Wave Simulations in Ponce de Leon Inlet Using Boussinesq Model

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Wave Simulations in Ponce de Leon InletUsing Boussinesq Model

Fengyan Shi1; James T. Kirby, M.ASCE2; Robert A. Dalrymple, F.ASCE3; and Qin Chen4

Abstract: An improved curvilinear grid model based on fully nonlinear Boussinesq equations is used to simulate wave propagPonce de Leon Inlet, Fla. We employ the nearshore bathymetry of Ponce de Leon Inlet and generate a stretched curvilinear grresolve shortwaves in the nearshore region and fit the complex geometry. Simulations of 18 cases with monochromatic input wTexel-Marsden-Arsloe spectral waves are carried out on the same scale as in the 1:100 scale physical model conducted at theEngineer Research and Development Center. Wave height comparisons, time series comparisons of surface elevation for monwave cases, and comparisons of power spectrum for spectral wave cases are made between numerical results and laboratoments. Comparison is also made between the probability distribution of surface elevation and skewness and asymmetry measuthe measured data and the Boussinesq model results. It is shown that for the computations of nonlinear wave transformation ovbathymetry, the Boussinesq model is able to predict nonlinear wave features and is thus a more accurate model than some cmodels in shallow water.

DOI: 10.1061/~ASCE!0733-950X~2003!129:3~124!

CE Database subject headings: Boussinesq equations; Numerical models; Finite difference method; Florida; Inlets, waterwa

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IntroductionBoussinesq models for surface gravity waves have been showprovide accurate tools for simulations of wave evolutioncoastal regions. Recent advances both in improved dispersionlationships in relatively deep water~Madsen and Sørensen 1992Nwogu 1993! and in computer technology allow the use oBoussinesq models in large nearshore regions. The incorporaof wave breaking and wave runup into Boussinesq mod~Karambas and Koutitas 1992; Scha¨ffer et al. 1993; Madsen et al.1997; Veeramony and Svendsen 2000; Kennedy et al. 2000; Cet al. 2000a! also allows these models to be applied to surf zonand swash zones. In addition, Boussinesq models with eitstructured curvilinear grids or unstructured grids have been devoped for application to complex nearshore domains. SørensenSørensen~2000! developed a finite-element Boussinesq modbased on the equations derived by Madsen and Sørensen~1992!.A finite-element model and a finite-difference curvilinear modbased on Beji and Nadaoka’s~1996! equations were developed byLi et al. ~1999! and Li and Zhan~2001!. Shi et al.~2001! devel-oped a finite-difference Boussinesq model in generalized curvil

1Research Associate, Center for Applied Coastal Research, UnivDelaware, Newark, DE 19716. E-mail: [email protected]

2Professor, Center for Applied Coastal Research, Univ. of DelawaNewark, DE 19716.

3Professor, Dept. of Civil Engineering, The Johns Hopkins Univ., Batimore, MD 21218.

4Professor, Dept. of Civil Engineering, Univ. of South Alabama, Mobile, AL 36688.

Note. Discussion open until October 1, 2003. Separate discussimust be submitted for individual papers. To extend the closing dateone month, a written request must be filed with the ASCE ManagiEditor. The manuscript for this paper was submitted for review and psible publication on December 11, 2001; approved on November 8, 20This paper is part of theJournal of Waterway, Port, Coastal, and OceanEngineering, Vol. 129, No. 3, May 1, 2003. ©ASCE, ISSN 0733-950X2003/3-124–135/$18.00.

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ear coordinates based on the fully nonlinear Boussinesq equaof Wei et al. ~1995!. Spatially varying and boundary-fitted gridwere adopted in the case studies of the model that showed, wcompared to the Cartesian version of the Boussinesq modelcurvilinear model has better efficiency and capability to deal wcomplex geometry in complicated nearshore domains. A literareview on a number of recent developments of Boussinesq mels and applications to nearshore wave propagation, surf zprocesses and wave-induced currents is given by Kirby~2002!.

Recently, a 1:100-scale physical model was constructed atU.S. Army Engineer Research and Development Center basethe nearshore bathymetry surveyed at Ponce de Leon Inlet,The inlet is flanked by a 1,295-m north jetty and a south jetty textends approximately 120 m from the mean low-water contoThe physical model was built to study proposed changes tonorth and south jetty configurations and, thus, a series of expmental studies of wave transformation was conducted inmodel. Smith and Harkins~1997! evaluated the performances othree numerical wave models—RCPWAVE, REF/DIF-1, aSTWAVE—in their studies using the experimental data. It wfound that model errors increase markedly with increases in wnonlinearity, because the three numerical models are basiclinear models in shallow water.

In this paper, a Boussinesq model is used to carry out a stof nonlinear wave propagation in Ponce de Leon Inlet. For wacomputations in such a large computational domain, withproximate dimensions of 2,50034,000 m~prototype! and with acomplicated geometry, the generalized curvilinear Boussinmodel shows some advantages. First, a curvilinear grid wstretched grid sizes is able to resolve a broad spectrum of wain the whole computational domain with good computationalficiency. Secondly, the grid can be a nonorthogonal grid, fittthe complicated geometry well. Starting with the curvilineBoussinesq model developed by Shi et al.~2001!, several im-provements are made in the present paper to deal with wbreaking, Smagorinsky subgrid turbulence mixing, and wa

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runup/rundown. Corrections for missing small terms are madfollowing Chen et al.~2000b!. A Texel-Marsden-Arsloe~TMA !spectral wavemaker is implemented in the model for simulationof irregular waves. The improved model is then used in 18 casethat include monochromatic waves and directional spectrawaves. Snapshots of instantaneous water surfaces illustrate warefraction, diffraction, reflection, and other nonlinear wave features in the complicated domain. The calculated wave heights acompared with measured data at gauge points in two measument arrays. Spectral analysis for both numerical results and mesurement data is carried out. The probability distributions of suface elevations and wave skewness and asymmetry for strononlinear-wave cases are computed, and comparisons are mbetween numerical results and measurement data to demonstrthe capability of the present model for nearshore nonlinear wave

Model Improvements

The curvilinear Boussinesq model of Shi et al.~2001! is furtherimproved for this study by implementing the corresponding curvilinear terms into the model, following the existing techniquesused for wave breaking, runup/rundown, and Smagorinsky-typsubgrid mixing~Chen et al. 1999a, 2000a!, and the corrections~Chen et al. 2000b! for missing small terms associated with vor-ticity convection in Wei et al.~1995!. The detailed description ofthose curvilinear representations can be found in the Appendix

A TMA shallow-water wave spectrum~Bouws et al. 1985! anda wrapped-normal directional-spreading function~Borgman 1984!are used to simulate a directional sea state in the present modThe combined spectrum function can be expressed as

S~ f ,h,u!5ETMA~ f ,h!G~u! (1)

In Eq. ~1!, ETMA is the TMA shallow-water frequency distributionas follows

ETMA~ f ,h!5ag2f 25~2p!24F~2p f ,h!e25/4~ f / f p!24

3ge@2~ f / f p21!2/2s2#(2)

in which f p is peak frequency,g presents a frequency spreadingparameter which is varied in both the laboratory experiments anthe present numerical simulations, anda and s are coefficientswhich may be found in Bouws et al.~1985!. In this paper,a5831023 and

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where su denotes circular deviation of the wrapped-normaspreading function. To avoid the computational underflow,N<10/su in the present paper. Fig. 1 shows a case of TMA diretional spectral distribution withf p50.1 Hz, Hmo51.0 m, g55,andsu520.

The directional spectrum is divided into 2,100 componenwith random phases and equal energy at each frequency bloThe source function technique~Wei et al. 1999! is then used foreach component and the final surface elevation function canwritten as

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in which x-axis is oriented along the main axis of the wavemakeand Dmn , kmn , and «mn are, respectively, the amplitude, wavenumber, and phase of a component.

Model Setup

The nearshore bathymetry at Ponce de Leon Inlet is complex wthe presence of a large ebb shoal, navigation channel, jetties,inlet leading to the Halifax and Indian Rivers. The bathymetryshown in Fig. 2, was represented in a 1:100-scale physical molocated at the U.S. Army Engineer Research and DevelopmeCenter. The north jetty was represented by an impermeable vecal structure with small stone sloped on either side. The soujetty was not included in the physical model. Wave data wecollected with 30 wave guages, shown in Fig. 2~dots!, at a sam-pling frequency of 25 Hz. Two gauges were placed immediateshoreward of the depth transition~G-1, G-2! and four were placedin the lee of the jetty~J-1–J-4!. Twenty-four gauges were placedin two linear arrays. One array was positioned across the oulobe of the ebb shoal~offshore array, O-1–O-12 from north tosouth! and another closer to shore~nearshore array, N-1–N-12

Fig. 1. TMA directional spectrum (f p50.1 Hz, Hmo51.0 m, g55andsu520)

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from north to south!. To avoid significant diffraction effectswithin the gauge array, incident wave directions were limited t630° from a line perpendicular to the wave generator. Wave rflections from the model boundaries were minimized by placement of energy-absorbing matting along both lateral boundarieThe bathymetry for numerical models used in Smith and Harkin~1997! is derived from a 1994 airborne lidar survey and partiameasurements in the physical model. This bathymetry is also usin the present Boussinesq model.

The numerical calculations are performed on the laboratoscale and the prototype scale is used for model description in tpaper to be consistent with Smith and Harkins~1997!. Aboundary-fitted grid is generated and shown in Fig. 3. The grigeneration method is based on Brackbill and Saltzman’s~1982!theory. The corresponding grid-generation software can be dowloaded from ^http://www.coastal.udel.edu/;fyshi/gridmk/gridhome.html&. In this study, we generate a nonorthogonal grithat fits the jetty and shoreline perfectly. The grid resolutionadjusted by using variable grid-density coefficients~Brackbill andSaltzman 1982!. A finer resolution is generated near the jetty, thebb shoal, coastlines and the inlet to resolve structures and shwaves in shallow water. The dimension of the generated grid4013781. The minimum grid size is 1.39 m near the shoreline othe right side of the jetty and the maximum grid size in the offshore is 5.4 m. The stretched curvilinear grid is able to resolvshort waves with a resolution of at least 14 calculation points pwave length in this study.

The numerical wavemaker is located in the same place asthe physical model. The width of the wave-generation regionabout one wavelength~peak wavelength for spectral wave cases!as used in Wei et al.~1999!. In constrast to the physical model,there is no waveguide at lateral boundaries in the numericmodel. To reduce wave diffraction from the wavemaker for obliquely incident wave cases, the length of wavemaker is largthan that in the physical model. A sponge layer is used behind tnumerical wavemaker to absorb waves behind the wavemaker.minimize reflected waves from lateral boundaries and the articial boundary inside the inlet, sponge layers are put at the saboundaries as in the laboratory experiments. Relatively thin~fivepoint! sponge layers are also used around the jetty and the inbeach to partially absorb waves. It is found that the thin sponlayers could not absorb waves efficiently on the front side of th

Fig. 2. Bathymetry at Ponce de Leon Inlet~depths in meters, circlesrepresent wave gauges!


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jetty. On the lee side, waves are weak and thus the wavesflected from the jetty do not significantly affect the results at thmeasurement gauges.

A total of 18 cases with monochromatic input waves and TMAspectral waves are carried out in this study. Table 1 presents aof incident wave conditions generated in the numerical modeThe wave heights listed in the table are significant wave heighHs for monochromatic waves andHmo for spectral waves. Therewere 26 cases carried out in the physical model study. Generaa typical case from the cases with the similar incident wave coditions, was chosen. The cases with very large wave heights weskipped because there was some difficulty in generating ideincident waves by using the numerical wavemaker. The two-wanumerical wavemaker is suspected as causing the problem,cause the wave heights generated inside the source region arelarge.

For monochromatic wave cases, we specify incident wavheight, wave angle, and wave period as the model input. Fspectral wave cases, we input wave-peak frequencyf p , Hmo ,main-incident wave direction, as well as TMA parametersg ands0 described previously. Each case is run for about 150–200 pewave periods. The time steps used in the model are 0.1 s forcases, which guarantees that the Courant numbers are less0.5 everywhere. On a Linux-based PC with a single CPU of 1GHz and a memory of 1 GB, it takes about 103 h to calculate 20waves with a wave period of 10 s. Surface elevations at the 3gauge locations are output every time step during each model rThe time series are used for data analysis after removing the fi500 s of the results.

Model Results and Model ÕData Comparisons

Monochromatic Waves

A total of nine monochromatic wave cases were simulated usinthe curvilinear Boussinesq model. These wave conditions inclu8, 10, and 15 s prototype wave periods with wave directionsthe negative y-axis and630 degrees from the negative y-axis~Cases 1, 3, 6, 7, 9, 11, 13, 15, and 17 from Table 1!.

Fig. 4 is a snapshot of surface elevation of normally incidenmonochromatic waves with a period of 10 s~Case 7!. It clearlyshows the wave reflection on the up-wave side of the jetty, wa

Fig. 3. Computational grid~note: real grid resolution is two timeshigher than that in this demonstration figure!


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diffraction on the leeward side, wave scattering from the tip of thjetty, and refractive wave focusing in the area to the right of thinlet mouth. Fig. 5 is a wave pattern of normally incident wavewith a long period of 15 s~Case 13!. The same reflection, diffrac-tion, and focusing features can be found in the figure. In additiothis long-period wave case illustrates a nonlinear feature repsented by the significant secondary wave crests shown to the rof the jetty.

Figs. 6 and 7 compare measured and modeled significant wheights at the two linear arrays for the two cases with normaincident waves. As demonstrated in Figs. 4 and 5, both measuments and numerical results show the focusing feature indicaby wave height amplifications in the offshore array and nearshoarray.

Model/data comparisons of time series of surface elevatioare carried out at measurement gauges along the offshorenearshore arrays. The gauge closest to the wave generator~G-2! is

Fig. 4. Snapshot of wave surface elevation~Case 7,H50.93, T510 s, andu50)

Table 1. Cases with Different Incident Wave Conditions

Case Hmo /Hs(m) T~s! u~°!a gb su(°)c

01 0.95 8 0 Monod —02 1.01 8 0 3.3 3003 1.15 8 230 Monod —04 1.05 8 230 3.3 3005 1.05 8 30 3.3 3006 1.32 8 30 Monod —07 0.93 10 0 Monod —08 0.95 10 0 5.0 2009 0.87 10 230 Monod —10 0.93 10 230 5.0 2011 0.74 10 30 Monod —12 0.84 10 30 5.0 2013 0.78 15 0 Monod —14 0.98 15 0 7.0 1015 0.90 15 230 Monod —16 0.73 15 230 7.0 1017 0.77 15 30 Monod —18 0.76 15 30 7.0 10aIncident wave angle.bFrequency-spreading parameter.cCircular deviation of wrapped-normal-spreading function.dMonochromatic wave.

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chosen as a reference for aligning time axes in laboratory anumerical simulations. Figs. 8 and 9 present model/data compasons of time series of surface elevations in Case 13 at wagauges along the offshore array~O-1, O-6, O-9, and O-12! andalong the nearshore array~N-1, N-6, N-9, and N-12!, respectively.Generally, good agreement is found in both figures, althoughslight discrepancy in timing of wave-crest arrivals is observedboth the offshore and nearshore arrays. It can be seen fromfigures that the secondary wave crests are clearly presentgauges O-6, O-9, N-6, N-9, and N-12 in both the measuremeand the numerical results. Similar results are obtained in compasons of time series of surface elevations for other monochromawaves cases and are not shown in the paper.

To demonstrate the model performance in representing wadiffraction, we make model/data comparisons at the four gaugin the lee of the jetty. Fig. 10 shows the comparisons of timseries of surface elevations in Case 13 (u50) at the fourgauges—J-1, J-2, J-3, and J-4. The strong diffraction effect of tlong-period waves (T515 s) is represented at gauges J-1 and Jin both the measurements and the numerical results. At J-3 aJ-4, waves with very small wave height are found in the numecal results, while no obvious wave signals are observed in tmeasurements. The weak wave signals in the numerical resare the reflected waves from the inlet beach on the opposite sof the jetty and can be clearly seen in Fig. 5. The thin sponglayer put on the inlet beach is not very efficient to absorb wave


Fig. 6. Wave height comparisons~Case 7,H50.93, T510 s, andu50; circles: measurements; crosses: numerical results!

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Fig. 8. Comparison of time series of surface elevations at offshorarray ~Case 13,H50.78,T515 s, andu50; dashed line: numericalresults; solid line: measurements!

Fig. 9. Comparison of time series of surface elevations at nearshoarray ~Case 13,H50.78,T515 s, andu50; dashed line: numericalresults; solid line: measurements!

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For the obliquely incident wave cases, wave diffraction canfound in the shadow regions, such as the inlet throat as showFigs. 11, 12, and 13, which present the fairly good model/dcomparisons of wave height in the obliquely incident wave ca~Cases 3 and 6!. It is shown that in the nearshore array, the loctions of maximum wave heights caused by wave focusing othe shoal shift to the right- or left-hand side correspondingwaves from the northwest or southeast. Wave focusing is notvious in the offshore array.

Spectral Waves

Similar to monochromatic wave cases, nine spectral wave care carried out in the numerical study~Cases 2, 4, 5, 8, 10, 12, 1416, and 18 from Table 1!. Here, we only analyze some typicaspectral wave features obtained from the numerical resultsthen make wave-height comparisons between model and msurement.

A spectral wave case with a peak period of 15 s~Case 14! isshown in Fig. 14. The figure shows that the short-crested waoffshore become long-crested as they propagate towards sand wave refraction and focusing are less obviously demonstrthan in the monochromatic wave cases.

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Fig. 10. Comparison of time series of surface elevations at fogauges on the lee side of the jetty~Case 13,H50.78,T515 s, andu50; dashed line: numerical results; solid line: measurements!


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Fig. 14 also demonstrates the good resolution of the stretccurvilinear grid for short waves in the nearshore region. As timprovement of dispersion relationships for Boussinesq equatiohave expanded their range of validity for wave calculations, tresolution of numerical models becomes of importance especiafor spectral wave calculations. In Case 14, the maximum wafrequency in the components employed in the model is 0.13 Hzprototype, thus, the minimum wavelength in the wave-generatregion~water depth513 m) is about 75 m that can be resolved bthe offshore grid with grid spacing of 5.4 m. Wavelengths becomsmaller as waves propagate into shallower water. The wave coponent with the maximum frequency becomes only 33 m longshallow water of 2 m and even shorter in the shallower water~ifwe do not consider wave breaking!. A finer grid is absolutelyneeded to resolve the short waves, including short harmowaves in the shallow-water region. The stretched curvilinear gwith the minimum grid spacing of 1.39 m in the present paper jusatisfies the needs.

The wave heightHmo comparisons for normally incident wavecases are shown in Figs. 15 and 16. The figures show good agment between data and model results and also show the smoowave-height distributions than that in the monochromatic wacases.

Compared with the monochromatic-wave cases with obliqueincident directions, the oblique spectral waves do not cleashow the effect of diffraction as demonstrated in Fig. 17 becauof a wide range of input wave directions. Figs. 18 and 19 compa


Fig. 13. Wave height comparisons~Case 6,H51.32,T58 s andu530; circles: measurements; crosses: numerical results!


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modeled wave heights to measured wave heights for peaincident wave direction of230° and 30°. General features inwave-height distributions for the spectral wave cases are similato those of the monochromatic wave cases, such as maximuwave heights shift left or right corresponding to the incident waveangles. As opposed to the monochromatic waves that cause wafocusing to be consistently positioned along the nearshore arrathe spectral waves show less wave focusing at the nearshore ardue to the superposition of focusing from a wide range of wavperiods and directions.

Discussion of Nonlinear Properties

The Boussinesq model is an ideal model for surface gravity wavsimulations, because it can predict wave refraction, diffractionreflection, and most importantly, it can present nonlinear wavfeatures that cannot be predicted by linear models. When wavpropagate from deep to shallow water, wave nonlinearity increases with the increase of the Ursell number~1953! Ur5HL2/h3, whereH is wave height;L is wavelength, andh iswater depth. The conventional models become invalid when thUrsell number exceeds 40~Ursell 1953!. For the cases with a longperiod and a large wave height,T515 s, H50.98 m, for ex-

Fig. 14. Snapshot of wave surface elevation~Case 14,Hmo50.98,Tp515 s, u50, g57.0, andsu510)

Fig. 15. Wave height comparisons~Case 14,Hmo50.98,Tp515 s,u50, g57.0, andsu510; circles: measurements; crosses: numericaresults!


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The nonlinear wave behavior manifests itself by the generatioof higher harmonics in the shoaling region. The Boussinesq prdiction of harmonic wave generation and triad interactions havbeen studied by Madsen and Sørensen~1992!, Nwogu ~1993!,Wei et al.~1995!, and Chen et al.~1999b!. Wei et al.~1995! dem-onstrated the importance of high-order nonlinear terms in thfully nonlinear Boussinesq equations for harmonic wave genertion. In Fig. 5, we find secondary wave crests due to the releasesuperharmonics generated by nonlinear shoaling in the nearshregion. Fig. 20 shows the power spectrum at three locations of thoffshore array in a spectral wave case~Case 14!. All three panelsshow the apparent second harmonic peaks. Fig. 21 shows tpower spectrum at three locations of the nearshore array thatdicates the energy transfers towards higher and lower frequencfrom the spectral peaks at the offshore array.

Another signature of wave nonlinearity may be indicated bythe probability distribution of surface elevation. For a linearwave, the observed sea-surface elevationh(t) is normally distrib-uted with meanh and variancem0 with the probability-densityfunction PDF~h! taking the form

Fig. 16. Wave height comparisons~Case 8,Hmo50.95, Tp510 s,u50, g55.0, andsu520; circles: measurements; crosses: numericaresults!

Fig. 17. Snapshot of wave surface elevation~Case 12,Hmo50.84,Tp510 s, u530, g55.0, andsu520)

130 / JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINE

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For a given data set, the probability-density distributionsurface elevation can be computed and its deviation, away fthe Gaussian distribution presented in Eq.~6!, is a measure ofwave nonlinearity of the process. Fig. 22 shows the compuPDF for both the measured data and numerical results at O-the offshore array. There are clear deviations with similar degin both the measured data and numerical results, away fromnormal distributions. Compared to the normal distribution, this an excess of extremely large positive values of the functionwell as a deficit of large negative values. This corresponds tofact that nonlinearity has sharpened the wave crests and flattthe troughs. In addition, there is a deficit of values just abovemean of zero and an excess of values just below. This ocbecause of the relatively narrower crests and broader troughs23 shows the computed PDF for waves at N-2 of the nearsarray. It is shown that the stronger nonlinearity in shallow wacauses a more significant deviation from the normal distribut

Similarly, wave skewness and asymmetry are also imporindicators of nonlinear wave behavior. As waves shoal towarbeach, both skewness and asymmetry increase~asymmetry be-comes increasing negative!. The prediction of wave skewness

lFig. 18. Wave height comparisons~Case 10,Hmo50.93,Tp510 s,u5230, g55.0, andsu520; circles: measurements; crosses: nmerical results!

Fig. 19. Wave height comparisons~Case 12,Hmo50.84,Tp510 s,u530, g55.0, andsu520; circles: measurements; crosses: numcal results!

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important at the viewpoint of sediment transport. Elgar and Gu~1985! have shown that the nonlinear Boussinesq-type model thused for the calculation of shoaling gravity waves is able to prdict the sea-surface-elevation skewness of the shoaling wfield. In this paper, wave skewness and asymmetry are compufrom time series of surface elevation at the offshore array anearshore array. Skewness is defined as

Fig. 20. Power spectrum at O-1, O-5 and O-8 of offshore arra~Case 14,Hmo50.98, Tp515 s, u50, g57.0, andsu510; solidline: measurements; dashed line: numerical results!

Fig. 21. Power spectrum at N-2, N-5, and N-9 of nearshore arr~Case 14,Hmo50.98, Tp515 s, u50, g57.0, andsu510; solidline: measurements; dashed line: numerical results!


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where^&5the averaging operator; and the meanh has been re-moved from the time series of surface elevation. Asymmetrydefined as

Asymmetry5^H~h!3&^h2&3/2

where H5the Hilbert transform and is representative of strocoupling and energy transfer across the frequency spectrumnegative values indicating a transfer of energy from low to hifrequencies as in typical shoaling processes.

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Fig. 22. Probability density distribution of surface elevation at Oof offshore array~Case 14,Hmo50.98,Tp515 s, u50, g57.0, andsu510; solid line: Gaussian distribution; dashed line: PDF distribtion of measured data or numerical results!

Fig. 23. Probability density distribution of surface elevation at Nof nearshore array~Case 14,Hmo50.98,Tp515 s,u50, g57.0, andsu510; solid line: Gaussian distribution; dashed line: PDF distribtion of measured data or numerical results!


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A shortwave case~Case 1! is shown in Fig. 24, which com-pares measured and modeled wave skewness and asymmSkewness is seen to increase as waves propagate from theshore array to the nearshore array. In both measured and modresults, large skewness values are found in the correspondstrong shoaling regions around the 800-m points of both the oshore array and the nearshore array. Asymmetry does not increapparently as waves approach nearshore in the shortwave cFig. 25 illustrates comparisons in a long-wave period case~Case14!. In the strong nonlinear wave case, both the numerical resuand measured data show that the skewness and asymmetrycrease rapidly as waves propagate from the offshore array tonearshore array. Fig. 25 also shows the overprediction of waskewness at nearshore array that is caused by unknown reas

To quantitatively measure the model/data agreement, RMerror is defined as~Smith and Harkins 1997!

RMS~%!5A1

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3100% (7)

where N5number of data points compared;Hni

5numerical-model wave height; andHmi5measured waveheight. To compare to the RMS errors shown in Smith and Hakins ~1997!, a summary of RMS errors is listed in Table 2 for thecases that were also presented in Smith and Harkins’~1997!paper. We also list the RMS differences in the measurements frthe physical model to measure the uncertainties contained inphysical-modeling measurements.

For the monochromatic wave cases, the overall RMS errofrom the Boussinesq model are much smaller, especially in tnearshore array, than that from the conventional models. Innearshore array, RMS errors from RCPWAVE and REF/DIF aover 50% in some cases, because those models are invalid instrong nonlinear wave region. For the spectral wave cases,Boussinesq results appear to be only marginally better thanSTWAVE results. The reason could be that for broad-band wa

Fig. 24. Comparisons of skewness and asymmetry~Case 01,H50.95, T58 s, u50; circles: measurements; crosses: numericresults!


etry.off-

eledingff-asease.

ltsin-

theve

ons.S

r-

omthe

rshetherethethetheve

cases, the STWAVE model is able to predict well wave heighHowever, we expect that the STWAVE model may not give rigspectral distributions in shallow water, because it cannot presome wave nonlinearities, such as wave harmonics. Althousome of the results from the present model are clearly in beagreement with the data, we could not comment on the moaccuracy, because a part of the numerical model bathymetrydeveloped from linearly interpolated depths between design teplates, not a postconstruction survey of the concrete beach~Smithand Harkins 1997!.

Conclusions

The improved curvilinear Boussinesq model based on the funonlinear equations was used to simulate nonlinear wavesPonce de Leon Inlet, Fla. The curvilinear model can resolvecomplex geometry and predict nonlinear wave transformationthe large computational domain. Eighteen cases were calculand fairly good agreement in the wave-height comparisons wobtained.

Nonlinear wave features were presented in the case studythe fully nonlinear Boussinesq model. Secondary wave-crestterns were clearly shown in the monochromatic wave cases. Stral analysis of spectral wave cases also showed the wave-entransfer from the original peak frequencies to the correspondharmonic frequencies. As another indicator of wave nonlinearthe probability distributions of wave surface elevations were coputed from both the measured data and numerical resultsshowed similar deviations from their Gaussian distributions. Tgood agreements because model and data comparisons ofskewness and asymmetry indicate the capability of the modepredict nonlinear wave transformation.

Strong wave reflection from the jetty was found in the numecal results, which may not occur in the physical experimenFurther work with a structure boundary condition is needed. W

alFig. 25. Comparisons of skewness and asymmetry~Case 14,Hmo

50.98,Tp515 s, u50, g57.0, andsu510; circles: measurementscrosses: numerical results!


arshore

Table 2. RMS Errors of Wave Heights

Case

RCPWAVE REF/DIF STWAVE BOUSSINESQ Gauge

Offshore Nearshore Offshore Nearshore Offshore Nearshore Offshore Nearshore Offshore Ne

01 17 25 15 16 — — 11.8 17.4 5.3 5.402 — — — — 9.6 16 8.0 14.5 0.7 1.407 27 66 26 45 — — 15.4 24.6 3.2 3.108 — — — — 9.5 15 10.6 15.7 1.6 2.209 13 57 14 58 — — 13.4 20.9 6.9 6.410 — — — — 9.8 17 5.9 13.2 2.2 3.211 24 50 17 53 — — 15.2 21.4 9.1 9.412 — — — — 13 5.6 8.3 10.1 1.4 2.113 27 38 25 23 — — 13.6 20.8 4.3 4.914 — — — — 12 23 11.4 17.5 3.8 4.0

of

ts

e

r isbe

k-

es

ine

ave

also encountered difficulty when generating large incident wavwith the numerical wavemaker. The wave-generation techniqneeds to be further improved by using the one-way numeriwavemaker recently developed by Chawla and Kirby~2000!,which eliminates the backward propagating wave in the souregion and thus reduces problems associated with breakingtwo-way waves in that region.

Acknowledgments

This work was sponsored by National Oceanographic PartnersProgram~NOPP, N00014-99-1-1051! and the Office of Naval Re-search, Base Enhancement Program~N00014-97-1-0283!. Thewriters would like to express their appreciation to Zeki Demibilek and Jarrell Smith, U.S. Army Engineer Research and Devopment Center, who supplied the data for the Ponce de Leon Imodel study.

Appendix. Curvilinear Implementation of WaveBreaking, Subgrid Mixing, Slot Technique, andTerms Associated with Vorticity Transport

The fully nonlinear Boussinesq equations with corrections fmissing small terms can be found in Chen et al.~2000b!. After ageneralized coordinate transformation~Shi et al. 2001!, the equa-tions can be written in the tensor-invariant forms

bh t11

Ag0

]

]xk ~Ag0Mk!50 (8)

Mk5LH ~h1h!uk1~h1h!Fza2

22

1

6~h22hh1h2!G

3F 1

Ag0

]

]xl ~Ag0ul !G ! k1~h1h!Fza11

2~h2h!G

3F 1

Ag0

]

]xl ~Ag0hul !G ! kJ (9)

]uk

]t1gh! k1ulu,l

k 1V1k1V2

k1V3k1Rb

k1Rsk50 (10)

where k, l 51 and 2; (u1,u2)5contravariant components of areference velocity at a reference elevationza ; (x1,x2)5new in-dependent variables in the transformed image domain;b and L


esue

cal

rceof

hip

r-el-nlet

or

5two dimensionless multipliers introduced for the treatmentshoreline run-up;Ag0 is the Jacobi value; ()k represents the co-variant spatial derivative while !k represents the contravarianspatial derivative;V1

k andV2k are the dispersive Boussinesq term

that can be found in Shi et al.~2001!; and V3k represents extra

terms associated with vorticity transport~Chen et al. 2000b!.In Eq. ~10!, Rb

k and Rsk represent the breaking terms and th

subgrid mixing terms, respectively.

Energy Dissipation Due to Wave Breaking andSubgrid Mixing

The energy dissipation due to wave breaking in shallow watemodeled by using the momentum mixing terms, which canwritten in a general form as

1

Hdiv~HnD! (11)

wheren5eddy viscosity localized on the front face of the breaing wave ~Kennedy et al. 2000; Chen et al. 2000a!; D5rate-of-strain tensor defined by

D51

2@gradu1~gradu!T# (12)

in which (gradu)T is the transpose of gradu. The tensor-invariantforms of the diffusion terms in generalized curvilinear coordinatcan be expressed as

Rbk5

1

HAg0

]Ag0nHDki

]xi 1nD ji G j ik (13)

whereG j ik 5Christoffel symbol of the second kind and

Di j 51

2~gk ju,k

i 1gkiu,kj ! (14)

The subgrid mixing terms are in the same form as shownEq. ~11! except thatn should be replace by the eddy viscosity duto the subgrid turbulencens

ns5cAg0uTu5cAg0~Ti j Ti j !1/2 (15)

wherec5subgrid mixing coefficient;T5rate-of-strain tensor ofthe velocity of the wave-generated currentU which may be ob-tained by averaging the instantaneous velocity over several wperiods.T can be written as

T51

2@gradU1~gradU!T# (16)


al.

h.’’

.

A.g,

.,e

’’

,

up.

rents.’’,

. A

The tensor-invariant forms of the subgrid mixing terms can bwritten as

Rsk5

1

HAg0

]Ag0nsHDki

]xi 1nsDji G j i

k (17)

where

Tji 51

2~gk jU ,k

i 1gkiU ,kj ! (18)

and

Ti j 51

2~gkiU , j

k 1gk jU ,ik ! (19)

Slot Technique

Following the rectangular version of the fully nonlinear Boussinesq model~Chen et al. 2000a!, a slot technique is implementedin the curvilinear model. The dimensionless multipliersb andLin Eqs.~8! and ~9! are defined exactly as in Chen et al.~2000a!.The implementation is also the same as in the rectangular versexcept that the slots are always along the curvilinear coordinalines.

Implementation of Extra Terms Associatedwith Vorticity Transport

Compared with the Boussinesq equations in Li~1999!, Chen et al.~2000b! or Hsiao et al.~2002!, the missing terms in Wei et al.~1995! are theO(m2) vorticity terms in the momentum equationsand can be expressed as

V35~u•¹za!@za¹~¹•u!1¹~¹•hu!#2¹za@~u•¹!~¹•hu!

1za~u•¹!~¹•u!# (20)

In a tensor-invariant form, Eq.~20! can be written as

V3k5ul

]za

]xl FzaS 1

Ag0

]Ag0ul

]xl D ! k1S 1

Ag0

]Ag0hul

]xl D ! kG2za! kFul

]

]xl S 1

Ag0

]Ag0hum

]xm D 1zaul]

]xl S 1

Ag0

]Ag0um

]xm D G(21)

Based on the definitions in the specific model implementatioin Shi et al.~2001!, we defineF7 andG7 for the new extra termswhich are expressed as

F752~Uza,j11Vza,j2

!1

g0@g22za~DU!j1

2g12za~DU!j2

1g22~DHU!j12g12~DHU!j2

#11

g0~za,j1

g222za,j2g12!

3@U~DHU!j11V~DHU!j2

1zaU~DU!j11zaV~DU!j2

#

(22)

and


e

-

ionte

n

G752~Uza,j11Vza,j2

!1

g0@2g12za~DU!j1

1g11za~DU!j2

2g12~DHU!j11g11~DHU!j2

#11

g0~2za,j1

g121za,j2g11!

3@U~DHU!j11V~DHU!j2

1zaU~DU!j11zaV~DU!j2

#

(23)

The definitions of DU and DHU are the same as in Shi et~2001!.

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Wave Simulations in Ponce de Leon Inlet Using Boussinesq Model

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