Gang Wang*,Zhong-in Sun,Jun-liang Gaoc,Xiao-zhou Ma

aJiangsu Key Laboratory of Coast Ocean Resources Development and Environment Security(Hohai University),Nanjing 210098,China

bNanjing Hydraulic Research Institute,Nanjing 210029,China

cSchool of Naval Architecture and Ocean Engineering,Jiangsu University of Science and Technology,Zhenjiang 212003,China

dState Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology,Dalian 116023,China

1.Introduction

Edge waves are the trapped modes of longshore periodic wave motion,which exist along straight or gently curving shorelines and on uniformly sloping,concavely or convexly curving offshore depth profiles.Several mechanisms have been shown to generate edge waves in nature.Small-scale edge waves are usually excited by incident wind waves or swells through an instability mechanism(Guza and Davis,1974).Large-scale edge waves can be generated by a moving storm or a nearshore earthquake(Gonzalez et al.,1995;Greenspan,1956;Chang,1995).Edge waves are also directly related to infragravity wave motions in the coastal zone(Sch¨affer,1994).

Edge waves play an important role in many phenomena in the nearshore region.Bowen(1969)and Bowen and Inman(1969)described how edge waves can control the longshore spacing of rip currents.Bowen and Inman(1971)investigated the generation of crescentic bars with an edge wave mechanism,and hypothesized that the same mechanism could control the growth and spacing of beach cusps.Chen et al.(2004)showed that the resonance of Hua-Lien Harbor was induced by edge waves through physical and numerical experiments.Nazarov and Videman(2010)analytically examined the existence of trapped waves along infinite arrays of threedimensional periodic structures.Wang et al.(2013)used the properties of the real wavenumber's correspondence to propagating waves and the imaginary wavenumber's correspondence to evanescent waves to examine the instability of linear edge waves.Their results show that the offshore wavenumber should change from an imaginary number in deep water to a real number in shallow water,which indicates that edge waveswould disintegrate in the cross-shore direction from evanescent waves to propagating waves.Seo(2015)derived analytical solutions of edge waves forced by landslides moving in the cross-shore direction and moving atmospheric disturbances parallel to the shoreline,respectively.

Many laboratory experiments have been performed to study the generation and properties of edge waves.Most of these experiments have been concerned with the generation of subharmonic edge waves by a monochromatic wave normally approaching the plane beach(Blondeaux and Vittori,1995).Buchan and Pritchard(1995),Ursell(1952),and Yeh(1983)generated edge waves directly at the beach with a wedgeshaped wave paddle,which was hinged offshore and oscillated in the longshore direction.Unfortunately,there are many physical limitations to edge-wave generation.The wavepaddle motion is limited by the stroke of the hydraulic ram,and the linear approximation of the paddle motion usually used for the generation of waves with asymptotical offshore decay also limits the experimental precision.

Significant progress in numerical wave models has been made in the last two decades.Within the area of Boussinesq equations,coupled with innovative extensions to the theoretical framework,modeling schemes have been shown to be accurate in predicting a wide range of nearshore hydrodynamic behaviors,including wave propagation and shoaling,wave-current interaction,wave breaking and the generation of nearshore circulation,and wave structure interaction with a range of additional actions.However,the Boussinesq models,which are widely used in thefield of coastal engineering,are rarely involved in the simulation of edge waves.To the best of the authors'knowledge,only Chen et al.(2004)have directly generated edge waves at open boundaries using Boussinesq models.Unfortunately,they did not quantitatively examine the simulation results with the analytical solution.

In order to implement the numerical simulation of edge waves,this study attempted to use the internal wave generation method in the Boussinesq model.The edge wave was directly generated by adding the sinusoidal mass transport to the mass conservation equation,and the simulation results were quantitatively examined with the analytical solution.This numerical technique was also used to investigate the fundamental behavior and nonlinear dynamics of edge waves.

2.Green's shallow-water solution and its internal generation method for extended Boussinesq equations

The main difficulty in edge wave simulation is inputting the desired signals directly into the model.Previous researchers mostly focused on Ursell's full-water solution(Ursell,1952)or Eckart's shallow-water solution(Eckart,1951),coupled with their nonlinear extensions.As viscous effects are important in boundary layers on a sloping beach(Mahony and Pritchard,1980),edge waves in such cases cannot be described simply by the irrotational motion of an inviscidfluid.Thus,the error between the target and numerically simulated waves is very large there.Green III(1986)provided an analytical solution of edge waves near a seawall.It is useful in numerical simulations,as there is no moving shoreline,and the linear assumption can be satisfied exactly near the shoreline.This section introduces Green's shallow-water solution and its internal generation method for the Boussinesq equations.

2.1.Green's shallow-water solution

A straight and long beach with a constant slope s=tanθ is considered,as shown in Fig.1.The shore runs in the y-direction;the x-direction is offshore,and the extended bottom and mean sea level meet at x=0.There is a vertical wall at the shore,where water depth h=h0.The waves are assumed to be long waves,and to travel along the shore.The water surface elevation η,the cross-shore velocity u,and the longshore velocity v are written respectively as

where ζ,U,and V are the free surface elevation,and crossshore and longshore velocities that depend only on x,respectively;t is time;k is the wavenumber;and ω is the angular frequency.Considering that the water depth can be written as h=sx,Eq.(1)can be substituted into the linear shallow-water equation to obtain

and

where g is the gravitational acceleration.Introducing the new independent and non-dimensional variable

and the new dependent variable Ψ(τ）,defined as follows:

Eq.(2)is transformed into a confluent hypergeometric equation:

Fig.1.Sketch of long beach on sloping bottom with a vertical wall.

where

The general solution for Eq.(7)is

where A and B are constants,and F(τ, α)and G(τ, α)are confluent hypergeometric functions of the first and second kinds,respectively.

The wave amplitude must approach zero at infinity,and it is known that exp(-kx)F(2kx,α)increases without limit for a large value of x,unless α is a non-positive integer.On the other hand,the function exp(-kx)G(2kx,α)is known to approach zero for a large value of x,and α is determined by the boundary condition at x0=h0/s,where there is no normalflux,with

Thus,it is necessary that A=0.

For a given angular frequency ω,the values of α and k,which satisfy Eqs.(8)and(10),can be easily found by evaluating the confluent hypergeometric function G using the routine developed by Zhang and Jin(1996).

2.2.Internal generation of edge waves for extended Boussinesq equations

Proper treatment of incident wave generation and radiation of outgoing waves are important in any numerical wave model.The usual method of generating incident waves and absorbing outgoing waves at the boundary at the same time is to use an internal generation line and a sponge layer.The target waves are generated inside the computational domain,while the radiated waves are absorbed in the sponge layer located on the outer boundary of the domain.This method has already been widely used in Boussinesq models for monochromatic and irregular waves(Lee et al.,2001).The same technique is used for edge waves.

Nwogu(1993)used the velocity variable at an arbitrary level to derive equations and obtained a new form of Boussinesq-type equations,in which the dispersion property can be optimized by choosing the velocity variable at an adequate level.The mass conservation equation is

and the associated momentum conservation equation is

where u=(u,v)is horizontal velocity at a reference elevation zα=-0.531h.

Waves are generated internally by adding a sinusoidal water mass along a straight line to Eq.(11).Lee et al.(2001)proposed a linear source function added to the mass equation,which is

where ηIis the water surface elevation of incident waves,and Cgis the group velocity of incident waves.They further conducted numerical experiments to illustrate its capability of generating linear and nonlinear waves.This technique is similar to the process of a wave-paddle sinusoid oscillating to produce the waves in physical experiments.

The internal wave generation line is usually arranged in the constant-depth region to produce regular and random waves,and the source function is consistent along the line.There is a constant slope for edge waves,and the wave amplitude decays in the offshore direction.In this case,the wave generation line is arranged offshore,which reproduces the process of a wedgeshaped wave paddle hinged offshore swinging in the longshore direction perpendicular to the beach surface,with the amplitude decaying asymptotically offshore.However,as there is no approximation in the offshore decay,this technique can produce the exactflow field associated with edge waves.

The linear source function for edge waves is

where a is the amplitude at the shoreline x0,and Cegis the group velocity for edge waves obtained from the dispersion relation in Eq.(8),which is

and ζ(x）can be further expressed as

3.Model tests

Numerical experiments were conducted to verify the technique described above.As the governing equations are nonlinear Boussinesq equations,the fundamental standing edge wave property of radiating outgoing waves of twice the edge wave frequency was also investigated.

The Boussinesq equations were solved on a staggered grid in order to eliminate the abrupt changes introduced by a line source.Thefirst-order spatial derivatives were discretized to O((Δx)4)using the five-point finite-difference method.The dispersive terms were discretized to O((Δx)2),the same as the actual dispersive terms.A fourth-order predictor-corrector method originally developed by Wei and Kirby(1995)was employed to perform time updating.All boundaries were treated to perfectly reflect the vertical walls.To absorb reflected waves,sponge layers were placed at the outside boundaries,as in Wei and Kirby(1995).

3.1.Mode-0 edge wave

We considered a computational domain of 3 m≤x≤65 m and-20 m≤y≤115 m,shown in Fig.2.The water depth at the shoreline h0was 0.1 m,and the constant slope in the xdirection was equal to 1/30.Two sponge layers with a width of 20 m were placed at both ends of the domain in the y-direction to absorb edge wave energy.Another sponge layer with a width of 30 m was arranged in the open-sea boundary to absorb the energy of radiated waves.The wave-generation line was located at y=0 m.

The period of the edge wave was 5 s.In order to strictly satisfy the linear shallow-water approximation and suppress sup-harmonic components,the wave amplitude at the shoreline was set at 1 mm.The corresponding wave parameters are listed in Table 1.The mode-0 edge wave was generated at the wave generation line.The grid size was Δx= Δy=0.152 m,and the time step was Δt=0.06 s.The model was run up to 240 s for simulation without encountering any stability problems.

Fig.3 shows a snapshot of water surface elevation η at t=80 s.The edge waves are generated at the internal line and then propagate toward the two ends.Sponge layers at both ends of the domain work quite well in dissipating wave energy.The wavefield has reached a quasi-steady state,and the motion is trapped near the shore and decays in the cross-shore direction.

Fig.4 shows the water surface elevation η along the shoreline at t=80 s.The corresponding wave amplitudes are very close to their target value.The simulated wavelength is 6.15 m,which is also very close to the theoretical value 6.21 m.However,a little oscillation in the wave amplitude along the y-axis has occurred.The offshore amplitude profile is plotted in Fig.5.Deviations are noted between the simulated result and the mode-0 analytical solution.The maximum amplitude is only considered to be 0.001 m,with a corresponding water depth of 0.1 m,and the nonlinear parameter ε=a/h is only 1/100.The disagreement of the offshore amplitude profile and the longshore amplitude oscillation could not be caused by the nonlinear effect.A potential cause for these deviations is the existence of higher-mode edge waves.As the energy can be transferred between different edge wave modes,in addition to the target mode(n=0),the other modes are also apparent.This hypothesis of higher modes existing in the simulations is supported by Yeh(1983),who experimentally investigated the fundamental behavior of progressive edge waves.He pointed out that the deviations between the experimental results and the theory for the offshore amplitude profile of the Stokes mode were caused by the existence of higher-mode edge waves,and the magnitude of each mode was further identified in his paper.

Fig.2.Sketch of computational domain for internal generation method of edge waves.

Table 1 Wave parameters for edge waves with a period of 5 s on a beach(s=1/30,with shoreline located at x0=3 m).

If we assume that the generated results for the mode-0 edge wave consist of a superposition of the fundamental mode and the composition of the higher modes(n=1,2),then,neglecting the initial phase of each mode,the water surface elevation η(x，y，t）at a distance y from the wave generator may be written as

where aiis wave amplitude at the shoreline,kiis the wavenumber,ζiis the offshore amplitude,and indices 0,1,and 2 denote the fundamental,first,and second modes,respectively.Eq.(17)can be rewritten as

The amplitude of the water surface elevation η(x，y，t）can be expressed as

Fig.3.Temporal variation of two-dimensional water surface profile for mode-0 edge wave at t=80 s.

Fig.4.Water surface elevation along shoreline at t=80 s for simulation of mode-0 edge wave with an amplitude of 1 mm(η is normalized by amplitude at wall).

It is evident that the wave amplitude Afluctuates along the y-axis.

Since our study was specifically aimed at generating clean progressive edge waves,it is necessary to identify the magnitudes of the higher-mode waves.The amplitude A along the line y=y0can be written as

where b denotes the tidal effect. N data points(x0，A0），(x1，A1），…，(xN-1，AN-1）are located along the line y=y0.A residual γ is defined as the quadratic difference between the values of the dependent variable Aiand the predicted values A(xi）:

Suppose that γ becomes zero so that the least square error has a minimum value.The parameters a0,a1,a2,and b can be solved by means of the first derivatives of γ.

Fig.5.Offshore amplitude profile along line y=37.7 m for mode-0 edge wave with an amplitude of 1 mm at shoreline on slope 1/30.

We investigated wave amplitudes along the line y=37.7 m,and each parameter was obtained by least squares approximation,which was a0=1.000 mm,a1=0.051 mm,a2=0.046 mm,and b=0.016 mm.An excellent agreement between the simulation results and the superposition of different modes shown in Fig.5 supports the fact that the complex wave pattern is caused by the effect of different edge wave modes.The magnitude of the higher modes is small,and the wavefields are dominated by the mode-0 edge wave near the shore.

The Boussinesq equations can predict the propagation of nonlinear,weakly dispersive waves with high accuracy,especially in shallow water.Therefore,the model is run to generate the mode-0 edge wave with an amplitude of 5 mm at the shoreline.The corresponding water surface elevation along the shoreline at t=80 s is plotted in Fig.6.The waves are generated at the internal line with symmetrical crests and troughs,and then they evolve into the profile with peaked crests and flat troughs at the farfield due to the nonlinear interaction.The amplitudes oscillate even more clearly along the shoreline.Fig.7 exhibits the offshore amplitude profile along the line y=37.7 m,and the magnitude of each mode is estimated by the least squares approximation,which is a0=4.78 mm,a1=0.28 mm,a2=0.18 mm,and b=0.05 mm.A strong agreement between the simulation results and the superposition of different modes indicates that the generation of the higher-mode waves is unrelated to the nonlinearity of the mode-0 edge wave,and the ratios of the amplitudes for higher modes to the amplitude of the mode-0 edge wave are similar to the results of simulation with an amplitude of 1 mm at the shoreline.As the wave amplitude decays offshore and the water depth increases at the same time,the nonlinear parameter ε=a/h is quite small in the offshore zone,so the nonlinear interaction is concentrated in the nearshore zone.

Fig.6.Water surface elevation along shoreline at t=80 s for simulation of mode-0 edge wave with an amplitude of 5 mm(η is normalized by amplitude at shoreline).

Fig.7.Offshore amplitude profile along line y=37.7 m for mode-0 edge wave with an amplitude of 5 mm at shoreline on slope 1/30.

3.2.Mode-1 edge wave

The mode-1 edge wave was generated at the source line.The simulation process and wave parameters were the same as for previous waves.The water surface elevation along the shoreline at t=150 s is shown in Fig.8.Compared with the previous simulation,there are larger oscillations in the wave amplitude along the y-direction.Evidently,the simulation time of 150 s is long enough for the motion to achieve a quasisteady state,and the sponge layers are long enough to effectively dissipate wave energy;the only reason for this oscillation is that the motion is modulated by other modes.

Fig.9 exhibits simulated amplitudes along y=37.7 m,together with the linear superposition of a0=0.31 mm,a1=0.64 mm,and a2=0.14 mm.They agree with one another,and the slight deviations may be caused by the existence of even higher modes(n＞2).Because of these highermode edge waves,the wave energy is distributed further into the offshore region.

As there is an energy transfer between different edge wave modes,not only the target mode but also its adjacent modes have appeared in the simulations.For the generation of the mode-0 edge wave,the main coexisting waves are the mode-1 and mode-2 edge waves;the ratios of the amplitudes for these modes to the amplitude of the mode-0 edge wave are less than 6%.For mode n=1,the generated edge wave is highly modulated by the adjacent modes,especially the mode-0 edge wave,whose amplitude reaches half the amplitude for the target mode n=1.The mode-2 edge wave is also present,and the simulation results are even more complicated.The generated wave is not only modulated by the mode-0 and mode-2 edge waves,but also by the mode-3 and mode-4 edge waves.However,in the special geometry a certain mode is excited and its magnitude is amplified enormously,and the other modes are restricted at the same time.This case was investigated by the numerical model.

3.3.Standing edge waves

Fig.8.Water surface elevation along shoreline at t=150 s for simulation of mode-1 edge wave with an amplitude of 1 mm(η is normalized by amplitude at wall).

Fig.9.Offshore amplitude profile along line y=37.7 m for mode-1 edge wave with a period of 5 s on slope 1/30.

Fundamental properties of standing edge waves were investigated numerically with the Boussinesq model.The topography and the wave period were the same as those in Section 3.1.A computational domain of 3 m≤x≤126 m and 0 m≤y≤9.3 m was used,and only a 30 m-wide sponge layer was installed at the open-sea boundary to absorb the energy of the radiated waves.The other boundaries were totally re-flected.The grid size was chosen as Δx= Δy=0.186 m in order to obtain a sufficient spatial resolution.The time step was chosen as Δt=0.06 s to guarantee a stable solution.The source line was placed at y=0.372 m to generate the mode-1 edge wave with an amplitude of 0.8 mm at the shoreline.Notice that the width of the computational domain is equal to the wavelength for the mode-1 edge wave,so the mode is expected to be excited.A total time of 600 s was considered.As there was no energy loss in our simulation,in order to keep the amplitude from increasing infinitely,the internal wavemaker stopped working after 180 s,and the numerical model continued to simulate the free oscillation process.

Fig.10.Temporal variation of a two-dimensional water surface profile for mode-1 edge wave at t=200 s.

Fig.11.Time history of free surface elevation for a fully reflected wall-bound domain at point of x=3 m and y=0.186 m.

The free surface elevation at t=200 s in the domain of 3 m≤x≤20 m and 0 m≤y≤9.3 m is shown in Fig.10.The wave is symmetrical in the longshore direction,and the offshore structure of the mode-1 edge wave can be detected easily.The time history of the free surface elevation at point x=3 m and y=0.186 m is shown in Fig.11.The free surfacefluctuation gradually increases from zero to its maximum value at 180 s,and then decreases slowly as there are no waves generated by the internal wavemaker and the wave energy is radiated offshore.Wave shapes are not symmetrical due to the nonlinear effect.The results for average amplitudes along the line y=0.186 m are between 180 s and 600 s.As shown in Fig.12,a strong agreement between the simulation result and the theoretical profile for mode-1 edge wave is found.It is known that the mode-1 edge wave can transfer energy to its adjacent modes.However,in this geometry,the mode-1 edge wave is amplified,and the other modes are restrained atthe same time.The theoreticaloffshore amplitude profile attenuates to zero for x ＞ 21 m,but the simulated amplitude remains a small,constant value.The reason for this is that the waves are not edge waves but outgoing waves.

The properties of nonlinear standing edge waves were investigated theoretically by Guza and Davis(1974)and Rockliff(1978),and experimentally by Yeh(1986).All investigators showed that nonlinear standing edge waves are not completely trapped;wave energy is radiated offshore in the form of high-harmonic outgoing waves.Thefirst and second harmonics from the simulations between 180 s and 600 s along the line y=0.186 m are shown in Fig.13.The fundamental harmonic component approximates zero at x＞21 m,but the second harmonic is approximately constant in the farfield.Obviously,these second-harmonic components should be the outgoing waves,as predicted by the theory.Also,the third harmonic component was detected to be radiated offshore in our study,although its magnitude was much smaller than that of the second harmonic.

Fig.12.Offshore profile at line y=0.186 m for mode-1 edge wave with a period of 5 s on slope 1/30.

Fig.13.Offshore amplitude profiles along line y=0.186 m forfirst and second harmonic components.

4.Conclusions

Edge waves were produced by the internal generation method in the extended Boussinesq model.In addition to the numerical experiments previously described,more wave conditions that included different wave frequencies,slope steepness,and wall positions were simulated.All these simulations indicate that the higher modes of edge waves appear pronounced as the nonlinearity of edge waves increases.The ratios of the amplitudes for mode-1 and mode-2 edge waves to the amplitude of the mode-0 edge wave vary with wave conditions and topography,although they are small and can be considered negligible nearshore.However,for the generation of the mode-1 edge wave,the simulation results exhibit a superposition of mode-1,mode-0,and mode-2 waves,and the amplitude of the mode-0 edge wave reaches half the value of the target mode-1 edge wave.It is known that the mode-0 edge wave,which is the Stokes edge wave for Ursell's general solution,is the most common and stable edge wave in experiments andfield measurements.This study shows that this wave can be produced by the internal wave generation method in Boussinesq models.Therefore,the present model is a proper numerical approach to edge waves in coastal engineering.

The numerical model was also used to investigate basic properties of standing edge waves.The nonlinear mechanics that produce radiating outgoing waves of twice the frequency of standing edge waves were well captured by the model.

Acknowledgements

We are grateful to the editors and the anonymous referees for their helpful comments and suggestions.

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