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1DH Boussinesq Modeling of Wave Transformation Over Fringing Reefs Ocean Engineering 47 (2012) 30–42 Contents lists available at SciVerse ScienceDirect Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng 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.
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