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We propose alternative forms of the Boussinesq equations which extend the equations of Madsen and Schäffer by introducing extra nonlinear terms during enhancement. Theoretical analysis shows that nonlinear characteristics are considerably improved. A numerical implementation of one-dimensional equations is described. Three tests involving strongly nonlinear evolution, namely, regular waves propagating over an elevated bar feature in a tank with an otherwise constant depth, wave group transformation over constant water depth, and nonlinear shoaling of unsteady waves over a sloping beach, are simulated by the model. The model is found to be effective.

In the past 3 decades, great strides have brought Boussinesq-type equations into the family of operational coastal wave prediction models because they possess both dispersion and nonlinear properties and require relatively little computation effort. The development of variants of the theory was trigged by improving linear and nonlinear properties of the model. Extensive reviews have been provided in [

Earlier works have concentrated on improving linear dispersion of classical Boussinesq equations (which are accurate only for weakly dispersive and mildly nonlinear water waves), thus allowing the model to treat a wider range of water depths. Extensive research results (e.g., [

Subsequent to the initial work on improved linear properties, the next topic to draw attention is the problem of improving model’s nonlinear accuracy. One of the critical steps in this process is to develop equations with fully nonlinear characteristics [

Efforts have been made to address this issue within second-order Boussinesq equations. Zou [

Alternative forms of Boussinesq equations with second-order full nonlinearity are presented. The model is based on the equations of Madsen and Schäffer [

A set of governing equations with second-order full nonlinearity was presented in [

The dispersion contained in (

The first step in the procedure is to apply the operator

All the terms introduced in (

Only linear properties, that is, dispersion and shoaling, are considered for optimization in the previously discussed method. In this section, we focus instead on improving model’s ((

Compared with (

In this section, we conduct Fourier analysis of enhanced equations (

Solving the first-order problem yields the dispersion relation of the model (

Comparisons of phase celerity (a) and group velocity (b) between equations (solid line) and Padé

Specific values of

Comparisons of shoaling gradient between the equations (solid line) and the analytical solution (dashed line).

Nonlinear property is analyzed by solving second- and third-order problems. Parameter

Comparisons of second-order (a), third-order (b) transfer functions, and amplitude dispersion (c) between the enhanced equations (solid line) and the original equations (dashed line).

The numerical scheme mainly follows procedures presented in FUNWAVE [

Governing equations are discretized using a staggered grid (by contrast, a uniform grid is adopted in [

The entire computation domain is enclosed by impermeable walls, wherein the horizontal velocity is set to zero. Sponger layers are placed in front of the solid walls to absorb wave energy propagating out of the computational domain. The incident waves are internally generated in the computational domain by adding a source function into the mass equation. The implicit filter [

To validate the proposed model, three tests involving strongly nonlinear evolution of waves are chosen for the simulation, including (a) 1-D regular wave trains propagating over trapezoidal bar feature in an otherwise constant depth tank, (b) nonlinear evolution of 1-D wave groups over constantly deep water, and (c) nonlinear shoaling of unsteady waves up to the breaking point over a mildly sloping beach. Both enhanced and original models will be used in the simulations, and the computed results will be compared. Available experimental data or analytical solutions will be adopted as reference.

Wave propagation over a submerged bar is an extremely complicated process. Nonlinear interactions transfer energy from the leading wave component to higher harmonics as waves propagate onto the front slope of the bar and begin to become steep. At the leeside of the bar, water becomes deep and nonlinear coupling of higher harmonics with the fundamental wave becomes progressively weaker. Therefore, each of the Fourier components is released as free waves with their own bound higher harmonics. Each component travels with a different speed, thus resulting in a fairly complicated process. Numerical prediction of this process requires higher-order linear and nonlinear accuracies of the model. Such prediction has been widely used as a benchmark test for validating Boussinesq-type models [

Here, Case (a) of Luth et al. experiments [

The experimental setup of Luth et al. [

Wave surface elevations of Luth’s experiment (circle) [

The enhancement only improves the nonlinearity of the model, whereas dispersion and shoaling remain unchanged. As such, checking the spatial distribution of higher harmonics for this case is more reasonable. The corresponding numerical results for the two models are plotted in Figure

The spatial variation of the harmonics of Luth’s experiment (circle) [

First harmonic

Second harmonic

Third harmonic

Fourth harmonic

Surface elevation

The experiment for nonlinear evolution of wave groups over a constant water depth [

In the experiment, a wave maker is driven by the following signal in a flume with a constant water depth of 0.6 m:

Numerical results from the enhanced and original models are presented in Figure

Experimental data (circle) of wave elevations of wave groups and the numerical results of the enhanced model (solid) and the original model (dashed).

Figure

Comparisons of amplitude spectra at

The asymmetric growth of the spectrum shown in the figures presents another nonlinear characteristic of nonlinear wave group evolution apart from the amplitude dispersion, that is, the generation of new free-wave components due to resonant wave-wave interaction. This is a third-order

The unsteady shoaling test is chosen to investigate the effect of the improvements made to the original equations. In this procedure, the most nonlinear case from [

Initial (a) and final (b) surface elevations, with crest and trough envelopes and underlying bathymetry.

Both the original and enhanced equations are used for simulation, and the numerical results are compared with the numerical solution from the potential flow results [

Comparisons of crest and trough envelopes between the enhanced model, the original model, and the potential flow results.

An alternative form of the Boussinesq-type model, which extends the equations in [

Analysis of the equations for monochromatic waves is as follows.

In one dimension, and on a horizontal bottom, (

(i) First-order equations

(ii) Second-order equations

(iii) Third-order equations

The authors would like to thank the finical support from the National Natural Science Foundation of China (Grant nos. 51009018 and 51079024), the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (Grant no. LP1105), National Marine Environment Monitoring Center, State Oceanic Administration (Grant no. 201206), and Key Laboratory of Water-Sediment Sciences and Water Disaster Prevention of Hunan Province (Grant nos. 2012SS02 and 2013SS02). Special thanks are given to three anonymous reviewers. Their valuable suggestions greatly help improving paper’s quality.