NUMERICAL MODELLING OF COASTAL PROTECTION BY CORAL REEFS USING A CFD MODEL

NUMERICAL MODELLING OF COASTAL PROTECTION BY CORAL REEFS USING A CFD MODEL Inigo J. Losada, Javier L. Lara, Maria Maza, Pelayo Menendez Environmental Hydraulics Institute IH Cantabria, Universidad de Cantabria C/Isabel Torres n 15, Parque Científico y Tecnológico de Cantabria, 39011 Santander, Spain losadai@unican.es ABSTRACT: One of the most relevant features linked to coastal hazards assessment is the increment of mean water level due to broken waves, called wave setup. In fringing reefs, wave set-up is enhanced inducing larger run-up at the beach. Despite the well known hydrodynamic processes occurring in natural sandy beaches, the role played by bottom vegetation in fringing reefs is very relevant and poorly understood. Although many solutions are used to protect the coast, including artificial man-made structures or natural solutions based on coral reefs or vegetated bottoms, their influence in reducing wave energy needs further analysis. A comparative analysis of different scenarios, such as natural reefs, manmade submerged rubble-mound breakwaters and vegetated reefs, is performed in this work with the aim of demonstrating the ability of vegetated reefs to reduce wave energy at the coastline. The coral reef geometry located in Playa del Carmen (Mexico) has been used as a case study. Single transient wave groups have been analysed. The model called IH2VOF, a Navier-Stokes solver, is used. Results have shown that wave setup and wave run-up are mainly influenced by wave breaking. It has been observed that the more relevant reduction on wave setup and wave run-up is observed when a fore reef is not present. The presence of a vegetated reef has been revealed as an efficient solution, especially for the largest wave height cases, decreasing the wave run-up at the beach. INTRODUCTION The risk of flooding and erosion is increasing for many coastal areas owing to global and regional changes in climate conditions together with increasing exposure and vulnerability. The prediction of coastal hazards is of great importance at the coastline in order to assess risk related with flooding or coastal erosion, among others. The fundamental dynamics involved in coastal areas are multiple and complex, including a broad range of temporal and spatial scales. Although great advances have been done in the last decades, there are still processes, which are not fully understood. One of the most relevant features linked to coastal hazards assessment is the increment of mean water level due to broken waves, called wave setup. It has been broadly studied in gently sloping barred beaches, as a result of alternating large and small wave groups which causes wave setup on the beach. This feature is varying in time, resulting in surf beat oscillations on the order of minutes with high influence on wave run-up on the beach. One of the main consequences of wave run-up increment is coastal flooding and damage induced by wave impact at the coastline. Building detached-to-coast fully or partially submerged breakwaters to defend the coastline from storm waves is nowadays a common practice. However, recent studies have proven that wave setup is enhanced in sloping reefs (Lara et al., 2011) due to an increment of infragravity wave amplitude after wave breaking. Roeber et al. (2015) describes a good example of wave setup enhancement in fringing reef environments, which studied wave breaking induced dynamics at Hernani beach (Philippines) during Typhoon Haiyan, resulting on violent surf beat creating devastating effects on the coast. Classical approaches in gentle or mild sloping beaches represent wave setup by means of partitioning wave forces (radiation stress gradients) into either pressure gradients or bottom

stresses. Reasonable results are obtained, even using linear wave theory. However, such as balance cannot explain wave set-up in steep sloped beaches or reefs, where non-linear effects increase. Typical reef-type bathymetries are characterized by a sudden reduction of water depth becoming into a flat and shallow area in front of the beach. Plunging violent wave breaking is mainly observed occurring across a much narrower region than the one observed in dissipative beaches. In addition, bottom friction plays a more relevant role than on sandy beaches (Lowe et al., 2005) due to the presence of corals and vegetation on natural fringing reefs. Coral reefs are important shallow water features in subtropical and tropical regions playing an important role not only on nearshore hydrodynamics but also on associated biodiversity and ecosystem functioning. Although several works have shown the increment of wave setup due to the geometry and roughness of both the reef and lagoon (e.g. Hoeke et al., 2011), there is a lack of understanding of the role play by vegetation on wave induced setup and run-up in beaches. The analysis of wave breaking induced processes in vegetated fringing reefs is the main motivation of this work. The analysis is mainly focused on wave setup and run-up. The efficiency on wave run-up reduction of different solutions, such as natural reefs, man-made submerged rubble-mound breakwaters and vegetated reefs is analyzed. In order to take into account nonlinear wave transformation due to wave breaking and wave interaction with reef bathymetry, a two-dimensional Navier-Stokes model is used. The model called IH2VOF has been previously validated for gravity and infragravity wave transformation in steep sloping beaches and reefs (Lara et al, 2011), wave interaction with submerged rubble-mound breakwaters (Garcia et al., 2004; Lara et al., 2006) and wave interaction with vegetation (Maza et al., 2013). The ability of the model to solve the two-dimensional wave flow as well as the different scenarios is used to achieve reliable comparisons on wave run-up reduction efficiency. The coral reef geometry located in Playa del Carmen (Mexico) is used as a case study. Instead random wave trains, single transient wave groups are analysed to study in detail individual long wave transformation along the reef and wave setup generation. Transient groups have been defined based on tropical storms characteristics on the area. The paper is organized as follows. First, a brief description of the numerical model is presented, including the details about the boundary conditions for gravity waves and bounded long wave generation. Second, a description of the case study is shown. Results are presented next. Finally, conclusions are drawn. NUMERICAL MODEL Equations IH-2VOF (Lara et al., 2011) is the tool used in this paper. IH-2VOF solves two-dimensional wave flow for hybrid domains in a coupled NS-type equation system, including clear fluid region, porous media flow (Lara et al., 2006) and flow interacting with vegetation (Maza et al., 2013). The movement of free surface is tracked by the Volume of Fluid (VOF) method. At the clear-fluid region, 2DV Reynolds Average Navier-Stokes (RANS) equations are considered. The flow inside the porous media is modelled by the resolution of the Volume- Averaged Reynolds Averaged Navier-Stokes (VARANS) equations, first presented by Hsu et al. (2002). The flow inside the vegetation field is modelled considering an additional friction inducing a loss of momentum, which is represented by a drag force (Maza et al., 2013). Flow is coupled by means of the pressure field, which is assumed to be continuous all along the numerical domain. The final form of the equations used in this work is presented below:

u i = 0 (1) x i 1+ c A u i (2) n t + u j u i = 1 p 0 + ν 2 u i 1 u i u!" j αν(1 n)2 β(1 n) u n 2 x j ρ x i n x j x j n 2 x j n 3 D 2 i u n 3 D i u F D,i 50 50 In the previous equations, variables t, u and p 0 denote time, Reynolds averaged velocity and effective pressure ( p 0 = p + ρg i x i, where p is pressure and g is gravitational acceleration) respectively, n is the porosity, ρ is the density, ν is the kinematic viscosity, i and j represent the horizontal and vertical direction, the over-bar represents volume averaged, single and double prime are Reynolds averaged fluctuation and volume averaged fluctuation respectively. Volume averaged velocity is called seepage velocity. Drag force ( ) due to the effect of vegetation on the flow (last term in equation 2), is modelled as: F D,i = 0.5 C D a N u i u i, where a is the width of the vegetation element perpendicular to the flow direction, C D is the drag coefficient, N is the number of plants per unit area and u i the averaged velocity within the vegetation patch. As can be seen, for a clear fluid porosity becomes unity and VARANS equations yield into the traditional RANS equations. Drag force is only considered for flow within the vegetation field. Turbulence is modelled in the F D,i clear fluid region using a standard k-ε equation model. It is volume-averaged within the porous medium as presented by Hsu et al. (2002). The effect of the vegetation field is considered by two additional terms accounting for the production of turbulent kinetic energy and the energy dissipation produced inside the vegetation field (Maza et al., 2013). Wave generation boundary conditions In order to reproduce gravity and infragravity wave hydrodynamics, a detailed treatment of the boundary conditions has to be carried out. It is of great importance for a correct assessment of wave setup and run-up to use second order wave generation. It includes not only gravity waves but also infragravity-bounded waves, as demonstrated by Torres et al. (2007) in natural beaches and Baldock et al. (2006) and Lara et al. (2011) for very detailed analysis for long wave transformation on steep sloped beaches in laboratory experiments. In addition, simultaneous wave generation and absorption problem is needed to avoid infragravity wave reflections at the boundary. Single transient groups were generated by linear superposition of gravity waves. Transient group constituted by N =256 individual wave components of equal amplitude was generated. The primary short wave components were uniformly spaced over a specified band of the frequency domain in order to obtain a top-hat spectral shape. A heuristic approach was used to focus the wave energy close to the breakpoint of the short waves. See Lara et al. (2011) for further details about second order wave generation. In this work, waves are generated in the numerical model using a direct forcing method, which allows reproducing a piston type wavemaker. In order to reproduce bounded long waves for both random wave trains and transient wave groups, the methodology proposed by Schaffer et al. (1996) have been followed. That method presents an advantage among other theories existing in literature because it considers both super-harmonic and sub-harmonic interactions, yielding a more stable second order wave train. In addition, it this wavemaker theory involves no simplified assumptions, such as shallow water, small evanescent-mode interactions, or narrow-banded wave spectra, making suitable for the wave conditions used in F D,i

this work. Furthermore, the numerical wavemaker is equipped with a control system for simultaneous wave generation and active wave absorption, which is able to absorb not only the gravity waves but also infragravity free and bounded waves. The methodology proposed by Schäffer & Klopman (2000) is followed to absorb waves traveling offshore. See Lara et al. (2011) for further details. CASE STUDY DESCRIPTION Area of Study The study has been locally applied in Playa del Carmen, a well-known tourist city located along the Caribbean Sea in the state of Quintana Roo in eastern Mexico. The coastline is protected from storm waves by a natural defence formed by a long coral reef parallel to the coast. Playa del Carmen site is a good example to develop this analysis since at least seven ocean storms took place during the last two decades, causing significant damage and a temporary drop in tourist arrivals at the coastline. It was mainly created by very large wave setup and consequently runup and flooding. Hurricane Wilma (October 2005) is an example of the most destructive recent episode directly linked with tropical cyclones. The bathymetry data used in this work to parameterize coral reef corresponds to a data collection field campaign by several institutions in 2008, 2010 and 2011 (II-UNAM, Marine and Limnology, UNAM Sciences, National Reef Park of Puerto Morelos, Cinvestav). Coral reef location was provided by reefbase (http://reefgis.reefbase.org/). The coral reef extension in front of Playa del Carmen is about 1 km long (see figure 2). It presents a shallow fore-reef of 3 m depth and a reef crest 300 m long, followed by a reef lagoon 5 m depth and 400 meters wide. The reef modelled in this work has been parameterized with an outer 1:2 slope. The beach is 1/15 sloped with a berm which is defined high enough to do not allow to be overtopped by waves. Case Date Duration Hs(m) Tp(s) 1 October 1998 3 days 6.25 13.00 2 August 2001 12 hours 5.02 12.06 3 September 2002 1 day 4.91 9.48 4 September 2004 1 day 5.96 16.34 5 July 2005 16 hours 6.65 13.05 6 October 2005 3.5 days 6.83 13.00 7 August 2007 1 day 9.98 13.60 Table 1. Characteristics of largest storms in Quintana Roo (Mexico) Wave climate Ocean storms in the Gulf of Mexico are closely linked to hurricane episodes. A pre-analysis of the last two decades has been performed to obtain the order of magnitude of wave height and wave period of the past events. Wave data for seven storms occurred in Quintana Roo, was obtained from GOW (Global Ocean Waves) database (Reguero et al., 2015). These data are presented in table 1.With the aim of improving the understanding on the propagation pattern for long waves induced by a transient short wave group over sloping bottom, seven transient wave groups where simulated with IH2VOF. Transient wave group characteristics were associated to the seven storms presented in table 1. Transient groups analysis is based on representing the most unfavourable wave group at the reef. Focused wave height at that location is defined equal to the maximum wave height for the 6-hours storm based on GOW database. The frequency band is defined as 6 s wide around peak wave period. Table 2 summarizes the spectral characteristics of the transient short wave groups. In this work, only

cases 4 and 7 have been simulated, which correspond to the largest period and wave height, respectively, in order to analyse the influence of wave period and wave height on wave setup. Regarding sea level, the most unfavourable situation is considered. Long-term mean sea level variations (Sea Level Rise, SLR) are taken into account to represent future scenarios. According to IPCC, SLR predicted in Quintana Roo mid-century (2050) is of the order of +0.56m and it can reach +0.74 m at the end of the century for RCP 8.5 projections. Finally, +0.74 m is used. Case Date Duration Hs max(m) T inf - T c - T sup (s) 1 October 1998 3 days 11.25 10-13- 16 2 August 2001 12 hours 9.00 9-12- 15 3 September 2002 1 day 8.84 6.5-9.5-12.5 4 September 2004 1 day 10.73 13.5-16.5-19.5 5 July 2005 16 hours 11.97 10.13.16 6 October 2005 3.5 days 12.29 10-13- 16 7 August 2007 1 day 17.96 10.5-13.5-16.5 Table 2. Wave focusing characteristics used to represent largest storms Numerical setup Figure 1: Different reef scenarios The numerical domain is defined 975 m long and 60 m high. An orthogonal and structured mesh is used. The cell size varies from 2 m x 0.5 m in the target areas (reef flat and run up zone) to 2.8m x 0.5 m in the transient parts (from generation to the fore reef ), in the horizontal and vertical directions respectively, in order to have enough resolution for quantifying wave transformation not only for gravity waves but also for infragravity waves released at the breaking point. The total number of cells is 650 (horizontal) x 240 (vertical). Several scenarios have been studied: 4 geometrical scenarios: Original reef, flat bottom without reef, including a submerged rubble-mound type artificial structure substituting the reef and including a submerged vegetation meadow instead of a coral reef. 2 sea level scenarios: without sea level rise and with sea level rise.

2 sea storms: transient wave groups. The submerged rubble-mound structure is defined to have a porosity of 0.5 with quarry stones of a nominal diameter of 2 m. it is 100 m long and 2 m high. Water depth over the porous breakwater is 3 m. Porous drag parameters are defined based on Lara et al. (2006) results. To analyze the effect induced by vegetation, the area initially occupied by coral reef is replaced by a Thalassia testudium meadow. This species is the most common seagrass present in the study area. Its width and density vary along the year and they are also influenced by some other factors such as water quality or nutrient availability. Therefore, a mean value of both parameters is considered in the present study. Leave width is equal to 2 cm and vegetation density is 600 shoots/m2 (van Tussenbroek, 1995). Vegetation is considered to be 0.5 m high. Resulting a water depth over the vegetation at the reef is of 2.5 m. Drag coefficient is calculated following Mendez and Losada (2004) formulation for irregular waves. Figure 2: Gauges location in the numerical flume Figure 3: Total (solid line) and infragravity (dashed line) wave time history along the reef for the transient wave group for storms 4 (left) and 7 (right) In total, 16 numerical tests were carried out but for the purpose of this paper, only those corresponding to the storm events with the highest wave period will be analysed (cases 4 and 7). The reason of choosing the highest wave period sea state is because the target variable to analyse (run up oscillations due to low frequency motion) is mainly governed by wave period. The numerical flume set-up for the experiments is designed with five free surface gauges, located to measure the time and spatial evolution of waves along the reef. Gauge 1 (x=25m) is placed in the generation region, in order to check the proper operation of wave paddles and its corresponding absorption mechanisms. Gauge 2 (x=280m) is located in the fore-reef, just

behind the reef crest. Gauge 3 (x=435m) is positioned over the reef flat. Gauge 4 (x=760m) measures wave s transformation processes over the lagoon and Gauge 5 (x=955m) is located at the shoreline in order to provide information on the run up oscillations. RESULTS IH2VOF model has been extensively validated for several wave dynamics. No further validation is presented here. Figure 4: Total (solid line) and infragravity (dashed line) wave time history along the reef for the transient wave group for storm 4 for different scenarios Actual geometry Figure 3 shows both the gravity and infragravity wave evolution along the reef for the transient wave group, which represent storms 4 (left) and 7 (right) in table 1. Total free surface (including gravity and infragravity components) is plotted in a solid line. Infragravity wave is presented in a dashed line. It has been obtained using a low-pass filter, which cancels the information from the gravity wave band. Free surface time history is presented at four locations along the reef. The lower panel represents the wave run-up at the beach. The comparative analysis of both transient groups shows the similar behavior for the wave group transformation along the reef. Gauge A plots the incident transient wave group propagating along the horizontal bottom. Wave group at gauge B shows a high-non-linear transformation, with small wave crests but large wave troughs, as a consequence of the non-linear interaction with the steep slope and the generation of a very strong wave set-down. It is consistent with observations from Nwogu et al. (2010). Gauges C and D shows wave pattern evolution along the reef and the lagoon, respectively. As can be seen, short waves break forming a bore, which propagates atop the long wave. It is released at the breaking point on the fringing reef and propagated as free long wave towards the beach. It can be seen that the long wave amplitude is about 2 m in the reef flat and around 1.5 m at the lagoon. Wave run-up appears to be about 7 m in the vertical and it is produced by the two largest waves in the group. The model is also able to reproduce the reflected long wave at the beach. It can be seen for storm 4 (left panel) at gauges C and D around 350 s and 400 s, respectively. The amplitude of the

reflected long wave is similar to the incident long wave as expected. It is observed for storm 4 (left panel) to be a small amplitude wave in gauge A at 420 s when propagating offshore. Different scenarios A comparative analysis of wave evolution along the reef for different scenarios is presented next. Transient wave groups for storms 4 and 7 are shown in figures 4 and 5 respectively. Figure 5: Total (solid line) and infragravity (dashed line) wave time history along the reef for the transient wave group for storm 7 for different scenarios Wave evolution for the two analyzed scenarios (scenario with a reef, black line, and without reef, red line), is presented in the left column in figures 4 and 5. As can be seen in both figures, the reef induces a wave setup enhancement along the reef. It is larger, not only at the reef flat, but also at the lagoon when the reef is present. Based on the model results, wave setup is reduced to almost half in the absence of a reef for both wave groups. This feature has also a large influence on wave run-up at the beach, which is strongly reduced if the reef is not present. The transient wave group, which corresponds to storm 7 (figure 5), shows a smaller run-up than the one observed for storm 4 (figure 4). It is due to the difference in wave breaking, which is the main mechanism to generate long wave release and wave setup. Consequently, wave induced bounded long wave release is similar but lower in magnitude. It can be also seen that short waves evolution along the reef shows larger wave height when the reef is not present, due to weaker energy dissipation by breaking along the reef flat. Results for the other two scenarios analyzed in this work, submerged rubble-mound breakwater and submerged vegetation, are presented in the central and right columns, respectively, in figures 4 and 5. The same pattern is observed for both storms in the two scenarios, yielding into a wave energy reduction due to the increment of bottom friction in the wave set-up and run-up. Larger reduction is observed in the case of a vegetated reef for storm 4 (figure 4). However, the long wave transformation processes and long wave release at the breaking point are shown to be similar for the two scenarios and for both transient wave groups. Influence of water level The influence of the water level on wave setup and run-up is discussed in this section. Figure 6 shows the magnitudes calculated numerically for maximum wave run-up at the beach and

maximum wave setup at the lagoon for the two water levels considered. Maximum wave setup is calculated as the maximum value reached by the free surface for long waves. It is obtained by filtering the free surface time history calculated numerically by a low-pass filter. Results for storm 4 and 7 are presented in the left and right panels, respectively. Elevation is given in meters. Figure 6: Comparisons of wave setup and wave run-up results for the different scenarios for storms 4 (left) and 7 (right) As can be seen in the results presented in figure 6, a large influence on wave setup and wave run-up due to the increment of water depth has not been observed, except for the case when vegetation is present in the fore reef. Bottom friction increment by vegetation induced a reduction in wave set-up and run-up about 0.5 m and 1.5 m, respectively, showing the ability of vegetated reefs to induce dissipation and consequently to reduce wave energy at the shoreline. However, the more relevant reduction on wave setup and wave run-up is seen for the case when the reef is not present. Reduction in such magnitudes is calculated to be almost the half. CONCLUSIONS A numerical analysis of wave breaking induced processes in fringing reefs has been carried out in this work. The analysis has been focused on calculating wave setup and wave run-up on different scenarios, such as natural reefs, man-made submerged rubble-mound breakwaters and vegetated reefs. The main objective has been to determine the non-linear wave transformation processes to demonstrate the ability of vegetated reefs to reduce wave energy at the coastline. The model called IH2VOF has been used. The ability of the model to solve the two-dimensional wave flow as well as the different scenarios is used to achieve reliable comparisons on wave run-up reduction efficiency. The coral reef geometry located in Playa del Carmen (Mexico) has been used as a case study. Single transient wave groups have been analysed to study in detail individual long wave transformation along the reef and wave setup generation. Results have shown that, wave setup and wave run-up are mainly influenced by wave breaking. Long wave energy released at the breaking point induces a strong wave setup, which enhance wave run-up at the beach. It has been observed that the more relevant

reduction on wave setup and wave run-up is observed when the reef is not present. The modification on wave breaking due to the changes on water depth modifies the infragravity wave release and transformation along the reef. Reduction for both values is calculated to almost the half have in reef absence. The submerged rubble-mound breakwater has generated a weak reduction on wave energy all along the reef and at the shoreline. However, the presence of a vegetated reef has been revealed as an efficient solution, especially for the largest wave height cases, decreasing the run-up in 1.5 m for the tested conditions. ACKNOWLEDGMENTS This work has been funded under the RETOS INVESTIGACION 2014 (grant BIA2014-59718-R) program of the Spanish Ministry of Economy and Competitiveness. REFERENCES Baldock, T. E. 2006 Long wave generation by the shoaling and breaking of transient wave groups on a beach. Proc. R. Soc. A 462, 1853 1876. Garcia, N., Lara, J.L. and Losada, I.J. (2004). 2-D Numerical analysis of near-field flow at low-crested permeable breakwaters. Coast. Eng., 51, 10, 991-1020. Hoeke, R., C. Storlazzi, and P. Ridd (2011), Hydrodynamics of a bathymetrically complex fringing coral reef embayment: Wave climate, in situ observations, and wave prediction, J. Geophys. Res., 116, C04018 Hsu, T.-J., Sakakiyama, T., Liu, P.L.-F., 2002. A numerical model for wave motions and turbulence flows in front of a composite breakwater. Coast. Eng., 46, 25 50. Lara, J.L., Garcia, N. and Losada, I.J. (2006). RANS modelling applied to random wave interaction with submerged permeable structures. Coast. Eng., Vol. 53 (5-6), 395-417, Lara, J.L., Ruju, A., Losada, I.J. (2011) RANS modelling of long waves induced by a transient wave group on a beach. Proc. R. Soc. A. 467 (2129), 1215-1242. Lowe, R. J., J. L. Falter, M. D. Bandet, G. Pawlak, M. J. Atkinson, S. G. Monismith, and J. R. Koseff (2005), Spectral wave dissipation over a barrier reef, J. Geophys. Res., 110, C04001, Maza, M., Lara, J.L., Losada, I.J. (2013) A coupled model of submerged vegetation under oscillatory flow using Navier-Stokes equations. Coast. Eng., 80, 16-34. Nwogu, O. and Demirbilek, Z. (2010). Infragravity Wave Motions and Runup over Shallow Fringing Reefs. J. Waterway, Port, Coastal, Ocean Eng., 10.1061 Reguero, B.G. Menéndez, M., Méndez, F.J., Mínguez, R., Losada, I.J. (2012) A global ocean wave (GOW) calibrated reanalysis from 1948 onwards. Coastal Eng., 65, 38 55 Roeber, V. and Bricker, J. D.: Destructive tsunami-like wave generated by surf beat over a coral reef during Typhoon Haiyan, Nature Commun., 6, 7854 Schäffer, H. & Klopman, G. 2000 Review of multidirectional active wave absorption methods. J. Waterw. Port. Coast. Ocean Eng. 126, 88 97. Torres-Freyermuth, A., Lara, J.L. and Losada, I.J. (2010). Numerical modeling of short- and long-wave transformation on a barred beach. Coast. Eng., Vol. 57 (3), 317-330. van Tussenbroek, B. I. (1995). Thalassia testudinumleaf dynamics in a Mexican Caribbean coral reef lagoon. Marine Biology, 122: 33-40.