1DH Boussinesq Modeling of Wave Transformation Over Fringing Reefs

Ocean Engineering 47 (2012) 30–42

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Ocean Engineering

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1DH Boussinesq modeling of wave transformation over fringing reefs

Yu Yao a, Zhenhua Huang a,b,n, Stephen G. Monismith c, Edmond Y.M. Lo a a School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore b Earth Observatory of Singapore (EOS), Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore c Department of Civil and Environmental Engineering, Stanford University, 473 Via Ortega, Stanford, CA 94305-4020, USA article info abstract

Article history: To better understand wave transformation process and the associated hydrodynamic characteristics Received 14 May 2011 over fringing coral reefs, we present a numerical study, which is based on one-dimensional (1D) fully Accepted 12 March 2012 nonlinear Boussinesq equations, of the wave-induced setups/setdowns and wave height changes over Editor-in-Chief: A.I. Incecik various fringing reef profiles. An empirical eddy viscosity model is adopted to account for wave breaking and a shock-capturing finite volume (FV)-based solver is employed to ensure the computa- Keywords: tional accuracy and stability for steep reef faces and shallow reef flats. The numerical results are Wave-induced setup compared with a series of published laboratory experiments. Our results show that with an appropriate Wave-induced setdown treatment of boundary conditions and a fine-tuned eddy viscosity model, the full nonlinear Boussinesq Boussinesq equations model can give satisfactory predictions of the wave height as well as the mean water level over various Coral reef hydrodynamics reef profiles with different reef-flat submergences and reef-crest configurations under both mono- Mean water level Wave breaking chromatic and spectral waves. The primary 1D wave transformation processes, including nonlinear shoaling, refection, breaking, generation of higher harmonics and infragravity waves, can also be reasonably captured. Finally, the model is applied to study the effects of reef-face slopes and profile shapes on the distribution of the wave height and mean water level over the fringing reefs. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction almost-constant wave-induced setup over the reef top where the water is shallow and the depth is nearly constant. The low- Wave interaction with fringing coral reefs has been a primary frequency motions due to the so-called surf beat or reef-flat focus of nearshore hydrodynamics over decades. The physics of resonance might exist on the reef flat as well (e.g., Pe´quignet waves on reefs are different from that of waves on normal coastal et al., 2009). Since fringing reefs shelter many tropical islands beaches in many respects, including the roughness of the sub- from the flood hazards associated with tsunamis, hurricanes, and strate and the variety of geometries. A typical fringing reef is high surf events (Roeber et al., 2010), the wave-induced setup and characterized by a seaward sloping reef face and an inshore wave-generated currents across reef flats may have profound shallow reef flat extending towards the coastline. Corals com- geological, ecological, engineering and environmental implica- monly grow to mean low tide levels and may impose a shallow tions (see Gourlay and Colleter, 2005). water control on the waves reaching reef flats. Similar to the wave Precise characteristics of wave dynamics and the energy transformation over a shallow shelf, ocean waves first shoal on a dissipation through wave breaking are controlled mainly by the fore-reef face and then break either on the reef face or on the reef morphology of reef profile (reef-flat water depth, reef-face slope, flat. As a result, a shoaling-induced setdown can be observed and reef-flat width) and incident wave conditions. Although the before the breaking point, accompanying a breaking-induced reef profile may vary from site to site, ridges or similar config- setup after the breaking point. The surfzone always extends over urations (‘‘reef rim’’ or ‘‘reef crest’’ in some papers) have been a certain distance on the reef flat, starting from the incipient frequently observed at the edges of coral reefs (Gourlay, 1996b; breaking point to the location where the wave breaking ceases. Jago et al., 2007; Hench et al., 2008). Ridges, consisting of coral After wave breaking, very short waves may reform on the reef colonies, rubble algal, etc., function like bars or submerged flat and propagate towards the shoreline, accompanying an breakwaters to filter a substantial portion of the incident wave energy; their effect on wave-induced setup has been investigated experimentally by Yao et al. (2009) using an idealized rectangular n Corresponding author at: School of Civil and Environmental Engineering, Nanyang ridge model. Technological University, 50 Nanyang Avenue, Singapore 639651, Singapore. Tel.: þ65 67904737. Hydrodynamics associated with waves on fringing coral reefs E-mail address: [email protected] (Z. Huang). is more complex than that on plane beaches: a typical fringing

0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.03.010 Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 31 coral reef involves a transition of bottom profile from deep to waves transforming over a reef profile that is similar to the one shallow waters over a long distance, as well as a porous reef used by Skotner and Apelt (1999), they confirmed the ability of surface which may provide high resistance to the waves above. their Boussinesq model in describing the variation of significant Numerical modeling of nearshore coral reef hydrodynamics faces wave height, the mean water level across the reef profile, the several challenges such as the steep reef-face slopes (e.g., Seelig, evolution of the wave spectrum, the generation of infragravity 1983; Gourlay, 1996a; Vetter et al., 2010), the complex config- oscillations and shoreline runups. More recently, Roeber et al. urations of reef crest and flat (e.g., Seelig, 1983; Hench et al., (2010) employed a shock-capturing Boussinesq-type model to 2008), the spatially-varied roughness of reef surface (e.g., Lowe simulate the solitary wave transformation over fringing reefs, et al., 2005). Also, the wave transformation usually needs to be which involved energetic wave breaking, bore propagation and modeled at a long time scale (several hundred waves) and a large the transition from subcritical to supercritical flows under an space scale (over the entire reef profile). initially dry reef crest. Over decades, analytical models frequently deal with the one- The main objective of this study is to implement and validate a dimensional horizontal (1DH) idealized reef profiles (a typical weakly dispersive and fully nonlinear depth-integrated Boussi- idealized reef profile has a plane sloping reef face and a horizontal nesq-type model1 to help interpret some of our previous labora- platform reef flat). Conventionally, in analogy to the wave-driven tory work (Yao et al., 2009) and other similar published work on alongshore flows and the wave-induced setup/setdown on bea- wave transformation over fringing reefs. The following three ches (e.g., Svendsen, 2006), analytical solutions based on the reasons made us to adopt the fully nonlinear Boussinesq equa- radiation stress concept introduced by Longuet-Higgins and tions in this study: (1) Skotner and Apelt (1999) speculated that Stewart (1964) had been used frequently in the past to study the use of weakly nonlinear equations might improve the predic- 1DH reef hydrodynamics (Gourlay, 1996a; Symonds et al., 1995; tion of wave-induced setdown and setup; (2) the presence of a Hearn, 1999; Gourlay and Colleter, 2005). In recent years, the ridge at the edge of the reef would cause the waves to be highly effects of complex bathymetry and different forcing mechanisms nonlinear in its vicinity; (3) the present fully nonlinear Boussi- have been modeled by using two-dimensional horizontal (2DH) nesq model is capable of simulating a wide range of long and and three-dimensional (3D) models to study both the waves and short wave problems (Lynett et al., 2002; Hsiao et al., 2005; the mean flows, and usually the radiation stress concept is used to Lynett, 2006), and the model has also been applied to wave couple the waves and the mean flows (Kraines et al., 1998, 1999; overtopping over a levee system by Lynett et al. (2010). Yao et al. Douillet et al., 2001; Luick et al., 2007). The modeling results (2011) reported a preliminary validation of the present model for presented by Lowe et al. (2009) look very promising, even though an idealized reef profile. This study will first report comprehen- the predicted the mean water level was not as accurate as the sive comparisons between the numerical simulations and the computed wave heights and currents. Compared to field studies, available published data for various wave conditions and different fewer numerical models have been applied to well-controlled, fringing reef configurations. The effects of the reef face shapes and small-scale laboratory investigations in the literature. sloping angles on the mean water levels and wave heights will The most advanced Navier–Stokes approaches, e.g., the RANS- also be investigated using the validated numerical model. based models (Lin and Liu, 1998; Losada et al., 2005; Lara et al., The remaining of this paper is organized as follows. In Section 2, 2008; Torres-Freyermuth et al., 2010), are well suited for simulat- the mathematical formulation, numerical scheme, boundary condi- ing breaking waves and wave-structure interactions in small tions and energy dissipation sub-model are described. In Section 3, confined regions. However, Navier–Stokes approaches are still numerical simulations are compared with available experimental very computationally expensive to run, especially for the near- data for four representative scenarios to show the robustness of the shore zones where a large number of grid points and a fine mesh model. In Section 4,wereportarevisitoftwo published numerical are needed to capture accurately the fine turbulence structures, works on wave transformation over ringing reefs. In Section 5,the thus its use is generally restricted to a small number of waves and validated model is applied to study the effects of the inclination of a small regions. Currently, applying Navier–Stokes models to field plane reef face and the shape of reef face on the wave dynamics over scale reef profiles does not seem feasible in practice. the fringing reefs. The main conclusions drawn from this study are Another type of prevailing model, which is more computa- given in Section 6. tionally efficient, is based on Boussinesq-type equations. This depth-integrated modeling approach employs a polynomial approximation to the vertical profile of velocity field, thereby 2. Description of the numerical model reducing the dimensions of a three-dimensional problem by one. It has been proved to be able to account for both nonlinear and 2.1. Governing equations dispersive effects at different degrees of accuracy. After the first introduction of Boussinesq equations by Peregrine (1967), con- Let x-coordinate be pointing in the direction of wave propaga- siderable efforts have been made in different ways to extend the tion with its origin at the toe of the reef face, and z-coordinate Boussinesq equations to deeper waters (e.g., Madsen and pointing upward with its origin at the still water level. The 1DH Sørensen, 1992; Nwogu, 1993; Wei et al., 1995; Lynett et al., equations in conservative form are expressed as 2002) and to surfzones or swashzones (e.g., Madsen et al., 1997; @H @Hua Kennedy et al., 2000; Veeramony and Svendsen, 2000). One of the þ þDc ¼ 0 ð1Þ @t @x pioneer studies of extending Boussinesq-type model to coral reef studies was conducted by Skotner and Apelt (1999), who studied @Hu @Hu2 @Z a þ a þgH þgHDx þu Dc ¼ 0 ð2Þ the wave-induced setup by monochromatic waves propagating @t @x @x a onto a submerged fringing coral reef which consisted of a relatively steeper reef face. They found that Boussinesq-type where H¼hþZ is the total water depth; Z is the water surface could simulate satisfactorily the patterns of mean water level elevation; h is the still water depth; g is the gravitational for 1DH reef profiles subjected to small waves, but there was a acceleration and ua[x,z¼za(x,t)] is a reference horizontal velocity tendency to underestimate the wave-induced setup as the incident wave height was increased. Nwogu and Demirbilek (2010) 1 Coulwave code V.2.0. was modified for this study. See Lynett and Liu (2008) used a different set of Boussinesq equations to study spectral for details of Coulwave. 32 Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 in the x-direction at a specified depth of za(x,t)¼0.531h (Nwogu, spectrum. The initial condition assumes no wave or current 1993). The Dc and Dx are the second order terms describing the motion in the computational domain. nonlinear dispersion. The complete derivation and the expressions for Dc and Dx can be found in Kim et al. (2009). For 1DH problems, there are two sources of energy dissipation: 2.4. Wave breaking (Rb) bottom friction and wave breaking. Adding the dissipation terms to the momentum equation gives It is well-known that the depth-integrated Boussinesq-type models cannot describe the overturning of a free surface and the @Hua þþRf Rb ¼ 0, ð3Þ detailed breaking process. Hence, several empirical models have @t been proposed for the wave breaking in surf and swash zones. The where Rf and Rb are ad-hoc dissipative terms accounting for the most common approach is to add an ad-hoc dissipation sub- bottom friction and the wave breaking, respectively. Sub-models model to the momentum equation. There are two primarily types for Rb and Rf will be discussed in Section 2.4 and 2.5, respectively. of breaking models: roller models (e.g., Schaffer¨ et al., 1993; Madsen et al., 1997; Veeramony and Svendsen, 2000) and eddy 2.2. Numerical scheme viscosity models (e.g., Zelt, 1991; Karambas and Koutitas, 1992; Kennedy et al., 2000). Even though, the two approaches stem The numerical solver for the above equations has been from different ideas and have different controlling parameters, described in details in Kim et al. (2009). A third-order Adams– their overall effect in momentum equation is similar: both require Bashforth predictor and a fourth-order Adams–Moulton corrector an energy dissipation mechanism and a trigger mechanism for the scheme are used for time marching. For the spatial discretization, initiation of wave breaking. One of basic requirements on empiri- a shock-capturing FV-based approach is used. The leading order cal breaking models is that they must ensure the conservation of terms are solved with the fourth-order MUSCL-TVD (monotone mass and momentum and preserve some nonlinear wave proper- upstream-centered scheme for conservation laws — total varia- ties in surfzones. In this study, the simple eddy viscosity-type tion diminishing) scheme, while for the second-order terms, a formulation proposed in Kennedy et al. (2000) is used to model cell-averaged finite volume method is implemented. Compared the wave energy dissipation caused by wave breaking. It has been with traditional finite difference (FD)-based methods, FV formu- proven that the adopted wave breaking model can adequately lations in conservative form are generally every stable and predict the energy dispassion for both spilling and plunging accurate, thus appropriate for the present problems which may breakers (Kennedy et al., 2000; Lynett, 2006; Roeber et al., have complex flow conditions and rapid bottom variations. The 2010). For completeness and the convenience of discussing our numerical scheme is accurate to O(Dt4) in time and O(Dx4)or numerical results, the breaking model in Coulwave is O(Dx2m2) in space, where m is the wave length scaling parameter summarized below. defined by m¼h/L, with L being the incident wave length. The 1DH form expression for Rb is given by

Rb ¼½nðHuaÞxx ð4Þ 2.3. Boundary and initial conditions where n is an empirical eddy viscosity and given by the following zero-equation turbulence model 2.3.1. Numerical boundary conditions

Two types of numerical boundary conditions can be applied at n ¼ BdHZt ð5Þ the two ends of the computational domain: the reflective (or no- flux) boundary condition and the radiation (or open) boundary where d is an empirical coefficient to correct both the mixing- condition. For the latter, sponge layers are frequently used to length and friction-velocity scales. The parameter B accounts for effectively damp the energy of outgoing waves. The sponge layer the trigger mechanism to ensure a smooth transition between is usually applied in a manner similar to that recommended by breaking and non-breaking states. The expression for B is given by 8 Kirby et al. (1998). For completeness, the sponge layer imple- Z n <> 1 Zt 2Zt mentation in Coulwave is given in Appendix A. Preliminary n n n B ¼ Zt=Zt 1 Zt oZt r2Zt ð6Þ numerical experiments (Yao et al., 2011) revealed that the :> 0 Z rZn damping term in the continuity equation would behave like a t t sink/source term, causing a significant change of the total amount n where Zt determines the onset and stoppage of the breaking of water in the computational domain and resulting in an process and is evaluated by incorrect mean water level. Therefore, the sponger layers were 8 used only in the momentum equation in this study. The moving < ðFÞ tt ZTn n Zt 0 Z ¼ n ð7Þ boundary can be simulated using the ‘‘slot technique’’ (Tao, 1983), t : ðIÞ tt0 ðFÞ ðIÞ Z þ n ðZ Z Þ 0rtt0 oT the Lagrangian method by Zelt (1991), or the extrapolating t T t t boundary algorithm (Lynett et al., 2002); the latter is used here where ZðIÞ is a threshold value at the breaking inception; ZðFÞ is a to describe wave runup and rundown processes. The moving t t saturated value for the breaking cessation; t is the time at which boundary algorithm is needed only when we study initially dry 0 the breaking event starts; tt is the age of breaking event; T* is reef flats. 0 the duration of the breaking event. Our preliminary numerical experiments have found that d is 2.3.2. Wave generation and initial conditions the key parameter for wave breaking on fringing reefs. Values of d Internal source methods are frequently employed as efficient ranging from 1.4 to 10 have been used in the published literature and accurate methods for numerical wave generations (e.g., Lin (Kennedy et al., 2000; Lynett, 2006; Lynett and Liu, 2008). and Liu, 1999; Wei et al., 1999; Hsiao et al., 2005). The method Our numerical experiments suggested that d¼2 work well for using a distributed source function in the continuity equation as all idealized fringing reefs without ridge (Yao et al., 2011). proposed by Wei et al. (1999) is adopted in this study. Irregular For fringing reefs with ridge, model calibration is needed to find waves are generated by summing up many regular waves with a suitable value of d. For the trigger-related parameters in Eq. (7), different frequencies, amplitudes and random phases for a given the following values suggested by Lynett and Liu (2008) were Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 33 adopted in this study, under-predicts the reflection coefficients for very steep slopes. pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi The fluctuation of reflection coefficient with b is due to the ZðIÞ ¼ a gH; ZðFÞ ¼ a gH; Tn ¼ a H=g ð8Þ t 1 t 2 3 multiple reflections between the two edges of the reef face. where a1¼0.65, a2¼0.08, a3¼8 for plane beaches (Kennedy et al. (2000) used the still water depth h instead of the instantaneous water depth H). 3. Comparison of numerical and laboratory results

2.5. Bottom friction (Rf) 3.1. Experimental and numerical settings

The bottom friction term Rf can be calculated by the following 3.1.1. Experimental setting quadratic friction law The laboratory experiments were conducted in a closed wave flume in the Hydraulics Laboratory, Nanyang Technological R ¼ fu 9u 9 ð9Þ f b b University, Singapore. The flume is 36 m long 0.55 m wide, where ub is the bottom velocity and can be evaluated from ua; f is and 0.60 m deep. The experiments were designed to study the an empirical friction coefficient, which can be related to Manning hydrodynamic characteristics of two types of reef-crest profile coefficient (n)by subjected to both monochromatic and spectral waves: (1) an idealized fringing reef without a ridge, and (2) a fringing reef with gn2 ¼ ð Þ f 1=3 10 an idealized ridge. We first tested an idealized fringing reef H model, which had a relatively steep reef-face slope (V:H¼1:6) Values of n for surfaces of commonly used materials can be and a 7 m-long horizontal reef flat. For the second reef profile, we found in any standard text book for hydraulics or fluid mechanics. placed a rectangular box (55 cm long, 50 cm wide, and 5 cm high) on the reef flat with its front face aligned to the reef edge to 2.6. Model validation for rapidly varying bathymetry mimic an idealized ridge (see Fig. 1). The dimensions of the ridge model were chosen to mimic the reef ridge existing on the Unlike beaches, which typically have mild slopes, a typical Moorea reef studied by Hench et al. (2008). The detailed experi- coral reef often forms a steep transition from the relatively deep mental settings and preliminary data results for the experiments to shallow waters. One major concern with applying Boussinesq examined in this study have been reported by Yao et al. (2009). models to fringing coral reefs is the relatively steep reef-face To measure the cross-reef wave transformation, 12 wave gages slopes; this is because derivatives of the water depth are included (G1–G12) were used and their arrangement is shown in Fig. 1: in the higher order terms of Boussinesq equations. G9–G12 were equally spaced over the reef flat with the first gage As a verification of the capability of the present model to deal being located behind the surfzone. For monochromatic waves, with rapidly varying bathymetry, we consider here a train of monochromatic waves propagating over a plane shelf. This 0.25 problem was first studied by Booij (1983), who investigated the FEM 0.225 accuracy of a mild-slope equation by comparing the predicted Present model reflection coefficients with finite element method (FEM) solu- 0.2 tions. Since then, this problem has become a benchmark against 0.175 which the accuracy of a hydrodynamic model can be verified. An example for testing Boussinesq-type models using this bench- 0.15 mark is given in Madsen et al. (2006). The bottom profile for this

R 0.125 benchmark problem consists of a plane slope connecting two C constant-depth regions, which is similar to our idealized fring- 0.1 ing reef model (shown in Fig. 1). The offshore water depth 0.075 is ho¼0.6 m, while the reef-flat water depth is hr¼0.2 m. We studied a train of monochromatic waves with a period of 0.05 2.0 s. The width of the reef-face slope varied from b¼0.1 m to 10 m, corresponding to the slopes of V:H¼4:1–1:25. To resolve 0.025 the steepest slope numerically, we used a grid size of 0.04 m and a 0 time step of 0.01 s for all simulations. 0 1 2 3 4 5 6 7 8 9 10 The computed reflection coefficients are shown in Fig. 2, b (m) together with the FEM solutions by Suh et al. (1997). It clearly shows that the present numerical results are accurate up to the Fig. 2. Variation of reflection coefficient (CR) with the slope width (b). Solid line: slope width of b¼0.3 m, i.e., a slope of V:H¼4:3, but slightly FEM solution of Suh et al. (1997); open circles: present model.

Fig. 1. Computational domain for waves propagating over a fringing reef (the exact locations of the wave gages for all the laboratory experiments are given in Table 2). 34 Y. Yao et al. / Ocean Engineering 47 (2012) 30–42

G4–G8 were located in the vicinity of the reef edge to measure the face was b¼2.1 m wide with its toe starting at L2¼16.35 m from surfzone waves; G1 and G2 were placed seaward of the reef slope the wave generation point. The reef flat, which was extended all to separate the incident waves from the reflected waves and G3 the way to the shoreward sponge layer, had a width of L3¼9.8 m; was placed on the slope to measure the shoaling waves. the elevation of the reef flat was fixed at 0.35 m above the flume For spectral waves, the arrangement of wave gages is slightly bottom for all experiments. The water depth in the deeper section different: G3 was moved further seaward to form a three-probe of the flume (i.e., h0 in Fig. 1) varied from 0.35 m to 0.45 m for the array for wave spectrum separation and G4 was located on the four cases studied here, giving a change of kh0 from 0.84 to 1.54 reef face to observe wave shoaling; the remaining gages (G5–G8) and a change of khr from 0.27 to 0.53 (see Table 1). Numerical were put close to the reef edge to capture the spectral waves in wave gages were placed at the same positions as their laboratory the surfzone. Exact locations of the wave gages will be specified counterparts in the flume. Their exact locations measured from later. At the shoreward end of the flume, waves were effectively the toe of the reef face for the four test cases are given in Table 2. damped by a porous wave absorber to reduce the wave reflection. A grid spacing about Dx¼0.03 m was found to be sufficient to A series of monochromatic and spectral wave conditions were discretize the computational domain for our selected waves, tested in the experiments with a range of reef-flat submergences. leading to a variable grid points per wave length N¼50–100. Spectral waves were generated from the widely-used JONSWAP The numerical model was run for 200 waves using a Courant spectrum with a peak enhancement factor g¼3.3. number of Cy¼0.50 for monochromatic waves and for 1000 waves with a Courant number of Cy¼0.35 for spectral waves, respec- 3.1.2. Testing cases tively, which yields a time step ranging from 0.004 s to 0.007 s. Four representative cases are simulated in this study and they Also, to ensure the numerical stability, all simulations described are summarized in Table 1. Case 1 is for monochromatic waves in this section used a 4-point filter to reduce the curvature of the over an idealized plane reef flat and used for validating the model rapidly varying bathymetry. To ensure that the transient effects parameters. Case 2 is identical to Case 1 except that a rectangular are insignificant in computing the wave height and the mean ridge is present to investigate the model’s capability of dealing water level were, the initial 75 wave cycles for monochromatic with a complex reef crest. In the numerical simulations, however, waves and the initial 100 significant wave cycles for spectral both seaside and leeside vertical faces of the ridge were modified waves were not used in the data analysis. For empirical para- to V:H¼1:1 slopes to ensure numerical stabilities. The application meters in the surfzone model, we used the suggested values in of the model to an initially dry reef flat, which corresponds to the Section 2.4, except d¼10 for the reef with ridge (Case 2). The conditions where a reef flat is exposed to air at low tides, is tested value d¼10 was obtained by calibrating the model against the by Case 3. For Case 4, the idealized reef profile subjected to experimental results; an explanation for the increased value of d spectral waves is studied. when a ridge exists at the reef crest is given in the next section. Since the wave flume and the reef model were made of glass and PVC, respectively, a constant Manning coefficient n¼0.01, as 3.1.3. Numerical settings suggested in the literature, was used to estimate the friction Referring to Fig. 1, the computational domain, in terms of the coefficient in Eq. (10). A summary of the model parameters for the dimensions of the flume and the location and shape of the simulated cases are summarized in Table 3. idealized reef model, were designed to reproduce the main aspects of the laboratory settings. The 1DH numerical domain is also 36 m long. Dissipative sponge layers, typically of a width of 3.2. Effects of ridge on wave breaking 1.25 times the incident wave length, (i.e., W¼1.25L, where L is the wave length), were placed at the two ends of the computational Videos of breaking waves were taken after the wave fields had domain to ensure that the outgoing waves can be absorbed reached steady states. Plunging breakers were observed in Cases satisfactorily. The incident wave field was generated by using 1–3; spilling breakers were predominant in Case 4. Fig. 3 shows the aforementioned internal source method. The internal source the snapshots when the lips of the breakers hit the water surface was placed close to the seaward sponge layer, i.e., L1¼0. The reef Table 3 Table 1 A summary of model parameters for all the simulations. A summary of the four simulated laboratory experiments. a Case no. Dx Cr W (m) d a1 a2 a3 a a Case T (s) H0 (m) hr (m) kho khr ka Wave type Reef no. crest 1 0.03 0.5 2.65 2 0.65 0.08 8 2 0.03 0.5 2.65 2 0.65 0.08 8 1 1.25 0.095 0.1 1.33 0.53 0.14 Monochromatic Plane 3 0.03 0.5 1.78 10 0.65 0.08 8 2 1.25 0.095 0.1 1.33 0.53 0.14 Monochromatic Ridge 4 0.03 0.35 3.73 2 0.65 0.08 8 b 3 1.0 0.101 0b 1.54 0 0.22 Monochromatic Plane S&A 0.04 0.5 4.10 10 0.65 0.08 8 c 4 1.67 0.087 0.05 0.84 0.27 0.09 Spectral Plane N&D 0.05 0.35 3.60 2 0.65 0.08 8

a a ¼ For spectral wave height, H0 and T refers to the deep water signiﬁcant wave W 1.25L, where L is the incident wave length. b height and peak wave period, respectively. S&A stands for Skotner and Apelt (1999). b c hr¼0 corresponds to an initial dry reef ﬂat. N&D stands for Nwogu and Demirbilek (2010).

Table 2 The distances of the wave gages (G1–G12) from the toe of reef face for the four simulated experiments (Unit: m).

Case no. G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12

1 4.35 4.1 0.95 2 2.35 2.65 2.95 3.25 3.65 5.25 6.95 8.75 2 4.35 4.1 0.95 2 2.35 2.65 2.95 3.25 3.65 5.25 6.95 8.75 3 4.35 4.1 0.65 1.25 2 2.35 2.75 3.15 3.65 5.25 6.95 8.75 4 4.35 4.25 4 0.95 1.45 2 2.45 2.95 3.65 5.25 6.95 8.75 Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 35

Fig. 3. Snapshots of the breaking waves over the reef crest: (a) Case 1 without ridge; (b) Case 2 with ridge.

0.15 for Cases 1 and 2. In the absence of ridge, the localized breakers 0.12 plunged onto the horizontal reef flat at certain location down- 0.09 stream of the reef edge. In the presence of a ridge, the breaking 0.06 H(m) point shifted seaward: the breaking waves first stroke the front 0.03 side of the ridge and then plunged onto the ridge top, accom- 0 panying a stronger wave reflection. -8 -6 -4 -2 0 2 4 6 8 10 The surfzone locations and widths were estimated from the 0.04 video recordings of the experiments. The observed surfzones 0.02 extended approximately from 0.1 m shoreward from the reef edge to 1.4 m shoreward from the reef edge for Case 1, and from 0

0.05 m seaward from the reef edge to 0.9 m shoreward from the MWL(m) -0.02 reef edge for Case 2, giving surfzone widths of 1.3 m and 0.95 m, respectively. The seaward shift of the breaking point for Case 2 is -8 -6 -4 -2 0 2 4 6 8 10 due to the alteration of the wave field by the ridge. The calculated 0.5 outer surfzone ranges approximately from 0.1 m seaward from 0.25 the reef edge to 2.0 m shoreward from the reef edge for Case 1, and from 0.3 m seaward from the reef edge to 0.4 m shoreward 0 from the reef edge for Case 2. Therefore, the surfzone width is Seabed Elev.(m) -8 -6 -4 -2 0 2 4 6 8 10 greatly reduced by the reef ridge. The predicted reduction of Distance from the toe of reef face (m) surfzone width by the reef ridge can be explained by the breaking model used in the simulations. With a ridge, the average surfzone Fig. 4. Variations of the wave height and mean water level (MWL) over the reef profile for Case 1. Solid lines: predictions by present model; open circles: water depth is 0.05 m, which is just 50% of that without ridge laboratory measurements. according to Eq. (8). Therefore, the breaking duration Tn which is proportional to the square-root of surfzone water depth, is 0.15 reduced by approximately 30%, indicating that the breaking process would complete within a shorter distance. 0.12 Based on our numerical simulations, both the breaking wave 0.09 0.06 heights and reformed wave heights are almost the same for these H(m) two cases: about 0.12 m for breaking wave height and 0.04 m for 0.03 the reformed wave height. Consequently, about the same amount 0 -8 -6 -4 -2 0 2 4 6 8 10 of wave energy in the incident waves needs to be dissipated 0.04 within a relatively narrower surfzone when a ridge exists. There- fore, when a ridge is present at the edge of a reef crest, the 0.02 correction factor d needs to be increased to adjust the incorrect 0 mixing length and friction velocity scales in the eddy viscosity MWL(m) model. Of course, other factors such as flow separation and vortex -0.02 shedding at both the sharp edges of the ridge may also contribute -8 -6 -4 -2 0 2 4 6 8 10 to a larger d, but we believe these contributions are not substantial. 0.5

0.25 3.3. Wave height, mean water level and wave reﬂection 0

Our main concern in this study is the mean quantities such as Seabed Elev.(m) -8 -6 -4 -2 0 2 4 6 8 10 wave heights and mean water levels. The cross-reef variations of Distance from the toe of reef face (m) wave height and mean water level, together with the bottom Fig. 5. Variations of the wave height and mean water level (MWL) over the reef proﬁles for the four experiments listed in Table 1, are displayed in proﬁle for Case 2. Solid lines: predictions by present model; open circles: Figs. 4–7 for the convenience of the following discussion. For each laboratory measurements. 36 Y. Yao et al. / Ocean Engineering 47 (2012) 30–42

0.15 values as well as the measured and predicted maximum setups on the reef ﬂat for all four cases; all the R-square values are larger 0.12 than 0.87, suggesting that the global agreements between numer- 0.09 ical results and the measurements are satisfactory for all cases. The maximum setdown locations (breaking points) were not

H(m) 0.06 captured by the limited measurement locations during the experiments, thus are not compared in the table with numerical 0.03 results. 0 -8 -6 -4 -2 0 2 4 6 8 10 0.04 3.3.1. Wave height The simulated wave heights agree reasonably well with the 0.02 measurements for all the cases. It can be observed that the wave 0 height decreases rapidly in the surfzone due to wave breaking.

MWL(m) -0.02 The good agreement between the simulations and the measurements indicates that the empirical breaking model can simulate -8 -6 -4 -2 0 2 4 6 8 10 the energy dissipation well. The modulation of the simulated 0.5 wave height seaward is due mainly to the partial wave reflection 0.25 from the reef model, which may produce a partial standing wave 0 pattern in front of the reef model. This phenomenon is enhanced -8 -6 -4 -2 0 2 4 6 8 10 by the presence of the ridge in Case 2, and was nearly invisible in Seabed Elev.(m) Distance from toe of forereef slope (m) the Case 4 for spectral waves due mainly to the fact that there is no organized partial standing wave pattern that can be formed in Fig. 6. Variations of the wave height and mean water level (MWL) over the reef front of the reef model. profile for Case 3. Solid lines: predictions by present model; open circles: Some minor discrepancies in the reformed wave heights over laboratory measurements. the reef flat can be observed for both Case 1 and Case 2, though 0.15 the scatter in the measurements can also be noticed. These 0.12 discrepancies may be attributed partly to the empirical value 0.09 used to define the saturated breaking cessation in Eq. (8). Since (m) s 0.06 the wave energy density is proportional to the wave height H 0.03 squared, the minor differences found in the reformed wave 0 heights do not have significant effect on the accuracy of the -8 -6 -4 -2 0 2 4 6 8 10 0.04 predicted energy dissipation. The reef flat with zero submergence in Case 3 was rarely 0.02 studied in the existing literature. For this case, wave breaking first 0 occurred on the reef face and then pumped water onto the reef MWL(m) -0.02 flat through overtopping. The very good agreement between the simulated and measured wave heights for this case is very -8 -6 -4 -2 0 2 4 6 8 10 impressive (shown in Fig. 6). As the flow on the reef flat might 0.5 be supercritical (Gourlay and Colleter, 2005), the stability and the 0.25 accuracy of the numerical simulation were achieved with the help of shock-capturing FV-based solver and the moving boundary 0 -8 -6 -4 -2 0 2 4 6 8 10 Seabed Elev.(m) algorithm. Distance from the toe of reef face (m) As shown in Fig. 7, the agreement between the numerical and experimental results is remarkable for Case 4 (spectral waves), Fig. 7. Variations of the significant wave height and mean water level (MWL) over suggesting that the proposed surfzone model also work favorably reef profile for Case 4. Solid lines: predictions by present model; open circles: laboratory measurements. for spectral waves over fringing reefs. For this case, the surfzone is wider and spilling breakers are predominant.

Table 4 2 o The R-square (R ) of mean water level, the measured maximum setup (Zr ) and the 3.3.2. Mean water level p predicted maximum setup (Zr ) on the reef flat. Overall, the model can give good predictions of the variation of mean water level for all cases, with all R-square values being Case no. R2 o (mm) p (mm) p= o Zr Zr Zr Zr larger than 0.85 as shown in Table 4. The setdowns over the reef face are due mainly to wave shoaling before breaking. The setups 1 0.87 7.1 5.9 0.83 2 0.93 18.0 13.3 0.74 over reef flat are all reasonably predicted by the simulations (see 3 0.99 25.6 24.3 0.95 Table 4). The setdowns in the water in front of the reef face are 4 0.96 8.3 7.7 0.93 related to the requirement on the conservation of mass for a closed flume: the wave-induced setup over the reef flat must be balanced by the setdown in the other part of the flume. The slight case, the model parameters were calibrated by using the R-square underestimation of the wave-induced setups over reef flat for value to minimize (through a trial-and-error procedure) the Case 1 and Case 2 was directly related to the under-prediction of p global error between the measured mean water level Zi and the local wave heights for these two cases. A comparison of the o the predicted mean water level Zi . The R-squareP is defined in predicted wave-induced setups over reef flat for these two cases 2¼ ¼ M ð p oÞ2 this studyP by R 1 Serr/Stot with Serr i ¼ 1 Zi Zi and reveals that a ridge-like reef crest can increase the wave-induced M o o 2 o Stot ¼ i ¼ 1 ðZi ZaÞ and Za being the mean of all measured setup by narrowing the surfzone as discussed in the previous mean water levels at M locations. Table 4 lists the R-square section. Since the cross-shore wave height evolutions for both Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 37 cases are well reproduced by the present model, the excellent 50 G2 0 agreements between the numerical and experimental mean water (mm) -50 η levels for Case 3 and Case 4 are not surprising according to the 0 1 2 3 4 5 6 7 8 t (s) radiation stress concept. 50 0 G3

(mm) -50 3.3.3. Reflection coefficient η 0 1 2 3 4 5 6 7 8 The calculated and measured reflection coefficients for the first t (s) harmonic waves are compared in Table 5. Satisfactory agreements 50 0 are obtained between the numerical and laboratory results, the G5 (mm) -50 minor differences are due to: (1) the small differences between η 0 1 2 3 4 5 6 7 8 laboratory and numerical boundary conditions (e.g., the reflection t (s) from paddle of wavemaker was not modeled in all numerical 50 0 simulations), and (2) the minor difference in the shapes of the G7 (mm) -50 ridges used in the experiments and the simulations (the sharp η 0 1 2 3 4 5 6 7 8 edges of the ridge have been smoothed in the numerical simula- t (s) tions for numerical stability considerations). The influence of the 50 0 ridge on the wave reflection is evident for Case 2; the enhanced G9

(mm) -50 wave reflection is expected since the ridge structure functions like η 0 1 2 3 4 5 6 7 8 a submerged breakwater, which has been widely used to reflect t (s) the wave energy for shore protections. For spectral waves, 50 frequency averaged reflection coefficients as defined by Goda 0

(mm) -50 G11 (2000) are used: η 0 1 2 3 4 5 6 7 8 pffiffiffiffiffiffiffiffiffiffiffiffi t (s) KR ¼ ER=EI ð11Þ Fig. 8. Time-series of surface elevations at six locations (G2, G3, G5, G7, G9, and G11) where EI and ER are the total energy contained in the incident and for Case 1. Dashed lines: laboratory measurements; solid lines: predictions by reflective spectra, respectively. The results for Case 4 show that present model. spectral waves generally have larger values of reflection coefficient than the equivalent monochromatic waves do; this is because low frequency wave components turn to have larger reflection 50 G2 0 coefficients than higher frequency components (Seelig, 1983). (mm) -50 η 0 1 2 3 4 5 6 7 8 t (s)

3.4. Wave transformation 50 G3 0

(mm) -50 3.4.1. Monochromatic waves η 0 1 2 3 4 5 6 7 8 The computed surface elevations at six locations for Case 1 and t (s) Case 2 are compared with the experimental data in Figs. 8 and 9, 50 0 respectively. It can be observed that very good agreements are G5 (mm) -50 obtained at most locations for these two cases. Wave shoaling η 0 1 2 3 4 5 6 7 8 makes the wave form asymmetric and skewed at G3 on the reef t (s) face; at this location, the simulation for Case 1 agrees better with 50 0 experiments than for Case 2 because the smoothed ridge profile G7 (mm) -50 near the reef edge may have slightly affected the reflected waves η 0 1 2 3 4 5 6 7 8 and the wave dispersion in the water above the ridge. G5 and G7 t (s) are located in the surfzone, thus breaking waves with saw-tooth 50 0 shape are observed there; a relatively larger error in the measured G9 (mm) -50 waves in the surfzone is expected because of the entrained η 0 1 2 3 4 5 6 7 8 bubbles in the surfzone water. On the reef flat (G9), both the t (s) numerical and laboratory results show that the reformed waves 50 0 are very small and they look like cnoidal waves. G11 (mm) -50 Fig. 10 is a comparison of the measured and simulated surface η 0 1 2 3 4 5 6 7 8 elevations for Case 3 (dry reef flat). The wave form is satisfactorily t (s) predicted seaward of the reef face (G2) and in the shoaling zone (G3); the reformed waves are quite small (G9 and G11), therefore, Fig. 9. Time-series of the surface elevations at six locations (G2, G3, G5, G7, G9, most of the wave energy has been dissipated by wave breaking and G11) for Case 2. Dashed lines: laboratory measurements; solid lines: predictions by present model. and bottom friction. 3.4.2. Spectral waves Table 5 A comparison of the measured and simulated wave spectra is The measured (Cm) and predicted (Cp ) reflection coefficients. R R presented in Fig. 11. A good agreement can be observed at G2. Case no. m p For spectral waves, individual wave breaking occurs either on the CR CR reef face or on the reef flat, thus at G5 (located at the seaward side of 1 0.07 0.03 thereefedge),thereisstillaconsiderableamountofwaveenergy 2 0.22 0.11 around the peak frequency. The numerical model slightly over- 3 0.03 0.08 predicts the wave energy at the peak frequency near the reef edge 4 0.12 0.10 (G5) as well as during the shoaling process (G4). Waves first shoal 38 Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 on the reef face and break in the shallow water above the reef flat, to both higher and lower frequency waves, and our predicted wave transferring part of the energy from the fundamental waves to spectra at G6–G9 and G11 are satisfactory. Shoreward of the higher and lower harmonic waves. The numerical model also surfzone, there is no notable difference between the computed satisfactorily captures the energy transfer from the peak frequency and measured wave spectra since the local water depth on the reef flat is sufficiently shallow to filter out most of short waves through bottom friction and the low-frequency motions on reef flat is 50 0 G2 probably due to the infragravity waves (Nwogu and Demirbilek,

(mm) -50 η 0 1 2 3 4 5 6 7 8 2010). t (s) 50 0 G3 4. Revisits of other numerical studies

(mm) -50 η 0 1 2 3 4 5 6 7 8 t (s) In this section, we revisit two other published laboratory and 50 numerical studies on wave transformation over fringing reefs. The 0 G5 purpose of these two revisits is to see if highly nonlinear (mm) -50 η 0 1 2 3 4 5 6 7 8 Boussinesq equations can improve the predictions by weakly t (s) nonlinear Boussinesq equations and to demonstrate the effects 50 of numerical boundary conditions on the mean water levels. 0 G7 (mm) -50 η 0 1 2 3 4 5 6 7 8 4.1. Revisit of Skotner and Apelt (1999) t (s)

50 Skotner and Apelt (1999) reported both experimental data and 0 G9 numerical simulations for setdowns and setups induced by regular (mm) -50 η 0 1 2 3 4 5 6 7 8 waves propagating onto a submerged fringing coral reef as described t (s) by Seelig (1983). The reef proﬁle consisted of a steep composite reef 50 face with an average slope of V:H¼1:12, which was followed by a 0 G11 small sharp ridge-like reef crest, and a 7 m-long horizontal reef ﬂat. (mm) -50 η 0 1 2 3 4 5 6 7 8 For their numerical simulations, they used the weekly nonlinear t (s) Boussinesq model originated by Nwogu (1993) with a roller-based surfzone model proposed by Schaffer¨ et al. (1993). Monochromatic Fig. 10. Time-series of the surface elevations at six locations (G2, G3, G5, G7, G9, and G11) for Case 3. Dashed lines: laboratory measurements; solid lines: waves were generated internally with the sponge layers being used predictions by present model. only in the momentum equation on both sides of the computational

x 10-3 x 10-3

2 2 G2 G4 .s) .s)

2 1 2 1

S(m 0 S(m 0 0 1 2 3 0 1 2 3 f(Hz) f(Hz) x 10-3 x 10-3 2 2

.s) G5 .s) G6 2 2 1 1 S(m S(m 0 0 0 1 2 3 0 1 2 3 f(Hz) f(Hz) x 10-4 x 10-4 5 5

.s) G7 .s) G8 2 2.5 2 2.5 S(m S(m 0 0 0 1 2 3 0 1 2 3 f(Hz) f(Hz) x 10-4 x 10-4 5 5

.s) G9 .s) G11 2 2.5 2 2.5 S(m S(m 0 0 0 1 2 3 0 1 2 3 f(Hz) f(Hz)

Fig. 11. Wave spectra at eight locations (G2, G4–G9, and G11) for Case 4. Dashed lines: laboratory measurements; solid lines: predictions by present model. Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 39

0.02 0.12 0.09 0.01 0.06 H(m) 0.03 0 0

MWL(m) -4 -2 0 2 4 6 8 10 0.02 -0.01 0.01 0 2 4 6 8 0

0.5 MWL(m)

-0.01 -4 -2 0 2 4 6 8 10 0.25 0.6

Seabed Elev.(m) 0 0.3 0 2 4 6 8 Distance from the toe of reef face (m) 0 -4 -2 0 2 4 6 8 10 Seabed Elev.(m) Distance from the toe of reef face (m) Fig. 12. Variation of the mean water level (MWL) over the reef proﬁle for the Test 5 and Test 6 of Skotner and Apelt (1999). Dashed line: prediction by Skotner and Apelt (1999); solid line: prediction by present model; open circles: laboratory Fig. 13. Variations of the wave height and mean water level (MWL) over the reef measurements by Skotner and Apelt (1999). Dark black circles and lines are for proﬁle for the test in Nwogu and Demirbilek (2010). Dashed lines: predictions by Test 5; Light black circles and lines are for Test 6. Nwogu and Demirbilek (2010); solid lines: predictions by present model; open circles: laboratory measurements by Nwogu and Demirbilek (2010).

Tp ¼ 1:5 s and hr¼0.031 m. Following their paper, our simulations domain to reduce the wave reflection. Six tests were simulated, were carried out using a grid spacing Dx¼0.05m,atimestep covering a range of breaker types. They found that their numerical Dt ¼ 0:01 s and an equivalent friction coefficient f¼0.011 which model could predict the trend of the mean water level reasonably was converted from their Chezy coefficient Cf¼30. Our physical well, but there was an increasing discrepancy in the magnitude of simulation time was 900 s, and the initial 100 s surface elevations setup or setdown when the height of incident waves was increased. werenotusedincomputingthemeanwaterlevels.Themajor The boundary conditions and numerical inputs same as theirs were difference in the numerical settings is: they used sponge layers in used in simulation with a grid size Dx¼0.04 m, Courant number both the continuity and momentum equations at the seaward

Cr¼0.5 and the friction coefficient as given in Table 2 of Skotner and boundary and a plane beach at the shoreward boundary while we Apelt (1999). For the correction coefficient in the breaking model, use the sponge layers only in momentum equation at both bound- we found the value d¼10 could give the best fit to the laboratory aries. Parameter d in the present breaking model needs to be data. Details of all the simulation parameters are given in our calibrate with experiments; we used the value suggested in Table 3. Section 2.4, i.e., d¼2 for this case. A summary of all the simulation All cases in their study have been simulated to see if our model parameters is given in Table 3. can provide an improved prediction. A sample comparison among The measured and computed significant wave heights (Hs)and the results of laboratory and numerical simulations of Skotner the mean water level variations are compared in Fig. 13.Both and Apelt (1999) and the present model is given in Fig. 12 for simulations predicted similar wave heights over the reef. However, Test 5 and Test 6, where the seafloor profile is also shown. the simulation of Nwogu and Demirbilek (2010) failed to capture Test 5 and Test 6 have the same dimensionless depth kh but the wave-induced setdown seaward of the reef face, which was well different wave slope ka. Our results show that the wave-induced predicted by the present simulation. Nwogu and Demirbilek (2010) setdowns predicted by the fully nonlinear Boussinesq model attributed the difference between the measurements and their agree slightly better with the measurements than does the simulations to the sponge layer damping term used in their weakly nonlinear Boussinesq model, but both models under- continuity equation. From the comparisons in Fig. 13, we can estimate the setup over the reef flat and give comparable results conclude that the mean water level over the reef is sensitive to in the surfzone. Skotner and Apelt (1999) speculated that the the types of the boundary conditions used and that the damping under-productions might result from the ignorance of high non- terms should not be included in the continuity equation when linear terms in their governing equations, but our results show simulating the mean water levels in a closed flume. that the inclusion of the higher order nonlinear terms in Boussi- nesq models does not necessarily improve the prediction of the mean water level over the reef flat. 5. Model application to different reef-face slopes and profiles

4.2. Revisit of Nwogu and Demirbilek (2010) The influences of offshore wave conditions and reef-flat submergences on the mean qualities such as wave height and the Nwogu and Demirbilek (2010) presented a combined laboratory wave-induced setups over reef flat have been investigated experi- and numerical study for a typical fringing reef without any ridge mentally by, e.g., Gourlay (1994, 1996a) and numerically by, e.g., structure. Their tests were run for a wide range of irregular sea Skotner and Apelt (1999) and Nwogu and Demirbilek (2010). states generated from JONSWAP spectra. Their numerical model was However, to the best of our knowledge, neither existing labora- based on the weekly nonlinear Boussinesq equations derived by tory nor numerical work has studied the influence of the inclina- Nwogu (1993). The effect of wave breaking was also parameterized tion angle of a plane reef-face or the shape of reef face. In this using an eddy-viscosity concept based on a one equation turbulence section, we apply the numerical model to examine a series of reef closure model. We revisit here a representative case with Hs¼7.5 m, faces numerically. We take Case 1 (a plane reef face with a slope 40 Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 of V:H¼1:6) in Section 3 as our benchmark against which the present of the ridge), similar observations (not reported here) comparisons and discussion are made. Two types of numerical were obtained. It seems that the slope of a plane reef face can tests were conducted: (1) plane reef faces with slopes varying have influence on the wave-induced setdown and setup only in from V:H¼1:1 to 1:20, and (2) combinations of two types of the the shoaling zone and surfzone. circular arc profile (convex and concave) and two values of arc curvature (0.075, 0.15). For the circular arc profiles, the chords are 5.2. Effect of the shape of reef-face profile the same as the reef-face length in Case 1. For both types of numerical tests, the numerical settings were the same as those The influences of the shape of reef-face profile are summarized used for Case 1. in Fig. 15, which reveals that seaward of the reef face, there is no notable difference among different simulations in both the wave height and mean water level. However, the location of the 5.1. Effect of the inclination angle of plane reef face breaking point is affected by the slope profiles: the concave profiles (in analogy to mild slopes) move the breaking point The effects of reef-face slope on wave height and mean water seaward, while the convex profiles (in analogy to steep slopes) level are summarized in Fig. 14. It shows that seaward of the reef move it shoreward. model, the fluctuation of wave height due to the partial standing The wave-induced setdown is larger for the convex profiles waves is amplified as the reef face becomes steeper, indicating an than for the concave profiles or plane profile. However, the wave enhanced wave reflection. The fluctuation of the seaward mean setup on the reef flat remains nearly unchanged as the reef-face water level for steeper slopes can also be observed. Similar profile varies from the concave configurations to the convex phenomenon can also been observed in the simulations of configurations: this may be because the wave breaking dissipa- Skotner and Apelt (1999) and Ranasinghe et al. (2009). Currently tion is almost the same for all cases. The near-constant reef-flat we cannot provide a convincing explanation for it. setup indicates that the shape of the reef face profile is not a key The predicted wave heights on reef flat are almost the same, parameter contributing to the wave-induced setup over the which is controlled mainly by the breaking cessation criteria in fringing reef and the results obtained for the plane reef face are the breaker model. Wave breaking occurs on the seaside of the representative of other shapes. reef edge and breaking wave heights are almost the same for all simulations. However, the breaking points are shifted seaward slightly as the slope varies from V/H¼1:1 to 1:20. 6. Conclusions On the reef flat, the wave-induced setups are about the same for all simulations. This is due to the fact that the total amount of Numerical experiments based on weakly dispersive, fully energy dissipated in the surfzone is about the same for different nonlinear Boussinesq-type equations with a FV-based solver have slopes. However, the spatial variation of the setup in surfzone is been performed to study the wave transformation over various dependent on the surfzone width, which is related to the slopes of fringing reef profiles. Comparisons with published laboratory reef face. The calculated wave setup increases more rapidly for measurements and other numerical studies lead to the following steeper slopes due to the reduction in the overall surfzone water conclusions: depth and the surfzone width. When waves shoal on a milder slope over a lager distance, the bottom friction may also affect the (1) In order to conserve the mass in a closed wave flume, the wave-induced setup, however, our additional tests by varying the numerical damping layer can only be used in the momentum Manning coefficient found that the friction dispassion is negligi- equation. ble. We have also adjusted the reef-face slope for Case 2 (in the

0.15 0.15 0.12 0.12 0.09 0.09 0.06 H(m) 0.06 H(m) 0.03 0.03 0 0 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 2 3 4 5 6 0.01 0.01 0.005 0.005 0 0 -0.005

-0.005 MWL(m) -0.01 MWL(m) -0.01 -0.015 -0.015 -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 2 3 4 5 6 0.5 0.5 0.25 0.25 0 0

Seabed Elev.(m) -4 -3 -2 -1 0 1 2 3 4 5 6 -4 -3 -2 -1 0 1 2 3 4 5 6 Distance from the toe of reef face (m) Seabed Elev.(m) Distance from the toe of reef face (m) Fig. 15. Variations of the wave height and mean water level (MWL) over reef Fig. 14. Variations of the wave height and mean water level (MWL) over reef profile with different reef-face profiles. Light black solid line: concave arc with the profile with different reef-face slopes. Light black solid line: V:H¼1:1; Dash-dot curvature¼0.15; Dash-dot line: concave arc with the curvature¼0.075; Dark line: V:H¼1:3; Dark black solid line: V:H¼1:6; Dotted line: V:H¼1:10; Dashed black solid line: plane slope; Dotted line: convex arc with thecurvature¼0.075; line: V:H¼1:20. Dashed line: convex arc with the curvature¼0.15. Y. Yao et al. / Ocean Engineering 47 (2012) 30–42 41

(2) The zero-equation eddy viscosity model, with its model where W¼xe xs is the sponge layer width. In this study, we parameters being calibrated using measurements, can reason- adopted c1¼10, c2¼0.01 and W¼1.25L as suggested by Lynett ably simulate the key characteristics (wave reflection, wave and Liu (2008). height variation, wave-induced setup/setdown) of both monochromatic and spectral waves over various fringing reef profiles and conditions, including the dry reef fat condition References and a ridge at the reef edge. (3) The reef crest profile, especially the existence of a ridge at the Booij, N., 1983. A note on the accuracy of the mild-slope equation. Coastal Eng. 7, reef edge, may significantly affect the wave-induced setup/ 191–203. setdown over the fringing reef. Douillet, P., Ouillon, S., Cordier, E., 2001. A numerical model for fine suspended sediment transport in the southwest lagoon of New Caledonia. Coral Reefs 20, (4) The increase of the wave-induced setup over the fringing reef 361–372. with a ridge located at the reef edge is due mainly to the Goda, Y., 2000. Resolution of incident and reflected waves of irregular profiles. in: reduction of the surfzone water depth and the surfzone width. Random Seas and Design of Maritime Structures. World Scientific Press, Singapore, pp. 356–361. (5) Both the inclination angle of plane reef face and the shape of Gourlay, M.R., 1994. Wave transformation on a coral reef. Coastal Eng. 23, 17–42. reef-face profile have negligible effects on the wave-induced Gourlay, M.R., 1996a. Wave set-up on coral reefs. 1. Set-up and wave-generated setup/setdown outside the shoaling zone and surfzone. The flow on an idealised two-dimensional reef. Coastal Eng. 27, 161–193. Gourlay, M.R., 1996b. Wave set-up on coral reefs. 2. Wave set-up on reefs with results for plane reef face are representative of other profiles. various profiles. Coastal Eng. 28, 17–55. 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