NON-LINEAR WAVE PROPERTIES AND INFRAGRAVITY WAVE MOTIONS

NON-LINEAR WAVE PROPERTIES AND INFRAGRAVITY WAVE MOTIONS SIMULATED BY A BOUSSINESQ MODEL. Barthélemy Eric a,b, Cienfuegos Rodrigo b & Grasso Florent a a INPG, LEGI, BP53X 3841 Grenoble cedex9, eric.barthelemy@hmg.inpg.fr b Pontificia Universidad Católica de Chile, DIHA, Vicuña Mackenna 486, casilla 36, correo 221, Santiago, Chile Résumé : Un modèle de type Boussinesq (SERR1D) incluant une paramétrisation du déferlement est utilisée pour reproduire les propriétés non-linéaires de la houle en zone de surf ainsi que les ondes basses fréquences associées. Les performances du modèle sont comparées à des experiences en canal de laboratoire. Le modèle est appliqué à la propagation d ondes irrégulières sur une bathymétrie qui correspond à un fond sédimentaire en équilibre. La génération des ondes infragravitaires est analysée. Abstract : The present work aims at investigating the ability of Boussinesq-type model (SERR1D) incorporating a breakingwave parameterization [Cienfuegos et al., 25] to reproduce nonlinear properties of surf zone waves and infragravity motions. We compare results produced by this fully non-linear wave model [Cienfuegos et al., 26a, 27] with laboratory data. The model is run to simulate irregular wave propagation which results are compared to a laboratory experiment of irregular waves propagating over an uneven loose bed bathymetry at equilibrium. The generation of infragravity waves is analyzed. Key-words : Boussinesq modelling, non-linear, surf-beat 1 Introduction. Over the last 15 years important practical improvements for Boussinesq-type models have been introduced extending their applicability range into deeper waters and into the surf and swash zones. The breaking processes have been modelled by means of an ad-hoc extra term added to the momentum conservation equation of the inviscid (non dissipative) set of Boussinesq equations [Zelt, 1991, Brocchini et al., 1992, Schäffer et al., 1993]. An alternative approach was recently proposed by Cienfuegos et al. [25, 26b] where a diffusive-type term is also introduced in the mass conservation equation. The present work aims at investigating the ability of this particular breaking-wave parameterization to reproduce nonlinear properties of surf zone waves and nonlinear energy transfer in the shoaling and surf zones. Our breaking wave model is compared to data of an experimental random wave test. We investigate model predictions of the non-linear properties, the nonlinear energy transfer in the frequency domain to higher harmonics and also, in a preliminary way, to sub-harmonic infragravity motions. 1

2 Model features. The depth-averaged mass and momentum conservation equations (Serre equations; Cienfuegos et al. [26a]) can be written in the following generic form, h t + x (hu) D h =, (1) u t + 1 u 2 2 x + g h x + Γ d 1 h D hu =, (2) where h is the water depth, u is the depth-averaged horizontal velocity, D h and D hu represent extra terms accounting for energy dissipation by breaking, Γ d groups all high-order dispersive (non-hydrostatic) terms, and g is the gravitational acceleration; variables x and t denote space and time coordinates. In surf zone Boussinesq models, a breaking criterion must be adopted to turn on/off breaking dissipation. The breaking initiation/cessation criterion is based on threshold values of the breaker front slope Φ [Schäffer et al., 1993]. The present model termed SERR1D includes a mass conservative diffusivity term in the continuity equation intended to mimic local mass redistribution inside the roller and mixing layer effects. The breaking induced terms read, D h = [ ] h ν h x x & D hu = [ ] (hu) ν hu x x In this model, mass and momentum diffusivity coefficients, ν h and ν hu, are written as, c d ν h = δ h tanφ Λ(X/l c d r) & ν hu = δ hu tan Φ Λ(X/l r) for X l r (4) where δ h and δ hu are order-one coefficients, c(x) is the wave celerity, d is the mean water depth for a given wave and Φ is the breaker angle. Λ is a self-similar shape function that locally distributes diffusivity over the breaker front ensuring that extra terms are conservative over a breaking event. This form derives from experimental results produced by Svendsen et al. [2] and writes as, ( ) [ ( ) ( ) ] 2 X X X Λ(X/l r ) = exp 1 1 + 1, l r l r l r with X a local coordinate system moving with the wave (X = is located at the crest and positive shorewar), and l r the roller horizontal length. The different parameters have been scaled using empirical and theoretical knowledge on quasi-steady shallow water breakers and do have a physical interpretation [see Cienfuegos et al., 25, for details]. In the present work we simplify the model taking δ hu and Φ as constants and assuming that δ h =.1δ hu. Finally, the roller length is related to local wave properties using Cointe and Tulin [1994] theory of steady breakers calibrated on hydraulic jumps in similarity with surf zone waves [see Cienfuegos et al., 24]. It follows that l r /d =.865 (1 γ), with the breaker index γ = H/d and H the local wave tan Φ height. The numerical values of model parameters were calibrated on the regular wave spilling breaking data reported by Ting and Kirby [1994]. The best agreement between computed and measured properties is produced for Φ = 13., γ =.78 and δ hu = 7.8. In what follows we will freeze the calibrated parameter values for SERR1D model and move on to a more realistic situation where random waves propagate over uneven bathymetries. 2 (3)

3 Experimental set-up. The experiments were carried out in the 36 m long and 55 cm wide LEGI flume equipped with a piston wave generator [Michallet et al., 27]. The water depth at rest is 55.3 cm. The mean overall slope is approximately 1/5 (see Fig. 1c). The sloping bottom consists of a loose material of low density (1.19 g cm 3 ) and with a median diameter of d 5 =.6 mm which allow for representative sheet-flow (shoaling zone) and suspension (surf zone) sediment transport regimes. Irregular waves are generated according to a JONSWAP spectrum (peak enhancement factor =3.3, peak frequency at f p =.5 Hz, low frequency cut-off at 3/5f p ). We checked that these waves conform to the expected spectrum and that they follow a Rayleigh distribution at 2 m downstream of the wave maker. The beach profile was formed by the breaking waves in about 1 hours and the beach profile did not change in the mean in the last hours. The form of the equilibrium profile of the beach exhibits a terrace as also found in nature [Black et al., 21]. The large waves tend to break where the bottom slope also breaks (x = 14.4 m) from a steep to mild a slope. 4 Model results & nonlinear properties. The numerical model is now applied to the random wave experiment described above for the quasi-equilibrium beach configuration. The wave gages in the experiment are located at 2 m, 5.77 m, 6.23 m, 9.77 m, 1.23 m, 13.77 m, 14.3 m, 17.77 m, 18.3 m, 21.3 m, 23.2 m and 23.8 m. The time serie of free surface elevations at the wave gauge located at x = 2. m is used to prescribe the incident random wave field at the seaward boundary in the numerical model. However the experimental signal was high-pass filtered in order to eliminate infragravity motions contained in the experimental data. A moving shoreline condition, adapted from Lynett et al. [22], is implemented on the right boundary [see Cienfuegos et al., 27, for details]. We perform computations over 14 s using SERR1D. Wave-average statistics such as the H rms, the mean water level, the left-right asymmetry and the crest-trough skewness [Kennedy et al., 2] are plotted in Fig. 1. The overall agreement between computed and measured free surface statistics is considered adequate for this challenging test. The mean water level (mwl) evolution in the shoaling and surf zone has a good trend but has a slight off-set compared to the measurements. There is a slight tendency to underestimate the horizontal asymmetry in the inner surf zone while the computed skewness is in good agreement with the measurements except very close to the shoreline. Finally, we perform a spectral analysis in order to study the nonlinear energy transfer inside the domain and results are presented in Fig. 2 (left column). For this spectral analysis we processed results obtained from a simulation in which the incident wave field was not highpass filtered. As expected, for the first wave gauge located at x = 2 m the prescribed narrow band JONSWAP spectrum is recovered centered at the peak frequency f p =.5 Hz [Michallet et al., 27]. In the gravity spectral range (f 3 Hz) close to the breaking point x = 14.3 m, the harmonic 2f p has a clear signature in the spectrum. The numerical model overpredicts the energy content in the gravity spectral range at the shoreline but globally energy dissipation by breaking is well reproduced. This over-prediction of the energy of the high frequency range can be attributed to the set-up over-estimation in the model which, increases the mean water depth along the profile. In the model broken waves tend therefore to be larger than in the experiment at a given location because propagating on slightly deeper waters. It is also noticeable that some energy is present at frequencies lower than that of the JONSWAP spectrum. This range is the so-called infragravity range. 3

H rms (m) depth (m).1.5 5 5 2 25.2.2.4 5 5 2 25 a c As & Sk mwl (mm) 4 2 2 4 5 5 2 25 1.5 1.5.5 5 5 2 25 b d Figure 1: irregular wave propagation on an equilibrium bathymetry (bottom profile 224). Waves conform to a JONSWAP spectrum (wave time serie VAGUE2s17mm112a): peak period T p = 2s, significant wave height H s =.17m prescribed at the wave maker, water depth at the wave maker: h =.553m. a) root mean square wave height H rms, b) mean water level, c) bottom profile as measured, d) (*): asymmetry As & (o) skewness Sk ; plain lines: simulations with SERR1D. The breaking point is indicated by a large tick on top of each panels at 14.4 m. Simulations forced with the wave signal at x = 2 m high-pass filtered. 5 Infragravity motions. In Fig. 2 (left column) two strong low frequency peaks at f 1 =.27 Hz and f 2 =.63 Hz contribute to the low frequency spectrum. Recall that for these simulations the complete experimental free surface displacement time serie was prescribed at the seaward boundary of the model. Analysis by Michallet et al. [27] suggest that the f 1 low frequency motion corresponds to a nearly standing wave which half wave-length is nearly equal to the length of the flume. This is also sustained by the spatial evolution of the energy content at this frequency since maximum values are reached near the two edges of the flume while at x = 14.4 m the energy of this frequency is much smaller. Antinodes of the standing wave are located near the channel edges, while the node is near its center. This standing wave produces a first harmonic at f 2 =.63 Hz frequency which as anti-nodes at the seaward boundary, the shoreline and the breaking point [Michallet et al., 27]. Results in Fig. 2 (right column) where forced with the experimental data high-pass filtered as in Fig. 1. This physically means that the incoming JONSWAP waves are bound-long wave free. In other words the set-down associated with the wave packets is eliminated as was done by Madsen et al. [1997]. It is noticeable that the physics contained in the model have the characteristics to generate infragravity motions. It is known that the two relevant mechanisms are the bound-wave release in the shoaling and breaking zone and the moving break-point mechanism [Janssen et al., 23, Grasso et al., 27]. The first one is strongly inhibited by the filtering process. While propagating onshore the wave packets should develop a certain amount of bound wave energy. However the propagation length being small the break-point mechanism should dominate. The simulated infragravity range is quite widespread at x = 2 m and 23.8 m and does not contain as strong specific peak frequencies as in the experimental data. This is not suprising since as pointed earlier the boundary conditions in the experiments are that of a closed basin while the numerical model is intended to reproduce a semi-infinite sea. Close to the breaking point, which is also the antinode of the f 2 frequency, the infragravity spectral range is much narrower and centered at f 2. This supports the idea that in the course of the experiments the bottom adapts to the wave forcing and to the surf-beat that in turn produces a feedback which selects the f 2 low frequency small amplitude wave. In addition it should be emphasized that 4

x=2 m infra gravity gravity 1 2 f 1 f 2 f p 1 2 x=14.3 m 2 f p 1 2 1 2 x=23.8m 1 2 1 2 Figure 2: spectral power density of the free surface displacements. (- -): experimental data; plain lines: simulations with SERR1D. Left column: computations forced with the complete signal recorded at x = 2 m; right column: computations forced with signal recorded at x = 2 m high-pass filtered. Computations with bathymetry of Fig. 1c. Top line: x = 2 m; middle line: x = 14.3 m; bottom line: x = 23.8 m. the infragravity motions at 23.8 m tend to be more energetic than the high frequency waves that have undergone a long run dissipation. As expected the swash zone motions are energetically dominated by a wide spectrum of low frequency motions. 6 Conclusions. The present new model was successfully applied to irregular wave propagation on a nature-like bathymetry obtained in the LEGI flume. Results on nonlinear wave properties are in good agreement with the experimental results and the model also succeeds in reproducing the nonlinear harmonic generation of the incident narrow band JONSWAP spectrum. Moreover it was shown that the physics to generate infragravity motions are comprised in the model and that the characteristics of the surf-beat with a anti-node at the breaking point are in qualitative agreement with the experimental data. References K.P. Black, R.M. Gorman, and K.R. Bryan. Bars formed by horizontal diffusion of suspended sediment. Coastal Eng., 47:53 75, 21. M. Brocchini, M. Drago, and L. Iovenitti. The modelling of short waves in shallow waters. Comparisons of numerical models based on Boussinesq and Serre equations. In 23th. Int. Conf. Coastal Eng., pages 76 88, Venice, Italy, 1992. R. Cienfuegos, E. Barthélemy, and P. Bonneton. Roller modelling in the context of undertow prediction. In 29th. Int. Conf. Coastal Eng., pages 318 33, Lisbon, Portugal, 24. 5

R. Cienfuegos, E. Barthélemy, and P. Bonneton. A new wave-breaking parametrization for Boussinesq-type equations. In 5th Int. Symp. Oc. Wave Meas. Anal., Madrid, Spain, 25. CD-ROM. R. Cienfuegos, E. Barthélemy, and P. Bonneton. A fourth order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I : model development and analysis. Int. J. Num. Meth. Fluids, 51(11):1217 1253, 26a. R. Cienfuegos, E. Barthélemy, and P. Bonneton. Nonlinear surf zone wave properties as estimated from boussinesq modelling: random waves and complex bathymetries. In 3th Int. Conf. Coastal. Eng., San Diego, USA, 26b. R. Cienfuegos, E. Barthélemy, and P. Bonneton. A fourth order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part II : boundary conditions and validation. Int. J. Num. Meth. Fluids, 53(9):1423 1455, 27. R. Cointe and M.P. Tulin. A theory of steady breakers. J. Fluid Mech., 276:1 2, 1994. F. Grasso, H. Michallet, and E. Barthélemy. Infragravity waves in modile-bed laboratory experiments. In Coastal Sediment 7, New Orleans, USA, 27. accepted. T.T. Janssen, J.A. Battjes, and A.R. Van Dongeren. Observation of long waves induced by short wave groups. J. Geophys. Res., 18:3252 3264, 23. A.B. Kennedy, Q. Chen, J.T. Kirby, and R.A. Dalrymple. Boussinesq modelling of wave transformation, breaking and runup. I : 1D. J. Waterw. Port Coastal and Oc. Eng., 126(1):39 48, 2. P.J. Lynett, T.R. Wu, and P.L.F. Liu. Modeling wave runup with depth-integrated equations. Coastal Eng., 46:89 17, 22. P.A. Madsen, O.R. Sørensen, and H.A. Schäffer. Surf zone dynamics simulated by a Boussinesq-type model : Part II. Surf beat and swash oscillations for wave groups and irregular waves. Coastal Eng., 32:289 319, 1997. H. Michallet, F. Grasso, and E. Barthélemy. Long waves and beach evolution profiles. J. Coastal Res., 27. to be published. H.A. Schäffer, P.A. Madsen, and R.A. Deigaard. A Boussinesq model for waves breaking in shallow water. Coastal Eng., 2:185 22, 1993. I.A. Svendsen, J. Veeramony, J. Bakunin, and J.T. Kirby. The flow in weak turbulent hydraulic jumps. J. Fluid Mech., 418:25 57, 2. F.C.K. Ting and J.T. Kirby. Observation of undertow and turbulence in a laboratory surf zone. Coastal Eng., 24:51 8, 1994. J.A. Zelt. The run-up of nonbreaking and breaking solitary waves. Coastal Eng., 15:25 246, 1991. 6