A Consistent Nonlinear Mild-Slope Equation Model

Abstract

A new nonlinear frequency-domain model based on the mild-slope equation is outlined. The model is an enhancement over previous work in that a closer correspondence between scaling of nonlinearity and horizontal variation of bathymetry is made relative to earlier models. This results in additional terms in the nonlinear summation terms of the model, as amplitude gradient terms are required in order to formulate a consistent model. From the resulting elliptic model, a parabolic approximation is developed in order to efficiently model the equations. Comparisons between the present model, previously-formulated models, and experimental data show that the present model does evidence improvement in performance over previous models.

1. Introduction and Scale Analysis

Ocean wave propagation models are a cornerstone of coastal and ocean engineering research and practice. Models based on the mild-slope equation (Berkhoff, 1973), the Boussinesq equations (Peregrine, 1967) and similar are used to simulate phase-resolved wave transformation processes over varying bathymetry in coastal regions.

A subset of phase-resolved numerical wave models involves various forms of the mild slope equation, which are generally (with some exceptions) cast in the frequency domain. Generally speaking, the majority of these frequency-domain models assume that the dependent variable (primarily the free surface elevation) has purely periodic behavior in time and a combination of periodic and slowly-varying behavior in space. The resulting equation is elliptic, which can be problematic to solve for general open coast problems (e.g., Kirby, 1986). By using the parabolic approximation (e.g., Radder, 1979), models for monochromatic linear waves can be developed; examples include Radder (1979), and Lozano and Liu (1980). Once introduced, the method was further extended to include other effects (e.g., Dalrymple et al. (1984) for frictional dissipation effects and Booij (1981), Liu (1983), and Kirby (1984) for wave-current interaction.)

By employing the WKB approximations for wave potential and free surface elevation, Yue and Mei (1980), and Kirby and Dalrymple (1983) extended the parabolic method to the case of second-order Stokes waves in constant depth and slowly-varying depth, respectively. The formulations can provide valid descriptions of the wavefield for deep water, where the Ursell number ($Ur=\frac{a}{k^2{h}^3}$, where a is the amplitude, k is the wave number, and h is the water depth) is less than one. However, for cases where the Ursell number is greater than one, Stokes wave theory becomes invalid (e.g., Dean and Dalrymple, 1991). This would include scenarios in shallow water (small kh) and/or high waves (large a). Initial models for nonlinear wave propagation in shallow water were proposed using the Boussinesq equations of Peregrine (1967); examples include Rygg (1988) (in time domain) and Freilich and Guza (1984), Liu et al. (1985), and Kirby (1991) (in frequency domain). These models show good agreement to data in cases where the waves are in shallow water.

Considerable effort has been expended in attempts to increase the dispersive range of weakly nonlinear models. These have generally been focused on both extended Boussinesq models and nonlinear mild-slope equations. Extended Boussinesq models were first developed to enhance the applicability of the Boussinesq-type models to deeper water by retaining the weakly-dispersive, weakly-nonlinear formulation of the classical Boussinesq equation while altering the dispersion relation to be more accurate in deep water when compared to linear wave theory (e.g., Madsen et al., 1991; Madsen and Sørensen, 1992; Nwogu, 1993). More recent developments have included extensions to highly nonlinear waves (e.g., Wei et al., 1995; Madsen and Schäffer, 1998; Gobbi et al., 2000). Other models treated the depth dependence of the solution via various Taylor series expansions (Agnon et al., 1999; Madsen et al., 2003; Fuhrman and Madsen, 2009), or multi-layer models (e.g., Lynett and Liu, 2004; Liu and Fang, 2016; Liu et al., 2018) to help increase the linear characteristics of these models to better resemble that of fully-dispersive linear theory. In a contrasting approach, nonlinear mild-slope equations retain the fully-dispersive behavior of linear mild slope equations while adding weak nonlinearity (e.g., Agnon et al., 1993; Kaihatu and Kirby, 1995; Tang and Ouellet, 1997; Eldeberky and Madsen, 1999; Janssen et al., 2006; Toledo, 2013; Ardani and Kaihatu, 2019). Several recent models in this regard were derived using the solution in terms of truncated Taylor series expansions and the dispersion operator (e.g., Bredmose et al., 2005; Vrecica and Toledo, 2016). Many of these studies build from work by Bryant (1973, 1974), who developed a mild-slope model representing a fully dispersive wave over a flat bottom by substituting spatially-periodic forms for the wave potential and free surface elevation which satisfy the Laplace equation and the bottom boundary.

Nonlinear mild slope equations in the frequency domain are generally developed in the form of a parabolic equation, which allows for more straightforward modeling of open coastal regions. The parabolic form can be developed with a judicious choice of modulation scales. Kaihatu and Kirby (1995) followed the scaling approach of Yue and Mei (1980), where scales for amplitude of free surface elevation and the modulation scale δ (where A_n is function of δx and δ^1/2y) were chosen as follows (using the small ordering parameter ε = ka, or wave steepness):

$$A_n~O\left(\epsilon \right),O\left(\delta \right)~O\left({\epsilon }^2\right)$$ (1)

$$\frac{\partial A_n}{\partial x}=O\left(\epsilon \delta \right)~O\left({\epsilon }^3\right),\frac{\partial A_n}{\partial y}=O\left(\epsilon {\delta }^{1/2}\right)~O\left({\epsilon }^2\right)$$ (2)

where $A_n$ is the complex amplitude of free surface elevation of nth frequency component. They also chose a scale for depth change ∇_hh (where ∇_h is a gradient operator in horizontal coordinates) as O(ε²) to retain terms of bottom boundary condition to second order in ε (following Kirby and Dalymple, 1983). Horizontal derivatives of depth-dependent wave characteristics (e.g., wave number k, wave celerity C, the group velocity C_g) would then be restricted to O(ε²):

$$\frac{\partial h}{\partial x}=O\left({\epsilon }^2\right),\frac{\partial h}{\partial y}=O\left({\epsilon }^2\right)$$ (3)

$$\frac{\partial \left(kC{C}_g\right)}{\partial x}=O\left({\epsilon }^2\right),\frac{\partial \left(kC{C}_g\right)}{\partial y}=O\left({\epsilon }^2\right)$$ (4)

These choices of scale should allow the terms discarded in parabolic approximation to appear as a term at fifth order in ε:

$${\left[{\left(C{C}_g\right)}_n{A}_{nx}\right]}_x=O\left({\epsilon }^3\delta ,\epsilon {\delta }^2\right)~O\left({\epsilon }^5\right)$$ (5)

The nonlinear terms in the model of Kaihatu and Kirby (1995) (those proportional to the square of the wave amplitude) are only at second order in ε, but additional terms can be included in the parabolic equation because they are at lower order than the first discarded term. The addition of these consistently-ordered terms was implicitly addressed by Tang and Ouellet (1997); however, they did not fully consider terms up to fourth order in ε. It is anticipated that models which are consistent with the ordering would be expected to more fully describe triad wave-wave interaction between frequency components.

In the present study, we extend the model of Kaihatu and Kirby (1995) to derive a parabolic nonlinear mild slope equation consisting of all the nonlinear terms up to fourth order in ε by following similar approach and choices of scale as Kaihatu and Kirby (1995). Kirby (1991) and Kaihatu (2001) derived permanent form solutions from their associated evolution equations as a means of validating model behavior in deep and shallow water. We also develop a numerical permanent form solution of the derived model and compare the results of permanent form solution with those of the Stokes third-order theory and the stream function theory (Dean, 1965). We then also perform some comparisons to laboratory data to demonstrate the model’s skill.

We emphasize that the resulting model will remain weakly nonlinear, treating up to the quadratic nonlinear terms in the free surface boundary conditions. However, unlike previous weakly nonlinear models, more consistent connection between the spatial modulation scales of the amplitude and those of the depth variations is made. The various scales involved will be described in a later section.

2. Derivation of equations

2.1. Boundary value problem

A Cartesian coordinate system (x, y, z) is selected, with z taken positive vertically upwards from the still water level. We consider a surface gravity wavefield over a varying bottom in a horizontal direction, and the varying water depth is h(x, y). For inviscid, incompressible and irrotational fluid, the dimensional water wave boundary problem that the wave potential ϕ(x, y, z, t) (where t is time) and the free surface elevation η(x, y, t) satisfies is given as:

$${{\nabla }_h}^2\varphi +{\varphi }_{zz}=0;-h\le z\le \eta$$ (6)

$${\varphi }_z=-{\nabla }_{h}h\cdot {\nabla }_h\varphi \text{;}z=-h$$ (7)

$$g\eta +{\varphi }_t+\frac{1}{2}{\left({\nabla }_h\varphi \right)}^2+\frac{1}{2}{{\varphi }_z}^2=0;z=\eta$$ (8)

$${\eta }_t-{\varphi }_z+{\nabla }_h\eta \cdot {\nabla }_h\varphi =0;z=\eta$$ (9)

where ${\nabla }_h=\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y}\right)$ is a gradient operator in horizontal coordinates, g is the gravitational acceleration, and subscripts indicate partial derivatives.

Following the method of Bryant (1973), we retain dimensional quantities while noting that the leading order nonlinearity is O(ε²). Using Taylor series about still water level, z = 0, the truncated boundary value problem is formulated as:

$${{\nabla }_h}^2\varphi +{\varphi }_{zz}=0;-h\le z\le 0$$ (10)

$${\varphi }_z=-{\nabla }_{h}h\cdot {\nabla }_h\varphi ;z=-h$$ (11)

$$g\eta =-{\varphi }_t-\frac{1}{2}{\left({\nabla }_h\varphi \right)}^2-\frac{1}{2}{{\varphi }_z}^2-\eta {\varphi }_{tz}+O\left({\epsilon }^3\right);z=0$$ (12)

$${\eta }_t={\varphi }_z-{\nabla }_h\eta \cdot {\nabla }_h\varphi +\eta {\varphi }_{zz}+O\left({\epsilon }^3\right);z=0$$ (13)

Using the method of Smith and Sprinks (1975), Kaihatu and Kirby (1995) assumed a superposition of solutions for wave potential with depth dependence function from fully dispersive linear theory:

$$\varphi \left(x,y,z,t\right)=\sum _{n=1}^N{f}_n\left(h,z\right){\tilde{\varphi }}_n\left(x,y,t\right)$$ (14)

and

$$f_n\left(h,z\right)=\frac{\cosh k_n\left(h+z\right)}{\cosh k_{n}h}$$ (15)

where f_n is depth dependence function, k_n is the wave number, and ω_n is the wave angular frequency. Subscript n indicates the nth frequency component. The wave number k_n is determined by the linear dispersion relation:

$${{\omega }_n}^2=g{k}_n\tanh k_{n}h$$ (16)

After combining Eqs. (12) and (13) to eliminate η, combined free surface boundary condition for ϕ only is given:

$${\varphi }_z=-\frac{1}{g}\left[{\varphi }_{tt}+{{\left({\nabla }_h\varphi \right)}^2}_t+\frac{1}{2}{{\left({\varphi }_z\right)}^2}_t-\frac{1}{2g}{{\left({\varphi }_t\right)}^2}_{zt}-{\varphi }_t{\varphi }_{zz}\right];z=0$$ (17)

Depth dependence function f_n satisfies the following set of equations:

$$f_{nzz}-{k_n}^{2}f=0;-h\le z\le 0$$ (18)

$$f_{nz}=0;z=-h$$ (19)

$$f_{nz}=\frac{{{\omega }_n}^2}{g};z=0$$ (20)

2.2. Green’s second identity

Applying Green’s second identity for ϕ and f, we have:

$${\int }_{-h}^0\left(f_n{\varphi }_{nzz}-{\varphi }_n{f}_{nzz}\right)dz={\left[f_n{\varphi }_{nz}-{\varphi }_n{f}_{nz}\right]}_{z=0}^{}-{\left[f_n{\varphi }_{nz}-{\varphi }_n{f}_{nz}\right]}_{z=-h}^{}$$ (21)

Using Eqs. (10), (11), and (17) – (20), (21) becomes:

$$\begin{gathered} -{\int }_{-h}^0\left(f_n{{\nabla }_h}^2{\varphi }_n+{k_n}^2{\varphi }_n{f}_n\right)dz \\ =\left(N.L.\right)-{\left[\frac{1}{g}{\varphi }_{ntt}{f}_n+\frac{{{\omega }_n}^2}{g}{\varphi }_n{f}_n\right]}_{z=0}+{\left[f_n{\nabla }_{h}h\cdot {\nabla }_h{\varphi }_n\right]}_{z=-h} \end{gathered}$$ (22)

where nonlinear term is:

$$\left(N.L.\right)={\left[-\frac{f_n}{g}\left\{{{\left({\nabla }_h{\varphi }_n\right)}^2}_t+\frac{1}{2}{{\left({\varphi }_{nz}\right)}^2}_t-\frac{1}{2g}{\left({\varphi }_{nt}\right)}^2{}_{zt}-{\varphi }_{nt}{\varphi }_{nzz}\right\}\right]}_{z=0}$$ (23)

Horizontal derivatives of potential function ϕ are written with $\tilde{\varphi }$ and f:

$${\nabla }_h\varphi ={\nabla }_h\left(\tilde{\varphi }f\right)=\left({\nabla }_h\tilde{\varphi }\right)f+\tilde{\varphi }\frac{\partial f}{\partial h}{\nabla }_{h}h$$ (24)

$${{\nabla }_h}^2\varphi =\left({{\nabla }_h}^2\tilde{\varphi }\right)f+2{\nabla }_h\tilde{\varphi }\frac{\partial f}{\partial h}{\nabla }_{h}h+\tilde{\varphi }\frac{{\partial }^{2}f}{\partial h^2}{\left({\nabla }_{h}h\right)}^2+\tilde{\varphi }\frac{\partial f}{\partial h}{{\nabla }_h}^{2}h$$ (25)

Substituting Eqs. (24) and (25) into Eq. (22):

$$\begin{gathered} -{\int }_{-h}^0\left({\nabla }_h\left[\left({\nabla }_h{\tilde{\varphi }}_n\right){f_n}^2\right]+{\tilde{\varphi }}_n{f}_n\frac{{\partial }^2{f}_n}{\partial h^2}{\left({\nabla }_{h}h\right)}^2+\tilde{\varphi }{f}_n\frac{\partial f_n}{\partial h}{{\nabla }_h}^{2}h+{k_n}^2{\tilde{\varphi }}_n{f_n}^2\right)dz \\ =\left(N.L.\right)-{\left[\frac{1}{g}{\tilde{\varphi }}_{ntt}{f_n}^2+\frac{{{\omega }_n}^2}{g}{\tilde{\varphi }}_n{f_n}^2\right]}_{z=0}+{\left[{\nabla }_h{\tilde{\varphi }}_n\cdot {\nabla }_{h}h\left({f_n}^2\right)+{\tilde{\varphi }}_n{f}_n\frac{\partial f_n}{\partial h}{\left({\nabla }_{h}h\right)}^2\right]}_{z=-h} \end{gathered}$$ (26)

Using Leibnitz’ rule, Eq. (26) becomes:

$$\begin{gathered} -{\nabla }_h\left(\left[{\int }_{-h}^0{f_n}^{2}dz\right]{\nabla }_h{\tilde{\varphi }}_n\right)-{k_n}^2\left({\int }_{-h}^0{f_n}^{2}dz\right){\tilde{\varphi }}_n-{\int }_{-h}^0\left(f\frac{{\partial }^{2}f}{\partial h^2}\tilde{\varphi }{\left({\nabla }_{h}h\right)}^2+f\frac{\partial f}{\partial h}\tilde{\varphi }{{\nabla }_h}^{2}h\right)dz \\ =(N.L.)-{\left[\frac{1}{g}{f_n}^2{\tilde{\varphi }}_{ntt}+\frac{{{\omega }_n}^2}{g}{f_n}^2{\tilde{\varphi }}_n\right]}_{z=0}+{\left[f\frac{\partial f}{\partial h}\tilde{\varphi }{\left({\nabla }_{h}h\right)}^2\right]}_{z=-h} \end{gathered}$$ (27)

The water depth varies gradually (i.e., |∇h| ~ O(α) << 1, where α is a parameter characterizing the bottom slope). Following Kirby and Dalrymple (1983), we assume O(α) ~ O(ε²) in order to discard bottom boundary term of O(ε³) (i.e., –∇_hh∙∇_hϕ in Eq. 11) in the bottom boundary condition, similar to the manner in which the cubic nonlinear terms in the free surface boundary conditions are discarded. Additionally, the left hand side of Eq. (11) is zero by virtue of Eq. (19) (i.e., ${\left.{\varphi }_z\right|}_{z=-h}=0$ or ${\left.{f_{nz}\tilde{\varphi }}_n\right|}_{z=-h}=0$), thus satisfying the bottom boundary condition. Then, Eq. (27) is simplified by discarding the terms of O(εα²) ~ O(ε⁵) which is at higher order than the leading order nonlinearity O(ε²):

$$-{\nabla }_h\left(\left[{\int }_{-h}^0{f_n}^{2}dz\right]{\nabla }_h{\tilde{\varphi }}_n\right)-{k_n}^2\left({\int }_{-h}^0{f_n}^{2}dz\right){\tilde{\varphi }}_n=(N.L)-{\left[\frac{1}{g}{f_n}^2{\tilde{\varphi }}_{ntt}+\frac{{{\omega }_n}^2}{g}{f_n}^2{\tilde{\varphi }}_n\right]}_{z=0}$$ (28)

Discarding terms of O(εα²) ~ O(ε⁵), the nonlinear terms can be also expressed in terms of $\tilde{\varphi }$ and f:

$$\left(N.L.\right)=-\frac{1}{g}{\left[\begin{array}{l}{\left\{\sum _l\sum _m\left[f_l{f}_m\left({\nabla }_h{\tilde{\varphi }}_l\right)\cdot \left({\nabla }_h{\tilde{\varphi }}_m\right)+f_m\frac{\partial f_l}{\partial h}{\tilde{\varphi }}_l\left({\nabla }_h{\tilde{\varphi }}_m\right)\cdot {\nabla }_{h}h+f_l\frac{\partial f_m}{\partial h}{\tilde{\varphi }}_m\left({\nabla }_h{\tilde{\varphi }}_l\right)\cdot {\nabla }_{h}h\right]\right\}}_t\\ +\frac{1}{2}{\left\{\sum _l\sum _m{f}_{lz}{f}_{mz}{\tilde{\varphi }}_l{\tilde{\varphi }}_m\right\}}_t-\frac{1}{2g}{\left\{\sum _l\sum _m\left(f_{lz}{f}_m+f_l{f}_{mz}\right){\tilde{\varphi }}_{lt}{\tilde{\varphi }}_{mt}\right\}}_t\\ -\frac{1}{2}\left\{\sum _l\sum _m{f}_l{f}_{mzz}{\tilde{\varphi }}_{lt}{\tilde{\varphi }}_m+f_{lzz}{f}_m{\tilde{\varphi }}_l{\tilde{\varphi }}_{mt}\right\}\end{array}\right]}_{z=0}$$ (29)

where two arbitrary frequency modes, l and m, influence the nth frequency mode via triad wave-wave interaction between frequency components.

Expressions with depth dependence function are calculated as follows:

$$f_n\left(0\right)=1$$ (30)

$$f_{nz}\left(0\right)=\frac{{{\omega }_n}^2}{g}$$ (31)

$$f_{nzz}\left(0\right)={k_n}^2$$ (32)

$${\int }_{-h}^0{f_n}^{2}dz=\frac{{\left(C{C}_g\right)}_n}{g}$$ (33)

$${\int }_{-h}^0{\left(f_{nz}\right)}^{2}dz=\frac{{{\omega }_n}^2}{g}\left(1-\frac{C_{gn}}{C_n}\right)$$ (34)

$$\frac{\partial f_n}{\partial h}\left(0\right)={\left[\frac{k_n\sinh k_{n}z}{{\cosh }^2{k}_{n}h}\right]}_{z=0}=0$$ (35)

The time dependency will be factored out by assuming periodicity in time:

$${\tilde{\varphi }}_n\left(x,y,t\right)=\frac{{\widehat{\varphi }}_n}{2}{e}^{-i{\omega }_{n}t}+\frac{{{\widehat{\varphi }}_n}^{*}}{2}{e}^{i{\omega }_{n}t}$$ (36)

Applying Eqs. (30) – (36) into Eqs. (28) and (29), elliptic form of nonlinear mild slope equation is obtained:

$$\begin{gathered} {\nabla }_h\cdot \left[{\left(C{C}_g\right)}_n{\nabla }_h{\widehat{\varphi }}_n\right]+{k_n}^2{\left(C{C}_g\right)}_n{\widehat{\varphi }}_n \\ =-\frac{i}{4}\left[\sum _{l=1}^{n-1}2{\omega }_n{\nabla }_h{\widehat{\varphi }}_l\cdot {\nabla }_h{\widehat{\varphi }}_{n-l}-\left({\omega }_l{k_{n-l}}^2+{\omega }_{n-l}{k_l}^2\right){\widehat{\varphi }}_l{\widehat{\varphi }}_{n-l}+\left\{\frac{{\omega }_l{\omega }_{n-l}{\omega }_n}{g^2}\left({{\omega }_l}^2+{{\omega }_{n-l}}^2+{\omega }_l{\omega }_{n-l}\right){\widehat{\varphi }}_l{\widehat{\varphi }}_{n-l}\right\}\right] \\ -\frac{i}{2}\left[\sum _{l=1}^{N-n}2{\omega }_n{\nabla }_h{{\widehat{\varphi }}_l}^{*}\cdot {\nabla }_h{\widehat{\varphi }}_{n+l}-\left({\omega }_{n+l}{k_l}^2-{\omega }_l{k_l}^2\right){{\widehat{\varphi }}_l}^{*}{\widehat{\varphi }}_{n+l}-\left\{\frac{{\omega }_l{\omega }_{n+l}{\omega }_n}{g^2}\left({{\omega }_l}^2+{{\omega }_{n+l}}^2-{\omega }_l{\omega }_{n+l}\right){{\widehat{\varphi }}_l}^{*}{\widehat{\varphi }}_{n+l}\right\}\right] \end{gathered}$$ (37)

Compared to the elliptic equation of Kaihatu and Kirby (1995), Eq. (37) is almost identical except that the nonlinear terms have a form of $-\omega k^2\widehat{\varphi }\widehat{\varphi }$, rather than $\omega \widehat{\varphi }{\nabla }_h^2\widehat{\varphi }$ as seen in Equation 22 in Kaihatu and Kirby (1995). This term arises from using –ϕ_tϕ_zz in the combined free surface boundary condition (Eq. 17), rather than ϕ_t∇_h²ϕ as used by Kaihatu and Kirby (1995). We explain how these different nonlinear terms in the elliptic equations have a significant effect on the parabolic equations in section 3.

2.3. Parabolic approximation

While the model equation (37) is a comprehensive form of the nonlinear mild-slope equation, its elliptic formulation makes it difficult to implement. As mentioned earlier, elliptic equation requires pre-specification of all boundaries, which makes it difficult for open coast problems. In addition, the dependent variables in the equations oscillate at the scale of the individual waves, requiring high spatial resolution. To overcome these obstacles, we make use of the parabolic approximation (Lozano and Liu, 1980; Radder, 1979) to develop the model.

Incorporation of the parabolic approximation begins by imposing the following form for the velocity potential:

$${\widehat{\varphi }}_n\left(x,y\right)=-\frac{ig}{{\omega }_n}{A}_n\left(x,y\right)e^{i{\int }_{}^{}{k}_n\left(x,y\right)dx}$$ (38)

where A_n is complex amplitude having the small spatial change (i.e., horizontal derivatives), and representing waves that propagate primarily in the +x (axis normal to shore) direction.

Following the scaling approach of Yue and Mei (1980), orders of A_n and its horizontal derivatives are chosen, with derivatives of depth-dependent characteristics (e.g., wave number k, wave celerity C, the group velocity C_g) restricted to O(δ) ~ O(ε²) due to the assumption of the bottom slope, O(α) ~ O(ε²):

$$A_n~O\left(\epsilon \right)$$ (39)

$$\frac{\partial A_n}{\partial x}=O\left(\epsilon \delta \right)~O\left({\epsilon }^3\right),\frac{\partial A_n}{\partial y}=O\left(\epsilon {\delta }^{1/2}\right)~O\left({\epsilon }^2\right)$$ (40)

$$\frac{\partial h}{\partial x}=O\left(\alpha \right)~O\left({\epsilon }^2\right),\frac{\partial h}{\partial y}=O\left(\alpha \right)~O\left({\epsilon }^2\right)$$ (41)

$$\frac{\partial \left[{\left(kC{C}_g\right)}_n\right]}{\partial x}=O\left(\alpha \right)~O\left({\epsilon }^2\right),\frac{\partial \left[{\left(kC{C}_g\right)}_n\right]}{\partial y}=O\left(\alpha \right)~O\left({\epsilon }^2\right)$$ (42)

Linear terms of Eq. (37) are written with complex amplitude:

$${\left[{\left(C{C}_g\right)}_n{A}_{nx}\right]}_x+2i{\left(kC{C}_g\right)}_n{A}_{nx}+i{\left(kC{C}_g\right)}_{nx}{A}_n+{\left[{\left(C{C}_g\right)}_n{A}_{ny}\right]}_y$$ (43)

but which have the following ordering:

$${\left[{\left(C{C}_g\right)}_n{A}_{nx}\right]}_x={\left(C{C}_g\right)}_{nx}{A}_{nx}+{\left(C{C}_g\right)}_n{A}_{nxx}=O\left(\epsilon \alpha \delta ,\epsilon {\delta }^2\right)~O\left({\epsilon }^5\right)$$ (44)

$$\left[2i{\left(kC{C}_g\right)}_n{A}_{nx}\right],\left[i{\left(kC{C}_g\right)}_{nx}{A}_n\right]=O\left(\epsilon \delta ,\epsilon \alpha \right)~O\left({\epsilon }^3\right)$$ (45)

$${\left(C{C}_g\right)}_{ny}{A}_{ny}=O\left(\epsilon \alpha {\delta }^{1/2}\right)~O\left({\epsilon }^4\right)$$ (46)

$${\left(C{C}_g\right)}_n{A}_{nyy}=O\left(\epsilon \delta \right)~O\left({\epsilon }^3\right)$$ (47)

and we note that the orders of terms here are entirely equivalent to that of Kirby and Dalrymple (1983), including the smaller term (46) in the third order solvability condition.

At this stage it would be opportune to discuss the ordering in detail, in particular as it relates to the balance between nonlinearity ε, bottom slope α, and wave modulation scale δ. In the prior section we mentioned that only quadratic nonlinearity will be included, thus establishing the leading order nonlinearity as O(ε²). To be aligned with the parabolic equation ordering established above (as well as in prior studies), terms up to O(ε⁴) and equivalent (e.g., O(εαδ^1/2) in Eq. 45) can be retained. Since we are apriori limiting the nonlinearity to the leading order of O(ε²), the established ordering allows the retention of not only products of amplitude A_n to represent this order of nonlinearity, but also associated derivatives with respect to x and y. This is possible due to the ordering of O(δ) ~ O(α) ~ O(ε²) established in accord with Yue and Mei (1980) and thus is a consistent incorporation of quadratic nonlinearity within the framework of a parabolic mild slope equation model.

As an additional note, it is also worth comparison to the development of the linear parabolic mild slope equation (Lozano and Liu 1980; Tsay and Liu, 1982; Liu 1986). In these linear models, the leading order is O(ε), which scales the amplitude A_n as above (Eq. 39). By assuming O(δ) ~ O(α) ~ O(ε), these linear models retain the associated parabolic terms up to O(ε^2.5), determined by substituting ε for α and δ in O(εαδ^1/2).

With this ordering established, we discard terms of O(εδα, εδ²) ~ O(ε⁵) or higher in (44):

$$\begin{gathered} 2i{\left(kC{C}_g\right)}_n{A}_{nx}+i{\left(kC{C}_g\right)}_{nx}{A}_n+{\left[{\left(C{C}_g\right)}_n{\left(A_n\right)}_y\right]}_y \\ =\frac{1}{4}\sum _{l=1}^{n-1}\left[R_1{A}_l{A}_{n-l}+R_2{A}_{lx}{A}_{n-l}+R_3{A}_l{A}_{n-lx}+R_4{A}_{ly}{A}_{n-ly}\right]e^{i{\int }_{}^{}\left(k_l+k_{n-l}-k_n\right)dx} \\ +\frac{1}{2}\sum _{l=1}^{N-n}\left[S_1{A_l}^{*}{A}_{n+l}+S_2{A_{lx}}^{*}{A}_{n+l}+S_3{A_l}^{*}{A}_{n+lx}+S_4{A_{ly}}^{*}{A}_{n+ly}\right]e^{i{\int }_{}^{}\left(k_{n+l}-k_l-k_n\right)dx} \end{gathered}$$ (48)

where the interaction coefficients are:

$$R_1=\frac{g}{{\omega }_l{\omega }_{n-l}}\left[{{\omega }_n}^2{k}_l{k}_{n-l}+\left(k_l+k_{n-l}\right)\left({\omega }_{n-l}{k}_l+{\omega }_l{k}_{n-l}\right){\omega }_n\right]-\frac{{{\omega }_n}^2}{g}({{\omega }_l}^2+{\omega }_l{\omega }_{n-l}+{{\omega }_{n-l}}^2)$$ (49)

$$R_2=-2i\frac{g{{\omega }_n}^2{k}_{n-l}}{{\omega }_l{\omega }_{n-l}}$$ (50)

$$R_3=-2i\frac{g{{\omega }_n}^2{k}_l}{{\omega }_l{\omega }_{n-l}}$$ (51)

$$R_4=-2\frac{g{{\omega }_n}^2}{{\omega }_l{\omega }_{n-l}}$$ (52)

and

$$S_1=\frac{g}{{\omega }_l{\omega }_{n+l}}\left[{{\omega }_n}^2{k}_l{k}_{n+l}+\left(k_{n+l}-k_l\right)\left({\omega }_{n+l}{k}_l+{\omega }_l{k}_{n+l}\right){\omega }_n\right]-\frac{{{\omega }_n}^2}{g}\left({{\omega }_l}^2-{\omega }_l{\omega }_{n+l}+{{\omega }_{n+l}}^2\right)$$ (53)

$$S_2=2i\frac{g{{\omega }_n}^2{k}_{n+l}}{{\omega }_l{\omega }_{n+l}}$$ (54)

$$S_3=-2i\frac{g{{\omega }_n}^2{k}_l}{{\omega }_l{\omega }_{n+l}}$$ (55)

$$S_4=2\frac{g{{\omega }_n}^2}{{\omega }_l{\omega }_{n+l}}$$ (56)

The wave numbers k_n in the phase functions (the complex exponential terms in Eq. 38) are functions of both x and y; however, the integration in the phase functions is performed only in the x direction in keeping with the parabolic approximation. Kaihatu and Kirby (1995) selected the method of Lozano and Liu (1980) to factor out the y-dependence from phase function by using the y-averaged wave number ${\overline{k}}_n\left(x\right)$ as a reference wave number:

$${\widehat{\varphi }}_n\left(x,y\right)=-\frac{ig}{{\omega }_n}{a}_n\left(x,y\right)e^{i{\int }_{}^{}{\overline{k}}_n\left(x,y\right)dx}$$ (57)

where the amplitude functions are related:

$$A_n\left(x,y\right)=a_n\left(x,y\right)\exp \left(i{\int }_{}^{}{\overline{k}}_n\left(x\right)-k_n\left(x,y\right)dx\right)$$ (58)

Substituting Eq. (58) into Eq. (48) yields:

$$\begin{array}{l}2i{\left(kC{C}_g\right)}_n{a}_{nx}-2{\left(kC{C}_g\right)}_n\left({\overline{k}}_n-k_n\right)a_n+i{\left(kC{C}_g\right)}_{nx}{a}_n+{\left[{\left(C{C}_g\right)}_n{\left(a_n\right)}_y\right]}_y\\ =\frac{1}{4}\sum _{l=1}^{n-1}\left[\begin{array}{l}{R}_1{a}_l{a}_{n-l}+R_2\left(i\left\{{\overline{k}}_l-k_l\right\}{a}_l{a}_{n-l}+a_{lx}{a}_{n-l}\right)\\ +R_3\left(i\left\{{\overline{k}}_{n-l}-k_{n-l}\right\}{a}_l{a}_{n-l}+a_l{a}_{n-lx}\right)+R_4{a}_{ly}{a}_{n-ly}\end{array}\right]e^{i{\int }_{}^{}\left({\overline{k}}_l+{\overline{k}}_{n-l}-{\overline{k}}_n\right)dx}\\ +\frac{1}{2}\sum _{l=1}^{N-n}\left[\begin{array}{l}{S}_1{a_l}^{*}{a}_{n+l}+S_2\left(-i\left\{{\overline{k}}_l-k_l\right\}{a_l}^{*}{a}_{n+l}+{a_{lx}}^{*}{a}_{n+l}\right)\\ +S_3\left(i\left\{{\overline{k}}_{n+l}-k_{n+l}\right\}{a_l}^{*}{a}_{n+l}+{a_l}^{*}{a}_{n+lx}\right)+S_4{a_{ly}}^{*}{a}_{n+ly}\end{array}\right]e^{i{\int }_{}^{}\left({\overline{k}}_{n+l}-{\overline{k}}_l-{\overline{k}}_n\right)dx}\end{array}$$ (59)

We also reduce Eq. (48) to one dimension for comparisons with unidirectional laboratory experiments:

$$\begin{array}{l}{A}_{nx}+\frac{{\left(kC{C}_g\right)}_{nx}}{2{\left(kC{C}_g\right)}_n}{A}_n\\ =-\frac{i}{8{\left(kC{C}_g\right)}_n}\left[\begin{array}{l}\sum _{l=1}^{n-1}\left[R_1{A}_l{A}_{n-l}+R_2{A}_{lx}{A}_{n-l}+R_3{A}_l{A}_{n-lx}\right]e^{i{\int }_{}^{}\left(k_l+k_{n-l}-k_n\right)dx}\\ +2\sum _{l=1}^{N-n}\left[S_1{A_l}^{*}{A}_{n+l}+S_2{A_{lx}}^{*}{A}_{n+l}+S_3{A_l}^{*}{A}_{n+lx}\right]e^{i{\int }_{}^{}\left(k_{n+l}-k_l-k_n\right)dx}\end{array}\right]\end{array}$$ (60)

3. Comparison with other models

Two features are apparent in the present model. First, most models (Freilich and Guza, 1984; Liu et al., 1985; Agnon et al., 1993; Tang and Ouellet, 1997; Eldeberky and Madsen, 1999 among many others) were developed under the assumption that the order of amplitude A_n is the same as that of the spatial gradient of depth (or O(ε)) as well as modulation scale δ (where A_n is function of δx and δ^1/2y). Because of this assumption, the discarded terms in parabolic approximation have the same order as the x-derivatives of amplitude in the nonlinear term A_xA. The calculation of scale under the assumption that A ~ O(ε), A_x ~ O(ε²), (CC_g)_x ~ ∇_hh ~ O(ε) was taken in the previous models (e.g., Kaihatu and Kirby, 1995):

$$A_{lx}{A}_{n-l}~{\left[\left(C{C}_g\right)A_{nx}\right]}_x~O\left({\epsilon }^3\right)$$ (61)

However, we can include the x-derivative nonlinear term A_xA in the parabolic equation, since it was assumed that amplitude is of lower order than depth change (or O(ε²)) and modulation scale δ. For the present study, Eq. (62) shows the comparison of order between the discarded terms in parabolic approximation and the nonlinear term A_xA which is one of the additional terms:

$${\left[\left(C{C}_g\right)A_{nx}\right]}_x~O\left({\epsilon }^5\right)<A_{lx}{A}_{n-l}~O\left({\epsilon }^4\right)$$ (62)

3.1. Kaihatu and Kirby (1995)

The parabolic model of Kaihatu and Kirby (1995) is:

$$\begin{array}{l}2i{\left(kC{C}_g\right)}_n{A}_{nx}+i{\left(kC{C}_g\right)}_{nx}{A}_n+{\left[{\left(C{C}_g\right)}_n{\left(A_n\right)}_y\right]}_y\\ =\frac{1}{4}\sum _{l=1}^{n-1}{R}_1{A}_l{A}_{n-l}{e}^{i{\int }_{}^{}\left(k_l+k_{n-l}-k_n\right)dx}+\frac{1}{2}\sum _{l=1}^{N-n}{S}_1{A_l}^{*}{A}_{n+l}{e}^{i{\int }_{}^{}\left(k_{n+l}-k_l-k_n\right)dx}\end{array}$$ (63)

where R₁ and S₁ are the same as Eq. (48).

To examine the effect of x-derivative nonlinear term A_xA, Eqs. (48) and (63), respectively, were simplified into one-dimensional equations for constant depth. The model of Kaihatu and Kirby (1995) (Eq. 63) is simplified into Eq. (64):

$$A_{nx}=-\frac{i}{8{\left(kC{C}_g\right)}_n}\left[\sum _{l=1}^{n-1}\left[R_1{A}_l{A}_{n-l}\right]e^{i\theta }+2\sum _{l=1}^{N-n}\left[S_1{A_l}^{*}{A}_{n+l}\right]e^{i\psi }\right]$$ (64)

and the present model (Eq. 48) is simplified into Eq. (65):

$$A_{nx}=-\frac{i}{8{\left(kC{C}_g\right)}_n}\left[\begin{array}{l}\sum _{l=1}^{n-1}\left[R_1{A}_l{A}_{n-l}+R_2{A}_{lx}{A}_{n-l}+R_3{A}_l{A}_{n-lx}\right]e^{i\theta }\\ +2\sum _{l=1}^{N-n}\left[S_1{A_l}^{*}{A}_{n+l}+S_2{A_{lx}}^{*}{A}_{n+l}+S_3{A_l}^{*}{A}_{n+lx}\right]e^{i\psi }\end{array}\right]$$ (65)

where interaction coefficients are the same as Eq. (48), and:

$$\theta ={\int }_{}^{}{k}_l+k_{n-l}-k_{n}dx$$ (66)

$$\psi ={\int }_{}^{}{k}_{n+l}-k_l-k_{n}dx$$ (67)

are denoted as phase mismatches. These mismatches serve to mitigate the interaction between triads of frequencies by detuning the interactions. The interactions between wave number (e.g., k_l + k_n–l – k_n) have the same order of magnitude as weak dispersion μ² (where μ = kh). In deep water, Kaihatu and Kirby (1997) argued that the phase mismatch can become large, which may violate the assumption of slow horizontal variation of amplitude.

To focus on the influence of x-derivative nonlinear term A_xA on phase mismatch in the numerical model, we created an artificial case where all the values of wave characteristics (e.g., A, h, ω, k) at the wave maker station (x = 0) are given, and then calculate the amplitudes at the next grid point with forward difference method. To isolate the impact of μ², phase mismatches θ and ψ are expressed as μ²κΔx (where κ is non-dispersive wave number and Δx is the step size of the model grid). The use of κ insures dimensional homogeneity with the phase mismatches, while the size of μ determines the relative depth. In this artificial case, A_x is the only unknown, and the other variables including A can be represented with constants K₁ and K₂ since these values are specified. Eq. (68) is derived from Eq. (64), which is the simplified version of the model of Kaihatu and Kirby (1995):

$$A_x=K_1\exp \left(i{\mu }^2\kappa \mathrm{\Delta }x\right)$$ (68)

For the present model (Eq. 65, which is the simplified version of Eq. 48), we obtain:

$$A_x=K_1\exp \left(i{\mu }^2\kappa \mathrm{\Delta }x\right)+K_2{A}_x\exp \left(i{\mu }^2\kappa \mathrm{\Delta }x\right)$$ (69)

Solving Eq. (69) of the present model for A_x results in:

$$A_x=\frac{K_1\exp \left(i{\mu }^2\kappa \mathrm{\Delta }x\right)}{1-K_2\exp \left(i{\mu }^2\kappa \mathrm{\Delta }x\right)}$$ (70)

While K₁ involves the square of the amplitude (i.e., AA or AA*), K₂ involves only a single amplitude. Therefore, it is reasonable to assume K₂ to be greater than K₁, in accordance with the ordering. Assigning values of K₁ = 1 and K₂ = 3 or 5, the real part of A_x for each model is represented as:

$$\mathrm{Re}\left(A_x\right)=\{\begin{array}{l}\cos \left({\mu }^2\kappa \mathrm{\Delta }x\right);\text{Kaihatu}\text{and}\text{Kirby}\text{(1995)}\\ \\ \frac{\cos \left({\mu }^2\kappa \mathrm{\Delta }x\right)-3}{-6\cos \left({\mu }^2\kappa \mathrm{\Delta }x\right)+10}\text{;}\text{Present}\text{model}\text{with}{K}_2=3\\ \frac{\cos \left({\mu }^2\kappa \mathrm{\Delta }x\right)-5}{-10\cos \left({\mu }^2\kappa \mathrm{\Delta }x\right)+26}\text{;}\text{Present}\text{model}\text{with}{K}_2=5\end{array}$$ (71)

Fig. 1 shows the real part of amplitude change depending on the dimensionless grid size κΔx in shallow water (μ = 0.5) and deep water (μ = 3).

Fig. 1.

Comparison of Re(A_x) between model of Kaihatu and Kirby (1995) and the present model: (Top) μ = 0.5; (bottom) μ = 3 (Dashed: model of Kaihatu and Kirby (1995); Dotted: present model with K₂ = 3; Dash-dot: present model with K₂ = 5).

It is desirable for the real part of the amplitude gradient A_x to demonstrate a low degree of sensitivity to the dimensionless step size κΔx. From Fig. 1, it is clear that the model of Kaihatu and Kirby (1995) shows a high degree of oscillation in deep water (μ = 3) as a function of κΔx. However, also from Fig. 1, we can see that the amplitude of the mismatch might not violate the assumption of slow varying amplitude in deep water, because the real part of exp[iμ²κΔx] will only oscillate between – 1 and 1 even with large μ².

In contrast, mismatches in the present model (with x-derivative nonlinear term A_xA) shows a greatly reduced sensitivity to the dimensionless grid size. Because exp[iμ²κΔx] is multiplied by A_x on the right-hand side of Eq. (69) and serves as an oscillating coefficient of A_x, A_x might oscillate to a lesser degree (or remain almost constant similar to the case of zero-mismatch) even in deep water (μ = 3). In the interest of generalization, the simplest example for N = 2 is provided in the appendix.

Another source of difference between the present model and that of Kaihatu and Kirby (1995) concerns the combined free surface boundary condition (Eq. 17). We note that the last nonlinear term in the combined free surface boundary condition (17) is the vertical derivative –ϕ_tϕ_zz. In previous studies (Agnon et al., 1993; Kaihatu and Kirby, 1995; Tang and Ouellet, 1997; Eldeberky and Madsen, 1999 among many others), the Laplace equation was used to trade the vertical derivative for horizontal derivatives, leading to the nonlinear term ϕ_t∇_h²ϕ. It was generally further combined with ∇_hϕ_t∙∇_hϕ, yielding ∇_h(ϕ_t∇_hϕ) in the combined free surface boundary condition, affecting the resulting form of the model (e.g., Eq. 23). To compare representation of vertical derivatives with that of horizontal derivatives, we develop two equations with complex amplitude A_n solely based on these derivatives:

$${{\nabla }_h}^2\varphi ={{\nabla }_h}^2{\left\{Af{e}^{i\left[-\omega t+{\int }_{}^{}kdx\right]}\right\}}_{z=0}=\left\{-k^{2}A+A_{xx}+2ik{A}_x+i{k}_{x}A+A_{yy}+A{\left[\frac{{\partial }^{2}f}{\partial h^2}\right]}_{z=0}{\left({\nabla }_{h}h\right)}^2\right\}{e}^{i\left[{\int }_{}^{}kdx-\omega t\right]}$$ (72)

$$-{\varphi }_{zz}={\left\{-A{f}_{zz}{e}^{i\left[-\omega t+{\int }_{}^{}kdx\right]}\right\}}_{z=0}=\left\{-k^{2}A\right\}{e}^{i\left[{\int }_{}^{}kdx-\omega t\right]}$$ (73)

Since water depth h and complex amplitude A_n vary gradually in the horizontal direction, higher order terms arise from horizontal derivative of wave potential ϕ. As a result, when we represent the nonlinear term with the horizontal derivative (i.e., ϕ_t∇_h²ϕ), we discard some of terms higher order than O(ε⁴), for example, the terms proportional to A_xx or (∇_hh)², while the representing with vertical derivative (i.e., –ϕ_tϕ_zz) allows the parabolic equation to include the term without any discarded term. This allows the entirety of the nonlinear term –ϕ_tϕ_zz (or ϕ_t∇_h²ϕ) to be retained. In addition, including fewer horizontal derivative terms, such as double derivative of complex amplitude with respect to y, can enhance model stability and reduce iterations required for numerical convergence.

3.2. Tang and Ouellet (1997)

The first model of Tang and Ouellet (1997) in dimensional form is:

$$\begin{gathered} 2i{\left(kC{C}_g\right)}_n{A}_{nx}+i{\left(kC{C}_g\right)}_{nx}{A}_n+{\left[{\left(C{C}_g\right)}_n{\left(A_n\right)}_y\right]}_y \\ =\frac{1}{4}\sum _{l=1}^{n-1}\left[\left(R_1+R_{11}+R_{12}\right)A_l{A}_{n-l}+R_4{A}_{ly}{A}_{n-ly}+R_5{A}_l{A}_{n-lyy}+R_6{A}_{lyy}{A}_{n-l}\right]e^{i{\int }_{}^{}\left(k_l+k_{n-l}-k_n\right)dx} \\ +\frac{1}{2}\sum _{l=1}^{N-n}\left[\left(S_1+S_{11}+S_{12}\right){A_l}^{*}{A}_{n+l}+S_4{A_{ly}}^{*}{A}_{n+ly}+S_5{A_l}^{*}{A}_{n+lyy}+S_6{A_{lyy}}^{*}{A}_{n+l}\right]e^{i{\int }_{}^{}\left(k_{n+l}-k_l-k_n\right)dx} \end{gathered}$$ (74)

where R₁, R₄, and S₁, S₄ are the same as Eq. (48), and:

$$R_{11}=-i\frac{g{\omega }_n{\omega }_l}{{\omega }_l{\omega }_{n-l}}{k}_{n-lx}$$ (75)

$$R_{12}=-i\frac{g{\omega }_n{\omega }_{n-l}}{{\omega }_l{\omega }_{n-l}}{k}_{lx}$$ (76)

$$R_5=-\frac{g{\omega }_n}{{\omega }_{n-l}}$$ (77)

$$R_6=-\frac{g{\omega }_n}{{\omega }_l}$$ (78)

and

$$S_{11}=-i\frac{g{\omega }_n{\omega }_{n+l}}{{\omega }_l{\omega }_{n+l}}{k}_{lx}$$ (79)

$$S_{12}=-i\frac{g{\omega }_n{\omega }_l}{{\omega }_l{\omega }_{n+l}}{k}_{n+lx}$$ (80)

$$S_5=-\frac{g{\omega }_n}{{\omega }_{n+l}}$$ (81)

$$S_6=\frac{g{\omega }_n}{{\omega }_l}$$ (82)

Tang and Ouellet (1997) did not include the x-derivative nonlinear term, of which coefficients are R₂, R₃, S₂, and S₃ as well. In addition, the model of Tang and Ouellet (1997) has a greater number of terms (e.g., terms with R₁₁, R₁₂, R₅, and R₆) than the present model. These additional terms resulted from the double horizontal derivative nonlinear term ϕ_t∇_h²ϕ discussed above, after discarding higher order terms. In addition, the Laplace equation (10) dictates that Eq. (72) should be identical to Eq. (73), thus the sum of terms except for –k²A in Eq. (72) must cancel. As a result, discarding some of the higher order terms in ϕ_t∇_h²ϕ might lead to incomplete consideration of ϕ_t∇_h²ϕ (or –ϕ_tϕ_zz). Paradoxically, the present equation does not include the terms present in Tang and Ouellet (1997), but the final term in the combined free surface boundary condition (i.e., –ϕ_tϕ_zz) insures full consideration of this boundary condition to the specified order.

3.3. Permanent form solutions

Kirby (1991) and Kaihatu (2001) derived a permanent form solution for the purpose of validating model behavior in deep and shallow water by comparison to analytic and numerical wave theories. For similar purposes, we also developed a numerical permanent form solution of the present model, and compared the results of this solution to those of Stokes third-order theory (e.g., Mei, 1983) and stream function theory (Dean, 1965). In addition, Kaihatu (2001) and Eldeberky and Madsen (1999) improved the models by introducing the second-order relationship between amplitudes of wave potential and those of free surface elevation in the dynamic free surface boundary condition. To investigate the effect of second-order correction, the relationship between amplitudes of ϕ and η was used (Kaihatu, 2001):

$$\eta =\sum _{n=1}^N\frac{B_n}{2}{e}^{i\left[{\int }_{}^{}{k}_n\left(x,y\right)dx-{\omega }_{n}t\right]}+\text{C.C.}$$ (83)

where B_n denotes amplitude of η, and C.C. denotes conjugate complex

$$B_n=A_n+\frac{1}{4g}\left[\sum _{l=1}^{n-1}I{A}_l{A}_{n-l}{e}^{i{\int }_{}^{}\left({\overline{k}}_l+{\overline{k}}_{n-l}-{\overline{k}}_n\right)dx}+2\sum _{l=1}^{N-n}J{A_l}^{*}{A}_{n+l}{e}^{i{\int }_{}^{}\left({\overline{k}}_{n+l}-{\overline{k}}_l-{\overline{k}}_n\right)dx}\right]$$ (84)

where

$$I={{\omega }_l}^2+{\omega }_l{\omega }_{n-l}+{{\omega }_{n-l}}^2-g^2\frac{k_l{k}_{n-l}}{{\omega }_l{\omega }_{n-l}}$$ (85)

$$J={{\omega }_l}^2-{\omega }_l{\omega }_{n+l}+{{\omega }_{n+l}}^2-g^2\frac{k_l{k}_{n+l}}{{\omega }_l{\omega }_{n+l}}$$ (86)

Following Kaihatu (2001), a redefined velocity potential and free-surface elevation were used:

$$\varphi =\sum _{n=1}^N\frac{-ig}{2{\omega }_n}{\tilde{a}}_n{f}_n\left(z\right)e^{i\left[{\int }_{}^{}n\left(k_1+\tilde{k}\right)dx-{\omega }_{n}t\right]}+\text{C.C.}$$ (87)

$$\eta =\sum _{n=1}^N\frac{{\tilde{b}}_n}{2}{e}^{i\left[{\int }_{}^{}n\left(k_1+\tilde{k}\right)dx-{\omega }_{n}t\right]}+\text{C.C.}$$ (88)

where k₁ is wave number of the first frequency mode, and $\tilde{k}$ is the difference between the linear wave number and non-dispersive wave number. Full details of the method can be found in Kaihatu (2001).

Assuming no change in energy flux, one-dimensional permanent form solutions for model of Kaihatu and Kirby (1995) were proposed (Kaihatu, 2001):

$$\left[n\left(k_1+\tilde{k}\right)-k_n\right]{\tilde{a}}_n+\frac{1}{8{\left(\omega C_g\right)}_n}\left[\sum _{l=1}^{n-1}{R}_1{\tilde{a}}_l{\tilde{a}}_{n-l}+2\sum _{l=1}^{N-n}{S}_1{\tilde{a}}_l{\tilde{a}}_{n+l}\right]=0$$ (89)

For the present model, we obtain one-dimensional equations for permanent solution:

$$\begin{gathered} \left[n\left(k_1+\tilde{k}\right)-k_n\right]{\tilde{a}}_n+\frac{1}{8{\left(\omega C_g\right)}_n} \\ \cdot \left[\begin{array}{l}\sum _{l=1}^{n-1}\left\{R_1+R_2\left(i\left[l\left(k_1+\tilde{k}\right)-k_l\right]\right)+R_3\left(i\left[\left(n-l\right)\left(k_1+\tilde{k}\right)-k_{n-l}\right]\right)\right\}{\tilde{a}}_l{\tilde{a}}_{n-l}\\ +2\sum _{l=1}^{N-n}\left\{S_1+S_2\left(-i\left[l\left(k_1+\tilde{k}\right)-k_l\right]\right)+S_3\left(i\left[\left(n+l\right)\left(k_1+\tilde{k}\right)-k_{n+l}\right]\right)\right\}{\tilde{a}}_l{\tilde{a}}_{n+l}\end{array}\right]=0 \end{gathered}$$ (90)

The specified wave height H provides an additional equation:

$$H=2\sum _{n=1,3,5}^N{\tilde{b}}_n=2\sum _{n=1,3,5}^N\left[{\tilde{a}}_n+\frac{1}{4g}\left(\sum _{l=1}^{n-1}I{\tilde{a}}_l{\tilde{a}}_{n-l}+2\sum _{l=1}^{N-n}J{\tilde{a}}_l{\tilde{a}}_{n+l}\right)\right]$$ (91)

We compare the phase speeds and free surface profiles from the permanent form solutions of Kaihatu (2001) and present model with those from the Stokes third-order theory and 15th-order stream function theory. Table 1 shows the condition used in comparison. In Figs. 2–5, we see that, while both permanent form solutions perform well, the present model has a better performance than model of Kaihatu and Kirby (1995). In particular, the present model outperforms that of Kaihatu and Kirby (1995) when both are compared to phase speed estimates from Stokes third-order theory for the condition of h = 9 m and H = 3 m. There seems to be little effect of second-order correction on the phase speed and free surface profile. The better fit appears at the smaller wave height. In the regimes of deep water, and shallow water, permanent form solutions of the present model agree favorably with those from the Stokes third-order theory, and 15th-order stream function theory, respectively.

Table 1.

Conditions for permanent form solution.

Theory	Comparison	T (s)	N	h (m)	H (m)
Stokes third-order	Phase speed	5	10	20–9	0.5, 1.0, 2.0, 3.0
	Free surface	5	10	20, 9	3
Stream function	Phase speed	10	15	10–1	0.1
	Free surface	10	15	10,1	0.1

Fig. 2.

Comparison of phase speed between permanent-form solutions and third-order Stokes theory: (a) H = 0.5 m; (b) H = 1 m; (c) H = 2 m; (d) H = 3 m (Dashed: present model; Dotted: present model with second-order correction; Dash-dot: model of Kaihatu and Kirby (1995); Solid: model of Kaihatu and Kirby (1995) with second-order correction).

Fig. 5.

Comparison of free surface profiles between permanent-form solutions and 15th-order stream function theory: (Top) h = 10 m; (Bottom) h = 1 m (Solid: stream function theory; Dashed: present model; Dotted: present model with second-order correction; Dash-dot: model of Kaihatu and Kirby (1995); Dash-cross: model of Kaihatu and Kirby (1995) with second-order correction).

4. Comparison to experimental data

We compare numerical results from the present model to both experimental data and previous models. The Crank-Nicolson method was adopted to model the two-dimensional parabolic equation (59). The implementation of the method is similar to that of Liu et al. (1985). We use the notation that ${\left[a_{n,j}^i\right]}^k$ represents the nth harmonic function at x (= iΔx, i = 1 to N_x) and y (= j Δy, j = 1 to N_y). Because the nonlinear terms need to be centered between i and i+1, iteration is required. This was performed as follows:

$${\left[a_{l,j}^{i+1}{a}_{n-l.j}^{i+1}\right]}^{k+1}=\{\begin{array}{l}{a}_{l,j}^i{a}_{n-l.j}^i;k=0\\ {\left[a_{l,j}^{i+1}\right]}^k{\left[a_{n-l.j}^{i+1}\right]}^k;k\ge 1\end{array}$$ (92)

where the superscripts k + 1 and k are the current and previous iterations, respectively. The initial guess (i.e., k = 0) are obtained from the previous x-level solutions. The iteration procedure is stopped and converged solutions are calculated when the relative error between two successive iteration solutions is less than 10⁻³:

$$\frac{\left|{\left(a_{n,j}^{i+1}\right)}^{k+1}-{\left(a_{n,j}^{i+1}\right)}^k\right|}{\left|{\left(a_{n,j}^{i+1}\right)}^k\right|}<{10}^{-3}$$ (93)

For cases where a sine wave was specified at the offshore boundary, the amplitude for the first harmonic was prescribed with the recorded input values at the wave maker station for each experiment, and the initial conditions for higher harmonic waves are set to zero.

A grid convergence procedure was followed, in which the grid size is continually reduced until further reduction does not make any difference in results. Table 2 shows grid sizes and the number of grids used for computation of each experiment in this study.

Table 2.

Grid sizes and the total number of grids for computation of each experiment.

Data	Chaplain et al. (1992)	Whalin (1971)	Berkhoff et al. (1982)
Δx (m)	0.094	0.24	0.083
Δy (m)	-	0.085	0.083
Grids	250	100×74	261×243

To quantitatively examine the performance of the models, we used the Index of Agreement (IOA) representing the ratio of mean square error and the potential error (Willmott, 1982). The IOA ranges from 0 to 1, with 1 corresponding to the ideal model:

$$I_a=1-\frac{\sum _{j=1}^{N_y}\sum _{i=1}^{N_x}{\left(a_{n,j}^i-a_{n.j,obs}^i\right)}^2}{\sum _{j=1}^{N_y}\sum _{i=1}^{N_x}{\left[\left|a_{n,j}^i-{\overline{a}}_{n.j,obs}^i\right|+\left|a_{n,j,obs}^i-{\overline{a}}_{n.j,obs}^i\right|\right]}^2}$$ (94)

where $a_{n,j}^i$ is the computed nth harmonic function from models at x (= iΔx) and y (= j Δy), and $a_{n,j,obs}^i$ is the observed nth harmonic function from experimental data at x (= iΔx) and y (= j Δy), and over bar indicates an average.

4.1. Chapalain et al. (1992)

Chapalain et al. (1992) conducted a laboratory experiment to model the transformation of long, small-amplitude waves in constant depth. The primary goal of this experiment was to investigate the energy transfer between a small number of harmonic components due to nonlinear interactions. The experiment was conducted in a wave flume that was 33.54 meters long and 1.3 meters deep. A wavemaker at the end of the flume generated monochromatic sinusoidal waves, which were allowed to evolve over the flat bottom. Regularly spaced wave gauges in the tank were used to record time series of the free surface elevation, and the resulting records were analyzed to determine the amplitudes of each harmonic. The experiment captured the phenomenon of recurrence, in which energy flowing from low frequencies to higher ones reversed this transfer to recapture the initial state. This phenomenon was first seen in water waves in the experiments of Boczar-Karakiewicz (1972); Mei and Ünlüata (1972) analyzed this phenomenon using shallow water wave theory. Table 3 shows the wave parameters, the number of harmonics, and the nonlinearity parameters calculated for this experiment. We used N = 5 harmonics for all cases, but the first four harmonics are compared with that of the experimental data.

Table 3.

Wave parameters of Chapalain et al. (1992), and the nonlinearity parameters.

Case	a0 (m)	T (s)	h (m)	kh	Ur
A	0.042	2.5	0.4	0.530	0.373
C	0.042	3.5	0.4	0.371	0.764
D	0.0355	2.5	0.3	0.454	0.574
H	0.035	3	0.4	0.436	0.460

Figs. 6–7 compare the numerical results for each case obtained by the models of Kaihatu and Kirby (1995) (hereinafter KK95 in figures and tables), Freilich and Guza (1984) (hereinafter FG84 in figures and tables), and the present model (both with and without the second-order correction) with the experimental data of Chapalain et al. (1992). For unidirectional wave, the Kadomtsev-Petviashvili (K-P) model of Liu et al. (1985) (hereinafter KP85 in figures and tables) is reduced to that of Freilich and Guza (1984), and the model of Tang and Ouellet (1997) (hereinafter TO97 in figures and tables) is also reduced to that of Kaihatu and Kirby (1995); thus, these other models are implicitly included. The agreement between experimental data and numerical solution of the present model is reasonable. The first-, second-, and third-harmonic wave amplitudes are in good agreement with experimental data, but the numerical model underestimates the fourth-harmonic component. For the first- and second-harmonic, the present model mostly outperforms the previous models in terms of the amplitudes and the recurrence distances, with the exception that Freilich and Guza (1984) is in closer agreement in Case C, where kh is small. For the fourth-harmonic, on the other hand, Kaihatu and Kirby (1995) agrees more favorably in most of the comparisons. The inclusion of the second-order correction with Eq. (84) increases the amplitude, leading to better agreement for the fourth harmonics. Since there is no bottom variation in this experiment, A_xA are the only terms of present model that are additional to that of Kaihatu and Kirby (1995). As seen in section 3, these added terms contribute to nonlinear dispersion effects despite the explicit use of the linear dispersion relation for wavenumber calculations.

Fig. 6.

Comparison of wave amplitudes between models and data of Chapalain et al. (1992) for Case A: (a) first harmonic; (b) second harmonic; (c) third harmonic; (d) fourth harmonic; for Case C: (e) first harmonic; (f) second harmonic; (g) third harmonic; (h) fourth harmonic (Solid: present model; Dashed: present model with second-order correction; Dotted: model of Kaihatu and Kirby (1995); Dash-dot: model of Freilich and Guza (1984); Circle: experimental data).

Fig. 7.

Comparison of wave amplitudes between models and data of Chapalain et al. (1992) for Case D: (a) first harmonic; (b) second harmonic; (c) third harmonic; (d) fourth harmonic; for Case H: (e) first harmonic; (f) second harmonic; (g) third harmonic; (h) fourth harmonic (Solid: present model; Dashed: present model with second-order correction; Dotted: model of Kaihatu and Kirby (1995); Dash-dot: model of Freilich and Guza (1984); Circle: experimental data).

4.2. Whalin (1971)

Whalin (1971) carried out a laboratory experiment to investigate wave convergence and wave refraction over a varying bottom, shown in Fig. 8. The bottom configuration consists of two regions of constant depth connected with a tilted cylinder; the cylinder acts as a refractive focal lens, and the amplified wave amplitude at the focus point expected to lead to energy transfer among wave frequencies. In Table 6, we summarize wave parameters, the number of harmonics, and the nonlinearity parameters calculated for this experiment. The values with subscripts 1 and 2 in Table 6 refer to the deep and shallow portion of the tank, respectively (h₁ = 0.457 m, h₂ = 0.152 m). The reflective lateral boundary conditions are used along the side-wall and the centerline of the wave tank:

$$\frac{\partial a}{\partial y}=0\text{at}y=0\text{and}3.048\text{m}$$ (95)

Fig. 8.

Wave tank bathymetry of Whalin (1971).

Table 6.

Wave parameters of Whalin (1971), the number of harmonics, and the nonlinearity parameters.

Case	a0 (m)	T (s)	N	k1h 1	k2h 2	Ur 1	Ur 2
31	0.0068	3	5	0.468	0.264	0.068	0.640
32	0.0098	3	5	0.468	0.264	0.098	0.922
33	0.0146	3	5	0.468	0.264	0.146	1.373
21	0.0075	2	3	0.735	0.402	0.030	0.305
22	0.0106	2	3	0.735	0.402	0.043	0.431
23	0.0149	2	3	0.735	0.402	0.060	0.605
11	0.0097	1	2	1.921	0.873	0.006	0.084
12	0.0195	1	2	1.921	0.873	0.012	0.168

Figs. 9–12 show wave amplitudes along the centerline of the wave tank for each case, comparing the values obtained by the models of Kaihatu and Kirby (1995), Tang and Ouellet (1997), the K-P model of Liu et al. (1985), and the present model with the experimental data of Whalin (1971). Since the case of T = 1 s shows obvious differences after incorporation of second-order correction (Eq. 84) only for the case of T = 1 s, Figs. 12 compares results with the second-order correction by model of Kaihatu and Kirby (1995), and the present model. We note that the case of T = 1 s was obviously outside the range of validity for Boussinesq so that we do not compare the K-P model of Liu et al. (1985) for the case of T = 1 s.

Fig. 9.

Comparison of wave amplitudes between models and data of Whalin (1971) for Case 31: (a) first harmonic; (b) second harmonic; (c) third harmonic; for Case 32: (d) first harmonic; (e) second harmonic; (f) third harmonic; for Case 33: (g) first harmonic; (h) second harmonic; (i) third harmonic (Solid: present model; Dashed: model of Kaihatu and Kirby (1995); Dotted: model of Tang and Ouellet; Dash-dot: K-P model of Liu et al. (1985); Circle: experimental data).

Fig. 12.

Comparison of wave amplitudes between models and data of Whalin (1971) for Case 11: (a) first harmonic; (b) second harmonic; for Case 12: (c) first harmonic; (d) second harmonic (Solid: present model with second-order correction; Dashed: model of Kaihatu and Kirby (1995) with second-order correction; Dotted: model of Tang and Ouellet; Circle: experimental data).

For the case of T = 3 s, no model shows a good agreement with first harmonic amplitudes, while the second- and third-harmonic amplitudes are well predicted by all the models. Frictional dissipation of the waves could partially cause this deviation at first harmonic amplitudes (Liu et al, 1985). None of the models discussed here include the effect of viscous damping in the boundary conditions, while there was a very small amount of wave damping from the viscous boundary layers in wave tank (Whalin, 1971). The present model uniformly overpredicts the first harmonic wave amplitudes, and underpredicts the third harmonic wave amplitudes; this behavior is also seen with the other models in this comparison.

For the case of T = 2 s, the amplitudes predicted by all nonlinear parabolic mild-slope equation models are nearly identical. For the cases of T = 1 s and 2 s, where the shallow water assumption might not be applicable, the results of the K-P model (Liu et al., 1985) show significant discrepancy between theory and experiment. As the initial amplitude a₀ decreases, the performance of the present model improves. In particular, the present model appears to outperform the other models for the cases of the smallest amplitude from T = 1, 2 s (i.e., Case 11 and 21). For example, the value of a₀ in Case 21 is the smallest than that of the other cases of T = 2 s, therefore, Case 21 has the smallest value of Ur, which might support the ordering of present study rather than that of Boussinesq-type model, based on O(Ur) ~ O(1).

Kaihatu and Kirby (1995) argued that case of T = 1 s is a severe test of their model due to the great value of kh (i.e., k₁h₁ = 1.921 at the wave maker corresponds to intermediate depth), causing considerable phase mismatches and severe amplitude change oscillation with increasing grid size. However, the present model shows better accuracy in case of T = 1 s, which could imply that the effect of phase mismatches is alleviated by x-derivative nonlinear terms A_xA. From Fig. 12 (T = 1 s), we note that the present model is generally in agreement with oscillating second harmonic amplitude. These oscillations in amplitude occur with wave propagating in x-direction; the amplitudes in nonlinear terms are thus variable in x and the cause for the oscillations in A. This is in contrast with the situation described in Fig. 1, where the oscillation is directly related to the grid size rather than the wave amplitude propagation characteristics. As explained in comparisons with model of Kaihatu and Kirby (1995) of section 3, the exponential function multiplied by A_x on the right-hand side of Eq. (48), might alleviate the effect of phase mismatch, as it serves as an oscillating coefficient.

4.3. Berkhoff et al. (1982)

Berkhoff et al. (1982) conducted a laboratory experiment over a varying bottom with an elliptic shoal situated on a 1:50 slope, shown in Fig. 13, to investigate the behavior of wave focusing by a submerged shoal. Unlike the experiment of Whalin (1971), wave propagation in this experiment would be adequately described with a linear wave model (e.g., Berkhoff et al., 1982). However, it has been shown (e.g., Kirby and Dalrymple, 1984; Suh et al., 1990; Wei and Kirby, 1995) that wave propagation processes in this experiment are better replicated by a nonlinear wave model.

Fig. 13.

Bathymetry and gauge Layout of Berkhoff et al. (Dashed: gauge transects).

We ran a parabolic linear mild slope equation (linearized form of Eq. 48), Kaihatu and Kirby (1995), Tang and Ouellet (1997), and the present model against the data of Berkhoff et al. (1982). Table 10 shows the wave parameters, the number of harmonics, and the nonlinearity parameters calculated for this experiment. The values with subscription 1 and 2 are the values corresponding to the initial condition and the values at the shoal crest, respectively (h₁ = 0.45 m, h₂ = 0.128 m). At the wave maker station, a sinusoidal wave was input to the model and the value of kh = 1.895 would likely lead to appreciable phase mismatch. We used N = 2 harmonics, but the first harmonics were used to obtain relative amplitudes and the relative amplitudes are compared with the results of the experimental data. To investigate the effect of phase mismatch in the present model, we used N = 3 as well. For frequency domain mild-slope equations, increasing N results in more nonlinear terms with phase mismatch. Therefore, when N = 3 is used, frequency domain mild-slope equations can be expected to show more obvious discrepancy between numerical and experimental results. The sum of odd number harmonics was used in calculating relative amplitude (i.e., a₁ + a₃) for N = 3. We apply reflective lateral boundaries; however, the width of the numerical model grid is wide enough so that sidewall reflections do not affect the wave processes near the shoal. These lateral boundary conditions are:

$$\frac{\partial a}{\partial y}=0\text{at}y=0\text{and}20\text{m}$$ (96)

Table 10.

Wave parameters of Berkhoff et al. (1982), the number of harmonics, and the nonlinearity parameters.

a0 (m)	h1 (m)	h2 (m)	T (s)	N	kh 1	kh 2	Ur 1	Ur 2
0.0232	0.45	0.128	1	2, 3	1.895	0.788	0.014	0.292

Fig. 14 show the results of the models using N = 2 (for nonlinear models) at the gauge locations. Although the high kh value would lead to appreciable mismatch, all mild slope equations models, including our model, agree favorably with the experimental data. This can be interpreted as a result of few nonlinear terms and the use of fundamental harmonics only, because N = 2 was used. Comparison of the IOA values at gauge transect 3–6 (Tables 11 and 12) show that the present model outperforms the other models used. This is likely due to the additional AA_x and A_yA_y terms in Eq. (48). While the linear model used here shows reasonable results at gauge transect 1, its performance worsens at succeeding gauges. Berkhoff et al. (1982) noted that gauge transect 6 is near an amphidromic point; this is also where the present model performs particularly well, capturing the reduction in amplitude seen in the data far better than other models.

Fig. 14.

Comparison of normalized wave amplitudes between models and data of Berkhoff et al. (1982) for N = 2: (a) gauge 1; (b) gauge 2; (c) gauge 3; (d) gauge 4; (e) gauge 5; (f) gauge 6; (g) gauge 7; (h) gauge 8 (Solid: present model; Dashed: Linearized Mild slope equation; Dotted: model of Kaihatu and Kirby (1995); Dash-dot: model of Tang and Ouellet (1997); Circle: experimental data).

Table 11.

Comparison of IOA between models at 1–8 Gauge of Berkhoff et al. (1982) for N = 2.

Gauge	Present	LMSE	KK95	TO97
1	0.9127	0.9416	0.9036	0.9312
2	0.9726	0.9200	0.9767	0.9083
3	0.9801	0.9701	0.9665	0.9728
4	0.9877	0.9427	0.9769	0.9833
5	0.9789	0.8476	0.9583	0.9445
6	0.9910	0.8082	0.8151	0.8377
7	0.9304	0.8205	0.9452	0.8703
8	0.8694	0.7353	0.9388	0.8448

Table 12.

Comparison of IOA between models at 1–8 Gauge of Berkhoff et al. (1982) for N = 3.

Gauge	Present	LMSE	KK95	TO97
1	0.9043	0.9416	0.9062	0.9332
2	0.9735	0.9200	0.9706	0.9099
3	0.9838	0.9701	0.9348	0.9686
4	0.9838	0.9427	0.9399	0.9623
5	0.9682	0.8476	0.8696	0.8965
6	0.9862	0.8082	0.6498	0.7241
7	0.9397	0.8205	0.8846	0.8405
8	0.8461	0.7353	0.7488	0.8166

The results using N = 3, shown in Fig. 15, validates the hypothesis that the predictions of Kaihatu and Kirby (1995) and Tang and Ouellet (1997) become worse than the results of N = 2. The results of present model, on the other hand, shows better prediction at several gauges. This result is a manifestation of the ability of the model to alleviate phase mismatches, as discussed earlier. The present model shows notably better prediction skill at several gauges. Additionally, there are no obvious improvements using the second-order correction with Eq. (84).

Fig. 15.

Comparison of normalized wave amplitudes between models and data of Berkhoff et al. (1982) for N = 3: (a) gauge 1; (b) gauge 2; (c) gauge 3; (d) gauge 4; (e) gauge 5; (f) gauge 6; (g) gauge 7; (h) gauge 8 (Solid: present model; Dashed: Linearized Mild slope equation; Dotted: model of Kaihatu and Kirby (1995); Dash-dot: model of Tang and Ouellet (1997); Circle: experimental data).

5. Summary and conclusions

Kaihatu and Kirby (1995) and Tang and Ouellet (1997) extended the model of Agnon et al. (1993) from one-dimensional frequency domain model to the two-dimensional frequency domain models, including quadratic nonlinear terms which represent triad wave-wave interaction between frequency components. One major limitation of previous models from the boundary value problem are the possibility of severe phase mismatches between wave frequencies in high relative water depth, which violates the assumption of slow variation in the horizontal direction.

In this study, we further extended the model of Kaihatu and Kirby (1995) by following the scaling approach of Yue and Mei (1980) and Kirby and Dalrymple (1983). Since the order of amplitude used is lower than the orders of both depth change and modulation scale δ (where A_n is function of δx and δ^1/2y), the x-derivative nonlinear term A_xA is retained in the model equation. In addition, horizontal derivative was replaced by vertical derivative in the combined free surface boundary condition, therefore, more triad wave-wave interaction between frequency components is taken into consideration in the model than previous models.

In the region of shallow water, it was shown that the results of present model are in closer agreement with that of experiment compared to frequency domain Boussinesq models. In the region of intermediate or deep water, it is shown that the present model outperforms the previous models from the boundary value problem. Experiment of Chaplain et al. (1992) is one-dimensional case for constant depth, therefore, the only differences between the present and the previous models (Kaihatu and Kirby, 1995 and Tang and Ouellet, 1997) are x-derivative nonlinear term A_xA. As a result, this is probably the effect of x-derivative nonlinear term A_xA, alleviating the effect of phase mismatch.

Future work will focus on using the model to simulate irregular wave processes. The consideration of breaking and surf zone processes will be required to properly generalize the model for situations where these phenomena are present. This therefore will necessitate the incorporation of wave breaking (e.g., Kaihatu and Kirby, 1995, Eldeberky and Battjes, 1996) into the model.

Fig. 3.

Comparison of free surface profiles between permanent-form solutions and third-order Stokes theory: (Top) h = 20 m; (Bottom) h = 9 m (Solid: third-order Stokes theory; Dashed: present model; Dotted: present model with second-order correction; Dash-dot: model of Kaihatu and Kirby (1995); Dash-cross: model of Kaihatu and Kirby (1995) with second-order correction).

Fig. 4.

Comparison of phase speed between permanent-form solutions and 15th-order stream function theory (Dashed: present model without second-order correction; Dotted: present model; Dash-dot: model of Kaihatu and Kirby (1995); Solid: model of Kaihatu and Kirby (1995) with second-order correction).

Fig. 10.

Comparison of wave amplitudes between models and data of Whalin (1971) for Case 21: (a) first harmonic; (b) second harmonic; (c) third harmonic; for Case 22: (d) first harmonic; (e) second harmonic; (f) third harmonic; for Case 23: (g) first harmonic; (h) second harmonic; (i) third harmonic (Solid: present model; Dashed: model of Kaihatu and Kirby (1995); Dotted: model of Tang and Ouellet; Dash-dot: K-P model of Liu et al. (1985); Circle: experimental data).

Fig. 11.

Comparison of wave amplitudes between models and data of Whalin (1971) for Case 11: (a) first harmonic; (b) second harmonic; for Case 12: (c) first harmonic; (d) second harmonic (Solid: present model; Dashed: model of Kaihatu and Kirby (1995); Dotted: model of Tang and Ouellet; Circle: experimental data).

Table 4.

Comparison of IOA between models for Case A and C of Chapalain et al. (1992).

Case	Harmonics	Present	Present with correction	KK95	FG84
A	1	0.9475	0.9408	0.7848	0.7563
	2	0.9344	0.9399	0.8466	0.8067
	3	0.6678	0.7081	0.7448	0.6595
	4	0.3286	0.4840	0.8706	0.3804
C	1	0.9900	0.9850	0.9365	0.9980
	2	0.9734	0.9811	0.9529	0.9873
	3	0.8691	0.8856	0.8522	0.9606
	4	0.8505	0.8968	0.8889	0.8230

Table 5.

Comparison of IOA between models for Case D and H of Chapalain et al. (1992).

Case	Harmonics	Present	Present with correction	KK95	FG84
D	1	0.9667	0.9598	0.8759	0.8761
	2	0.9776	0.9793	0.9536	0.9077
	3	0.7547	0.7704	0.7745	0.7214
	4	0.4679	0.6233	0.7757	0.4787
H	1	0.8711	0.8787	0.9342	0.9228
	2	0.9602	0.9684	0.9026	0.9504
	3	0.8237	0.8387	0.7541	0.8720
	4	0.6101	0.7451	0.9009	0.6120

Table 7.

Comparison of IOA between models for T = 3 s of Whalin (1971).

Case	Harmonics	Present	KK95	TO97	KP85
31	1	0.6077	0.6375	0.6330	0.6120
	2	0.9168	0.9389	0.9280	0.9298
	3	0.8905	0.9376	0.9251	0.9179
32	1	0.4944	0.5771	0.5659	0.5213
	2	0.9551	0.9712	0.9629	0.9692
	3	0.9316	0.9727	0.9621	0.9622
33	1	0.5217	0.6917	0.6676	0.5994
	2	0.9633	0.9725	0.9672	0.9664
	3	0.9304	0.9794	0.9718	0.9799

Table 8.

Comparison of IOA between models for T = 2 s of Whalin (1971).

Case	Harmonics	Present	KK95	TO97	KP85
21	1	0.9460	0.9493	0.9461	0.8909
	2	0.9653	0.9534	0.9543	0.9263
	3	0.9639	0.9320	0.9310	0.9189
22	1	0.9519	0.9516	0.9481	0.8932
	2	0.9660	0.9629	0.9656	0.9534
	3	0.9789	0.9845	0.9861	0.9851
23	1	0.7601	0.7976	0.7951	0.6538
	2	0.9168	0.9077	0.9151	0.9034
	3	0.9530	0.9531	0.9566	0.9548

Table 9.

Comparison of IOA between models for T = 1 s of Whalin (1971).

Case	Harmonics	Present	Present (W/C)	KK95	KK95 (W/C)	TO97
11	1	0.9418	0.9417	0.9387	0.9387	0.9360
	2	0.8688	0.8834	0.8768	0.8795	0.8834
12	1	0.8455	0.8458	0.8320	0.8315	0.8316
	2	0.8507	0.9032	0.8751	0.8734	0.8908

Highlights.

• Consistent ordering by bathymetric change leads to a consistent nonlinear model

• Additional terms alleviate highly variable model behavior caused by phase mismatch

• Consistent nonlinear model shows better comparison to data than earlier models

Appendix

In section 3 we illustrated the impact of the additional terms developed by the consistent ordering on the phase mismatch, which was often mentioned (e.g., Kaihatu and Kirby, 1995) as a drawback of their nonlinear mild slope equation. In this appendix we illustrate the impact of these additional terms on the phase mismatch for the case of N = 2, or two frequency components. At the wave maker station (x = 0), the equations can be simplified since all values are given, with the x-derivatives of amplitude the only unknown.

(1). Kaihatu and Kirby (1995) (Eq. 64)

When N = 2 and a grid step size of Δx, the one-dimensional equations of Kaihatu and Kirby (1995) can be written for a constant depth:

$$A_{1x}=-i\frac{2S_1}{8{\left(kC{C}_g\right)}_1}{A_1}^{*}{A}_2\exp \left[i\left(k_2-2k_1\right)\mathrm{\Delta }x\right]$$ (A.1)

$$A_{2x}=-i\frac{R_1}{8{\left(kC{C}_g\right)}_2}{A_1}^2\exp \left[i\left(2k_1-k_2\right)\mathrm{\Delta }x\right]$$ (A.2)

The phase mismatches can be simplified up to second order in μ, similarly to what was done in section 3:

$$k_2-2k_1={\mu }^2\kappa +O\left({\mu }^4\right)$$ (A.3)

Except for x-derivatives of amplitude and phase mismatches, all the variables are given values, so the equations can be written with complex number constant K combining all given values (these are distinguished by subscripts and primes):

$$A_{1x}=K_1\exp \left[i{\mu }^2\kappa \mathrm{\Delta }x\right]$$ (A.4)

$$A_{2x}={K_1}^{\prime }\exp \left[-i{\mu }^2\kappa \mathrm{\Delta }x\right]$$ (A.5)

(2). Present model (Eq. 65)

When N = 2, the one-dimensional form of the present model can be written for a constant depth as follows:

$$A_{1x}=-\frac{i}{8{\left(kC{C}_g\right)}_1}\left\{2S_1{A_1}^{*}{A}_2+2S_2{A_{1x}}^{*}{A}_2+2S_3{A_1}^{*}{A}_{2x}\right\}\exp \left[i\left(k_2-2k_1\right)\mathrm{\Delta }x\right]$$ (A.6)

$$A_{2x}=-\frac{i}{8{\left(kC{C}_g\right)}_2}\left\{R_1{A_1}^2+R_2{A}_{1x}{A}_1+R_3{A}_1{A}_{1x}\right\}\exp \left[i\left(2k_1-k_2\right)\mathrm{\Delta }x\right]$$ (A.7)

and again, combining all known values into a complex constant K:

$$A_{1x}=\left(K_1+K_2{A_{1x}}^{*}+K_3{A}_{2x}\right)\exp \left[i{\mu }^2\kappa \mathrm{\Delta }x\right]$$ (A.8)

$$A_{2x}=\left({K_1}^{\prime }+{K_2}^{\prime }{A}_{1x}\right)\exp \left[-i{\mu }^2\kappa \mathrm{\Delta }x\right]$$ (A.9)

where K₂ʹ includes both terms with R₂ and R₃

Substituting Eq. (A.9) into (A.8) to eliminate A_2x:

$$\left(1-K_3{K_2}^{\prime }\right)A_{1x}=K_1\exp \left[i{\mu }^2\kappa \mathrm{\Delta }x\right]+K_2{A_{1x}}^{*}\exp \left[i{\mu }^2\kappa \mathrm{\Delta }x\right]+K_3{K_1}^{\prime }$$ (A.10)

If A_1x has terms with exp[iaμ²κΔx], then a can take the values 1, – a + 1, or 0. Therefore, A_1x can be set as follows with the complex number coefficients C:

$$A_{1x}=C_1\exp \left[i{\mu }^2\kappa \mathrm{\Delta }x\right]+C_2\exp \left[i\frac{{\mu }^2}{2}\kappa \mathrm{\Delta }x\right]+C_3$$ (A.11)

Finally, A_2x can be also obtained:

$$A_{2x}=\left({K_1}^{\prime }+C_3{K_2}^{\prime }\right)\exp \left[-i{\mu }^2\kappa \mathrm{\Delta }x\right]+C_2{K_2}^{\prime }\exp \left[-i\frac{{\mu }^2}{2}\kappa \mathrm{\Delta }x\right]+C_1{K_2}^{\prime }$$ (A.12)

Substituting Eqs. (A.11) and (A.12) into Eq. (A.8), the coefficients C can be obtained from the given amplitude:

$$\left(1-K_3{K_2}^{\prime }\right)C_2=K_2{C_2}^{*}$$ (A.13)

$$\left(1-K_3{K_2}^{\prime }\right)C_1=K_1+K_2{C_3}^{*}$$ (A.14)

$$\left(1-K_3{K_2}^{\prime }\right)C_3=K_3{K_1}^{\prime }+K_2{C_1}^{*}$$ (A.15)

Finally, unlike model of Kaihatu and Kirby (1995) (i.e., Eq. A.4), present model contains the term C₃ which has the effect of a buffer against high degrees of oscillation. Additionally, terms proportional to A_1x exp[(iμ²κΔx)/2] appear; these terms have longer cycle than exp[iμ²κΔx], and can thus also alleviate the impact of phase mismatch. This appendix therefore shows that the present model is less sensitive to the grid size κΔx in this artificial case than the model of Kaihatu and Kirby (1995) for the case of two distinct frequencies, thus extending the related discussion in Section 3.