Revisiting study on Boussinesq modeling of wave transformation over various reef profiles

Ke-zhao FANG*, Ji-wei YIN, Zhong-bo LIU Jia-wen SUN, Zhi-li ZOU

1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, P. R. China

2. National Marine Environment Monitoring Center, State Oceanic Administration, Dalian 116023, P. R. China

3. Heilongjiang Province Navigation Investigation and Design Institute, Harbin 150001, P. R. China

Revisiting study on Boussinesq modeling of wave transformation over various reef profiles

Ke-zhao FANG*1,2, Ji-wei YIN3, Zhong-bo LIU1, Jia-wen SUN2, Zhi-li ZOU1

1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, P. R. China

2. National Marine Environment Monitoring Center, State Oceanic Administration, Dalian 116023, P. R. China

3. Heilongjiang Province Navigation Investigation and Design Institute, Harbin 150001, P. R. China

To better understand the complex process of wave transformation and associated hydrodynamics over various fringing reef profiles, numerical experiments were conducted with a one-dimensional (1D) Boussinesq wave model. The model is based on higher-order Boussinesq equations and a higher-accuracy finite difference method. The dominant energy dissipation in the surf zone, wave breaking, and bottom friction were considered by use of the eddy viscosity concept and quadratic bottom friction law, respectively. Numerical simulation was conducted for a wide range of wave conditions and reef profiles. Good overall agreement between the computed results and the measurements shows that this model is capable of describing wave processes in the fringing reef environment. Numerical experiments were also conducted to track the source of underestimation of setup for highly nonlinear waves. Linear properties (including dispersion and shoaling) are found to contribute little to the underestimation; the low accuracy in nonlinearity and the ad hoc method for treating wave breaking may be the reason for the problem.

wave-induced setup; wave-induced setdown; Boussinesq model; wave breaking; reef

1 Introduction

Fringing reefs are commonly found in the tropics and subtropics. A typical fringing reef is characterized by a composite seaward sloping reef face with an abrupt transition to an inshore shallow reef platform extending towards the shoreline. This specific bathymetry significantly modifies the wave transformation process and makes the associated hydrodynamics far more complex than on normal coastal beaches in many respects (Demirbilek and Nwogu 2007; Nwogu and Demirbilek 2010; Monismith et al. 2010; Yao et al. 2012). Intense wave breaking typically occurs on the reef face, enhancing the oxygen content and circulation that support the coral ecosystem (Achituv and Dubinsky 1990). Also, by dissipating wave energy, fringing reefs provide protection for the shore and tropical shelterislands from the flood hazards induced by tsunamis, hurricanes, and high surf events (Roeber et al. 2000). However, wave-induced setup and low frequency waves emerging from the surf zone can induce extensive flooding with large and variable wave overwash, especially under high-energy wave conditions (Demirbilek and Nwogu 2007). Due to its profound geological, ecological, and environmental significance, there is an increasing amount of interest in investigating the hydrodynamic processes associated with waves occurring on fringing coral reefs (Demirbilek and Nwogu 2007; Massel and Gourlay 2000; Nwogu and Demirbilek 2010; Monismith et al. 2010; Yao et al. 2012).

Numerical simulations are widely used to predict the significant wave transformations over such bathymetry. However, numerical modeling of nearshore reef hydrodynamics is a challenging task owing to the steep reef face slopes, large and spatially-varied roughness of the reef bottom, and complicated reef profile configurations (Massel and Gourlay 2000; Yao et al. 2012). The most advanced Navier-Stokes models (Lara et al. 2008; Huang and Lin 2012; Hu et al. 2012) are well suited for the purpose because they have shown satisfactory accuracy in describing wave transformation before, during, and after wave breaking, even in a complicated nearshore environment. However, for the present, they remain an expensively computational approach, especially when the fine grid and long-term simulation are needed in a reef environment. Alternatively, numerical models built upon Boussinesq equations have the potential to handle these nearshore processes with the characteristics of nonlinearity and frequency dispersion, which play a key role in accurately describing wave motions in a reef environment (Yao et al. 2012; Shermert et al. 2011). The prevailing Boussinesq model is more computationally efficient for large spatial and temporal scales, in contrast to the expensive Navier-Stokes approaches. Detailed reviews of Boussinesq equations have been provided by Kirby (2002) and Madsen and Fuhrman (2010).

To the best of our knowledge, the published numerical results for waves and associated hydrodynamics from Boussinesq models are still limited. Skotner and Apelt (1999) developed a Boussinesq model based on the weakly nonlinear equations derived by Nwogu (1993) to compute the mean water level (MWL) of regular waves propagating onto a submerged coral reef, and the numerical results were compared against their measurements. With the model, they accurately computed the setdown and setup of regular waves of small incident wave heights, but there was a tendency to underestimate the wave setup as the incident wave height increased. Demirbilek and Nwogu (2007) and Nwogu and Demirbilek (2010) used variant forms of Nwogu’s (1993) Boussinesq equations to numerically investigate the infragravity motions in the wave runup process over fringing coral reefs. The computed wave height, MWL, time series of surface elevation, runup, and energy spectrum were compared against the experimental data. It was found that the developed model was able to describe complex changes of the wave spectrum over the reef flat due to nonlinear wave-wave interactions and wave breaking as well as wave runup at the shoreline. Yao et al. (2012) used the Boussinesqmodel, based on the fully nonlinear version of Nwogu’s equations, to validate their previous and other published experiments. Their results show that the fully nonlinear Boussinesq model can give satisfactory predictions of the wave height as well as the MWL over various reef profiles with different reef-flat submergences and reef-crest configurations under both mono-chromatic and spectral waves. The primary one-dimensional (1D) wave transformation processes, including nonlinear shoaling, reflection, breaking, and generation of higher harmonics and infragravity waves, can also be reasonably captured. Roeber et al. (2010) recently solved Madsen and Sørensen’s (1992) Boussinesq equations using the finite volume method and further used the model to simulate solitary wave propagation over an initially dry reef flat.

All the above Boussinesq models acquired satisfactory results to some extent, but there still exist some problems that need further investigation. First, studies over recent decades have already shown the enrichments of higher-accuracy Boussinesq equations, but the performance of these newly developed or improved equations on modeling reef hydrodynamics needs further examination. Secondly, the setup over the reef flat is underestimated for all the published results regarding the Boussinesq modeling of highly nonlinear regular waves, and this discrepancy has not been explained thoroughly. Hence, the objective of this study was twofold: the first was the application and validation of a Boussinesq wave breaking model, with a relatively high accuracy in both linear and nonlinear properties, in modeling reef hydrodynamics. The second was to revisit the underestimation of the setup from the point of view of linear and nonlinear accuracy embodied in the Boussinesq equations.

2 Model descriptions

2.1 Governing Boussinesq equations

Zou and Fang (2008) presented alternative forms of higher-order Boussinesq equations with full nonlinearity accurate up to the second order. The resulting equations are enhanced to obtain better dispersion and shoaling properties using the method proposed by Madsen and Schäffer (1998). In the end, the 1D form of the mass conservation equations is written as

where η( x ,t ) is the water surface elevation; d(x ,t)is the local water depth, and d(x ,t) = h(x )+ η (x ,t), with h(x ) denoting the still water depth; anduis the depth-averaged velocity. The subscripts x and t denote the partial derivatives with respect to x space and time t, respectively. The momentum equation is

where g is the gravitational acceleration, and

In Eqs. (2) and (3), the following operators are used (Zou and Fang 2008):

where F is the function that needs to be operated. The coefficients α1,α2,β1, and β2are set to be 1/9, 1/945, 0.146, and 0.002 after the dispersion and shoaling properties of equations are optimized. This set of equations has a Pade [4,4] approximation of the exact dispersion, and is applicable even in a deep water limit (hL0= 1.0, whereL0is the typical wave length). For the convenience of the following discussion, linear properties (including dispersion and shoaling) and the nonlinear property (the second-order harmonic) are plotted in Fig. 1, and the corresponding results for Nwogu’s (1993) equations are also presented for comparison. All the Stokes-type reference solutions and the theoretical solution of shoaling can be found in Madsen and Schäffer (1998). Further details about the equations for the present model and Nwogu’s equations were described by Zou and Fang (2008) and Nwogu (1993), respectively.

Fig. 1 Comparison of phase celerity C, group velocityCg, shoaling coefficientαs, and second-order harmonica2for present model and Nwogu’s (1993) equations (the superscript * indicates the reference solution from the Stokes wave theory)

For 1D problems considered in the present study, wave energy dissipation was mainly caused by wave breaking and seabed bottom friction. These two mechanisms dominate the reef environment due to the intense wave breaking on the reef face and the fact that waves are prone to be affected by the bottom of a reef flat covered with largely distributed shallow water. All of them are accounted for by introducing ad hoc terms into Eq. (2) as follows:

where Rband Rfdenote the effects of wave breaking and bottom friction, respectively. The bottom shear stress is given by a quadratic term written in terms of the combined velocity due to waves and currents as follows:

where Cfisthe bottom friction coefficient, with the value of 0.005. The breaking term is based on the eddy viscosity concept (Kennedy et al. 2000) and is as follows:

where ν is the eddy viscosity, andCbris the breaking strength coefficient, with a value of 2.0. The parameter B controls the occurrence of wave dissipation, which is expressed as follows:

The wave breaking criteria change in a linear trend once breaking events occur:

where T*is the transition time, and T*= 5(h g )12; t0is the time when wave breaking starts; and t is the time during wave breaking.are the critical values for wave breaking initiation and ceasing. The recommended range of values is 0.35 to 0.65 for γ1and 0.05 to 0.15 for γ2, respectively. The lower limit of the coefficient γ1is found to be more suitable for bar/trough beaches with a relatively coarser grid resolution, whereas the upper limit shows an optimal agreement for waves breaking on monotonic sloping beaches (Kennedy et al. 2000; Kirby 2002). For the present model, γ1= 0.45 and γ2= 0.15 are found to give the optimum numerical results.

2.2 Numerical scheme and boundary conditions

The governing equations were discretized on a staggered gird system and numerically solved with the finite difference method. A six-order predictor-corrector Adams-Bashforth-Moulton integration scheme was adopted to perform time marching. The independent variable of the wave surface elevation η could be directly solved through the continuity equation, whereas the other independent variable u was obtained by solving a pentagonal linear system.

The entire computation was enclosed by impermeable walls, where the horizontal velocity was set to be zero. Sponger layers were placed in front of the solid walls to absorb wave energy. The internal wave maker was used for generating waves in the computation domain (Kennedy et al. 2000), and the source function had an identical form to that in Gobbi and Kirby (1999).

This numerical implementation mainly followed the FUNWAVE model (Kennedy et al. 2000). A more detailed description for the present model is also referred to in Fang et al. (2011).

3 Numerical results and discussion

The developed model was used to reproduce the available laboratory experiments for wave propagation over different reef profiles, including experiments conducted by Skotner and Apelt (1999), Demirbilek and Nwogu (2007), and Yao et al. (2012). The computed results of the present model were compared with measurements for model validation, and also compared against the numerical results from other Boussinesq models to show the effect of linear and nonlinear accuracy of Boussinesq equations on the numerical results.

3.1 Revisiting of Skotner and Apelt’s (1999) experiment

Skotner and Apelt (1999) presented the results of a combined laboratory and numerical investigation into the setdown and setup induced by regular waves propagating over a submerged coral reef. The reef profile consisted of a composite reef face with an average slope of 1:12, which was followed by a sharp ridge-like reef crest and a 7 m-wide reef flat, as shown in Fig. 2. The incident wave conditions for six tests in the experiments are listed in Table 1, where A0, H0, and T0are the wave amplitude, mean wave height, and wave period, respectively, whereas h0and hrdenote water depths in an offshore region and on the reef flat, respectively. For numerical simulations, they used a weakly nonlinear Boussinesq wave model derived by Nwogu (1993) and a surface roller model proposed by Schäffer et al. (1993). Yao et al. (2012) also simulated tests 5 and 6 using a Boussinesq wave model developed by Kim et al. (2009), which was based on the fully nonlinear version of Nwogu’s (1993) equations and the eddy viscosity concept (Kennedy et al. 2000). In this subsection, the numerical results from these two models are simply referred to as SA99 and YHML12, respectively.

Fig. 2 Reef profile in Skotner and Apelt’s (1999) experiment

Table 1 Test series of Skotner and Apelt’s (1999) experiments

All cases listed above were simulated using the grid size Δx= 0.04 m and time step Δt= 0.01 s. Following Skotner and Apelt (1999) and Yao et al. (2012), the model was run for 250 wave periods, and the MWL was extracted from the computations during 20 wave periodsimmediately prior to the program termination. The computed MWLs are compared with the experimental data, SA99, and YHML12, as shown in Fig. 3. For all cases, the model predicts the right variation trend of MWL, i.e., there exists a setdown before the breaking point and a setup after the breaking point. SA99 also captures a similar trend. However, it overestimates the setdown for all tests except tests 2 and 3, and underestimates the setup significantly for tests 2, 4, and 6. The present model gives overall better numerical results than those from a weakly nonlinear model (SA99). This is expected, as the present model has a higher accuracy of linear and nonlinear properties than the weakly nonlinear model used in Skotner and Apelt (1999) for all cases considered, as seen from Fig. 1. Compared with YHML12 for tests 5 and 6, the present model predicts identical results of setdown but presents a slightly better setup over the reef flat. The present simulation demonstrates overall better numerical results than weakly nonlinear and fully nonlinear versions of Nwogu’s (1993) equations.

Fig. 3 Comparison of computed MWL from Boussinesq wave models with experimental data for six tests in Skotner and Apelt’s (1999) experiment

It is worthwhile to note that the discrepancy between the numerical results from three models and measurements increases as the wave nonlinearity increases (see tests 2, 4, and 6). Skotner and Apelt (1999) speculated that the discrepancy might result from ignorance of higher-order nonlinear terms in the governing equations. However, YHML12 and SA99 both underestimate the setup over the reef flat and give comparable results in the surf zone (see tests 5and 6 in Fig. 3). Yao et al. (2012) argued that inclusion of the higher-order nonlinear terms in Boussinesq models does not necessarily improve the MWL over the reef flat. Our numerical simulations seem to also support this conclusion because the present model is based on full nonlinearity up to the second order but still underestimates the setup over the reef flat for highly nonlinear incident waves. It should be noted that h0L0varies from 0.146 to 0.260 forall cases considered. In this range, the linear properties (the phase celerity, group velocity, and shoaling coefficient) for the present model exactly follow the analytical solutions, as shown in Fig. 1. Hence, we conclude that the underestimation of the setup on the reef flat cannot be attributed to the low accuracy of linear properties. Fig. 1(d) shows that the second-order harmonic of the present model deviates from the theoretical solution, which may be the reason for the underestimation.

3.2 Revisiting of Yao et al.’s (2012) experiment

Yao et al. (2012) conducted experiments of waves propagating over fringing reefs with monochromatic and spectral waves. Two representative cases for regular waves were simulated in this study and are listed in Table 2. Test 1 was for monochromatic waves over an idealized plane reef flat with a steep reef face slope of 1:6. Test 2 was identical to test 1, except that a rectangular ridge was present on the top of the reef flat to mimic a reef crest profile. The reef profiles in Yao et al.’s (2012) experiment are shown in Fig. 4. A grid size Δx = 0.03 m and a time step Δt= 0.01 s were used for simulation in this study. Following Yao et al. (2012), the model was run for 200 wave periods and the last 125 wave periods were used for data analysis.

Table 2 Test series of Yao et al.’s (2012) experiment

Fig. 4 Reef profiles in Yao et al.’s (2012) experiment

The computed cross-reef variations of wave height H and MWL for tests 1 and 2 with the present model are compared with the experimental data and the numerical results from Yao et al. (2012) in Fig. 5. The generally good agreement between the computed results and measurements shows that the developed higher-order Boussinesq model with an eddy viscosity submodel can simulate the energy dissipation well for the bathymetry considered. Compared with the numerical results from Yao et al. (2012), we can see that the two models have almost identical performances before wave breaking. The discrepancy occurs after wave breaking over the reef flat, where the present model presents predictions closer to the measurements.

We note that wave conditions listed in Table 2 are comparable to those highly nonlinear ones in Skotner and Apelt (1999). However, the underestimation of setup over the reef flat forYao et al.’s (2012) experiment was negligible. The main difference comes from the reef configuration, i.e., the reef face slope in Skotner and Apelt (1999) was mild (1:12), while a steeper one (1:6) was adopted in Yao et al. (2012). A mild slope strengthens the shoaling effect and adequately enables the wave height to increase, showing that the effect of nonlinearity is stronger on a mild slope than on a steep slope. This implies that the nonlinearity embodied in the present model is insufficient for accurately predicting the setup over a reef flat for strongly nonlinear waves, as mentioned in section 3.1.

Fig. 5 Comparison of computed wave height and setup from Boussinesq wave models with experimental data from Yao et al.’s (2012) experiment

3.3 Revisiting of Demirbilek and Nwogu’s (2007) experiment

Demirbilek and Nwogu (2007) presented the results of a combined laboratory and numerical investigation of irregular wave propagation over fringing coral reefs. The reef profile was identical to that used by Skonter and Apelt (1999), but without any ridge configuration, as shown in Fig. 6. Wave gauges were located at distances of −1.11, −0.92,−0.59, 2.75, 3.68, 4.22, 4.80, 6.97, and 9.14 m from thetoe of the reef, which can be referred to in Demirbilek and Nwogu (2007) for details. The representative case 48 for spectrum waves with significant wave height Hs= 7.5 cm, peak period Tp= 1.5 s, and hr= 0.031 m was simulated with the present model. A time series of surface elevation at the location with the greatest water depth (gauge 1) was put into the model to generate the corresponding internal wave signal for the desired wave. In the simulations, the grid size Δx= 0.05 m and time step Δt= 0.01 s were used. Following Nwogu and Demirbilek (2010) and Yao et al.’s (2012) work, the present simulation lasted for 900s and simulationrecorders from 100 s to 900 s were used for data analysis.

Fig. 6 Reef profile in Demirbilek and Nwogu’s (2007) experiment

The simulated significant wave height H and MWL are compared with the measurements in Fig. 7, where the numerical results from Nwogu and Demirbilek (2010) and Yao et al. (2012) are also plotted. The predicted wave heights from these three models are almost identical and in good agreement with the experimental data. Meanwhile, MWL shows clear differences: Nwogu and Demirbilek’s model fails to predict the setdown before wave breaking, but obtains a series of zero values. Nwogu and Demirbilek (2010) attributed that to the use of a Rayleigh damping term in the mass conservation equation, which acted like a sink/source term to disturb the mass balance in the closed numerical wave tank, and therefore led to the incorrect MWL. This was further confirmed by Yao et al. (2012), who recommended the use of damping terms only in the momentum equations. However, their numerical results (as shown in Fig. 7) were indeed improved. For the present model, the damping terms are included both in mass and momentum equations, while the setdown before the breaking point is well captured. The different performances of damping terms in the three models may be caused by different levels of the accuracy of the mass continuity equation. The mass conservation equation (Eq. 1) in the present model is expressed by the depth-averaged velocity, and therefore is exact, while the mass equations in Nwogu and Demirbilek (2010) and Yao et al.’s (2012) models are formulated by the velocity at a certain water column and only approximated to the second order.

Fig. 7 Comparison of experimental data with numerical results for case 48 in Demirbilek and Nwogu’s (2007) experiment

Spectral densities (Sf) of the water surface elevation at the selected gauges (gauges 3, 5, and 6 through 9) from the measurements and simulations are compared in Fig. 8. The redistribution of wave energy inthe frequency domain is intuitively seen in the process of wave propagation from the offshore deep water zone to the shallow water breaking zone. At gauge 3, wave energy is concentrated around the incident wave spectral peak frequency fp= 0.67 Hz, except for a relatively small amount of long-period energy, which could be partially due to spurious long waves being generated at the wavemaker to compensate for the linear wavemaker transfer function (Nwogu and Demirbilek 2010). As the waves propagate over the reef face, namely at gauge 5, the wave energy at the peak frequency decreases due to the bottom friction. Meanwhile, waves of higher frequencies begin to emerge due to the nonlinear interaction between wave components and the bathymetry variation. The infragravity energy at gauge 5 is also amplified relative to the offshore gauge. At gauges 6 and 7, the spectral peak energy decreases due to wave breaking and bottom friction and is transferred to those of thelower and higher frequencies. As waves propagate over the reef flat, it can be seen that the energy spectra at gauge 8 and at the middle and end of the reef flat are dominated by low-frequency wave motions from about 0 to 0.5 Hz, with most of the incident wave energy at the peak frequency dissipated by wave breaking and bottom friction. Overall, the present model can reasonably reproduce the decreasing spectral peak due to energy dissipation and the energy and frequency transfer, although some discrepancies exist.

Fig. 8 Comparison of computed and measured wave energy spectra at gauges 3, 5, and 6 through 9 for case 48 in Demirbilek and Nwogu’s (2007) experiment

Time histories of the computed surface elevation from 400 to 500 s at selected gauges (gauges 2 through 9) are compared to the measured data in Fig. 9. For the first five gauges (gauges 2 through 6), the predicted wave shape is in excellent agreement with the experimental data, as well as the wave phase, showing the model’s ability to describe the nonlinear effects of shoaling and front steepening as waves propagate from the offshore area to the reef face. The discrepancy begins to emerge after wave breaking. However, the model is still able to predict the highly asymmetric profile of post-breaking waves in relatively shallow water depth, and the amplitude and phase of the low-frequency motions on the reef flat. Considering that wave breaking is simply treated via the ad hoc method in the Boussinesq model, we cannot expect that the details of post-breaking waves are accurately captured, as has been mentioned by other researchers, e.g., Nwogu and Demirbilek (2010).

4 Conclusions

A numerical model based on 1D higher-order Boussinesq equations was used to study wave propagation over different reef profiles. The model presented has better accuracy of linear and nonlinear properties than other models that have been used for the same purpose. Three laboratory experiments covering a wide range of incident wave conditions and reef profileswere simulated. The computed results, including mean values of wave height and MWL, time series of surface elevation, and energy spectrum were compared with the measurements, as well as the published numerical results from other Boussinesq models. From the numerical results, conclusions may be drawn as follows:

Fig. 9 Comparison of surface elevation time histories of experimental data and numerical results for case 48 in Demirbilek and Nwogu’s (2007) experiment

(1) With appropriate treatment of boundary conditions and carefully tuned parameters in the eddy viscosity submodel, the developed model can present an overall agreement with the measurements for both monochromatic and spectral waves over different reef profiles.

(2) Both the present and other Boussinesq models tend to underestimate the setup over a reef flat for highly nonlinear waves. Numerical experiments show that the linear properties (dispersion and shoaling) contribute little to the underestimation. The fact that the nonlinearity embodied in the Boussinesq equations is insufficient for highly nonlinear waves may be the reason for the problem. However, this needs further confirmation through simulations using Boussinesq models with accurate nonlinear properties. We should also note that all Boussinesq models use the ad hoc method to treat wave breaking, which may be the main source of error for the underestimation.

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(Edited by Ye SHI)

This work was supported by the National Natural Science Foundation of China (Grants No. 51009018 and 51079024) and the National Marine Environment Monitoring Center, State Oceanic Administration, P. R. China (Grant No. 210206).

*Corresponding author (e-mail: kfang@dlut.edu.cn)

Received Jan. 4, 2013; accepted Jul. 15, 2013