₀ the frequency coordinate of zero point of T(ω, θ)on the frequency axis

6. T_l the local positive extremum of T(ω, θ)

7. ω₃ the frequency coordinate of T_l

8. θ₃ the angular coordinate of T_l.

Formally, the dependencies of all of these parameters of T(ω, θ) on the parameters of spectrum S(ω, θ), that is, on ω_p, θ_p, S_p, δ, and A(ω) must be determined. The problem posed is analogous to that considered earlier for deep water (Polnikov 1989), but it is new for the case of finite depth.

4. Results of the calculations

Results of the calculations are presented in Figs. 1–6 and in appendix B for the six variants of spectral shapes given in Table 1.

In the figures, the one-dimensional functions of NET [i.e., T(ω) = ∂S(ω)/∂t ≡ ∫ [∂S(ω, θ)/∂t] dθ] are given in units of dimensional constant C presented above. With the aim to compare curves of T(ω) for different magnitudes of depth, the values of T(ω) are normalized by multiplication of ∂S(ω)/∂t to a certain power function of the kind tanhⁿ(k_p0h) (using the simplified relationship k_p0 = ω²_p/g). The fitting of parameter n has shown that the values of normalized extrema T⁺, T⁻ become comparable, for different depth h, when n has values in the range n ≅ 4–10. For this reason, the functions ∂S(ω)/∂t[tanh¹⁰(k_p0h)] are presented in Figs. 1–6.

In appendix B, as an example of direct numerical results, the two-dimensional NET functions are presented in a tabulated form for the first run given in Table1. The values of T(ω, θ) are given at the grid points (ω_i,θ_j) in percentage of the difference R = T⁺ − T⁻.

From these numerical data, the representative Tables2–7 have been constructed, describing the dependencies of the main parameters of T(ω, θ) on the shape parameters of the spectrum under the integral.

5. Analysis and parameterization

Analysis of the results presented above demonstrates the principal properties of the NET for the finite depth case. The most important properties are as follows:

1. The extremal values of the NET, T⁺ and T⁻, grow with diminishing depth h approximately as function tanh⁻ⁿ(k_p0h). The value of n changes within the rangen ≅ 4–6 depending on the spectral shape. The smaller the parameters δ or A(ω), the greater n.

2. The parameters ω₀, ω₁, ω₂, ω₃ of the two-dimensiona lNET function T(ω, θ) are complicated functions of depth and spectral shape.

3. The one-dimensional curve T(ω) and two-dimensional surface T(ω, θ) shift to lower frequency asdepth approaches zero (k_p0h → 0).

4. The size of the negative domain Ω⁻ of T(ω, θ) (along the frequency axis) becomes smaller as k_p0h → 0.Due to this, extremum T⁻ grows faster than extremum T⁺.

5. The local positive extremum T_l becomes relatively large (with respect to T⁺) while k_p0h → 0.

6. The angular spreading function of T(ω, θ) is less changeable than the frequency dependence and is similar to the form typical of the deep water case.

Some of these properties can be seen, to some extent,in the figures in a previous paper (Hasselmann and Hasselmann 1985), but they are formulated here in physical terms for the first time.

The relatively wide variety of spectral shapes considered here permits construction of a more or less reasonable parameterization of the two-dimensional function T(ω, θ).

Taking into account only the last list of parameters, one can propose the following parameterization of the different parts of representation (14).

For function Φ(ω, θ), taking into account the previous experience of parameterization of the angular spreading function for deep water

View Expanded

where

View Expanded

The fitting coefficients are

View Expanded

For points ω₀, ω₁, ω₂, ω₃, the following expressions have been found:

View Expanded

6. Discussion

First, the properties of the NET described above reflect only the main features of the two-dimensional function T(ω, θ) topological dependence on the spectral shape parameters. The description of fine details [suchas dependence of coordinates of the local extremum (ω_l,θ_l) and its intensity T_l on values of ε, δ, A(ω), or the specification of property 6] needs additional, more exactcalculations of T(ω, θ).

Regarding numerical accuracy, some attention should be paid to the remarkably anomalous results for the values ε ≡ k_p0h ∼ 1. For these values, the error ofapproximation (4) is greatest. This fact warrants caution in conclusions regarding the range ε ∼ 1.

Second, it is interesting to understand why the high-frequency local extremum, T_l, becomes greater than that for low frequency T⁺ for the case of small values ofk_p0h (k_p0h < 0.3). One may expect that for such small values of k_poh, the four-wave kinetic equation approximation is not valid for a description of the nonlinearwave dynamics, and a more accurate approximation needs to be derived. This point is to be investigated theoretically and numerically in more detail.

Further theoretical investigations should also clarify the role of three-wave interactions in the description of nonlinear properties of waves in finite depth and shallow water cases. Present papers dealing with this issue(Abrue et al. 1992; Madsen and Sorensen 1993) arerather questionable.

Nevertheless, the results obtained here give additional understanding of the role of nonlinearity in wave dynamics for the case of finite depth. They permit one to construct an appropriate analytical parameterization of the NET. In our proposal, the error of the parameterization, presented by Eqs. (14)–(17), is less than 15% in the description of extremal values of the numerical function T(ω, θ).

At present, this parameterization is being used for numerical experiments with a wind wave model for shallow water, constructed recently by the author and colleagues (Polnikov and Sychov 1996). We have found that for rather shallow water (k_p0h ≤ 0.3), the influence of nonlinearity becomes very strong and exceeds the effects of refraction and shoaling. This means that a reasonable numerical result cannot be obtained by modeling without correct use of the nonlinear evolution termin the source function. Investigation of this term is currently in progress.