DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2010.13.729 DYNAMICAL SYSTEMS SERIES B Volume 13, Number 4, June 2010 pp. 729 738 LARGE-SCALE VORTICITY GENERATION DUE TO DISSIPATING WAVES IN THE SURF ZONE Philippe Bonneon, Nicolas Bruneau and Bruno Caselle Universié Bordeaux 1, CNRS, UMR 5805-EPOC avenue des Faculés, Talence, F-33405, France Fabien Marche Universié Monpellier 2, Insiu de Mahémaiques e de Modélisaion de Monpellier CC 051, Place Eugene Baaillon, 34095 Monpellier cedex 5, France Absrac. In his paper, we invesigae he mechanisms which conrol he generaion of wave-induced mean curren voriciy in he surf zone. From he verically-inegraed and ime-averaged momenum equaions given recenly by Smih [21], we obain a voriciy forcing erm relaed o differenial broken-wave energy dissipaion. Then, we derive a new equaion for he mean curren voriciy, from he nonlinear shallow waer shock-wave heory. Boh approaches are consisen, under he shallow waer assumpion, bu he laer gives explicily he generaion erm of voriciy, wihou any ad-hoc paramerizaion of he broken-wave energy dissipaion. 1. Inroducion. In he nearshore, alongshore variaions in waves and wave-induced currens are ubiquious. These variaions can be due o alongshore inhomogeneiies in he inciden wave field or in he local bahymery. As shown heoreically by Peregrine [18], non-uniformiies along he breaking-wave cres drive verical voriciy. The voriciy ha is being discussed here is no he small scale voriciy caused direcly by wave breaking and subsequen urbulen moions, bu he voriciy in he form of quasi wo-dimensional eddies (usually called macrovorices ) wih horizonal scales larger han he local waer deph. The mos frequenly observed nearshore macrovorices are rip curren circulaions. Rip currens are shore-normal, narrow, seaward-flowing inense currens ha originae wihin surf zone, exend seaward of he breaking region, and are associaed wih horizonal eddies. These macrovorices play a major role in circulaion and mixing processes in he nearshore. Sudies described in Peregrine [18] and Brocchini e al. [4] draw aenion o he way in which non-uniformiies along he bore cress lead o generaion of verical voriciy. They proceeded o a direc analysis of voriciy, a he wave s ime scale, modeling he breaking even as he developmen of a surface and curren disconinuiy in he non-linear shallow waer equaions (also called Sain Venan equaions). These breaking wave processes induce wave-averaged curren and mean voriciy. The aim of he presen paper is o invesigae wave-averaged mean flow voriciy due o differenial wave breaking in he surf zone. Mean verical voriciy equaions are 2000 Mahemaics Subjec Classificaion. 74J15, 74J40, 76B15, 7605. Key words and phrases. Surf zone, Shallow waer, Wave-induced curren, Rip curren, Voriciy, Shock, Sain Venan equaions. 729
730 P. BONNETON, N. BRUNEAU, B. CASTELLE AND F. MARCHE Figure 1. Definiion skech for he surf zone. derived from boh he verically-inegraed and ime-averaged momenum equaion given by Smih [21] and he nonlinear shallow waer shock wave heory. 2. Wave-averaged model. The nearshore circulaion is generally deermined by he deph-inegraed and ime-averaged equaions of mass and momenum (see Phillips [19] ). In his secion, we analyze he mechanism of verical voriciy generaion in he framework of his classical approach. 2.1. Mass and momenum conservaion equaions. The verically-inegraed mass and horizonal momenum budges are examined. For he convenience of verical inegraion, he verical coordinae z is reaed separaely from he horizonal ones (x 1, x 2 ). As shown in figure 1, he fluid is bounded beween he bed, z = d(x 1, x 2 ), and he free surface elevaion, z = ζ(x 1, x 2, ). For simpliciy, we consider periodic waves of period T and, in his secion, urbulence is negleced. Assuming a ime scale separaion beween waves and currens, he horizonal flow velociy, v i (x 1, x 2, z, ) can be separaed ino mean, v i, and wave, ṽ i = v i v i, componens, where he ime operaor (.) is defined as: (.) = 1 +T T (.)dτ. Verical inegraion of he mass equaion, combined wih kinemaics boundary condiions and subsequen ime inegraion resul in h + M j = 0, (1) where h is he mean waer deph and M i = ζ d v i dz is he oal horizonal momenum. Assuming ha he mean horizonal velociy U i = v i is deph-uniform and ha he mean pressure is hydrosaic, he horizonal momenum equaions can be wrien as (see Phillips [19] ) M i + ( ( Mi Mj )/ h ) + g h ζ x i = S ij, (2)
LARGE-SCALE VORTICITY GENERATION 731 where g is he graviaional acceleraion, ρ he fluid densiy and S ij is he radiaion sress which can be expressed as S ij = ζ d ( P ρ δ ij + ũ i ũ j ) dz 1 2 g h 2 δ ij M i Mj h, (3) wih P he pressure. The radiaion sress represens he excess momenum flux ha resuls from wave moions. The classical approach, based on equaions (1) and (2), has been used in many nearshore applicaions. However, he radiaion sress encompasses differen wave processes. In paricular, he radiaion sress gradien combines non-dissipaive wave effecs as well as dissipaive effecs due o wave breaking, which alone can creae mean flow voriciy. Recenly, Smih [21] presened a reformulaion of his heory. The oal momenum, Mi = hu i + ζ ζ ṽi dz, is splied ino mean curren momenum, hu i, and wave momenum, Mi = ζ ζ ṽi dz. By subracing he waves momenum budge (based on linear wave heory and an ad-hoc paramerizaion of he broken-wave energy dissipaion) from he oal, Smih [21] obained a new se of equaions hu i h + hu j = M j (4) + ( hui U j ) + g h ζ x i = F i, (5) where he wave force F i acing on he mean flow can be wrien as F i = k id bm σ + M j ( U j U i ) U M j J i h, x i x i wih, σ he inrinsic wave frequency, k i he wave vecor, D bm he broken-wave energy dissipaion, E he wave energy and J k = E. The broken-wave energy sinh(2k h) dissipaion D bm is generally esimaed from he analogy beween a breaking-wave and a hydraulic jump (see Thornon e Guza [20]) and can be expressed as D bm = g H 3 4T h, (6) where H is he wave heigh. The firs erm in he wave force expression is associaed wih he dissipaion of wave momenum due o wave breaking. This loss of wave momenum is direcly ransfered o he mean flow. We will see in he nex secion ha his dissipaive erm conrols he generaion of mean curren voriciy. The Smih s model is equivalen o he sysem (1) and (2), bu allows a beer undersanding of he exchanges beween wave momenum and mean curren momenum. 2.2. Voriciy equaion for he mean curren. Bühler [7] presened a general heoreical analysis of wave-driven currens and vorex dynamics due o dissipaing waves. He idenified a dissipaive force wihin he radiaion-sress convergence, which conrols he mean-curren voriciy generaion. In his secion, we use he non-conservaive form of he mean horizonal momenum equaion (5) o explicily sae he dissipaive force due o wave-breaking in he nearshore, and hen we derive he mean curren voriciy equaion. Equaion (5) can be rearranged in a non-conservaive form U + (U. )U + g ζ = G, (7)
732 P. BONNETON, N. BRUNEAU, B. CASTELLE AND F. MARCHE where he wave force G acing on he mean flow can be wrien as G = De k + M h ( U) J + T u, wih e k = k/ k. The dissipaive force D is given by D = D bm and, using (6), can hc φ be wrien D = g H 3 4c φ T h, (8) 2 ( where c φ is he norm of he phase velociy. The las erm, T ui = ν Ui + Uj x i ), is added o paramerize he urbulen momenum diffusiviy of he mean curren. For simpliciy, we consider in his paper a consan eddy viscosiy ν. Noe ha his choice of parameerizaion is no energeically consisen (see Gen [10]). A beer choice, regarding consisency, would be he viscous formulaion proposed by Marche [16], which was asympoically derived from he 3D Navier sokes equaions wih free surface. However, he presen derivaion of an auonomous voriciy equaion can no be exended o such a formulaion. The equaion for he mean flow verical voriciy, ω = U2 U1 x 2, is obained sraighforward by aking he curl of equaion (7), which yields ω +. (ωu T) = ν 2 ω + De k, (9) where U T = M h = U + M h is he mean ranspor velociy and he cross produc is reaed as a scalar. In equaion (9), he firs erm on he righ-hand side expresses diffusion of he mean curren voriciy and he second one is a voriciy producion erm, which can be approximaed by D e k. This erm is acive in presence of dissipaive waves, when he gradien of D is no parallel o he wave vecor. This resul is in agreemen wih he sudy of Peregrine [18], who showed ha verical voriciy generaion is associaed wih non-uniformiies in bores. The equaion (9) provides a simple and efficien model o undersand he generaion of vorical moions in he surf zone, such as longshore currens or rip currens. Explanaion of rip curren generaion following he classical radiaion sress approach (see Caselle and Bonneon [8] or MacMahan e al. [14] ) is difficul because a large par of he wave driving force (gradiens in he radiaion sress) does no generae currens as i is irroaional. In our approach, he roaional par of he wave driving force is clearly idenified, which allows a beer explanaion of wave-induced vorical rip currens. A qualiaive explanaion of rip curren dynamics is as follow. Due o refracion, wave breaking is more inense over he shoal. This differenial wave breaking (see figure 2) induces a circulaion wih shoreward currens over he shoals and a seaward curren (called rip curren) over he lower par of he bahymery. To describe more qualiaively his phenomenon, we presen in figure 3 a numerical simulaion of wave-induced currens and voriciy over a ransverse bar and rip morphology which is ypically observed on he aquianian coas (see Caselle and Bonneon [8]). The compuaions were performed wih a numerical model coupling he specral wave code SWAN (Booij e al. [3]) wih he flow model MARS 2DH, which solves he equaions (4) and (7) (see Bruneau e al. [5] for more deails). We observe in figure 3a ha he wave-induced voriciy erm, D e k, is inense close o he rip channel, wih a clockwise forcing in he upper par and an ani-clockwise forcing in he lower par. This is due o srong alongshore variaions in he bahymery close
LARGE-SCALE VORTICITY GENERATION 733 Figure 2. Schemaic represenaion of rip circulaion induced by wave breaking over shoals. o he rip channel. This wave forcing generaes wo main circulaion cells on boh sides of he channel (see 3b), which are associaed wih shoreward currens over he bars and a seaward curren in he rip channel. The mean voriciy field (figure 3b) is srongly correlaed o he voriciy forcing erm (figure 3a). 3. Shock-wave model. We showed in he previous secion ha he dissipaive force D plays a key role in he surf zone circulaion. However, his force has been inroduced in he Smih s heory in an ad-hoc way, by adding a dissipaive breaking erm in he wave-acion conservaion equaion. In his secion, we show ha we can explicily derive such a dissipaive force, from he shock-wave heory for Sain Venan (SV) equaions, wihou any ad-hoc paramerizaion. Indeed, SV equaions are a good approximaion o wave moion in he surf zone (see Hibber and Peregrine [12], Kobayashi e al. [13] or Bonneon [2]). 3.1. 1D cross-shore mean flow equaions. The one-dimensional SV equaions are given by h + hu = 0 (10) hu + (hu 2 + 12 ) x gh2 = gh d, (11) 1 where u(x 1, ) = 1 ζ h d v 1 dz is he deph-averaged cross-shore velociy. Following he concep of weak soluions (Godlewski and Raviar [11], Whiham [22]), we can approximae he broken-wave soluion (figure 4a) by inroducing a disconinuiy (see figure 4b) saisfying jump condiions based on mass and momenum conservaion across he shock: c b [h] + [hu] = 0 c b [hu] + [hu 2 + 1 2 gh2 ] = 0,
734 P. BONNETON, N. BRUNEAU, B. CASTELLE AND F. MARCHE Figure 3. Numerical simulaion of wave-induced circulaion over a ransverse bar and rip sysem. Offshore waves: H s = 1.5 m and T = 9 s; hin lines: isobahs beween -12 m (offshore) and 0 m (landward); bold line: shoreline. (a) Voriciy forcing erm D e k ; (b) voriciy field wih superimposed mean ranspor velociy vecor field U T.
LARGE-SCALE VORTICITY GENERATION 735 Figure 4. Definiion skech. (a) Cross-secion of a broken-wave in he surf zone; (b) shock represenaion. c b is he broken-wave celeriy, H he wave heigh, h he waer deph and subscrip 1 and 2 indicae values respecively ahead and behind he shock. where he brackes [ ] indicae a jump in he quaniy and c b is he shock velociy. A convenional noaion is o use subscrip 1 and 2 for values ahead and behind he shock respecively (see figure 4b). So he jump condiions can also be wrien in he form ( gh2 u 1 c b = (h 2 + h 1 ) 2h 1 ( gh1 u 2 c b = (h 2 + h 1 ) 2h 2 ) 1 2 ) 1 2 (12). (13) Mahemaically, he composie soluion, composed of coninuously differeniable pars saisfying equaions (10) and (11), ogeher wih jump condiions (12) (13), can be considered as a weak soluion of he SV equaions. Like in he preceding secion, he flow is separaed ino mean and wave componens: u = ū + ũ. Time averaging he conservaive mass equaion gives: T M = M dτ h + hū = M, (14) where M = ζũ is he wave momenum. To derive he mean curren momenum equaion we develop he expression of he gradien M, where M = 1 2 u2 + gζ, ( ) ( +T ) s = s = M dτ + d s dx 1 M( s ) + +T + s M dτ ( ) + +T M dτ + s M dτ d s dx 1 M( + s ), where s (x 1 ) is he ime a which he wave fron (or he shock) is locaed in x 1. In coninuous pars of he flow, he momenum equaion (11) is equivalen o he
736 P. BONNETON, N. BRUNEAU, B. CASTELLE AND F. MARCHE following equaion: u + M = 0. (15) Inside inervals [, s ] and [+ M s, +T] he wave soluion is coninuous and so can be evaluaed from (15), which yields T M = and finally s u dτ +T + s u dτ + 1 [M] = T ū c b + 1 ([M] c b [u]), (16) c b ū + ū ū + g ζ = D J, (17) where J = 1 2ũ2 and D = 1 c b T ([M] c b[u]). D is deermined by using shock condiions (12) and (13) and wries D = g (h 2 h 1 ) 3. (18) 4c b T h 2 h 1 where H = h 2 h 1 is he wave heigh. Equaions (14) and (17) are consisen, under he shallow waer assumpion, wih he sysem (4, 7) derived in secion 2. In equaion (17), he dissipaive force D is now explicily derived from he shock wave approach. I can be applied eiher o sauraed breakers (H = h 2 h 1 ), as in classical approaches (see Thornon e Guza [20] and Bonneon [1]), and o non-sauraed breakers (h 2 h 1 < H, see figure 4b). We can see ha equaion (8) is an approximaion of equaion (18) limied o sauraed breakers and based on esimaing he broken wave celeriy c b by he linear phase velociy c φ, which is a crude esimae of c b (see Bonneon [1]). For saionary cross-shore mean flows we can obain a new equaion for waveinduced mean waer level increase (wave seup) ( ) g ζ = D J + 1 M 2. (19) 2 h 2 The main conribuion for he wave seup is due o he dissipaive force D. Equaion (19) is ineresing from a physical poin of view because, conversely o he classical heory based on radiaion sresses (Phillips [19]), his equaion provides a simple and explici relaion beween wave seup and energy dissipaion. Bonneon [2] showed ha equaion (19) can represen an alernaive o he classical radiaion sress mehod for compuing wave seup in he surf zone. 3.2. 2D mean flow equaions. To exend he previous approach o a wo-dimensional one, we consider he SV equaions in a curvilinear coordinae sysem based on rays and heir orhogonals. The rays are he orhogonal rajecories of he successive posiions of he wavefron. The velociy field, u = (u 1, u 2 ), wries u = V e k. Time averaging of he wo-dimensional SV equaions gives h +.( hū) =. M (20) ū + (ū. )ū + g ζ = De k J ω(ũ e z ), (21) where ω is he wave componen of he verical voriciy and J = 0.5 V 2 = 0.5(ũ2 1 + ũ 2 2 ).
LARGE-SCALE VORTICITY GENERATION 737 From he non-conservaive momenum equaion (21) i is sraighforward o ge he equaion for he mean flow voriciy ω +. ( ωū + ωũ ) = (De k ). (22) Equaion (22) is consisen, under he shallow waer assumpion, wih equaion (9), wih in paricular he same voriciy forcing erm, (De k ), relaed o differenial wave dissipaion. However, in he presen approach he dissipaive force D is explicily given by he nonlinear shock-wave heory. 4. Conclusion. In his paper, we have invesigaed he mechanisms which conrol he generaion of wave-induced mean curren voriciy in he surf zone. From he verically-inegraed and ime-averaged momenum equaions given recenly by Smih [21], we obained a voriciy forcing erm relaed o differenial broken-wave energy dissipaion. Then, we derived a new equaion for he mean curren voriciy, from he nonlinear shallow waer shock-wave heory. Boh approaches are consisen, under he shallow waer assumpion, bu he laer gives explicily he generaion erm of voriciy, wihou any linear assumpion and ad-hoc paramerizaion of he broken-wave energy dissipaion. Furher work is required o evaluae he predicive capabiliy of he shock-wave approach in comparison wih recen large-scale laboraory experimens (Caselle e al. [9]) and field measuremens (Bruneau e al. [6]). In hese experimens, macrovorices are mainly generaed by wave-bahymery ineracions above srongly varying bahymery. The numerical simulaion of such opographically conrolled macrovorices requires he use of high order robus wellbalanced schemes (Marche e al.[15, 17]). This is currenly under invesigaion. Acknowledgmens. This work has been suppored by he ANR MahOcean, he projec ECOS-CONYCIT acion C07U01, he projec MODLIT (SHOM and RELIEFS/INSU) and was also performed wihin he framework of he LEFE-IDAO program (Ineracions e Dynamique de l Amosphère e de l Océan) sponsored by he CNRS/INSU. REFERENCES [1] P. Bonneon, Wave celeriy in he inner surf zone, Proc. 29h In. Conf. on Coasal Eng., 1 (2004), 392 401. [2] P. Bonneon, Modelling of periodic wave ransformaion in he inner surf zone, Ocean Engineering, 34 (2007), 1459 1471. [3] N. Booij, R. C. Ris and L. H. Holhuijsen, A hird-generaion wave model for coasal regions 1. Model descripion and validaion, J. Geophys. Res., 104 (1999), 7649 7666. [4] M. Brocchini, A. Kennedy, L. Soldini and A. Mancinelli, Topographically conrolled, breakingwave-induced macrovorices. Par 1. Widely separaed breakwaers, J. Fluid Mech., 507 (2004), 289 307. [5] N. Bruneau, P. Bonneon, B. Caselle, R. Pedreros, J-P. Pariso and N. Sénéchal, Modeling of high-energy rip curren during Biscarrosse 2007 field experimen, Proc. 31s In. Conf. on Coasal Eng., 1 (2008), 901 913, doi: 10.1142/9789814277426-0076. [6] N. Bruneau, B. Caselle, P. Bonneon, R. Pedreros, R. Almar, N. Bonneon, P. Breel, J-P. Pariso and N. Sénéchal, Field observaions of an evolving rip curren on a meso-macroidal well-developed inner bar and rip morphology, Coninenal Shelf Res., 29 (14) (2009), 1650-1662, doi:10.1016/j.csr.2009.05.005. [7] O. Bühler, On he voriciy ranspor due o dissipaing or breaking waves in shallow-waer flow, J. Fluid Mech., 407 (2000), 235 263. [8] B. Caselle and P. Bonneon, Modeling of a rip curren induced by waves over a ridge and runnel sysem on he Aquianian Coas, France, C. R. Geosciences, 338 (2006), 711 717.
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