Surf zone currents and vorticity on beaches with rip currents

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1 Surf zone currents and vorticity on beaches with rip currents Eric Barthélemy 1 Leandro Suarez 2 Hervé Michallet 1 2 Grenoble University, Laboratoire de Ecoulements Géophysiques et Industriels, France 2 Pontificia Universidad Católica de Chile, Departamento de Ingeniería Hidráulica y Ambiental, Chile 2 mars 2015 E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

2 Lay out 1 Introduction Context Circulation and vorticity E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

3 Lay out 1 Introduction Context Circulation and vorticity 2 Experimental and model set-up Non-linear shallow water equations Physical model Model validation E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

4 Lay out 1 Introduction Context Circulation and vorticity 2 Experimental and model set-up Non-linear shallow water equations Physical model Model validation 3 Diagnosis tools Vorticity equation Orders of magnitude E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

5 Lay out 1 Introduction Context Circulation and vorticity 2 Experimental and model set-up Non-linear shallow water equations Physical model Model validation 3 Diagnosis tools Vorticity equation Orders of magnitude 4 Bowen s solution (1969) E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

6 Lay out 1 Introduction Context Circulation and vorticity 2 Experimental and model set-up Non-linear shallow water equations Physical model Model validation 3 Diagnosis tools Vorticity equation Orders of magnitude 4 Bowen s solution (1969) 5 Conclusions E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

7 Introduction E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

8 Context Nearshore environment : neashore, interface between the ocean and the land : mainly sediment exchanges E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

9 Context Nearshore environment : neashore, interface between the ocean and the land : mainly sediment exchanges large variety of geological coastal features : beaches, cliffs, deltas, barrier islands, tidal inlets, spits... E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

10 Context Nearshore environment : neashore, interface between the ocean and the land : mainly sediment exchanges large variety of geological coastal features : beaches, cliffs, deltas, barrier islands, tidal inlets, spits... strong interactions between waves and these coastal features : erosion, morphodynamics E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

Context Nearshore environment : neashore, interface between the ocean and the land : mainly sediment exchanges large variety of geological coastal features : beaches, cliffs, deltas, barrier islands,

11 Context Nearshore environment : neashore, interface between the ocean and the land : mainly sediment exchanges large variety of geological coastal features : beaches, cliffs, deltas, barrier islands, tidal inlets, spits... strong interactions between waves and these coastal features : erosion, morphodynamics we focus on sandy beaches E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

12 Nearshore scales Nearshore zones associated to beaches are characterized by, shallow waters 10 m to 0 m at the shore line L 100 m bottom bathymetry strongly influences wave propagation & dynamics : shoaling and breaking intense forcing by breaking : wave breaking turbulence : horizontal and vertical mixing/diffusion boundary layers & sediment motion at the scale of the surf zone : longshore, vorticity / rips and beach morphology E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

Mean current : longshore possible on alongshore uniform beach ( y h = 0) oblique waves waves lose alongshore momentum (radiation stress) with the breaking process

13 Mean current : longshore possible on alongshore uniform beach ( y h = 0) oblique waves waves lose alongshore momentum (radiation stress) with the breaking process momentum transfered to a longshore current balanced by bottom friction sheared current (mixing) E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

channel cut beach cusps induced rips, off-shore morphology induced rips, structure induced rips

14 Mean circulation : rip currents possible with frontal waves (shoreline normal rays) induced by alongshore gradients in wave breaking localized off-shore jet morphological signature : rip channel / channel cut beach cusps induced rips, off-shore morphology induced rips, structure induced rips macro-vorticity : coherent large scale vortices E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

Vorticity generation mechanism waves are essentially non-rotational differential breaking : gradients in wave height H along the crest of the wave [Peregrine, 1998 ; Bonneton, 2010] either induced by

15 Vorticity generation mechanism waves are essentially non-rotational differential breaking : gradients in wave height H along the crest of the wave [Peregrine, 1998 ; Bonneton, 2010] either induced by the topography or inhomogeneity in the forcing wave field transformation of 3D vorticity into vertical vorticity vorticity shed by each waves how does it organize at the large scale? [Ph. Bonneton] E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

16 therefore, at i = 0, the longshore component of the velocity v is given by Bowen s solution (1969) v(0, y) = -o linearized and both Longuet-Higgins omponents of the velocity equations are therefore zero at the shoreline, the line at which balance between vorticity production and the water depth d goes to ero. friction The free solution outside the surf zone must be patched to the forced solution at lhe breakers mus therefore have the same longshore dissipation in terms of radiation stress & d breakers variation. likethe linear solution waves subject o the boundary condition thai mus remain bounded constant vorticity source term ; constant ß is given by breaker index γ = H/h in the surf zone ; plane beach (z, ) = Q(Xz + 1)e - sin (36) hen the neck of the rip at the breaking point ( t 2 free motion of Ox the -- - vortex Qh xe : sin ky The longshore D ωcomponent Dt + ω u of = velocity 0 is given by 1 0 QX' e_xx sin Xy this defines a time scale of vortex overturning v(x, 'y)- : x tan /? Ox :=-tan/? so that, at any given value of y, the longshore velocity outside the surf zone is unidirectional and decays monotonically seaward. The con- T v = L U stants to be evaluated. As numerical methods must be used to examine the effects of the nonlinear terms, it is conveniento treat the linear problem at the same time. o I =0 0 /I = XXs 0 '/2 Xx BREAKER LINE Fig. 6. A linear solution using bottom œri ion. The offshore dependence of the transport stream function (Xx), where -- 0 at the position of maximum set-up, is shown above the full solution, (xx) ß sin xy. E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

of vorticity : numerical model & Bowen Numerical simulations with SURF-WB (Marche et al.

17 Objectives validate the use of nonlinear shallow water models for circulation modelling extract orders of magnitude of the mean flow variables : currents, breaking generated macro-vorticity scaling laws for the level of vorticity : numerical model & Bowen Numerical simulations with SURF-WB (Marche et al., 2007) MODLIT experiments at LHF-ARTELIA wave basin (Michallet et al., 2013) E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

18 Model validation by experiments E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

19 Model choice A NSW numerical model (Marche et al., 2007, Guerra et al. 2014) is chosen to study the nearshore circulation : dissipation due to wave breaking is embedded in the equations and the numerical scheme, in accordance with the shock theory a time domain model : drifter analysis wet-dry interfaces are treated by solving Riemann problems at the interfaces. E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

20 The shallow-water model 2DH Hydrodynamic model SURF-WB [Marche et al., 2007 ; Guerra et al., 2010] Q = ( h hu hv ), F (Q) = Q t + F x + G y = S(Q) ( hu hu Fr 2 h 2 huv ), G(Q) = ( hv huv hv Fr 2 h 2 ) S(Q) = 0 h z S Fr 2 x fx h z S Fr 2 y fy h(x, y, t) : water depth u(x, y, t), v(x, y, t) : vertically averaged velocities F, G : flux vectors S : sources terms quadratic friction law : S = C f h u u E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

21 Model validation : laboratory rip experiment Michallet et al LHF wave tank 30 mx30 m mobile sand bed, d 50 = 164 µm h 0 = m at the wavemaker scaled experiment : length similitude 1/10, time similitude 1/3 film! E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

Model validation : laboratory rip experiment Michallet et al. 2013 x: Cross-shore direction (m) 0 5 10 15 20 25 60 parallel segmented Wavemaker Sliding rail Bathymetric survey area Beach : z >0.

22 Model validation : laboratory rip experiment Michallet et al x: Cross-shore direction (m) parallel segmented Wavemaker Sliding rail Bathymetric survey area Beach : z >0.765 m LHF wave tank 30 mx30 m mobile sand bed, d 50 = 164 µm h 0 = m at the wavemaker scaled experiment : length similitude 1/10, time similitude 1/3 film! y : Longshore direction (m) 60 parallel segmented wavemaker : irregular waves 3 fixed wave gauges at X = 5 m mobile wave gauges and ADV on the sliding beam E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

23 Experiment : bathymetry beach initially uniform in y ; H s = 18 cm and T p = 3.5 s ; breaking point depth : h b 0.2 m non-uniform wave forcing : induces non-uniform bathymetries. Wave height H s = 0.18 m thus H 0.01 m [Michallet et al., 2013, JGR] essai t=00 :00,t=26 :00,t=51 :40 E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

24 Experiment : bathymetry beach initially uniform in y ; H s = 18 cm and T p = 3.5 s ; breaking point depth : h b 0.2 m non-uniform wave forcing : induces non-uniform bathymetries. Wave height H s = 0.18 m thus H 0.01 m [Michallet et al., 2013, JGR] essai t=00 :00,t=26 :00,t=51 :40 t=00 :00 E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

25 Experiment : bathymetry beach initially uniform in y ; H s = 18 cm and T p = 3.5 s ; breaking point depth : h b 0.2 m non-uniform wave forcing : induces non-uniform bathymetries. Wave height H s = 0.18 m thus H 0.01 m [Michallet et al., 2013, JGR] essai t=00 :00,t=26 :00,t=51 :40 t=00 :00 t=26 :00 E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

Experiment : bathymetry beach initially uniform in y ; H s = 18 cm and T p = 3.5 s ; breaking point depth : h b 0.2 m non-uniform wave forcing : induces non-uniform bathymetries. Wave height H s = 0.

26 Experiment : bathymetry beach initially uniform in y ; H s = 18 cm and T p = 3.5 s ; breaking point depth : h b 0.2 m non-uniform wave forcing : induces non-uniform bathymetries. Wave height H s = 0.18 m thus H 0.01 m [Michallet et al., 2013, JGR] surf zone width : L x 10 m rip current channel : L y 10 m essai t=00 :00,t=26 :00,t=51 :40 t=00 :00 t=26 :00 t=51 :40 E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

27 Chapter 4 Circulation in the nearshore zone Validation : time series Free surface elevation cross-shore distance (m) 7.3 m 9.3 m 11.3 m 13.3 m 15.3 m 17.3 m Time (s) Figure 4.18 Free surface elevation time-series in a cross-shore profile at y = 10 m for 6 different wave gauges located at different cross-shore positions for experiment 30 (t=21:00-26:00). Experimental results (line) and numerical model (dashed line) longshore position : y = 10 m different cross-shore gages model and experiments in phase no secondary crests in the model : no dispersion model predicts breaking already at 9.3 m, not in the experiments E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

28 Validation : time averaged velocities ADV measured velocities (approx 5 cm above bed level) numerical model : vertically averaged velocities, JONSWAP forcing E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

29 Validation : time averaged velocities ADV measured velocities (approx 5 cm above bed level) numerical model : vertically averaged velocities, JONSWAP forcing E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

30 Validation : time averaged velocities ADV measured velocities (approx 5 cm above bed level) numerical model : vertically averaged velocities, JONSWAP forcing E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

31 Validation : time averaged velocities ADV measured velocities (approx 5 cm above bed level) numerical model : vertically averaged velocities, JONSWAP forcing strong cross-shore velocities : rip current directed offshore. U 0.1 m/s E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

32 Model validation : circulation and vorticity film! E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

33 Diagnosis tools & orders of magnitude E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

$(2011)] vorticity source term : curl of the dissipative force D t ω + (ω u) = (D e k ) e z ( ω ũ) β t h + (u h) = M ( u h ) e z D the dissipation force : D = D b hc with little refraction : B.$

34 Mean vorticity equation starting point : the nonlinear shallow water equations separation of wave orbital motions and large scale motions [Smith (2006) ; Bonneton et al. (DCDS-S 2010) ; Bruneau et al. (2011)] vorticity source term : curl of the dissipative force D t ω + (ω u) = (D e k ) e z ( ω ũ) β t h + (u h) = M ( u h ) e z D the dissipation force : D = D b hc with little refraction : B. Castelle (2013) (D e k ) D e k E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

35 Non dimensional numbers ω Ω u U Ω L x y 1 t 1ˆT L ( ) U ˆT t ω + ω u = L ( ) ( ) ( ) ( ) ( ) U ˆT D U ˆT βl u (D e k ).e z e z L ΩU L U h h ˆT is a time scale of the average motion : wave grouping ( ) Strouhal number : U ˆT L in which L/U is overturn time scale of the vortex. D βl in the experiments : ΩU 1 to 10 and U h 1 to 10 E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

36 Dissipative force D= F/ρ Ch ' C 3 g H3 γ 2 4 C T h0 4T energy flux F(x, y ) shock theory H ' 0.05 to 0.1 m h0 ' 0.1 m in the surf zone Tp = 3.5 s γ ' 0.6 O (D) 0.02 E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

37 Wave scale vorticity weak shock : h+ H C u(y) H(y) g h(x) h(x) u h u = 0 wave scale vorticity : ω u g y h H y 0.14 L. Suarez (2014) film E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

38 Results and discussion E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

39 Vorticity order of magnitude In our experiments/model friction, production and advection of similar magnitude Vertical vorticity magnitude Ω Ω 2 C 4TL γ3 = Γ r (ω u) (D e k ).e z U Ω L γ = H h Enstrophy E is the rms of the vorticity : E = ω 2 ds The surface S is surfzone wide and contains only one vortex (experiment 30) S ǫ Γ r L. Suarez (2014) T (s) E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

40 Wave period dependency γ is a function of m and T choice of γ m g T 1/2 h γ' 2 [Raubenheimer et al., 1996 ; Bonneton, 2001] L. Suarez (2014) E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

41 Bowen s model extension steady and vorticity production - friction balance (u h) = 0 u h = ψ e z ( ) u 0 = (D e k ). e z β e z h ( ) 1. h 2 ψ = 1 β yd dissipation force computed with the shock theory non constant γ quadratic friction as in SURFWB : but constant β = C f u frontal waves : (D e k ). e z y D Streamfunction equation ( ) 1. h 2 ψ = m3 g 2 T 2 32 β 1 h 2 yh E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

42 0 Numerical implementation standard elliptic equation : matlab PDE toolbox the choice of the size of the domain L x and L y : ratio rip spacing to surfzone width ( 1.6) y =1 non-dimensional equations h(x, y) = m x (1 ɛ (cos 2πy + ɛ cos 4πy)) the boundaries are streamlines : ψ = bathymetry: ε = 0.3; L= 1; x b = y =0 x = 0 D 0 x = x b D = 0 x = L longshore (m) h = cross shore (m) E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

43 Numerical results trick : h(x = 0) = 0.01 m vortex dipole because of the bathymetric variations vortex centers very close to the breaking line (Bowen) E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

44 Numerical results : dependency on alongshore variability 0.4 alongshore variability parameter χ (Ruessink et al., 2001) : E A 0.15 B C 0.1 D E F 0.05 G H χ 2 b Enstrophy versus χ 2 L. Suarez (2015) χ 2 = 1 L x L y χ = ( ) h h 2 dx dy h ɛ 1 + ɛ 2 ɛ 2 the numerical NSW enstrophy E varies linearly with χ 2 (varies with different bathymetries) the streamfunction model gives : Ω ɛ 1.1 streamfunction model rip maximum velocity is also linearly depend of ɛ : not surprising since U rip Ω L y lidated with a wide set of data, of free surface and eproduce E.B., thel.s., energy R.C. dissipation (LEGI / PUC) gradients related to Surf zone currents 2 mars / 43

45 Conclusions NSWE useful tool for the dynamics of bathymetry induced circulation ; dissipation and dissipation gradients correctly modeled even though secondary crests (harmonics) not reproduced orders of magnitude : a balance between large scale friction dissipation and vorticity production by waves (Bowen s hypothesis) physics cannot be captured if the ratio wave height to depth is considered constant in the surf zone we improved Bowen s model : the variant gives good scalings compared to NSW. The circulation strongly depends on the slope, the period, the bathymetric gradients and the water depth closed analytical solutions of the model variant? transient motion measured by a Strouhal number : group forcing, instabilities, Very Low Frequency motions? NSW time domain model E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

46 Thank you for your attention, Acknowledgments : PhD grant of L. Suarez : CONICYT (Chili, PhD grant No ) Escuela de Ingeniería de la PUC Other fundings : the project MODLIT (DGA/SHOM INSU/RELIEFS) the project BARCAN (LEFE) l ANR BARBEC the project SOLi (MANU-LEFE) to Ph. Bonneton, B. Castelle, & F. Marche, for fruitful discussions. E.B., L.S., R.C. (LEGI / PUC) Surf zone currents 2 mars / 43

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