Jia-yi Xu,Shu-xue Liu,Jin-xuan Li,Wei Jia

State Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology,Dalian 116024,China

Abstract:The wave transformation over slope topography is widely studied,but most of studies are for common coastal slopes.This paper presents an experimental study of 2-D regular and irregular wave transformations on a simplified coral reef with a steep slope of 1:5 and a horizontal reef flat,focusing on the characteristicsof thewavesthat break on the reef flat.The analyzed resultsshow that the estimates of the wave breaking made by using the well-known previous formulae do not agree completely with the experimental results.When the waves break on a reef flat,the relative breaking wave height ( H / d )b is related to the incident deep-water wave steepness and the relative water depth d b/L0 .Hence,a new criterion for breaking waves on a reef flat is proposed.Furthermore,in view of the fact that the local Ursell number is commonly used to parameterize the wave nonlinearity,the relationships among the skewness,the asymmetry,and the local Ursell number are also presented.Experimental data confirm that when the Ursell number is greater than 30,the absolute values of the skewnessand the asymmetry on a reef flat are greater than those on a steep slope.

Key words:Coral reef bathymetry,wavebreaking,waveheight attenuation, representative wave parameter,nonlinear parameters

Introduction

Coral reefs can be found in many tropical and subtropical regions.In studies,they are usually represented as a steep slope and a reef flat with a shallow water.As a wave propagates over the steep slope,the wave profile becomes steep,and the wave breaks when the wave height reaches a certain limit.The breaking waves make the sea level rising,the resulting pressure gradient drives the flow across the reef,and these wave-driven currents are directly related to the sediment transport.These wave processes also have a direct impact on the coral reef ecosystems[1-5].Therefore,it is important to study the wave propagation characteristicson the reef terrain for the evaluations of the coastal processes and for the design of coastal structureson the reef flats.

There were some field studies of the wave transformation on the coral reefs.Lowe et al.[1]conducted a field experiment to measure the surface wave dissipation on a barrier reef.The rates of dissipation of breaking waves and the bottom friction were estimated.Roeber and Bricker[2]made field measurements to investigate the setup of the water surface generated by breaking waves near the fringing-reef-protected town during the Typhoon Haiyan.Lentz et al.[3-4]examined the transformation of surface gravity waves and the current dynamics across a platform coral reef in the Red Sea.The optimal parameters of the wave transformation model and the quadratic drag coefficient were presented.These studies focused mainly on the wave dissipation,with few studies of wave-breaking criteria on the coral reefs.

Most previous studies of the wave breaking were for common coastal slopes.The most commonly used breaking criterion was proposed by Goda[6], together with the relationship between the breaking-wave height, the relative water depth and the slope gradient.Goda[6]reanalyzed the original regular and irregular breaking wave data and proposed reasonable parameter values for the breaking criterion on a gentle slope.Kamphuis[7]modified the typical breaking criteria based on the experimental data of regular and irregular waves,and three different slope gradients were considered in his experiment.Rattanapitikon and Shibyama[8]made a modification of the slope effect term in the breaking criteria proposed by earlier researchers,based on a great number of experimental and field data.Dally et al.[9]analyzed the spatial variations of the energy flux associated with wave breaking in the surf zone and developed an analytical solution for the wave-height attenuation on a plane slope.However,it is in doubt whether the breaking criteria and the empirical formulae used for plane beachesarevalid for coral reefsaswell.

Because of the shoaling and the nonlinear energy transfer,the wave profiles gradually change with the decreasing water depth.The nonlinear properties of the wave profiles can be described by certain parameters such as the skewness and the asymmetry.These two parameters are related to the sediment transport and are effective in describing the wave process[10-12].Rocha et al.[13]investigated the crossshore variation of the skewness and the asymmetry of irregular waves propagating over a sloping beach.Peng et al.[14]investigated the evolution of irregular wavesover a low-crested breakwater with a very steep slope and proposed some empirical relationships among the skewness,the asymmetry,and the Ursell number on both sidesof the breakwater.Dong et al.[15]investigated the effects of the bottom slope on the nonlinear transformation of irregular waves and proposed an empirical formula for the relationship among the skewness,the asymmetry,and the local Ursell number based on the data measured on various slopes.The results show that the skewness and the asymmetry areclosely related to the Ursell number.

So far,most proposed theories and empirical formulae for wave breaking and energy dissipation in the literature were for common coastal slopes[16-17].The characteristics of the waves propagating on coral reefs are not well studied because the attenuation and the transformation of the waves on steep slopes are very rapid and complex. This paper investigates the characteristics of the wave propagation on a coral reef topography based on physical experiments.The wave heights and the energy dissipation are analyzed.The applicability of previous empirical formulae for coral reefs is examined,and a new breaking criterion for breaking waves on a reef flat is proposed.In addition,the relationships among the skewness,the asymmetry,and the local Ursell number on a steep reef slope and a horizontal reef flat are presented.The results of this paper may apply for studying other terrain with steep slopes and for verifications of corresponding numerical models.

1.Experimental setup

1.1 Wave flume and experimental arrangement

The experiment is conducted in a wave flume at the State Key Laboratory of Coastal and Offshore Engineering (SLCOE),Dalian University of Technology,China.The flume is 69.0 m long,2.0 m wide and 1.8 m deep.Figure 1 shows a sketch of the experimental setup.The flume is equipped with an irregular-wave maker and with a wave absorber installed at the other end to mitigate the wave reflection.

The reef topography usually sees a sudden decrease of the water depth from the deep water to the reef flat and could be considered as a steep slope terrain.The field study by Mei and Gao[18]shows that the fore steep slope of the coral reef is in the range of 1:12-1:4,followed by a reef flat.Buckley et al.[19]constructed a fringing reef with a 1:5 reef slope,a horizontal reef flat and a 1:12 beach to investigate the effect of the bottom roughness on the setup dynamics.Yao et al.[20]rebuilt an idealized fringing reef model consisting of a steep fore reef slope(1:6),a horizontal platform and a reef crest in his experiment to study the wave-induced setup.To simulate the sudden decrease of the water depth from the deep water to the reef flat,the coral reef topography is simplified as a steep slope (β = 1:5),followed by a reef flat in the present experiment.The present study is focused on the wave breaking on the suddenly changing topography and its effect on the wave transformation,therefore,the surface of the experimental topography is assumed smooth.The influence of the permeability and the roughness on the wave energy dissipation can be further separately studied.

Fig.1 Experimental setup(10-2m)

The wave data are recorded as a time series of the water surface levels measured by wave gauges at 18 locations along the wave flume.The wave gauges 1-3 are located in front of the steep slope,the wave gauges 4-10 are located on the slope,the wave gauge 11 is at the edge of the reef flat,and the wave gauges 12-18 are arranged on the reef flat.In the following discussion,=0 m x is defined as the mean position of the wave gauge1.

Three water depths are considered.The deep water depths1( )h in front of the steep slope are 0.625 m,0.715 m and 0.835 m,and the shallow water depths on the reef flat2( )h are 0.125 m,0.215 m and 0.335 m (the water depth ratio,ε = h2/ h1=0.2,0.3 and 0.4, respectively).

1.2 Wave conditions

Both regular and irregular waves are considered in the experiments.To investigate the effect of the incident-wave steepness on the wave transformation on the topography,the incident-wave height is gradually increased over a specified wave period.Table1 lists the variouswaveconditions.For irregular waves,the frequency spectrum used in the experiment is the JONSWAPspectrum

wheresH andsT are the significant wave height and the significant period,respectively,pT andpf are the peak period and the corresponding peak frequency,respectively,and γ is the peak enhancement factor, with a value of 3.3 in this study.

Wave surface elevations are measured by the wave gauges with a sampling interval of 0.02 s.For regular waves,the data length of each record is 1 024 when the wave periods are 1.0 s and 1.3 s.When the wave periods are 1.6 s and 2.0 s,the data length of each record is 2 048.For irregular waves,the data length of each record is 8 192.To determine the breaking position,the breaking-wave type,and the breaking-wave height,the entire process of the wave propagation in the main surf zone(the steep slope and the area within 3.0 m of the edge of the reef flat)is videotaped.

In the statistical analysis,the mean wave heights(periods)are considered for regular waves,and the significant wave heights(periods)are considered for irregular waves.

2.Experimental resultsand discussions

2.1 Determination of breaking point and breakingwave height

The position and the type of breaking waves are determined from the video pictures and by observation during the experiment.Three types of wave breaking (the spilling,the plunging,and the surging)could be determined by gradually increasing the incident-wave height.Figure 2 shows typical examples of breaking waves.The breaking point and the breaking-wave height are determined according to the following rules:

(1)For the spilling type,the location where the turbulent whitewater is spilled down the surface of the wave isregarded as the breaking point.

(2)For the plunging type, when the wave frontapproaches the vertical point,then curls over and drops onto the trough of the wave,the wave breaks,and this location is defined as the breaking point.

Table 1 Experimental wave parameters

(3)For the surging type,the breaking point is defined as the location where the lower part of the wave crest develops spray.

Fig.2 (Color online)Examples of breaking waves

For regular waves,the breaking point for a wave train is quite obvious,and the breaking-wave height could be accurately determined as the mean wave height at the breaking point.However,in the case of irregular waves,not all individual waves in a wave train would break during the propagation at a specified position and the percentage of breaking wavesisnot a fixed value.In thisstudy, thenumber of waves in an irregular wave train is in the range of 100-160,we assume that if there are more than 10 waves breaking in a wave train,the irregular wave is regarded as a breaking wave.This point can be determined by re-examining the recorded tapes,combining with the measured wave time series by the wave gauges.In fact,the breaking point defined in this way generally conforms to the description by Kamphuis[7],i.e.,the breaking point is usually the position where the largest wave height along the wave propagation is located.The significant wave height at the breaking point is defined as the breaking-wave height for the irregular wave.

2.2 Wave-height evolution along thetopography

Figures 3 and 4 show the wave-height evolution for different incident-wave heights with the same period for regular and irregular waves,respectively.When the waves propagate from the deep water to the shallow water,the wave profiles become steep,and the wave heights gradually increase.The higher waves break on the reef flat when their heights reach the point defined by the breaking criterion.The breaking points gradually shift towards the slope with the increase of the incident-wave height and finally move onto the slope,meanwhile,the wave breaking changes from the spilling type to the plunging type.After the wave breaking,the wave heights decrease rapidly and remain almost constant on the reef flat beyond the surf zone.This phenomenon indicates that wave breaking is the main cause of the wave energy dissipation.The large-period irregular waves break on the slope mostly as the plunging type breaking,and the wave-breaking position is a region,not a point as in the cases of regular waves.

Fig.3 (Color online) Variation of wave height along the reef bathymetry for regular waves with different incident-wave heights( ε= 0.3, Tm =1.6 s )

2.3 Evolution of wave energy spectra

Fig.4 (Color online)Variation of significant waveheight along thereef bathymetry for irregular waveswith different incident-wave heights( = 0.3ε,=1.6 ssT )

Fig.5 (Color online)Comparisons of wave energy spectra at different positions along the reef bathymetry for regular waves( ε= 0.3, Tm =1.6 s )

Figures 5 and 6 show the typical spectral evolution of regular and irregular waves.In these figures,the spectra at the wave gauges 4,15 and 18 represent the incident-wave spectrum,the wave spectrum after wave breaking,and the wavespectrum on thereef flat,respectively.In Fig.5(a),because of the lower incident-wave height,no breaking wave and no significant energy change are observed in the main frequency range during thewave transformation.However,when the wave propagates to the shallow region,a significant increase of the energy is observed in the second and third harmonic frequency ranges due to the nonlinear wave interaction.Moreover,the wave energy in the range of higher harmonic frequencies remains almost constant when the wave propagates on the reef flat.However,in the case of larger incidentwave heights,as shown in Fig.5(b),breaking waves can be observed.There is an obvious attenuation between the wave spectra at the wavegauges 4 and 15

Fig.6 (Color online)Comparisons of wave energy spectra at different positionsalong the reef bathymetry for irregular waves( ε= 0.3,Ts =1.6 s )

due to the wave breaking.The energy in the main frequency range decreases sharply,with the higher-harmonic wave energy observed.However,the higher-harmonic wave energy disappears as soon as the wave propagates to the wave gauge 18.Similar phenomena are observed for irregular waves,as shown in Fig.6.The higher-order frequency energy is observed when the wave propagates to the shallow water,the main-wave peak energy decreases sharply because of wave breaking,and the wave energy shifts from the higher-order frequencies to lower-order frequencies beyond the surf zone.

2.4 Breaking criteria on coral reef terrain

2.4.1 Breaking criteria on steep reef slope

So far,several criteria were proposed to describe the wave breaking.However,most of them are applicable to the waves that propagate in a constant water depth or along a gentle slope.The most commonly used breaking criterion was proposed by Goda[6],based on many experimental studies for gentle slopes

wherebH is the breaking-wave height,θis the slope degree,=0.17A,=11B for regular waves,A=0.12,B=11 for irregular waves.

However,in the present experiment,the reef slope is steep.To check the applicability of Eq.(2)for the present topography,Eq.(2)is fitted using the experimental data for the wave breaking on steep slopes.Table 2 lists the fitted parameters and also presents the values proposed by Goda[6]for comparison.The results show that the values of the parameter A remain the same,but the values of the parameter B for the steep slope(1:5)are much smaller than that for the gentle slope.Figure7 showsa comparison of the experimental data and the fitted curve of Eq.(2)for breaking waves on a steep slope.The fitted curves of the empirical formula proposed by Goda[6]agree well with the experimental results measured on the steep slope,which means that Eq.(2)proposed by Goda[6]is applicable to steep slopes if appropriate parametersare used.

Table 2 Values of parameters A and B

Note that the value of the parameter B for irregular waves is larger than that for regular waves in this study,unlike the results given by Goda[6].This discrepancy needs to be further investigated by experimental studiesfor steeper slopes.

2.4.2 Breaking criteria on a horizontal reef flat

As mentioned above,the waves break on a horizontal reef flat when the incident-wave heights are appropriate.If Eq.(2)is still used to describe these cases,the slope degree can be taken as 0.Therefore,Eq.(2)can be simplified as

Fig.7 (Color online)Comparisons of experimental data and the fitted curves of Eq.(2)for a steep slope

Fig.8 (Color online)Comparisons of experimental data and the fitted curves of Eq.(3)on a reef flat

Fig.9 (Color online)Variation of relative breaking-wave height ( H / d )b against H 0/L0 for different values of d b/L0

2.5 Wave height evolution on the reef flat after wave breaking

As shown in Figs.3 and 4,the wave heights decrease within a region after the wave breaking and then remain almost constant on the reef flat.Experimental data of breaking waves on the reef flat are used to analyze the wave-height evolution after the wave breaking.

Dally et al.[9]assumed that when a wave breaks in a region where the bottom becomes horizontal,according to the shallow-water linear wave theory,the decay of the wave height in the constant water depth can be defined as

wherebH is the breaking-wave height,x is the distance from the breaking point,2h is the water depth on the reef flat and Γ and k are dimensionless coefficients.

Fig.10 Comparisons of measured ( H / d )b and the values calculated by Eq.(7)on the reef flat

In this paper,Eq.(8)is used to describe the wave-height evolution after the wave breaking on the reef flat.Figures 11 and 12 show examples comparing the measured wave heights on the reef flat and the fitted curves of Eq.(8)with proper dimensionless coefficients in different cases.The curves clearly agree well with the experimental data,which means that Eq.(8)as proposed by Dally et al.[9]is appropriate for describing the wave evolution after the wave breaking on a reef flat,but the values of Γ and k should be properly determined.

In fact,Dally et al.[9]suggested that Γ=0.35 and =0.20k according to the experimental data by a topography consisting of a steep slope (1:5)and an extended horizontal flat area.Further,they extended the formula to analyze their experimental studies for gentler slopes 1:30,1:65 and 1:80 and different values of Γand k were given corresponding to the slopes.

Fig.11 Comparisons of measured wave heights and the fitted curve of Eq.(8) with appropriate parametersΓand k on a reef flat for regular waves

Fig.12 Comparisons of measured significant wave heightsand the fitted curve of Eq.(8)with appropriate parameters Γand k on a reef flat for irregular waves

In this paper,the further analysis of the experimental data shows that the values of Γand k are related to the water depth ratio.To extend the applicability of the results,three cases of depth ratiosε are considered,as discussed earlier.Because the differences of the values of Γand k in cases with the same water depth ratio( )ε are small,average values of Γand k are used for each εin thepresent study (Table 3).It can be seen that the values of Γand k clearly increase with the increase of ε.As shown in Table 3,when =0.4ε ,the values of Γand k are almost identical to the results given by Dally et al.[9].Figure 13 shows a comparative plot of the measured wave heights and the calculated ones by Eq.(8).The calculated wave heights clearly agree well with the experimental data.Hence,with the appropriateΓ and k,Eq.(8)as proposed by Dally et al.[9]can be used to calculate the wave height after the wave breaking on a reef flat.

Table 3 Average values of Γand k for different εvalues

Fig.13 Comparisons of measured wave heights and the wave heights calculated by Eq.(8)

2.6 Estimating the representative wave height for irregular waves

where( )f H is the probability density function of the wave height ( )H and0m is the zeroth moment of the wave spectrum,which isdefined as

If the probability distribution of the wave heights obeys the Rayleigh distribution,the representative wave heights can be obtained by using the probability function.ThenmH ,sH andrmsH can be expressed as:

Hence,mH andsH can be expressed with respect tormsH as:

where M is the total number of individual waves identified by the zero-crossing method.

Fig.14 (Color online) Relationships between the dimensionless heights H m /H rms and H s /H rms versus H rms/d .The results calculated by the method of Rattanapitikon and Shibayama[8]are also plotted for comparison

wheremk ,sk andrmsk are the proportional coefficients ofmH ,sH andrmsH ,respectively.If the probability distribution of the wave heights obeys the Rayleigh distribution,the coefficientsmk ,sk andrmsk should be 2.51,4.00 and 2.83,respectively (see Eqs.(11)-(13)).

Dong et al.[15]indicated that the relationships between the real wave heights and the results derived from the Rayleigh distribution are primarily affected by the local water depth.According to Dong et al.[15],the proportional coefficient k can be expressed as a function of the relative wave height

2.7 Variationsof nonlinear parameters

As mentioned above,when a wave propagates from the deep water to the shallow water,the wave profile becomes steep.The skewness and the asymmetry of the wave can be used to describe the lack of symmetry of the wave profile relative to the horizontal and vertical axes,respectively and to indicate the degree of the wave nonlinearity.The skewnessand the asymmetry can be defined as

Fig.15 (Color online) The relationship between k m / s /rms and H m /d

where S is the wave skewness,A is the wave asymmetry,ηis the wave surface level,ηis the mean wave surface level,H denotes the Hilbert transform,and isthe mean operator.

Another parameter used to describe the wave nonlinearity is the Ursell number.Peng et al.[14]pointed out that the local Ursell number calculated based on the peak period does not contain any information about the change of the wave period.The mean wave period could be used to calculate the local Ursell number instead.Hence,the local Ursell number can be defined as

wheresH is the significant wave height,h is the local water depth andmL is the local mean wavelength calculated based on the mean wave period.

Peng et al.[14]proposed a quantitativerelationship between the skewness,the asymmetry,and the Ursell number based on the experimental data collected in the DELOS project:

Figure 16 shows the relationships among the skewness,the asymmetry,and the local Ursell number obtained in this study for breaking waves,together with the plots of Eqs.(30)and (31)by Peng et al.[14]for comparison.On the steep slope,the skewness is positive and constant at a small Ursell number,it increases with the increase of the Ursell number and finally reaches its maximum value.The asymmetry fluctuates around zero at a small Ursell number,with the increase of the Ursell number,it becomes negative,and goes on decreasing.On the reef flat,the skewness is almost constant,and the asymmetry decays with the increase of the Ursell number,but when the asymmetry values on the reef flat become greater than zero,the values of the local Ursell number varies in a small interval around 30.Moreover,when the Ursell number is greater than 30,the absolute values of the skewness and the asymmetry on the reef flat are greater than those on the steep slope at the same Ursell number.

The results calculated by the equations proposed by Peng et al.[14]agree basically with the analyzed results in the deep water,but the analyzed wave skewness is overestimated on a steep slope and the analyzed wave asymmetry is slightly underestimated.The reason may be that the formula proposed by Peng et al.[14]is obtained based on the experimental data collected on a 1:3 slope,steeper than the slope used in thisstudy.

Fig.16 (Color online) Measured asymmetry ( )A and skewness ( S )versus local Ursell number (U r)for irregular waves

3.Conclusions

The wave transformation along a simplified coral reef bathymetry is experimentally studied in this paper.The variation of the wave heights and the frequency spectra along the bathymetry,the wave-height evolution after the wave breaking,the estimation of the representative wave height,and the relationships between the wave nonlinearity parameters are analyzed.A new breaking criterion for breaking waves on a horizontal reef flat is proposed.The final conclusions can be summarized as follows:

After the wave breaking on a reef flat,the wave-height evolution can be well described using the formula proposed by Dally et al.[9].The dimensionless coefficientsΓ and k for different water depth ratiosεare analyzed and shown in Table 3.

The wave-height distribution deviates slightly from the Rayleigh distribution for the coral reef bathymetry.The analyzed relations between the representative wave heights agree basically with the resultsof Rattanapitikon and Shibayama[8].

Relationships among the analyzed skewness,asymmetry,and local Ursell number on a coral reef are also presented in this study.When the Ursell number is greater than 30,the absolute values of the skewness and the asymmetry on the reef flat are greater than those on the steep slope for the same Ursell number.Moreover,the formulae proposed by Peng et al.[14]overestimate the wave skewness and slightly underestimate the asymmetry on the steep slope.