Boussinesq modeling of a rip current system

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1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 14, NO. C9, PAGES 2,17-2,37, SEPTEMBER 1, 1999 Bussnesq mdelng f a rp current system Qn Chen, Rbert A. Dalrymple, James T. Krby, Andrew B. Kennedy, and Merrck C. Haller Center fr Appled Castal Research, Unversty f Delaware, Newark Abstract. n ths study, we use a tme dman numercal mdel based n the fully nnlnear extended Bussnesq equatns [We et al., 199] t nvestgate surface wave transfrmatn and breakng-nduced nearshre crculatn. The energy dsspatn due t wave breakng s mdeled by ntrducng an eddy vscsty term nt the mmentum equatns, wth the vscsty strngly lcalzed n the frnt face f the breakng waves. Wave run-up n the beach s smulated usng a mvng shrelne technque. We emply quas furth-rder fnte dfference schemes t slve the gvernng equatns. Satsfactry agreement s fund between the numercal results and the labratry measurements f Haler et al. [1997], ncludng wave heght, mean water level, and lngshre and crss-shre velcty cmpnents. The mdel results reveal the tempral and spatal varablty f the wave-nduced nearshre crculatn, and the nstablty f the rp current n agreement wth the physcal experment. nsghts nt the vrtcty asscated wth the rp current and wave dffractn by underlyng vrtces are btaned. 1. ntrductn Hggns [197a, b], amng thers, frm the theretcal fundatn fr understandng nearshre crculatn gen- Mdelng surf zne hydrdynamcs, ncludng transerated by wave breakng. A recent lterature revew n frmatn f surface waves, crss-shre and lngshre surf zne hydrdynamcs was made by Svendsen and currents, and lw-frequency mtns, s f great nter- Putrevu [199]. n the lterature, mst f the numercal est fr many reasns. Recently, advances have been mdels fr wave-nduced nearshre crculatn are based made n extendng the Bussnesq equatns frm a set n the depth-ntegrated, tme-averaged (ver a shrt f equatns vald nly fr surface waves wth very small wave perd) cnservatn laws f mass and mmenwave numbers t numercal mdels that are nw capable tum. Radatn stresses due t the shrt wave mtn f mdelng wave prpagatn frm deep water t shalare taken as a frcng n the mmentum equatns fr lw water [see, e.g., Madsen and S rensen, 1992; Nwgu, the mean flw, and the tme-averaged mass flux f the 1993; We et al., 199]. Wave breakng n surf znes s shrt wave mtn s ncluded n the mass equatn fr als ncrprated nt Bussnesq mdels by Karambas the nearshre crculatn. The effect f the underlyng and Kuttas [1992], SchSffer et al. [1993], Madsen et current feld n the wave transfrmatn can be taken al. [1997], Svendsen et al. [199], and Kennedy et al. nt accunt by an teratve prcess [see, e.g., Brke- [1999], amng thers. On the ther hand, advances n meer and Dalrymple, 197], but t s tme cnsumng cmputer technlgy nw permt the use f Bussnesq n practce. The mssn f wave-current nteractn, mdels fr large nearshre regns and allw the avhwever, can nt be justfed n the case f strng cureragng f mdel results t predct mean flws n the rents. nearshre, ncludng lngshre and rp currents. The n ths study, we emply a tme dman numercal wave blckage by strng ppsng currents s als smmdel based n the fully nnlnear Bussnesq equatns ulated by Chen et al. [1998], usng a Bussnesq mdel ntrduced by We et al. [199] t nvestgate the fully fr the fully cupled wave and current mtn. Lteracupled nteractn f surface waves wth rp currents ture revews n recent advances n Bussnesq mdelng and the nearshre crculatn generated by wave breakf nearshre surface gravty waves are gven by Krby ng n a barred beach wth a rp channel. n sectn 2, [1997] and Madsen and Schaffer [1999]. we present the gvernng equatns ncludng addtnal The cncept f radatn stress ntrduced by Lnguetterms t accunt fr wave breakng, subgrd turbulent Hggns and Stewart [191] and the pneerng wrk mxng, bttm frctn, and shrelne run-up. Sectn n lngshre currents by Bwen [199] and Lnguet- 3 descrbes the numercal smulatn f wave-nduced Cpyrght 1999 by the Amercan Gephyscal Unn. Paper number 1999JC /99/1999JC ,17 nearshre crculatn n a bar/trugh beach wth a rp channel. We present the spatal and tempral varatn f the cmputed wave feld, the underlyng current feld averaged ver tw wave perds, and the vrtcty feld btaned frm bth the averaged velcty feld

2 2,18 CHEN ET AL.: BOUSSNESQ MODELNG OF RP CURRENTS and the nstantaneus velcty f the cmbned wave and current mtn. The numercal results are cm- pared wth labratry measurements, ncludng wave heght, mean water level, and lngshre and crss-shre currents alng several transects. n sectn 4, an deal bathymetry s ntrduced t study the effects f bathymetrc nnunfrmty n rp stablty and vrtex structure asscated wth the rp current. Cmparsn wth the same set f labratry data s presented. Sectn s devted t the study f wave refractn/dffractn by a rp current. Results frm bth the Bussnesq mdel and the refractn/dffractn (REF/DF)mdel based n the parablc apprxmatn f the mld-slpe equatn shw a smlar dffractn pattern f the surface wave feld by the underlyng rp current. The mplcatn f the dffracted wave feld wth respect t the nearshre crculatn s dscussed. Fnally, we summarze the fndngs n sectn. 2. Mdel Frmulatn 2.1. Gvernng Equatns The extended Bussnesq equatns f We et al. [199] are wrtten n terms f the velcty vectr us= (us, v ) at a reference elevatn z n the water clumn and the free surface elevatn /relatve t the stll water level. The equatn fr cnservatn f mass may be wrtten as where 3r/t + V.M = (1) { [ z 2 1 (h2 _ hr/+ r/2 ) M-A(h+r/) us+ 2 v v) v v. (2) n whch h s the stll water depth, h s the stll water depth at the ffshre lmt f the slt, the subscrpt t dentes tme dfferentatn, and V s the hrzntal gradent peratr. n addtn, / and A are tw dmensnless multplers ntrduced fr the treatment f shrelne run-up as descrbed n sectn 2.2. The asscated mmentum cnservatn equatn s u t + (us.v) us + gv /+ V + V2 - Rb - R8 + Rf = (a) where g s the gravtatnal acceleratn and V1 and V2 are the dspersve Bussnesq terms The addtnal terms, Rb, Rs, and R, representhe effects f wave breakng, subgrd lateral turbulent mxng, and bttm frctn, respectvely, as detaled n sectns 2.3 and 2.4. The fully nnlnear Bussnesq equatns have mprved dspersn characterstcs n the case f large wave number and nnlnearty n shallw water. n cnnectn wth surface waves and wave-nduced cur- rents, t s wrth mentnng that the equatns are sutable fr mdelng wave-current nteractn, as shwn by Krby [1997] A Treatment f Mvng Shrelnes T smulate swash mtns, t s necessary fr the mdel t nclude a treatment f the sea-land nterface. nstead f trackng the wetted and dry cells durng wave run-up/run-dwn n the beach, we treat the entre cmputatnal dman as an actve flud dman by emplyng an mprved versn f the slt r permeable-seabed technque prpsed by Ta [1984] fr the smulatn f wave run-up. The rgnal slt technque has been used by Madsen et al. [1997] n a Bussnesq mdel frmu- lated n terms f mass flux and free surface elevatn. The basc dea behnd ths technque s t replace the sld bttm, where there s very lttle r n water cverng the land, by a prus seabed, r t assume that the sld bttm cntans narrw slts. Ths allws the water level t be belw the beach elevatn. Fgure 1 llustrates a beach wth the presence f the slt. The replacement f the sld bttm by narrw slts results n a mdfcatn f the mass equatn shwn by (1), where and _{ + _ 1, r/> v < z* () q- "7 (e x (n-z*) -X(n+z*)) h --e h, r/ _< z* A -- v_z. v - + (z*+h) -]-W +'7 ( 1- e _x ( +z*), rl> ) ( - )n Here -/ -, T- ' s the relatve wdth f slt wth respect t a unt wdth f beach, X s the parameter fr the smth transtn frm unty t, and h s the ffshre stll water depth where a slt begns. (7) (4) stll water level h A A-A Fgure 1. Schematc f a beach wth the presence f a narrw slt.

3 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS 2,19 Madsen et al. [1997] shwed that, even thugh a very narrw slt wdth s used, there s stll abut a 1% errr n the cmputed maxmum run-up n cmparsn wth the analytcal slutn by Carrer and Greenspan [198]. Ths s attrbuted t the addtnal crss-sectnal area ntrduced by the narrw slt, because the maxmum run-up s very senstve t the ttal vlume f mass at the run-up tp. n cntrast t Ta's [1984] frmulatn, whch des nt cnserve mass n the presence f a slt, we retan an equvalent crss-sectnal area f a unt wdth f beach, leadng t the mprvement n the smulatn f run-up as shwn by Kennedy et al. [1999]. The mdfed seabed elevatn z* may be expressed as 1- +hø 1- + (8) n whch z s s the elevatn f the sld seabed. The ptmal values f and h are fund t be.2 and 8, respectvely, whch gve the best agreement wth the analytcal slutn by Carrer and Greenspan [198]. Fr smulatns f wave run-up n steep slpes, hwever, a larger slt wdth and a lcalzed flter may be needed t avd numercal nstablty. Chen et al. [1999] verfed the Bussnesq mdel wth the mprved permeable-seabed technque aganst the labratry experment n sltary wave run-up n a crcular sland descrbed by Lu et al. [1991 Gd agreement between the cmputed and measured maxmum run-up was fund A Treatment f Wave Breakng Attempts have been made n the lterature t ntrduce treatments f wave breakng nt Bussnesq mdels. They may be gruped nt eddy vscsty [e.g., Zelt, 1991; Karambas and Kuttas, 1992; Kennedy et al., 1999] and "rller" breakng mdels [e.g., Schffer et al., 1993; $vcndscn et al., 199]. Wth respect t energy dsspatn due t wave breakng, these tw types f mdels gve smlar results as demnstrated by Svendsen et al. [199]. Fr smplcty, we chse an eddy vscsty type mdel. Fllwng Kennedy et al. [1999], the energy dsspatn due t wave breakng n shallw water s mdeled by ntrducng the mmentum mxng terms: (9) frnt face f the breakng wave. t shuld be emphaszed that the lcalzatn f the eddy vscsty s f mprtance fr mdelng nnlnear waves. n cntrast, a glbal eddy vscsty wuld smear the asymmetry and skewness f the breakng waves n a nnphyscal man- ner. We defne eddy vscsty as B*:l(h + v)v.m (11) n whch s a mxng length ceffcent wth an emprcal value f = , 1.8. The quantty B that cntrls the ccurrence f energy dsspatn wth a smth transtn frm t s gven by 1, r/t >_, v, < v? B- v? ' - (12) Fgure 2 llustrates wave breakng n a barred beach and the mmentum mxng asscated wth the rller. n analgy t the rller mdel by Schffer et al. [1993], we determne the nset and cessatn f wave breakng usng the parameter /, whch s defned as lt - "tt _( ) +-W-c'tt t-t r_( ') - t ( ) ), t-t - <T* (13) where T* s the transtn tme, t s the tme when wave breakng ccurs, and t- t s the age f the breakng event ß The value f qt ( ) s chsen between.3 h and. h, whle the values f V F) and T* are.1 h nd /g, respectvely. The cnstructn and verf- catn f the breakng mdel are detaled by Kennedy et al. [1999]. The lwer lmt f the emprcal ce - cent () s fund t be mre sutable t bar/trugh beaches, whle the upper lmt gves ptmal agreement fr waves breakng n mntne slpng beaches. Chen et al. [1999] descrbe the mplementatn and verfcatn f the breakng mdel n tw hrzntal dmensns. / Dstance Cg Rller Rller + x + {t,[(h + rl)v ]y}y) (1) Barred Beach where superscrpts x and y represent the drectns n the hrzntal plane, subscrpts x and y dente spatal dfferentals, and s the eddy vscsty lcalzed n the Fgure 2. beach. Schematc f wave breakng n a barred

4 2,2 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS 2.4. Subgrd Turbulent Mxng and Bttm Frctn Bussnesq mdels are based n vertcally ntegrated mass and mmentum equatns. Hwever, the grd sze nvlved wth the smulatn f surface waves s usually smaller than the typcal water depth. The hrzntally dstrbuted eddy vscsty resultng frm subgrd turbulent prcesses may therefre becme an mprtant factr nfluencng the flw pattern f the wave-generated current feld. n the absence f the subgrd mdel n the gvernng equatns, the underlyng current feld generated by wave breakng may becme s chatc that n realstc flw pattern can be recgnzed. n the present study, we utlze the Smagrnsky-type subgrd mdel [Smagrnsky et al., 19] t accunt fr the effect f the resultant eddy vscsty n the underlyng flw. 1 : + V (( [( + 1 ) (14) +"s [(h +,)U,]y} {,s[(h + V)V,]y}y) (1) where y s the eddy vscsty due t the subgrd turbulence. = y n Whch U and V are the velcty cmpnents f the tme-averaged underlyng current feld; Ax and y are the grd spacng n the x and y drectns, respectvely; nd c s the mxng ce cent wth a default value f.2. n the curse f smulatn, we btan the underly- ng current feld by averagng the nstantaneus velcty ver tw wave perds and update accrdngly. As usual, the bttm frctn s mdeled by the quadratc law Rr = u. lu.[ (17) where the frctn ceffcent s chsen t be f =. x 1-3 n the present smulatn. Ths value s abut 2 rders f magntude larger than the frctn ceffcent used by Zelt [1991] n hs Bussnesq mdel fr sltary wave run-up n a 1:2 slpng bttm, but t s abut 1 rder f magntude smaller than the bttm frctn ceffcents used t cmpute lngshre currents generated n labratres (e.g., k. x 1-2 f Kbayash et al. [1997]). Ntce that us s the velcty vectr fr the cmbned wave and current mtn. Under feld cndtns, wng t the varablty f hydrdynamc and mrphlgc characterstcs, spatally varable frctn ceffcents are lkely t be used. Fllwng We et al. [199], quas furth-rder fnte dfference schemes are used t slve the gvernng equatns descrbed abve. The numercal mdel has been extensvely tested aganst labratry measurements by We et al. [199], We [1997], Kennedy et al. [1999], and Chen et al. [1999]. We shall use ths mdel t study wave-nduced nearshre crculatn and vrtex structures asscated wth rp currents. 3. Smulatn f Rp Currents 3.1. Tpgraphy and Mdel Setup A beach wth ffshre sand bars ncsed by rp channels s a cmmn type f bathymetry n many castal regns. Fgure 3 shws the bathymetry and the cnturs f the water depth used fr the smulatn f rp currents n the present study, whch was nspred by a physcal experment cnducted at the Unversty f Delaware [see Haler et al., 1997]. T save cmputatnal effrts, nly half f the expermental tpgraphy s used n the numercal smulatn, assumng symmetry abut the crss-shre center lne f the physcal wave basn. We cnstruct the tpgraphy n the bass f the ba :hymetrc data frm the labratry experment. The numercal wave basn s 19 m lng and 9.1 m wde. As depcted n Fgure 3a, a 1.8 m wde rp channel nterrupts a submerged bar n a beach wth a 1:3 slpe. The water depth f the ffshre fat bttm s.373 m. The averaged water depths n the crest and at the ffshre te f the bar are.48 and.1 m, respectvely. Each f the bar sectns spans 1.2 m n the crss-shre drectn and 3. m n the lngshre drectn. As shwn n Fgure 3b, the bathymetry s slghtly asymmetrc. We use a nrmally ncdent, mnchrmatc wave tran wth a perd f 1. s and wave heght f 4.8 cm as nput t the mdel. Waves are generated by the surce functn technque as detaled by We et al. [1999]. The wave generatn s lcated nternally at x = 4. m, and a dampng spnge layer s put behnd the surce lne t absrb utgng waves reflected by the submerged bar and the slpng beach. The grd sze s. and.1 m n the crss-shre and lngshre drectns, respectvely. The tme step s chsen t be.2 s Mdel Results The Bussnesq mdel prvdes a tme seres f free surface elevatn and velcty cmpnents fr the cmbned wave and current mtn. These results are used t btan many useful quanttes, ncludng radatn stress, mean water level, wave heght, underlyng current feld, and vrtcty feld asscated wth the wavenduced nearshre crculatn. Fgure 4 llustrates the cmputed mean water level averaged ver tw wave perds, and the lcatns f wave breakng after 2 s have elapsed n the smulatn. Obvusly, wave breakng causes a crss-shre varatn

5 ß ß ß ß. CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS 2, lo (a) Tpgraphy ß. ' -...:... ß -.,,,-...,..._..... ß "?' '--'..:':-:..-',.,.,.'.-.7,:.&, :.: ',,:z-:'"?7:.:. ' ' ß :... '"?'"'" '""' ' " '' " ' ''""'' '---"'"';';;"; ":'.' ' ' "--... the rp channel, wave breakng manly ccurs near the shrelne. The dfference n the breakng lcatns results n the lngshre gradent f the mean free surface. The resultant pressure gradent s beleved t be ne f the drvng frces fr rp currents n barred beaches as suggested by Dalrymple [1978]. n Fgure 4b, the dashed lnes dente the ftprnt f the submerged bar. Frm the cntur map f the (a) Water Level Averaged ver 2 Wave Perds at the 2th secnd y (m) x (m) (b) Cnturs f Stll Water Depth n Centmeters J, SHORE,, x (m) (b) Cnturs f Mean Water Level n Mllmeters,, - -,... 8 (c) Transects f Wave and Current Measurements ', E >,4 8, x (m) 2 lo 1 x (m) Fgure 3. Tpgraphy f a barred beach wth a rp channel and the lcatns f measurements: (a) tpgraphy, (b) cnturs f stll water depth, and (c) transects f wave and current measurements. The dashed lnes n Fgure 3c dente the bar ftprnt. (c) Cmputed Lcatns f Wave Breakng :. :. : :.;:. : : : ::: ";m.: : '. :";. 2 ; >..:- - - : : :. / : : : ' :½ : : : : :.: :.; : -': ' : :: 1?" ' " ': ':' " ' '. :..' ' ' ; : : j r. * 2;; r : :::?:. :::: : : :. : :: :: : : :: :. :. : : ::.. : :; : :.. J ; : :... ' ':'" " - : ::: :::::;:...:. : :.? ::::: :: ::: ½;J :: :. ;: : :; :: ;: :::.::-:.'.' :: : ::: -: ;;%½ '1.::::::: ::::.:'.*.'.:. ' f radatn stresses that creates the setup f the mean free surface. On the ther hand, lngshre varatns f radatn stresses als exst wng t the presence f the rp channel. The shaded areas n Fgure 4c dsplay the cmputed lcatns f wave breakng ver tw wave perds. We ntce that waves break n the submerged bar and near the shrelne n the barred beaches, whle at Crss-shre Dstance (m) Fgure 4. Mdeled mean free surface and the lcatns f wave breakng (dented by shadng) averaged ver tw wave perds (t s): (a) mean water level, (b) cnturs f water level, and (c)lcatns.

6 ß 2,22 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS Free Surface Elevatn at t=2 s.8!.4.2 (m) --1! b) The Underlyng Current Feld. >, ,<---: (1/s) - - ' Vrtcty 1!. O/s)! t Crss-shre Dstance (m) -. Plate 1. A snapsht f (a) the cmputed free surfacelevatn, (b) the underlyng current feld and asscated mean vrtcty, and (c) the nstantaneus vrtcty feld at t - 2 s.

7 CHEN ET AL.: BOUSSNESQ MODELNG OF RP CURRENTS 2,23 Free Surface Elevatn at t= s Free Surface Elevatn at t=8 s t t 1!! t '--1 t t 1 1 >'4 "' 1! 1 1 lo 1 b) The Underlyng Current Feld The Underlyng Current Feld v ::::::::::::::::::::::::."'1... r_... :...: :... :..::: ::::'...,. :,,,,;:, tl; :::::::::::::: :... ß ::O. 'rf/ :::::::::'::::::1::7:.1:... :-:... :. t -,-_ -- qq=.... :.: ;-.:;: : :.. _-=1,t.,-, "..-' ::::::::::::::::::::::::::: '*"' :1,!:... ' -;:...,,".,;g 'l;',. ' ' "'... :::::::::::::::....-"T-' j::, ;--. t -. ß.- ::::::::::::::::::::::::::::::::::::......!s :::!::: F:-:?: ::::, lo ::::::::::::::::::::::::::::;:::..,,-;.11:: :.,, ' :",' :l--,... :: :: ::: ::: ::..: ::':::::::t:;;;;...,..,..,,.,,, :.'.. ;' ::;:::::::: ::::::::::::::::::::::::::::::::::...! %,.... 1,;.?.... ::::1';"-... ::::;c.... ;;;... t,,.' ,, '"'"',,,.',,...:'.,k.,,--...,'_,--,,,.. ::::::::::... :.. _.,., - ;_.::...',.'',.?./.p"f./.,. X½.- /'f_., ".;:.. ;_,*,, ß f,.,,.....,,,,..,...,. ß,[ ' ',.., _ 'f. ', '... _';,;,.) t ß,, :....,,( :.,-:-..,,!._? :... :. :.:::::: :::::: :ø' 1..., _-_LJ:F - :.:'. '_. :'.' _ : '-... :::=;:l p,. 'ø ''-ss; ==================================== [-;,,... ':: : ::: : : :::: :! ::!! -: :..';'-:.... ::::-:' :' :::: : :! ".:.:":."* :?,.' 1,,' ' lo 1 c) vrtcty vrtcty 8! v E >'4 2! 2'-'! Crss-shre Dstance (m) Crss-shre Dstance (m) Plate 2. Snapshts f (a) the cmputed free surfacelevatn, (b) the underlyng current feld, and (c) vrcccy at (left) t - s and (rght) t - 8 s.

8 2,24 CHEN ET AL' BOUSSNESQ MODELNG OF RP CUR. R. ENTS Free Surface Elevatn at t=1 s Free Surface Elevatn at t=12 s 8!! 1! >'4! 2 2 v E >'4 : '., b) The Underlyng Current Feld The Underlyng Current Feld...!:.' l?..e!' ;... :' : : ::,.:::::::... ::::::::::::::::::::::::::::::::::::::::::::::::: ;;:.;....,,.; : ß,:.:.._., : t lz: ,-,...;";:... '::::,,.-.. ;:.-. h,,.,.... ::::::::::::... : 1, --'"": 7-., ;" ' '... :..... v v... ß t,,,, :::::::::::... :: ;;. llpl _ l l.,... 2-[.'... >'4 ':':::: ===================== ll:.:: [,: >'4 '... :::ltt &; ¾x'x <::::: :.:. - :::::::::::: J, :: ' :::::: ::-:' :::::::::::::::::::::::.;; _x... *s,... ::::::::::::::::::::- r -..,,.-:--- ':)?$3T. :' -'. ::::::::::::: -';!!!!!!!!!!!!!;!!!- '"': : ';' 7r. :::-.:::: ' :... '!!!!!!;!;!:.":.: :,:,:.!.:.. '... Jz,, $d '...,..... ======================== ß......,t,,,.:..-. ':... - :::::::::::::::::::::::::::::::::::::::::,;.. ;.>_,.,, ::::::::::::::::::::::::::::::::::::::::::::::::: ========================================== '1 1 C) Vrtcty! 8 8 Vrtcty >' Crss-shre Dstance (m) j......! 1 1 Crss-shre Dstance (m) Plate 3. Snapshts f (a) the cmputed free surface elevatn, (b) the underlyng current feld, and (c) vrtcty at (left) t - 1 s and (rght) t - 12 s.

9 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS 2,2 mean free surface, t s ntceable that tw depressns f mean water level appear n the edges f the rp channel. They are lcated at the eyes f the vrtex pars as shwn n Plate 1. The mean water levels n bth sdes f the rp channel are nt equal because f the slght asymmetry n the bathymetry. Plate 1 presents a snapsht f the cmputed free surface elevatn, the underlyng current feld, and the vrtcty develpng n the bar crest and near the shrelne at the 2th s f smulatn. The vrtcty s defned as f -- vx - Uy, where (u,v) can be ether the nstantaneus velcty cmpnents (u,v ) r the underlyng current (U, V) averaged ver tw shrt wave perds. The waves prpagate frm left t rght. n Plate la, blue dentes the wave trugh and yellw and red are the wave crest. The unt f the clr bar fr the free surface elevatn s n meters, whle that fr vrtcty s s -1. The dashed lnes nce agan representhe bar ftprnt. Owng t the effects f nnlnear shalng, the wave crest becmes peaky n the bar crest. t s ntceable that the wave heght n the rp channel s hgher than the wave heght n the bar crest because f wave-current nteractn and the absence f wave break- ng n the channel. We btan the underlyng current feld drven by the breakng waves by averagng the mdeled flud partcle velcty ver tw wave perds. The velcty, as ne f the dependent varables f the extended Bussnesq equatns, s lcated at the elevatn f z h n the water clumn. Assumng a farly depth unfrm underlyng flw, the cmputed velcty f the current feld can be taken as the depth-averaged velcty f the flow. n Plate lb, we ntce that a par f vrtces develps n the edges f the rp channel. As mentned abve, the eddes cause the depressn f the mean water level wth the magntude f abut half the maxmum setup f mean free surface n the present case. The mechansm f the vrtex prductn s attrbuted t the strng velcty shear asscated wth the rp current n the area between the channel and the submerged bar. We cmpute the nshre mass flux due t the wave mtn based n the mdeled results and fnd that the nshre mass flux f wave mtn ver the submerged bar s much greater than that n the rp channel as expected. Ths lngshre gradent f mass flux als cntrbutes t the vrtex generatn. A cmparsn f Plates lb and c shws that the vrtcty n Plate lb cmputed frm the underlyng current feld s n agreement wth that drectly calculated frm the nstantaneus velcty f the cmbned wave and current mtn as depcted by Plate lc. The absence f spatal perdc features at the scale f the wave length n the nstantaneus vrtcty plts mples that there s a separatn between the underlyng rtatnal flw and the wave mtn. n addtn t the vrtex par n the edges f the rp channel, vrtces als appear n the bar crest and near the shrelne behnd the rp channel. The vrtcty n the crest f the bar rgnates frm the lngshre nnunfrmty f breakng n the bar crest because f the perturbatns n the bathymetry. Ths s smlar t the mechansm f vrtex generatn due t nnunfrmtes n bres as suggested by Peregrne [1998]. The vrtex generatn and vrtcty transprt may be explaned by the vrtcty equatns derved frm the mmentum equatns (3). Takng the curl f (3) leads t, + (u.v) - - V.u +VxRb + VxR8 - VxRf + O(/ 2 gh -7) ( 8) n whch f s the vertcal cmpnent f the vrtcty vectr at the reference elevatn z and f = Vxu. Ntce that / s the measure f wave dspersn and/ << n the surf zne, where the water depth h s much smaller than the wave length L. Clearly, 7xRb s the surce f vrtcty caused by the lngshre varatn f wave breakng. The bttm frctn and subgrd mxng serve as the dsspatn agent f vrtcty. Owng t the nnzer depth varatn f vertcal velcty, the frst term n the rght-hand sde f the vrtcty equatn prvdes the mechansm f vrtex stretchng. T llustrate the spatal and tempral varablty f the the rp current and nearshre crculatn, we present a tme sequence f the cmputed wave feld, underlyng current feld, and vrtcty n Plate 2 and Plate 3. The tme nterval between each snapsht f the mdel results s 2 s. As depcted by the sequence f cmputed flw feld, the mdeled rp current meandered at the rp channel ext. Hallet et al. [1997] bserved that the lngshre cmpnent f the current measured at the rp channel ext scllated wth an averaged perd f 17 2 s n the physcal experment. Owng t the cmputatnal lmts, hwever, the numercal mdel dd nt run lng enugh t allw fr a spectral analyss f the rp scllatn. Several nterestng phenmena are bserved frm the tme sequence f the mdel results. Frst, the wave feld behnd the submerged bar s n lnger lng crested. The develpment f the rp current and crculatn cells causes the wave crest t be refracted as llustrated by the cmputed free surface magery. Because f the release f hgher harmncs after the wave passage ver the submerged bar, secndary crests are fund n the surf zne when we examne the crss-shre free surface prfles n the barred beaches. Secnd, vrtces cntnue t grw n the bar crest and near the shrelne and they merge as shwn n Plate 2c, left. Frm Plate 2c, rght, we ntce that the frst par f vrtces s shed ffshre, fllwed by a secnd par. The rp current s accmpaned by strng vrtces. Thrd, the cmputed flw felds averaged ver tw wave perds shw that the rp current s unstable and scllates n the rp channel. As tme ges n, the vrtcty feld n the surf zne becmes very cmplex and vrtces seaward f the channel ext appear t mve alngshre rather than beng

10 2,2 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS drectly transprted ffshre (Plate 3c, rght). Ths becmes mre clear at the later stage f smulatn and s cnfrmed by the mean current feld averaged ver a lng perd f tme as shwn n the next subsectn. The rp nstablty results n the scllatn f rp current and prevents the vrtces frm beng cnvected far ffshre. The vanshng f rp currents away frm the ext, f the rp channel culd be f sgnfcance, n practce. t mples that sedments r pllutants mght nt be transprted ffshre by the rp current tself. nstead, they may stay n the area seaward f the submerged bar r be transprted away by ther mechansms. As bserved n the labratry experment, flaters deplyed n the surf zne flwed ut thrugh the rp channel but mst f them dd nt mve far away ffshre. They returned t the surf zne ver the sub- merged bar by the surface drft f the waves and prb- n4 ably by eddes asscated wth the rp current. m2 m ' x(m) y=7.m, Barred Beach _ y=4.m, Rp Channel x (m) - - _ - y=l. lm, Barred Beach Mdel/Data Cmparsn The labratry experment n rp current generatn wth a smlar tpgraphy by Hallet et al. [1997] prvdes an deal test case fr mdel verfcatn. The physcal wave basn s twce the sze f the numercal wave basn and has tw rp channels. A detaled descrptn f the labratry experment s gven by Hallet et al. [1997]. n the fllwng, we shall cmpare tmeaveraged prpertes f the cmbned wave and current mtn predcted by the numercal mdel and measured frm the physcal experment. t s wrth mentnng that the labratry data were averaged ver the last half f the data cllectn perd (819 s) t elmnate the effects f transents n the wave basn. Owng t the cmputatnal lmt, the numercal results are averaged ver 18 wave perds after the frst wave arrves at the shrelne. Ths averagng tme s abut 1 rder f magntude shrter than t s n the physcal experment. Fgure and Fgure present the cmparsn f the cmputed and measured wave heght and mean water level alng three crss-shre transects. Tw transects (y and 7. m) are lcated n the barred beach near the sdewalls f the numercal wave basn, whle the ther (y - 4. m) s alng the center lne f the rp channel n the crss-shre drectn. The crcles represent the labratry data, whle the numercal results are dented by stars. We cmpute the rt-mean-square wave heght (Hrms) and the mean water level by averagng the numercal results ver 18 wave perdsø Despte a slght dscrepancy alng the transect f y = 1.1 m, whch mght result frm the use f nly half the expermental wave basn by the numercal mdel, the agreement between the cmputed results and measurements alng the crss-shre transects f the barred beach and the rp channel s satsfactry. We als ntce that the wave heghts at the ffshre te f the submerged bar are smewhat verestmated. Nevertheless, the Bussnesq mdel predcts reasnably well the decay f wave heght due t wave breakng and the re Crss-shre Dstance (m) Fgure. Cmparsn f cmputed (stars) and measured (+.2 cm) (crcles) wave heght. sultant setup f the mean water level n cmparsn wth the labratry measurements. Next, we cmpare the cmputed mean current wth measurements as shwn n Fgure 7 and Fgure 8. The velcty cmpnents are cnsdered alng fur lngshre transects. The transect f x = 1 m s seaward f the channel ext, and x: 13 m s shreward f the submerged bar. The ther tw transects (x = 11.2 and 12.3 m) crss the rp channel. We btan the mdeled current speed by averagng the cmputed nstantaneus flud partcle velctes ver 18 s. Fgures 7 and 8 shw that the mdeled velcty cmpnents agree reasnably well wth the measurements. Frst, the Bussnesq mdel captures well the spatal varatn f the tme-averaged, measured rp current. Fr nstance, the mdel predcts the same sgnature f vanshng rp current seaward f the channel ext (.e., transect. = 1 m) and the cnvergence f the rp current near the channel entrance (.e., transect x = 12.3 m). Secnd, the maxmum crss-shre and lngshre currents are als predcted crrectly by the numercal mdel as llustrated by the gd agreement alng the x = 12.3 and 13. m transects, respectvely. The Frude number at the rp neck s abut.2, whch causes sgnfcant effects t the surface waves. A cmparsn f the cmputed and measured mean current feld s presented n Fgure 9. The flw pattern predcted by the numercal mdel s smlar t the measured velcty feld. The rp nstablty dmnshes the mean current n frnt f the channel ext because f the meanderng f the rp current. As the tpgraphy s nt perfectly symmetrc abut the center lne f

11 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS 2,27-1. [ [ y=7.m, Barred Beach, y=4.m, Rp Channel , [, y=1.1 m, Barred Beach x (m) X (m) Crss-shre Dstance (m) Fgure. Cmparsn f cmputed (stars) and measured (crcles) mean water level. the rp channel, the mean rp current averaged ver a lng perd f tme has a slght bas tward ne sde f the channel. There s a shreward mean flw and secndary crculatn clse t the shrelne. Ths can be attrbuted t the larger wave setup at the shrelne shreward f the rp channel n cmparsn wth that shreward f the submerged bar. Ntce that the averagng tme (18 s) fr the numercal data cvers abut a perd f the very lw frequency mtn (> 1 s) and abut 1 perds f the rp scllatn (17 -,- 2 s) bserved n the labratry exper- ment f Haler et al. [1997]. Bth numercal results and labratry measurementsuggest a large degree f tempral and spatal varablty n the rp current system. As an attempt t measure the degree f varablty, we cmpute the standard devatn f the tme seres f mean surface elevatn and crss-shre and lngshre currents. The tme seres s btaned by tme averagng the nstantaneusurface elevatn and partcle velcty every tw wave perds frm bth numercal results and labratry measurements. Fgure 1 depcts the rt-mean-square (RMS) values f the cmputed and measured mean water levels alng three crss-shre transects,.e., y - 7., 4., and 1.1 m. The RMS values f the tempral varatn f wave setup agree reasnably well wth the measuredata. Fgure l l and Fgure 12 shw the cmparsn f the RMS values f the cmputed and measured crss-shre and lngshre currents alng fur lngshre transects. Althugh the numercal results appear t verpredct the degree f tempral varablty n the shrt-waveaveraged currents near the rp channel ext, the verall agreement s satsfactry. 4. Effects f Bathymetrc Unfrmty n Rp Stablty 4.1. An deal Tpgraphy n agreement wth bservatns n the physcal experment, the numercal results as shwn n the precedng sectn ndcate that the rp current s unstable. One may ask whether the rp current s stable f there are n perturbatns n the bathymetry and n the mean free surface. Ths sectn s therefre devted t answerng ths questn. We cnstruct an deal tpgraphy by averagng the crss-shre prfles f the barred beach frm the bathymetrc data taken durng the labratry experment. Ths prcess flters ut the perturbatns and leads t a lngshre unfrm barred beach wth a rp channel n the mddle f the bar as shwn n Fgure 13. The basc cnfguratn f the bathymetry and the numercal mdel setup are dentcal t thse descrbed n the prevus sectn Delayed Rp nstablty The rp current generatn and asscated nearshre crculatn are smulated usng the deal bathymetry. nterestng phenmena are bserved frm the numercal x= 1m O..._ -.2 ' ' 2.2.2rn, _ ',, ' x ='12.3m , -.2 ' ' Lngshre Dstance (m) Fgure?. Cmparsn f cmputed (sld lnes) and measured (-+-. m/s) (crcles) crss-shre mean velcty.

12 ß 2,28 CHEN ET AL' BOUSSNESQ MODELNG OF RP CURRENTS.2, x= 1m O -.2 ' x = 11.2m 2 x= 12.3m ,, [,, -.2 ' ' Lngshre Dstance (m) Fgure 8. Cmparsn f cmputed (sld lnes) and measured (+. m/s) (crcles) lngshre mean velcty. At the early stage f smulatn, fr nstance, at the 2th s as shwn n Plate 4c, left, a par f vrtces smlar t the case wth the expermental tpgraphy s develpng n the edges f the rp channel (see Plate lc). Hwever, n vrtces appear n the bar crest wng t the absence f bathymetrc perturbatn n the crest f the submerged bar n the deal tpgraphy. We ntce the frst par f vrtces s shed ffshre, fllwed by ther pars. n cntrast t the case wth the expermental bathymetry, the rp current remans stable n the frst 12 s f smulatn. The rp current flws straght ffshre as depcted by Plate c, left. An nterestng phenmenn f wave dffractn by the rp cur- rent s als bserved n the same fgure, whch shall be dscussed n detal n the next sectn. After abut 2 mn f smulatn, the nstablty starts develpng and the rp current becmes unstable as llustrated n Plate c, rght. The rp current cannt mantan a flw straght ffshre. t meanders and sclates back and frth n the rp channel as the smulatn tme ges n. Crculatn cells behnd the submerged bar are n lnger symmetrc. n the present smulatn, althugh there are n bathymetrc perturbatns, there s actually a perturbatn n the gemetry f the numercal wave basn because the mdel tpgraphy s nt perfectly symmetrc abut the center lne f the rp channel when Ay -.1 m s used. The lack f bathymetrc perturbatns delays the rp nstablty, but the slght gemetrc asymmetry eventually leads t an unstable rp current as demnstrated n Plate. Our numercal experment wth a perfectly symmetrc deal bathymetry cnfrms the effect f the gemetrc pertur- experment. t turns ut that the spatal and tempral varatn f the rp current and the vrtex structure are sgnfcantly dfferent frm thse utlzng the expermental bathymetry. Plate 4 and Plate present the wave felds, the underlyng current felds, and the asscated vrtcty felds at fur dfferent nst nts based n the numercal results. (a) Cmputed Mean Current Feld ß :,.,.,...% ::::. m/s 'ø-..s:..,!! ß.. ::..,-:: ::::: : ß. 7 ß :... ß:.-...,1,, /..-:,,.,,...,,tt,... :::;.... ½,-; r-,.,_.,, (b) Measured Mean Current Feld 2,,, y=7.m, Barred Beach { ' O,, x (m) y=4.m, Rp Channel , y=l.lm, Barred Beach x (m) :'!!!!!!! :: : l':--'" ;... ::::::::::::::::::::::::::::::::::: Crss-Shre Dstance (m) Fgure 9. Cmparsn f the (a) cmputed and (b) measured mean current felds Crss-shre Dstance (m) Fgure 1. Cmparsn f the rt-mean-square values f cmputed (stars) and measured (crcles) mean water level. 14.

13 CHEN ET AL' BOUSSNESQ MODELNG OF RP CURRENTS 2,29.2 [! [ ].2 x= lrn, , x = 11.2m x = 12.3m ,,,,,,,,.1 x = 13.m Lngshre Dstance (m) Fgure 11. Cmparsn f the rt-mean-square values f cmputed (sld lnes) and measured (crcles) crss- shre current. drectn, althugh t slghtly ncreases the setup nsde the rp channel. The farly gd agreement between the cmputed and measured mean water levels suggests that the lack f bathymetrc perturbatns des nt markedly change the lng-term-averaged frcng f pressure gradent fr the rp current generatn. Cmparsns f the cmputed and measured mean velcty cmpnents alng fur lngshre transects are gven n Fgure 1 and Fgure 17. The fur transects are lcated n the rp channel area, whch are dentcal t thse n the case f expermental bathymetry. The mdeled mean velcty cmpnents are btaned by averagng the numercal results ver 18 s. Far agreement between the cmputed mean currents and measurements s bserved. n cmparsn wth the mdel results f expermental tpgraphy, the deal bathymetry generates a brader but weaker rp current at the channel entrance as shwn by the crss-shre velcty prfle at x = 12.3 m. Because the same average tme n the case f expermental bathymetry s used t cmpute the mean current, the delayed rp nstablty wng t the lack f bathymetrc perturbatn leads t an verpredctn f the rp strength n frnt f the channel. As the rp current s als unstable n the case f deal bathymetry, we expect that the persstence f the rp current near.2,,,,,,,,.1 x=1m n- p batn and suggests that the accumulatn f cmputer rund-ff errrs may als prvde the seed fr the rp nstablty f the smulatn s su cently lng Smlar Mean Quanttes.2,.1 x=11.2rn ' Fr the deal bathymetry, Fgure 14 and Fgure 1 gve cmparsns f the rt-mean-square wave heghts and mean water levels alng three crss-shre transects dentcal t thse n the case f expermental bathymetry. The cmputed Hrms and mean water level are agan averaged ver 18 s. Frst, we ntce that the.2.1, x = 12.3m cmputed wave heght near the channel ext s much ' greater than the measurements. Ths s attrbuted t the effects f the strng rp current n the wave transfrmatn seaward f the rp channel. As shwn n the next fgures, the strength f the rp current near the O.2,,,,,,,, channel ext s verpredcted wng t the delayed rp nstablty. Despte the verestmates f wave heght n the rp channel, the agreement between the cmputed.1 x = 13.m results and measurements alng the transects crssng the submerged bars s farly gd. The dfference n Lngshre Dstance (m) wave heght n the rp channel des nt appear t sub- Fgure 12. Cmparsn f the rt-mean-square values stantally affect the agreement f the mean water level f cmputed (sld lnes) and measured (crcles) lngalng the center lne f the channel n the crss-shre shre current.

14 2,3 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS Free Surface Elevatn at 2 s Free Surface Elevatn at 8 s t -, t L! 1 1 t t lo 1 b) The Underlyng Current Feld The Underlyng Current Feld :::::::::::::::::::::!!!!! [! E! /:: : E:::: ::::::::::::1::--':: t:::::. :: ::::::::::::::::::::::::::::::::::... ::::: ß ::::: :: :::_ _',... ß.O J.....'"'":1: '"'.,,1... ß T " - -"-: :;'- -'... ::::::::::::::::::::::::::::::::::::::::::::::: :... :...!... ::[,., ' ;,.:':'':::'::::::.::... :... :;.. "-O..' h/:....:::::':::: ::" J"; 2,,' ::.:... :.::::::::::::'.::..., -,,,,'....,.,...,... ::... 1 :::::::::::::::::::::::::::::::::::::::::: :: lo 1 >'4... ' ' ' "" ' ß ' '--_---...?.. =... ::::::::::::.:::: :::... :'.: :--, ;,..,... Lk G., :::::::::::... :: :_'_. _'- _,?:' '.. _"¾ t... ::::::::::::::::::::::::::::::::::::::::: ½., ::::::::::::::::::::::::::::::::::::::... t ' '" 2" -" -.::::::::::::::::::... :-:;:::.:: ':3::."...;,,... _:; -!:: ::::!::::j:::x:::;;::: lo 1 Vrtcty! Vrtcty! >'4!! 1 Crss-shre Dstance (m) 1 E --1! ,..,...,..,, u Crss-shre Dstance (m) j Plate 4. Cmputed (a) wave feld, (b) underlyng current, and (c) vrtcty feld at (left) t - 2 s and (rght) t - 8 s.

15 CHEN ET AL' BOUSSNESQ MODELNG OF RP CURRENTS 2,31 a) Free Surface Elevatn at 12 s Free Surface Elevatn at 18 s '!. 1!! 1 1 ' 1! 1 1 b) The Underlyng Current Feld :::::::.2-::::::::::::::::::::::::::::::::: ',-'7' :::::::::::::::::::::::::::::::::::: _ :X ::.:..:::.:::::.:::... :::::.:::2.. ;,.,::.'r.4::::::::!!!!:::::[:; ß...._... ß tlfl '- s '- : ::::,, ;::_-. :::::::;:;.::::::;;...:...,,. {!!.":!!:::... :::":'::: ;::'" :'?, t{,... ::':'::"::... ::'::1:::;2..'.' -- : ::: ;;::::; ::: ;;;; ;;:;;:: :;::*.;: (.'J;::: ; 77. ::::::::::::::::::::::::::::::::::::::::::::::: 1 1 The Underlyng Current Feld :;;:::;::::':;:::.' :::::::::.:;, ;]::" :: :::... ' ' """,,' :: ''... : '.:.'..':; ::, :::.!!!!!:"l;:!!!!l ::::::::::::::::::::::::::::'"'-':. " _ _'.J :. s' :: ,, -_.. - ' ;;,:,'..; -.% :::::::::::::::::::::::::::::::::::: ===================== ::::::::::::::::::::::::::::::::::::::::::::: :::::..:::::::::::::::::::::::.:::: ':-:-...',;c % '...,... _, r.,,: 1 1 c) Vrtcty Vrtcty!! t..--. J ' 1 ' 1 2!! t! Crss-shre Dstance (m) Crss-shre Dstance (m) Plate. Cmputed (a) wave feld, (b) underlyng current, and (c) vrtcty feld at (left) t - 12 s and (rght) t- 18 s.

16 , 2,32 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS a) Surface Elevatn Gven by REF/DF1.4.2 land (m) lo b) Surface Elevatn Gven by Bussnesq Mdel --. J -1! t! 4 8 lo (m) c) :!... 8 ' ::' : ' Underlyng Current Feld and Vrtcty ß ::::'"::!::: : '-':::..:... '.. :'::... :!", :,,. E ' ' "': >,4... :: ß: ß: -::::::....-:::::..: :.:: :--..-'-,.-. ", :-- :....,...::::.: :::: ::... -::... :::.',,,.._:.-. :--., ::::':::::::. ::':::.:::::.::::::.:::r,,.:..:., 2 '-::'::::::' ' =============================.::... :': : ::!:,..::-:t, lo Lngshre Dstance (m) 1.. (1/s) Plate. Wave refractn/dffractn by a rp current. (a) Surface elevatn predcted by the REF/DF mdel, (b) surface elevatn predcted by the Bussnesq mdel, (c) underlyng current feld and vrtcty.

17 CHEN ET AL' BOUSSNESQ MODELNG OF RP CURRENTS 2, (a) Tpgraphy 1 y (m) x (m).:. 2 bathymetrc data s larger than that f deal bathymetry wth the same length f smulatn tme. n cntrast, the deal bathymetry leads t smewhat better agreement n terms f lngshre current at x = 12.3 m. n shrt, althugh small bathymetrc perturbatns may sgnfcantly alter the spatal and tempral varatn f the rp current and the asscated vrtex structures, they may nt substantally change the mean characterstcs f the nearshre crculatn n the sense f averagng ver hundreds f waves. Wave Refractn/Dffractn by Rp Currents (b) Cnturs f Stll Water Depth n Centmeters SHOR,, 1 1 x (m) Fgure 13. (a) deal tpgraphy and (b) cnturs f water depth. the channel ext wll be dmnshed after a lnger smulatn tme wng t the rp scllatn. Cnsequently, the agreement between the mdeled and measured wave heghts n the rp channel s als expected t mprve. T establsh a quanttatve measure f mdel/data agreement, we use the ndex prpsed by Wlmtt [1981] as fllws: d Jn--1 [y(j) - x(j)]2 (19) Ejn=l [ly(j) - 21 q-x(j) - l] 2 where x(j) are the measured data, Y() are the cmputed results frm the numercal mdel, and ß and are the mean value f x(j) and y(j), respectvely. When d = 1, t ndcates a perfect agreement, whle d = means a cmpletely dsagreement. Table 1 lsts the values f the ndex d fr Hrms, mean water level, and crss-shre U and lngshre V currents n the cases f expermental and deal bathymetry. Frst, a cmparsn f the ndex fr Srms cnfrms that the expermental bathymetry gves better agreement f Srms than des the deal bathymetry. Secnd, wth respect t /, the numercal results agree well wth the measurements n the case f expermental bathymetry, whch s als better than the agreement f the deal bathymetry. Fr crssshre currents, the ndex n the case f usng the survey 2 One f the advantages f the Bussnesq mdel s that wave-current nteractn s autmatcally taken nt accunt by the mdel. We have fcused n the wavenduced currents n the prevus sectns. Frm the numercal results, we bserve the sgnfcant effects f the underlyng current n the surface wave transfrmatn. One f the effects s wave refractn/dffractn by the vrtces asscated wth the rp current. n the lterature, surface wave scatterng by sld bdes, such as cylnders, has been well studed usng the ptental flw thery. Hwever, gravty wave scatterng by cylndrcal flud vrtces has nt drawn much attentn. Frm a theretcal pnt f vew, bth scatterng phenmena are dfferent. When waves prpagate n the nhmgeneus meda wng t the presence f rp currents, cmbned wave dffractn and refractn wll ccur. n ths sectn, we use the Bussnesq mdel and the REF/DF mdel [Krby, 198] t examne the refractn/dffractn effects f a rp current n the surface wave feld. Gven a current feld by averagng the results f the Bussnesq mdel wth the use f an deal bathymetry, the REF/DF mdel based n a parablc apprxmatn f the mld-slpe equatn [Krby and Dalrymple, 1983] s used t cmpute the wave feld wth the cmbned refractn and dffractn. Plate presents the wave feld predcted by REF/DF and the Bussnesq mdel wth the underlyng current feld n the deal tpgraphy. n cmparsn wth the results f Bussnesq mdel as depcted by Plate b, REF/DF gves a smlar pattern f dffracted waves by the rp current accmpaned wth vrtex pars. The patches f ht clrs n the wave crests llustrate the varatn f the wave feld n the lngshre drectn. Smlar t the results gven by the weakly nnlnear versn f REF/DF that ncrprates the ampltude effect n the dspersn relatn f a mnchrmatc wave, a lnear versn f REF/DF als predcts the dffractn pattern. Ths ndcates that the dffractn by the vrtces s essentally a lnear prcess. Fgure 18 gves a clser lk at the crss-shre varatn f free surface elevatn alng the center lne f the rp channel, and Fgure 19 llustrates the lngshre nnunfrmty f the wave heght caused by the rp current. The dashed lnes represent the Bussnesq results,

18 2,34 CHEN ET AL' BOUSSNESQ MODELNG OF RP CURRENTS ' 4 m2 O ,, y=7.m, Barred Beach _ --,. O O O x (m), 'E --.. y=4.m, Rp Channel x (m) y=l.lm, Crss-shre Dstance (m) O Barred Beach Fgure 14. Cmparsn f cmputed (stars) and measured (crcles) wave heght n the case f deal bathymetry. 14. n the ffshre area befre x = 8. m, the free surfaces predcted by bth mdels are n agreement because waves are essentally lnear n that regn. As expected, dfferences are bserved when waves becme mre nnlnear because f the shalng n the slpng beach. REF/DF fals t predct the skewness and asymmetry f the shalng waves. Weaker dffractn effects n the area far away frm the vrtces n the REF/DF result cmpared wth the Bussnesq mdel may be attrbuted t the parablc apprxmatn f REF/DF. Nevertheless, the sgnature f lngshre varatn f wave heght s captured by bth mdels. Because a jet-lke rp current s unstable n nature [Hallet et al., 1997], any perturbatns resultng frm lngshre nnunfrmtes f ncmng waves and tpgraphy wll lead t rp scllatn. The dffracted waves wll cause nnunfrmty f the radatn stresses n the lngshre drectn and may cntrbute t the cmplexty f the crculatn pattern behnd the submerged bar. T llustrate the dffractn effect n vrtex generatn, Fgure 2 shws the cmparsn f the lngshre varatn f the wave heght and vrtcty n the crest f the bar predcted by the Bussnesq mdel. Obvusly, the varatn f the wave heght n the bar crest s cncdent wth the varatn f the vrtcty as the whle the sld lnes are the REF/DF results. Bth mdels predct smlar lngshre nnunfrmty f wave heght wng t the dffractn by the rp accmpaned wth vrtex pars. The wave length f the dffracted waves n the lngshre drectn s cmparable t the crss-shre wave length. O.2,,,,,,, l x: 1m -.2 ' ' ' ' ' ',, , y=7.m, Barred Beach, x(rn) 14. L x: 11.2m On c) _, ("1 _.21,,, ,,,,,, y=4.m, Rp Channel x: 12.3m g v - ; x(m),, y:1.1 m, Barred Beach Crss-shre Dstance (m) Fgure 1. Cmparsn f cmputed (stars) and measured (crcles) mean water level n the case f deal bathymetry. -.2,, ,,,,, x: 13.m -.2,,,,, Lngshre Dstance (m) Fgure 1. Cmparsn f cmputed (sld lnes) and measured (crcles) crss-shre mean velcty n the case f deal bathymetry.

19 ,=, CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS 2,3.2, x= 1m -.2 ' 1 O.2,,,,,, x = 11.2m -.2 ' ' ' ' ' O.2,,,,,, x = 12.3m O -.2 ' ' ' ' ' ' ,,,,, -.2 ' Lngshre Dstance (rn) Fgure 17. Cmparsn f cmputed (sld lnes) and measured (crcles) lngshre mean velcty n the case f deal bathymetry. crrelatn ceffcent s equal t.92 fr bth curves at <_ x <_ 2. m. Ntce that the flw feld s symmetrc abut the center lne f the rp channel, whle the vrtcty feld s antsymmetrc. n addtn, we als fund that the wave length f the lngshre varatn s cmparable t that f the crss-shre wave mtn. A cmparsn f Plates lc and 4c, left, suggests that the bathymetrc perturbatns n the crest f the bar results n the prductn f vrtcty there. We bserve frm an anmatn f the vrtcty feld that cncentrated vrtcty appears n the bar crest nce vrtex pars n the rp channel are shed ffshre n the case f deal tpgraphy. The presence f vrtcty n the submerged bar away frm the rp channel s smlar t Free Surface at y=4.m. REFDF' -; BOUSSNESQ Mdel: '1 tt -.[ --, 1' ' ' ' ' Crss-shre Dstance (m) Fgure 18. Cmparsn f the crss-shre varatns f free surface predcted by the refractn/dffractn (REF/DF) mdel (sld lne) and the Bussnesq mdel (dashed lne). what we bserved at the early stage f smulatn befre vrtex sheddng n the case f expermental tpgraphy wth bathymetrc perturbatns n the bar crest (e.g., Plate lc). Althugh the mechansm f vrtcty generatn n bth cases s the lngshre nnunfrmty f breakng n the bar crest, the lngshre varatn f wave heght n the case f deal bathymetry s attrbuted t the dffractn effect f wave feld by the rp current rather than the bathymetrc perturbatns. The cmplex crculatn pattern shreward f the submerged bar results frm the cmbned effects f the rp channel and wave refractn/dffractn by the rp current. Owng t the lack f small bathymetrc perturbatns n the case f deal bathymetry, the nfluence f wave dffractn by the rp current s mre prfund because f the mre rganzed vrtex n cmparsn wth thse n sectn 3. Hwever, n bth cases f sectns 3 and 4, the dmnant mechansm f vrtcty generatn s the lngshre nnunfrmty f wave breakng wng t the presence f the rp channel E,.8 Wave Heght at x=9rn. REFDF' -; BOUSSNESQ Mdel: - -, J,,,,,, 14 Table 1. Wlmtt's [1981] ndex as a Measure f Mdel/Data Agreement ndex Hrms U V Bathymetry..4.2 /,' d expermental d deal Lngshre Dstance (m) Varables are defned as fllws: Hrms, rt-mean-square wave heght;, mean water level; U, crss-shre current; and V, lngshre current. Fgure 19. Cmparsn f the lngshre varatns f wave heght predcted by the REF/DF mdel (sld lne) and the Bussnesq mdel (dashed lne).

20 2,3 CHEN ET AL.' BOUSSNESQ MODELNG OF RP CURRENTS > lo Lngshre Varatn f Wave Heght at t= s,,,, Lngshre Dstance (m) Lngshre Varatn f Vrtcty at t= s,,,, \ Lngshre Dstance (m) Fgure 2. Lngshre varatns f (tp) wave heght and (bttm) vrtcty near the bar crest.. Summary and Cnclusns Acknwledgments. Ths study has been supprted by A numercal mdel based n the fully nnlnear Bussthe Offce f Naval Research, Base Enhancement Prgram nesq equatns [We et al., 199] has been extended and Castal Dynamcs Prgram, thrugh research grants t nclude wave breakng and mvng shrelnes fr N and N smulatn f wave transfrmatn and wave-nduced nearshre crculatn. Fully cupled wave/current n- References teractn s taken nt accunt by the Bussnesq equatns. The mdel nt nly predcts the nearshre prpagatn f nnlnear surface gravty waves but als gves the underlyng unsteady flw generated by wave breakng. The current feld s btaned by tme averagng f the cmputed flud partcle velcty ver tw wave perds, whle the vrtcty feld s cmputed drectly frm the nstantaneus flud partcle velctes. The mdel results are cmpared wth the measure- ments frm a labratry experment cnducted by Hallet et al. [1997] n rp current generatn n a barred beach wth rp channels. Farly gd agreement s bserved between the labratry data and the cmputed lngshre and crss-shre currents, mean water level, and the rt-mean-square wave heght alng several transects n the surf zne. The verfed numercal mdel s emplyed t nvestgate the spatal and tempral varablty f rp currents and asscated vrtces. The effect f bathymetrc unfrmty n rp nstablty and wave refractn/dffractn by the underlyng rp current accmpaned wth vrtex pars are als studed. n agreement wth the physcal experment and theretcal predctn by Hallet et al. [1997], the numercal results ndcate that the rp current s unstable. The rp nstablty results n an scllatng rp current and the alngshre mvement f vrtces asscated wth the rp current. t s fund that the bathymetrc unfrmty f the barred beach n the lngshre drectn may delay the nset f nstablty and cause sheddng f vrtex pars ffshre. Althugh bathymetrc perturbatns can sgnfcantly alter the spatal and tempral varatn f rp currents and vrtex structures, they may nt change the mean characterstcs f the cmbned wave and cur- rent mtn averaged ver a lng perd f tme, f the rder f several hundred wave perds, because f the unstable nature f the rp current. Wave dffractn by the rp current s bserved frm the numercal experment. A parablc mdel, REF/DF [Krby, 198], fr cmbned refactn and dffractn s als used t examne the effects f the vrtces. The refracted/dffracted waves by the underlyng current feld cause nnunfrmty f the radatn stresses n the lngshre drectn and may cntrbute t the cmplexty f the crculatn pattern behnd the submerged bar. Vrtcty generatn n the bar crest results frm the lngshre nnunfrmty f breakng caused by e- ther dffractn effects f the rp current r bathymettc perturbatns. The numercal results btaned n ths study are prmsng. Further verfcatn f the bserved phenmena n the numercal smulatn by physcal experments s requred. Brkemeer, W. A., and R. A. Dalrymple, Nearshre water crculatn nduced by wnd and waves, paper presented at Sympsum n Mdelng Technques, Am. Sc. f Cv. Eng., San Francsc, Calf., 197. Bwen, A. J., The generatn f lngshre currents n a plane beach, J. Mar. Res., 73, , 199. Carrer, G. F., and H. P. Greenspan, Water waves f fnte ampltude n a slpng beach, J. Flud Mech.,, 97-19, 198. Chen, Q., P. A. Madsen, H. A. Sch/ffer, and D. R. Basc, Wave-current nteractn based n an enhanced Bussnesq apprach, Castal Eng., 33, 11-4, Chen, Q., J. T. Krby, R. A. Dalrymple, A. B. Kennedy, and A. Chawla, Bussnesq mdelng f wave transfrmatn,

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