1 Introduction

The main objective of this study is to evaluate the validity and performance of the formulations for depth-induced wave breaking and dissipation processes in very shallow waters.For this purpose,the results of the Simulating WAves Nearshore(SWAN)model(Booij et al.1999)are compared with the data obtained from a series of field measurements conducted at the southern coast of the Caspian Sea.

The SWAN model is briefly described in Section 2 and various formulations of wave breaking and dissipation terms implemented in the model are introduced and discussed.Field measurement program and results are presented in Section 3.The overall evaluation methods are discussed in Section 4 and the simulation results in comparison with the field data are evaluated in Section 5.The study is summarized and concluded in the final section.

2 Model Description

In this study,the phase-averaged third-generation wave model SWAN version 41.01 was employed to simulate spectral evolution of waves in shallow water in the surf zone.The model numerically solves the conservation of wave action density equation as follows:

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即m+1时刻预测值为Pm+1,将最后L-1个数据和测试集中的第m+1个数据合并为新的数据并输入网络,得到m+2时刻的输出Pm+2,依次类推得到预测序列为

Therefore,in the rest of the simulations conducted in this study,triad interaction was deactivated for wave breaking modeling.

where N=E/σ is action density,and E is energy density dependent on relative frequency σ and mean direction θ.In addition,cx,cy,cθand cσ represent the wave group velocities in x,y,θ and σ directions,respectively.The S on the right side denotes the total sum of sink and source terms and can be written as follows:

where Sin stands for wave generation by wind;Snl denotes energy exchange between different frequencies by nonlinear triad and quadruplet wave-wave interactions;and Sdis is the dissipation term that includes depth-induced breaking,bottom Friction,vegetation or mud-fluid effects,and white capping.More descriptions of theoretical and numerical backgrounds of SWAN can be found in the works by Zijlema and van der Westhuysen(2005),Ris et al.(1999),and the Wamdi Group(1988).

The spectral form of the bore model,introduced by Battjes and Janssen(1978),hereafter referred to as BJ78,is the default method of SWAN to incorporate the depth-induced wave breaking and its corresponding dissipation term in the wave action balance equation.In this approach,a Rayleigh distribution for random wave heights is assumed within the surf zone and the total dissipation rate is estimated by

Nonlinear triad interactions in the shoaling zone have been observed and reported in several studies(Beji and Battjes 1993;Dong et al.2008;Elgar et al.1990;Elgar et al.1993;Elgar et al.1995;Hasselmann 1962;Sénéchal et al.2002).A disagreement occurs on nonlinear triad interaction for breaker waves in the surf zone.Herbers et al.(2000)showed a close interplay between depth-induced breaking and nonlinear triad interactions in the surf zone.By contrast,Eldeberky(1996),Sénéchal et al.(2002),and Becq(1998)showed that severe wave breaking destroyed the triad interaction in shallow waters.Mahmoud of et al.(2016)reported,using the bispectral method on the same data set applied in the present study,that triad interaction was decreased dramatically during wave breaking and almost no energy transfer occurred through nonlinear wave-wave interaction.Despite this observation,all of the embedded triad interaction schemes in SWAN were activated and evaluated in comparison with the present data set.

那天一脸凄风苦雨的赵仙童回家，看到呆坐在电脑前愁眉不展的砖子，扔了一颗手雷给他：诌不出狗屁就别诌了，弄饭！

由图7可知，随着硅藻土用量的增加，72%vol红枣白兰地的透过率先上升后趋于稳定，当硅藻土用量为0.6 mL/100 mL时，透过率为99.972%，此时澄清效果最佳。由于硅藻土澄清剂中含有微量的金属离子，可能会与酒中的物质发生反应。由图8可知，最大吸收波长未发生移动，硅藻土未引起72%vol红枣白兰地色泽的改变。

where H rms is the root-mean-square wave height.

BJ78 suggested γBJ=0.8.Subsequently,γBJ=0.73 was proposed on the basis of a vast range of field and laboratory studies by Battjes and Stive(1985),hereafter referred to as BJ78,0.73.However,using a constant value for γ and employing a Rayleigh distribution for broken waves are questionable.The value of γ and its dependency on bed slope,depth,wave number,significant wave height,wave steepness,and sea condition have been studied and modified by several researchers(Ruessink et al.2003;Battjes and Janssen 1978;Battjes and Stive 1985;Alsina and Baldock 2007;Baldock et al.1998;Janssen and Battjes 2007;Katsardi et al.2013;Lippmann et al.1996;Thornton and Guza 1983;Goda 2004;Rattanapitikon et al.2003;and Robertson et al.2013).

Different results based on wave conditions have been reported.Several studies showed that the BJ78 model overestimates the dissipation of locally generated waves(Bottema and van Vledder 2009;Goda 2009;Groeneweg et al.2009;Van Vledder and Groeneweg 2009;Salmon et al.2015).By contrast,Salmon et al.(2015)showed that the dissipation of swells is underestimated by the BJ78 model.

Nelson(1987),hereafter referred to as NEL87,defined the breaker index as a function of bottom slope β:

where γBJ varies from 0.55 for horizontal bottom to 1.3 for very steep slopes.The effect of the slope was experimentally evaluated in other studies,e.g.,Goda and Morinobu(1998).

Thornton and Guza(1983)(hereafter TG83)included a weighting function to modify Rayleigh distribution for broken waves,thereby perm itting higher wave heights.Their proposed bulk dissipation equation is as follows:

whereThey assumed γ=0.42 and TG αTG=0.5 or 3.4 for laboratory and field conditions,respectively.

They suggested usingγ0=0.54,a1=7.59,a2=-8.06,and a3=8.09 in theirβ-kd model,hereafter BKD15.Based on this formulation,at shallow water depthsthe breaker index is only controlled by local bottom slope(Eq.(9)),whereas for deeper water,γβ-kd is mostly dependent on kd variations.In the transition depths,both β and kd are effective in breaker index evaluation.Other prior studies such as Goda(2010)emphasized that breaker index is governed by both the beach slope and the relative water depth.

where and Hb=γBd.The preceding modified version of Baldock et al.(1998)is the foundation of wave dissipation calculation according to certain breaker index formulations such as that by Ruessink et al.(2003),hereafter referred to as RUE03.In that formulation,the variable γB is suggested as a function of relative depth kp d,

本区位于燕山山地和太行山山地，是京津冀的生态屏障，也是平原地区众多城市的水源涵养区。植被退化，水土流失严重，水源涵养能力差。主导基础功能为水源涵养和土壤保持，水土保持的重点是建设生态清洁小流域，通过生态修复、生态治理和生态保护三道防线，调节径流、涵养水源、改善水质、控制面源污染；同时要维护和提高土地生产力，减少入库泥沙，加强土壤保持功能。

where the breaker index in their study varied between 0.48 and 0.86 for 0.25＜kpd＜0.75.At very shallow waters,the values of kp d approach zero and solitary wave behavior dominates.Fenton(1990) implied that for such waves,the breaker index is insensitive to relative depth.Salmon et al.(2015)proposed a new parameterization for breaker index dependent on both relative depthin which is the lower-order mean wave number,as used in white capping(Wamdi Group,1988)and local bottom slope β as follows:

Baldock et al.(1998)replaced the clipped Rayleigh distribution with Rayleigh distribution.Thereafter,the final bulk energy dissipation was corrected by Alsina and Baldock(2007)and Janssen and Battjes(2007)as follows:

The simulated spectra for the second stationary storm at ST1 and ST2 as representative cases are depicted in Fig.7.Each spectrum consists of an average of 54 frequency-energy distribution spectra obtained every 20 mins in the 18 h duration of the storm.Figure 7 shows that the activation of triad interaction schemes had no effect on the simulated spectra except at low frequencies where energy content was slightly amplified using the Benit(2009)scheme.Thus,we conclude that triad scheme activation is not impressive in the simulated results during wave breaking and this behavior is consistent with our observations reported by Mahmoud of et al.(2016).

3 Field Study

The Caspian Sea is the largest enclosed water body in the world.Offshore and nearshore wave characteristics of this sea have been inspected in several field and numerical studies,e.g.,Najafi-Jilani and Nik-Khah(2011)and Rusu and Onea(2013).The physical environment of the Caspian Sea isslightly similar to that of the Black Sea(Rusu 2009).

In this study,field measurements were conducted on a straight single-barred beach over a shore perpendicular transect located in the west of Nowshahr Port on the southern coast of the Caspian Sea(Fig.1).Tidal water-level variation was neglected due to its small variations(less than 10 cm).

Bathymetry along the shore perpendicular transect was recorded using a single-beam echo sounder at the beginning and end of the field campaign,which showed negligible changes during the measurement period.The recorded beach profile and the location of measurement stations are shown in Fig.2.The profile exhibits a single-bar system with seaward slope of~0.025.The beach slope at the shallow water nearshore is mild.

Water-level fluctuations were recorded by five synchronized pressure sensors deployed within the surf zone as shown in Fig.2.The measurements were continuously performed from March 3 to 16,2014.Details of Instrumentation are presented in Table 2.The water level was measured continuously in ST1,ST2,and ST5 while in ST3 and ST4,data retrieval was mandatory after a 2-day measurement due to data logger capacity.Therefore,the water level was recorded for three 2-day cycles at ST3 and a 2-day cycle at ST4.The data acquisition rate was set to 4 Hz for ST1 and ST2,and 2 Hz for ST5,while sampling rates of ST3 and ST4 were set to 1 Hz.A lso,time series of current profiles were measured using ADCP in ST5.Data gathering was set to record 20 min averaged velocity within each hour at 25 cm bins.No significant current was observed in the study area.The hourly averaged current intensities were 0.09 and 0.02 m/s in the long-shore and cross-shore directions,respectively.Thus,no effective interactions between currents and waves were expected to occur.

The 2D spectra of energy distributed on frequency-direction space and measured in ST5 were exerted as the boundary condition for numerical modeling.According to Met Ocean data(http://www.bocmetocean.com/forecast_maps.php),two storms generated in the central part of the Caspian Sea approximately 600 km away from the study area and passed over the study area during the deployment period.The first was from March 8,2014,to March 10,2014.The duration of the second storm was prolonged,starting on March 13,2014,and ending on March 16,2014(Fig.3).Three periods of wave shoaling were the other considerable events during this period.At all stations,Fourier analysis was performed for each burst of 20 min duration data.Notably,no wave breaking was observed for offshore waves until ST5 in the total period of measurements.The wave characteristics of both storms were similar with significant wave height of approximately 1.4 matST5.On the other hand,the measured peak wave periods were in the range of 8.2 to 9.0 s at all stations during both storms,except at ST1 where high-energy infragravity waves(fi≤0.05Hz)with peak period up to 56 s were observed.Due to grow th of infragravity waves and low-frequency harmonics(0.05＜fi≤0.1Hz),a second peak with considerable energy intensity comparable to wind waves was observed at this station.This finding is consistent with the observations of Marshall and Stephenson(2011).Therefore,the energy of infragravity waves was removed by high-pass filtering to compare the field observations and numerical modeling results in the wind wave range of the energy spectrum.

Table 1 Characteristics of the computational domain and the activated(1)and deactivated(0)options in the model setting

Coord. Δx(m) Δy(m) Δθ/(o) f low f high nf nx ny np Cart. 4 5 15 0.0039 0.5 52 120 76 9120 Wave Wind Tide Current Wh.-cap. Quad. Triad Bottom friction Setup Breaking 1 0 0 0 0 0 1.0 1 1 1

Fig.1 Study area with respect to a Caspian Sea,b Nowshahr Port,and c the isobaths close to the assumed transect

As shown in Fig.4,from 11:00 on March 13 to 5:00 on March 14,negligible variations of significant wave height were recorded at the offshore station(ST5)and consequently at all nearshore stations(ST4 to ST1),which is an indication of stationary wave condition during this period.Visual observation at the site indicated that no wave breaking occurred at the seaside of ST5.Normally,this 18 h observation was recognized as a suitable period to visualize some of the results such as depicted averaged energy spectra shown in Fig.5.

During observed storms,the wind was weak and the maximum wind speed at 10 m above the sea surface level was less than 5 m/s based on a coastal synoptic station located 10 km west of the study area.This observation proves the presence of non-locally generated energetic waves during the measurement period.

1）唐孝祥教授在《近代岭南建筑美学研究》( 2002)一文中首次提出“文化地域性格”一词；而后在中国建筑工业出版社刊出的《岭南近代建筑文化与美学》（2010）与《建筑美学十五讲》（2017）中指出，岭南建筑的文化地域性格包括三个主要层面：地域技术特征、社会时代精神、人文艺术品格。

Fig.2 Beach profile changes and the positions of five measurement stations along the transect with respect to the coastline

The directional wave and current data recorded by ADCP at ST5 were applied as offshore boundary conditions for wave simulation using SWAN during the measurement period.

4 Evaluation Method

The error in predicting significant wave height was evaluated by calculating the following five statistical indices:root-meansquare error(RMSE),bias(B),their normalized forms(NRMSE and NB),and index of agreement(IA)as follows:

where Hmo and Hob denote the modeled and observedsignificant wave heights,respectively,and over-bar denotes average value.

Table 2 Details of instrumentations and measurement stations

Stations Instrument Sampling rate(Hz)Depth/m Distance from shore/m Duration ST1 RBRvirtuoso 4 0.8 35 3/4-3/16 ST2 RBRvirtuoso 4 1.4 103 3/4-3/16 ST3 DST-centi Star-Oddi 1 2.5 135 3/5-3/7/2014,3/9-3/11,3/12-3/14 ST4 DST-centi Star-Oddi 3.2 245 3/5-3/7 ST5 ADCP 2 4.8 310 3/3-3/16 1

In addition to significant wave height,the model performance in reproducing the correct shape of the wave spectrum is important.In the present study,the energy content close to the spectral peak was chosen as a characteristic feature of the wave spectrum in the comparison between modeled and observed data.The energy content close to the spectral peak frequency was estimated by the following parameter:

where f1＜fp＜f2,where fp is the spectral peak frequency and f1 and f2 have close values to fp chosen heuristically based on the frequency resolution(Δf)of the simulation.This approach is similar to the technique used by Hanson and Phillips(2001).Here,values of f1=0.83fp and f2=1.1fp were applied to both observed and modeled wave energy spectra.The shaded area in Fig.6 shows the selected range close to the spectral peak frequency of the wind-generated part of the wave spectrum.A minimum cut-off frequency f m in=0.05Hz was chosen to eliminate the effects of secondary infragravity peak observed in the shallow stations.Finally,the error in the wave spectrum close to the spectral peak was computed for each station using the following equation:

Fig.3 Time series of significant wave height at ST5

where ef p,ob and ef p,om are the observed and simulated energy contents,respectively.The average of hourly spectra for each station during the 18 h of stationary wave condition was calculated to be compared with the model results.

5 Results and Discussion

5.1 Triad Interaction Scheme

where B is a tunable coefficient,f is mean frequency,）is maximum local wave height,d is still water depth, is difference between mean and still water level,and γ is breaker index.Qb is the fraction of broken waves obtained from the following equation:

Fig.4 Time series of significant wave height variations at five stations during the stationary storm

Fig.5 Averaged energy spectra at measurement stations during the stationary storm

Lumped triad approximation(LTA)of Eldeberky(1996),which is an adaptation of the discrete triad approximation(DTA)of Eldeberky and Battjes(1995),is the default scheme in the SWAN model(version 41.01).The primary results of this study and several other studies such as Benit(2009)showed that the default LTA scheme overestimated the transferred energy to super harmonics.Therefore,the stochastic parametric model based on Boussinesq equations(Becq-Girard et al.1999)and the empirical distributed co-linear approximation(Benit 2009)were utilized to include the triad interaction effects in the simulations conducted in this study.In a separate simulation,a triad interaction option was deactivated so that we could judge the severity and existence of the triad within the wave breaking zone.

In this study,the non-stationary 2D mode of SWAN was applied using spatial grid sizes of 5 and 4 min the cross-shore and long-shore directions,respectively.The logarithmic scale for frequencies with a ratio of f1=+/fi=1.1 was used.The lowest and highest frequencies were set to 0.04 and 0.5 Hz,respectively.Bottom friction was modeled using a friction coefficient of c bottom=0.038m2s-3.A sensitivity analysis on the bed friction coefficient showed the negligible effect of this parameter for the short distances encountered in field observations.A similar conclusion was reported by Gorrell et al.(2011)and Thornton and Guza(1983).Default settings and coefficients of the model were employed except for depthinduced dissipation formulations,which are the main focus of this study.Also,wave setup computation was activated.The various embedded triad schemes were utilized and evaluated to include positive and negative triad interaction effects.The characteristics of computational domain and model settings such as geographical space,directional space and time resolution,and number of grid points are presented in Table 1.

本文使用相关软件对上面提到的理论模型，即智力资本、战略柔性和创新能力3个因素之间的相互作用关系进行了分析，假设检验结果如表5所示。

(1)淀粉膜：淀粉膜是将稀碱溶液改性后的淀粉作为基材，然后添加一定量甘油所形成的果蔬保鲜膜，其材料来源广泛，制作技术低下，价格也比较低廉，因此在目前的果蔬保险行业也获得了良好的应用效果。在进行淀粉膜材料的选择过程中，豌豆淀粉膜还具备有更加优越的阻油性，也就能够发挥出更好的果蔬保鲜效果。

在玉米的施肥管理中，要根据玉米的不同阶段施不同的肥料。从授粉至乳熟期是玉米吸收养料的第2高峰时期，玉米在这一阶段要大量施用氮元素和磷元素，氮元素的需要量占氮元素总量的1/3，磷元素的需要量也占总量的1/3。在玉米开花时要重视补追攻粒肥，保障玉米的生长速度和质量。高产的玉米田补追肥可以占追肥总量的15%-20%，此时要注意，如果在前期所施肥料较多的情况下，也可以不追肥。同时，对玉米施肥中的生长情况建立相应的数据库，加强对玉米的智能化管理。

5.2 Significant Wave Height Prediction

Fig.6 The selected range close to the spectral peak

Fig.7 Comparison of two embedded triad schemes and no triad scheme

The available depth-induced wave breaking and dissipation formulations within SWAN with the default values for tunable parameters were applied to simulate the observed storms discussed in the “Field Study”section.Simulated wave parameters such as significant wave height and peak wave period were compared with observed values at all stations.

The results of the error indices for the predicted significant wave heights are illustrated in Table 3.Also,we conclude that the predicted peak periods were not sensitive to the depth induced wave breaking configuration used in SWAN.

根据表1、表2分别确定各个溃坝生命损失间接影响因素对风险人口死亡率的影响程度ni（取值详见表5）及其对风险人口死亡率的影响程度权重系数ti。通过公式（3）计算，分别得到白天溃坝和夜间溃坝情况下的 m2=0.525和 m2=0.602。

The accuracy of various models is not the same at different stations(different depths).For instance,the configuration of RUE03 is more successful in deep stations(ST3 and ST4)than in shallow stations(ST1 and ST2).Finally,we can conclude that RUE03 represented the best results with average error of~7.90%as NRMSE and agreement index of~0.35.Furthermore,BJ78 with constant γ=0.55(BJ78,0.55)and TG83 showed appropriate results of 8.05%and 8.83%as NRMSE,and agreement indices of 0.30 and 0.38,respectively.

根据《山西省运城市第二次水资源调查评价报告》中伍姓湖与地下水互补关系微弱的结论，加之历时较短故不统计湖周边浸润带的散发量、湖面降水量和湖水渗漏补给量。

观察两组血液流变学、患者满意度。患者满意度采取我院自制问卷调查，主要分为三个等级——十分满意、基本满意、不满意，十分满意+基本满意=总满意概率。

5.3 Spectral Peak Energy

A comparison between measured and simulated energy spectra for stationary waves(in the second storm)at all stations using TG83,which was shown as an appropriate option for predicting significant wave heights,is depicted in Fig.8.Although the simulated wave spectra at deeper stations(ST4 and ST3)were predicted effectively,the simulated values of energy content close to peak frequencies at stations ST2 andST1 were significantly overestimated.This is an indication of the fact that the energy dissipation at these stations was underestimated by the model close to the peak period.

Table 3 SWAN skill in prediction of significant wave height at all shallow stations using different configurations for breaking term

Depth/m Model RMSE/m NRMSE/% B/m NB/% AI 2.5(ST4) BJ78,0.73 0.13 12.1 -0.11 -10.9 0.23 BJ78,0.55 0.11 10.1 0.10 9.3 0.35 BKD15 0.10 9.5 -0.08 -8.1 0.28 RUE03 0.06 5.6 0.03 2.6 0.29 NEL87 0.11 10.9 -0.10 -9.5 0.17 TG83 0.10 9.8 0.09 8.8 0.37 2(ST3) BJ78,0.73 0.15 15.0 -0.13 -13.6 0.22 BJ78,0.55 0.10 9.6 0.08 8.1 0.37 BKD15 0.12 12.6 -0.11 -10.9 0.28 RUE03 0.06 6.0 0.02 1.6 0.30 NEL87 0.16 16.1 -0.15 -14.7 0.16 TG83 0.09 8.9 0.07 7.0 0.39 1.4(ST2) BJ78,0.73 0.20 32.0 -0.19 -31.7 0.20 BJ78,0.55 0.03 4.6 -0.01 -1.4 0.31 BKD15 0.13 21.3 -0.13 -20.9 0.27 RUE03 0.04 6.2 0.02 2.6 0.38 NEL87 0.29 47.9 -0.29 -47.7 0.14 TG83 0.04 6.9 -0.03 -5.3 0.42 0.8(ST1) BJ78,0.73 0.16 36.3 -0.15 -35.4 0.21 BJ78,0.55 0.03 7.9 -0.01 -2.7 0.17 BKD15 0.11 25.4 -0.10 -24.2 0.27 RUE03 0.05 12.8 0.04 8.6 0.44 NEL87 0.23 53.3 -0.23 -52.7 0.16 TG83 0.04 9.7 -0.03 -6.4 0.36 Absolute average BJ78,0.73 0.16 23.85 0.15 22.90 0.21 BJ78,0.55 0.07 8.05 0.05 5.38 0.30 BKD15 0.12 17.18 0.11 16.03 0.27 RUE03 0.05 7.90 0.03 4.20 0.35 NEL87 0.20 32.05 0.19 31.15 0.16 TG83 0.07 8.83 0.06 6.88 0.38

Since the modeled peak periods are not sensitive to the breaking configurations embedded in SWAN,the simulated time series of peak period variations using RUE03 configuration as a representative case are compared with field observations at all stations within the second storm in Fig.9.In these comparisons,the effects of infragravity waves were removed by high-pass filtering,but the impact of low-frequency harmonics is still mainly evident at ST1.

The agreement between simulated and measured peak periods is good at ST5-ST3,partially acceptable at ST2,and very poor at ST1.However,the wave breaking was not effective for peak periods from offshore until ST3 and to an extent at ST2.To calculate the energy content near the spectral peak at ST1,we assumed that the wind wave peak period at this station is equal to the peak period at deeper stations to avoid peaks in the low frequency part of the spectrum.

We attempted to evaluate the performance of the model in predicting the energy content close to the spectral peak by applying all the configurations embedded in SWAN.Table 4 presents the calculated errors for all of the configurations mentioned at stations ST4-ST1 by means of Eq.(18)throughout all wave breaking periods in the field measurements.

Formulations such as RUE03 and TG83 are more successful than the others at transitional depths(stations ST3 and ST4),but none of the existing wave breaking and dissipation options in SWAN yield appropriate results at very shallow stations(ST1 and ST2).Thus,we can conclude that the dissipation rates in very shallow water were underestimated by all configurations available in SWAN.The best results with a minimum averaged relative error of 30%are produced by RUE03.This formulation represented the lowest value of the breaker index,and consequently,the highest dissipation rate at shallower depths was obtained.

Fig.8 Comparison of energy spectra between measurements and TG83 configuration during the second storm

Fig.9 Comparison of time series of peak period between measurements and RUE03 configuration during the second storm

5.4 Improvement in Prediction of Spectral Shape

Based on the results presented,SWAN is unable to reproduce the correct shape of the wave spectrum in very shallow waters.However,some depth-induced wave breaking formulations can accurately predict significant wave height.This discrepancy is due to the inability of SWAN to transfer spectral energy to higher and lower frequencies during wave breaking.Ruessink et al.(2003)reported that most of the phase-averaged wave models neglect the energy transfer from spectral peak to lower frequencies(0.05-0.10 Hz)due to wave breaking.Masselink(1998)concluded that wave breaking is responsible for primary wave decomposition into higher-frequency components(0.25-0.40 Hz).None of these two mechanisms have been implemented and coded in SWAN.According to the preceding observations,the amount of redistributed energy due to wave breaking becomes more significant in very shallow water,which results in higher error in the prediction of spectral peak energy at ST1 and ST2.Gorrel et al.(2011)pointed out that SWAN was not successful in modeling harmonic grow th and change in wave shape spectra in shallow water accurately because of inaccurate nonlinear energy transfers.However,a study by Mahmoudof et al.(2016)demonstrated that the nonlinear triad interaction was not responsible for observed inaccuracy in the grow th of lower and higher harmonics in very shallow waters of Nowshahr.

Among all embedded depth-induced wave breaking formulations in SWAN,RUE03 was the best option for shallow water wave breaking modeling in the studied area.The averaged error rates of this formulation were 7.08%and 30%for significant wave height and spectral peak energy prediction,respectively.The results indicate that the breaker index decreased with shore ward depth attenuation but two separate formulations were necessary to reproduce the two aforementioned parameters.As shown in Table 4 and mentioned in the preceding sections,γ formulations are mostly inaccurate in very shallow water stations(ST1 and ST2).To remedy this issue,decreasing γ in the range of shallow water depth improves the model accuracy.Thus,it was attempted to tune the γ formulation in RUE03 to improve the results.The final formulation is

with 2.2%averaged error for spectral peak energy prediction during observed storms in the field experiment(Table 5).The performance of Eq.(19)for the stationary wave period is presented in Fig.10.The gradient of Eq.(19)is close to the relationship presented by Ting(2001)(γB=1.53kpd+0.17)based on laboratory experiments.The breaker index resulted in a range of 0.05＜γb＜0.67 through the proposed formulation(Eq.(19))when kpd varies between 0.08 and 0.51.This formulation is applicable to the range of water depth,which represents positive values for the breaker index.However,the lowest value of the breaker index is controlled as γmin=0.3 by the mandatory model.In other words,according to this formulation,the breaker index is constant in very shallow waters.This result is consistent with the findings of Fenton(1990)implying that the breaker index is insensitive to relative depth in very shallow waters.

6 Summary and Conclusion

In this study,the depth-induced wave breaking and dissipation on a sandy beach at the west of Nowshahr Port in the southern coast of Caspian Sea were investigated.Field data were acquired at five stations on a single-barred shore perpendicular transect during measurements that lasted for 2 weeks.Two storms originally generated at the central part of the Caspian Sea(more than 600 km away from the study area)were observed with significant wave height~1.4 m and peak period~9 s.The wave breaking periods were used to evaluate the performance of several wave breaking formulations in SWAN.

Using different embedded triad schemes in SWAN,we found that simulated triad interactions were negligible and no considerable energy was transferred from peak frequency to lower/higher frequencies by nonlinear interactions.

Several default breaking formulations coded in SWAN were employed,and the time series of the simulated significant wave heights were compared with measured wave data.We also found that the formulation by Ruessink et al.(2003)represented the most accurate results with less than 10%normalized RMSE at all stations.Also,the findings of Thornton and Guza(1983)and Battjes and Janssen(1978),with constant breaker index as γ=0.55,were appropriate formulations for significant wave height prediction.However,the simulated peak wave periods were insensitive to depth induced wave breaking formulations.

Table 4 Estimated relative errors in the prediction of spectral peak energy content during fortnight observations

Models Error in ST1 Error in ST2 Error in ST3 Error in ST4 Average%BJ78,0.73 118.91 87.93 30.78 31.63 67.31 BJ78,0.55 69.92 47.98 9.92 9.12 34.24 BKD 104.59 73.42 29.02 28.69 58.93 RUE03 49.56 40.17 14.19 15.23 29.79 NEL87 149.21 110.07 32.75 29.91 80.49 TG83 73.23 56.61 10.81 9.65 37.58

Table 5 The error of spectral peak energy prediction using Eq.19

Error in ST1 Error in ST2 Error in ST3 Error in ST4 Average%1.06 2.57 0.91 4.50 2.26

Fig.10 Comparison of spectral peak energy measurements and model prediction using Eq.(19)

Despite the success of the model in predicting significant wave height,we observed that the aforementioned configurations yielded considerable overestimation in predicting the spectral peak energy content,specifically in very shallow stations.Decreasing the breaker index in these stations resulted in more accurate predictions.All other embedded formulations in SWAN for depth-induced wave breaking were applied,leading to the conclusion similar to that of Ruessink et al.(2003)with an averaged relative error of 30%as the most successful configuration from this point of view.This discrepancy could be interpreted as the inability of the current version of SWAN(41.01)to transfer energy content from peak frequency to lower and higher frequencies because of wave breaking,as mentioned by Ruessink et al.(2003)and Masselink(1998).In other words,this process is different from the triad interaction and has a key role in energy distribution for very shallow waters,which is ignored by SWAN.

The results implied that the Ruessink et al.(2003)formulation,which decreases the breaker index with shoreward depth reduction,was the most accurate option for wave modeling in the study area.Furthermore,two separated breaker index formulations were necessary to predict the significant wave height and spectral peak energy.Therefore,the Ruessink et al.(2003)formulation was modified to attain accurate spectral peak energy prediction.Thus,we found that γB=max(0.3,1.47kpd-0.08)was the most successful option with 2.2%average error.

Acknowledgments The authors are grateful to M.Jafari,A.Aynali,and H.Bagheri for their assistance in conducting the field measurements.Special thanks to M.Allahyar and the Ports and Maritime Organization for their support in the field measurements.

在精神文化类，洪江古商城作为古商业之都、湘商之源，囊括娱乐文化、镖局文化及医药文化等多种文化，保留着“吃亏是福”的商业警语及“鱼龙变化”“里仁为美”“外圆内方”等从商精髓。黔阳古城文创特色突出，文学作品和书法艺术极具研究价值。高椅古村蕴含生态哲理及人文特质的古建文化，显著地包含传统曲艺、雕刻剪纸等民间艺术。荆坪古村更偏向孝节文化等礼制感知。

Funding Information This study was supported by Iranian National Institute for Oceanography and Atmospheric Science.

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