Numerical Study of the Impact of Climate Change on Irregular Wave Run-up Over Reef-Fringed Coasts

Wave hydrodynamics over fringing reefs is largely controlled by the reef surface roughness and hydrodynamic forcing. It is believed that climate change will result in a net increase in the water depth over the reef flat, a degrading of the surface roughness of coral reefs and changes in extreme incident wave heights. For an accurate assessment of how climate change affects the safety of reef-fringed coasts, a numerical study of the impact of climate change on irregular wave run-up over reef-fringed coasts was carried out based on a Boussinesq wave model, FUNWAVE-TVD. Validated with experimental data, the present model shows reasonable prediction of irregular wave evolution and run-up height over fringing reefs. Numerical experiments were then implemented based on the anticipated effects of climate change and carried out to investigate the effects of sea level rise, degrading of the reef surface roughness and increase of extreme incident wave height on the irregular wave run-up height over the back-reef beach respectively. Variations of run-up components (i.e., spectral characteristics of run-up and mean water level) were examined specifically and discussed to better understand the influencing mechanism of each climate change-related effect on the run-up.


Introduction
Run-up of waves is often expressed as a vertical distance between the shoreward edge of water and still water level consisting of two components: wave setup and fluctuations (Ruggiero et al., 2004). It plays a critical role in accessing the susceptibility of coastal beaches to wave-driven hazards, governing swash zone sediment transport and the design of shore protection structures (Chen et al., 2010;Zhang and Liu, 2009). Investigations of run-up dynamics have typically taken place on mild beaches for decades (e.g., Guza and Thornton, 1982;Raubenheimer et al., 1995). Different from mild beaches, fringing coral reefs surrounding many tropical and subtropical continental coastlines and atolls are characterized by abrupt bathymetric change in the fore-reef slope and shallow reef flat. Due to this high-energy dissipative bathymetry, fringing reefs have been reported as efficient buffers to wind wave energy (Hardy and Young, 1996;Lowe et al., 2005); extensive wave inundations, however, still frequently occur during extreme wave events at coastal areas behind coral reefs (e.g., Jaffe and Richmond, 1993;Tajima et al., 2014;Yamano et al., 2007). Previous studies on reef hydrodynamics have revealed that wave run-up over reef-fringed beaches is mainly contributed from wave setup and infragravity waves. Incident seaswell waves are breaking strongly over the steep fore-reef face, resulting in a significant setup over the reef flat (Gourlay, 1996). Infragravity waves can be generated by the time variation of the breaking point resulting from the wave groups (Péquignet et al., 2009) and increase from the reef edge to the shore (Nwogu and Demirbilek, 2010). Moreover, the infragravity wave components may be trapped by the reef-beach system under certain conditions, forming resonant oscillations on the reef flat, so that wave run-up over reef-fringed coasts can be further enhanced (Nwogu and Demirbilek, 2010;Péquignet et al., 2009). Direct field observations (Cheriton et al., 2016) clearly demonstrate the overall shoreline variations along the reef-fringed coasts during storm events are dominated by the low-frequency waves, particularly the enhanced infragravity waves. Therefore, reef-fringed coasts are still vulnerable to ex-treme wave events and protecting structures have been developed along the heavily urbanized and highly engineered coasts behind coral reefs (Yao et al., 2016).
Nowadays, climate change is believed to be occurring as a long-term rise in the average temperature of the Earth's climate system. Ongoing and anticipated effects of global warming include sea level rise and regional precipitation changes. According to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC, 2007), the global sea level will rise by up to 60 cm by the end of 21th century in response to ocean warming and glaciers melting. Several researchers further revealed the possibility of future sea level rise of up to 2 m by 2100(e.g., Vermeer and Rahmstorf, 2009;Grinsted et al., 2010). However, vertical accretion rates of coral reefs are reported to be about 1− 4 mm/year (Montaggioni, 2005), which is only an order of magnitude smaller than the anticipated rates of sea level rise. Therefore, sea level rise will result in a net increase in the water depth over the reef flat. It is also believed that the above-average seawater temperatures is the leading cause of the coral bleaching (Michael, 2016), resulting in a degrading of the surface roughness of coral reefs. Moreover, predicted regional precipitation changes will induce more frequent extreme weather events and drive increases in extreme incident wave heights . These effects of climate change may have a significant impact on reef hydrodynamics as wave-reef interactions are largely controlled by the reef surface roughness (Lowe et al., 2005;Pomeroy et al., 2012;Péquignet et al., 2011;Zhu et al., 2004) and hydrodynamic forcing (Nwogu and Demirbilek, 2010;Péquignet et al., 2009;Pomeroy et al., 2012). Zhu et al. (2004) indicated the bottom friction coefficients on the reef flat of Yonshu Reef in South China Sea is about 10 times that of the sand or silt bottom. Field study of the dissipation of wave energy over Ipan reef, Guam (Péquignet et al., 2011) clearly found 20% of incident wave energy is dissipated by bottom friction over the fore-reef face and reef flat. The relative submergence (the ratio of the water depth over the reef flat to the incident wave height) not only affects the wave setup (Gourlay, 1996) but also the generation (Péquignet et al., 2009) and dissipation (Pomeroy et al., 2012) of infragravity wave. Based on anticipated effects of climate change, Quataert et al. (2015) has used an X-beach model, a nonlinear shallow water model, to further check the climate change-related effects on wave hydrodynamics over fringing reefs. These studies have improved the understanding of the impact of climate change on wave characteristics over fringing reefs.
However, so far, the impact of climate change on wave run-up process over reef-fringed coasts has not been fully investigated. As emphasized by recent studies (Cheriton et al., 2016;Ning et al., 2019b), run-up is the primary process connecting reef flat water dynamics and the beach face and by comparison with the wave spectra over the reef flat, it is found that low frequency wave components are further enhanced while high frequency wave components are further suppressed during run-up motions. Therefore, an accurate assessment of how climate change affects the safety of reeffringed coasts must consider shoreline run-up. Moreover, a specific investigation of variations of the run-up components (i.e., spectral characteristics of run-up and mean water level) in response to each climate change-related effect may provide coastal engineers and managers better guidance for improving the protection structures or planning adaption activities. In light of these, a numerical study of the impact of climate change on irregular wave run-up over reeffringed coasts was thereby carried out in this study. The adopted numerical model is a fully nonlinear and dispersive Boussinesq-type model, FUNWAVE-TVD (Shi et al., 2012). Boussinesq-type model has become the most pervasive tool of modelling wave hydrodynamics over fringing reefs (e.g., Gao et al., 2019;Liu et al., 2019;Ning et al., 2019aNing et al., , 2019bRoeber and Cheung, 2012;Su et al., 2015;Yao et al., 2012;Zhang et al., 2019) due to its accuracy and efficiency. The present model is the most recent implementation of the well-known Boussinesq model, FUNWAVE initially developed by Kirby et al. (1998). In this paper, experimental data of irregular wave transformation and run-up over fringing reefs were first used to verify the present model. Then the model was used to explore the impact of climate change on irregular wave run-up over reef-fringed coasts, including the effects of sea level rise, coral bleaching and increase of extreme wave heights. Variations of the run-up components (i.e., spectral characteristics of run-up and mean water level) in response to sea level rise, coral bleaching and increase of extreme wave heights were studied respectively to better understand the influencing mechanism of each climate change-related effect on the run-up.

Numerical model
All numerical simulations in this study are performed by using the fully nonlinear and dispersive Boussineq-type model, FUNWAVE-TVD (Shi et al., 2012). It was developed based on a more complete set of fully nonlinear Boussinesq equations with the vertical vorticity correction proposed by Chen (2006), incorporating a time-varying reference elevation (Kennedy et al., 2001). The governing equations of the model are rearranged into a conservative form and solved by a hybrid finite-difference and finitevolume TVD scheme. With the hybrid numerical scheme, wave breaking can be treated as shock waves by switching the Boussinesq equations to the nonlinear shallow water equations when the local ratio of the surface elevation to the water depth exceeds a certain threshold, 0.8 (Tonelli and Petti, 2009). This shock-capturing capability is the desirable property for a model that aims at simulating nearshore reef hydrodynamics (Fang et al., 2016) because of its satisfaction of conservation laws across flow discontinuities (Roeber and Cheung, 2012). In addition, the third-order Strong Stability-Preserving (SSP) Runge−Kutta scheme is employed for the time stepping, in which case an adaptive time step is implemented following the Courant-Friedrichs-Lewy condition. The bottom friction is calculated by a quadratic friction term R incorporating a Manning coefficient expressed as: where g is the gravitational acceleration, n is the Manning coefficient, h is the still water depth, is the surface elevation and is the horizontal velocity at the reference elevation. Although reef surface roughness is spatially inhomogeneous, the Manning coefficient with a constant value has been used as a convenient parameter to characterize the surface roughness of coral reefs in previous numerical studies (e.g., Gelfenbaum et al., 2011;Kunkel et al., 2006). Wave run-up can also be handled quite conveniently with a mirror boundary condition applied to the TVD scheme. More details about the present model can be referred toShi et al. (2012).

Model validation
The present model is firstly validated with the experimental data of irregular wave transformation and run-up over fringing reefs. The laboratory experiments were conducted in a two-dimensional wave flume, which is 35 m long, 0.6 m wide, and 0.8 m high, of the Ocean Experiment Hall at Zhejiang University. Fig. 1 gives the sketch of the experimental setup. An idealized reef-beach model with a 1:3 fore-reef slope, a 4.93 m horizontal reef flat, a 1:11.9 beach slope and smooth steel surface was implemented in the wave flume. Water surface fluctuations were recorded at eight wave gauges (G1−G8). Gauges G1−G3 were placed in front of the physical model to measure the incident waves and wave reflection. G4 and G5 were placed over the forereef slope. G6 was placed at the reef edge. G7 and G8 were placed in the middle and at the end of the reef flat. Moreover, a 2.0 m wave gauge G9, whose measuring wire projected no more than 2 mm above the model surface, was laid parallel on the back-reef slope to measure the run-up height. The run-up height was defined as the vertical dis-tance between the surface elevation and still water level and accordingly, the data of G9 were transformed to the run-up height based on the geometrical relationship. Conditions of wave cases for model validation are listed in Table 1. Incident irregular waves were run with significant wave heights H S , 4 cm and 6 cm, spectral peak period T P , 1.25 s (i.e. a peak frequency f p =0.8 Hz) and water depths over the reef flat h r , 3 cm. The incident irregular waves generated by the wave maker, which has an active absorption, followed a JONSWAP spectrum with a peak enhancement factor γ=3.3. The model geometry, incoming waves and water levels over the reef flat were designed based on the Froude similarity with a geometric scale factor of 1:100. Therefore, at the prototype scale, the fore-reef slope angle, reef-flat width, water depth over the reef flat all fall within the common ranges of reported reef dimensions summarized in Quataert et al. (2015). The incident significant wave heights and spectral peak period were set to represent the typical wave conditions during the severe weather. It should be mentioned that the present model has been utilized in previous studies (Ning et al., 2019b;Su et al., 2015) for irregular wave processes over fringing reefs. The model shows reasonable prediction of the spatial variations of significant sea-swell wave height and infragravity wave height but the infragravity wave height over the reef flat tends to be underestimated. The relative submergence (h r /H s ) used in previous studies (Ning et al., 2019b;Su et al., 2015) was relatively large (e.g., larger than 1 in Ning's study), in which case primary waves break on the outer reef flat and the generation mechanisms of infragravity waves involves bound-released long waves (Longuet-Higgins and Stewart, 1964) and breakingforced long waves (Symonds et al., 1982). On the other hand, the water over the reef flat of most reported fringing reefs are shallow, usually smaller than 4 m (Quataert et al., 2015), and the relative submergence should be smaller during extreme wave events, in which case primary waves break on the fore-reef or reef edge and the generation mechanism of infragravity waves is mostly ascribed to wave breaking (Symonds et al., 1982). Therefore, for a more realistic account of surf zone processes in a reef environment during extreme wave events, this study adopted a smaller relative submergence to test the model performance.
Numerical setup for model validation was implemented based on the laboratory experiments. Irregular waves were generated by an internal wave maker, following the method of Wei et al. (1999), at 12.75 m from the toe of the fore-reef slope. The whole computational domain was 45 m long with a sponge layer at the offshore boundary. A minimum total water depth was applied to detect the end point of moving shoreline, in which case if the calculated total water depth is smaller than a specified threshold, 0.001 m, the cell is considered as a dry cell. The Manning coefficient has been proved to represent the surface roughness of the present physical model well for wave attenuation associated with the bottom fiction (Ning et al., 2019a). Therefore, the value of Manning coefficient was set to 0.012 for the smooth steel based on standard hydraulic books.
Spatial distribution of significant sea-swell wave height H ss , infragravity wave height H IG and mean water level for measured and predicted results are demonstrated in Fig. 2 and Fig. 3. H ss and H IG were estimated respectively as: where S(f) is the wave spectrum calculated from the last 100 s experimental or numerical results by Fourier transformation. The typical dominant periods of ocean swells are 1 to 25 s in reality. Therefore, based on the similarity principle of this study, 0.5f p was used to divide S(f) into the sea-swell band and infragravity band. Moreover, although grid resolution is important for the present model to simulate the wave breaking accurately (Shi et al., 2012), the grid size dx=0.02 m (about 106 grid points per wave length) was directly adopted in this study since this grid size has shown enough accuracy in Ning et al. (2019b) and the incoming wave lengths in Ning et al. (2019b) and the present study are more or less the same. In order to quantify the model performance, the model skill is expressed as (Willmott, 1981):

Q
where P i and Q i represent the predicted and measured data, is the mean of measured data. As seen in Figs. 2 and 3, predicted results agree well with the experimental data. Spatial distribution of H ss and mean water level are well captured by the present model. H IG on the reef flat still tends to be underestimated for both cases but the underestimation seems not large compared with the previous studies (Ning et al., 2019b;Su et al., 2015). The model skill for infragravity wave height prediction is larger than 0.89, which is better than that demonstrated in Su et al. (2015).This indicates the shock-capturing scheme may perform better for the infragravity wave evolution over fringing reefs when the relative submergence is small and generation of infragravity waves is mainly due to wave breaking. Figs. 4 and 5 show the measured and predicted wave spectra at gauges G3 and G6−G8 for both cases. It can be seen that the spectrum peak, shape as well as the energy transform are reasonably predicted by the present model. Fig. 6 shows the measured and predicted run-up heights for both cases. Spectral analysis was also performed for the run-up motions and similar to H ss and H IG , significant seaswell run-up height (R ss ), infragravity run-up height (R IG ) were calculated from the run-up spectrum as: where S R (f) is the run-up spectrum. Moreover, 2% highest run-up height (R 2% ), which is the mean of top 2% of timevarying run-up heights, was selected to characterize the maximum run-up height. As seen in Fig. 6, shoreline run-up motions are dominated by the infragravity waves and the model reasonably reproduced the spectral characteristics of the run-up motions and 2% highest run-up height. R IG and R 2% tend to be underestimated in Case B since H IG near the shoreline is underestimated as seen in Fig. 3.

Numerical experiments
The validated model was used to further access the impacts of sea level rise, degrading of the reef surface roughness and increase of the extreme wave height on the irregular wave run-up over the back-reef beach. A series of numerical experiments at the prototype scale were conducted by varying water level over the reef flat, Manning coefficient, and significant incident wave height. The numerical experiment setup is shown in Fig. 7. Irregular waves were generated initially at x=1500 m, following the JONSWAP spectrum, and the whole computational domain was 4500 m with a grid size dx=2 m. A sponge layer with the length of 1000 m was arranged at the offshore boundary to absorb reflected waves. Three groups of wave cases were implemented and conditions of these groups are listed in Table 2, where H s is the significant incident wave height, h r is the Fig. 4. Measured and predicted wave spectra at G3 and G6−G8 for Case A.    LIU Wei-jie et al. China Ocean Eng., 2020, Vol. 34, No. 2, P. 162-171 167 θ β θ β water depth over the reef flat, w is the reef flat width, tan is the fore-reef slope angle, tan is the back-reef slope angle, n 1 is the Manning coefficient for the horizontal bottom in front of the fore-reef slope, n 2 is the Manning coefficient for the fringing reef and n 3 is the Manning coefficient for the back-reef beach. Only one parameter varies in each group while other parameters are kept unchanged so that the effects of sea level rise, degrading of the reef surface roughness and increase of the extreme wave height are demonstrated independently. Values of w, tan , tan were set based on reported reef dimensions as summarized in Quataert et al. (2015). Moreover, water depths over the reef flat are usually 0−4 m (Quataert et al., 2015) while studies have revealed the possibility of future sea level rise of 2 m by 2100 (e.g. Vermeer and Rahmstorf, 2009;Grinsted et al., 2010). Therefore, based on this projected change in sea level, the maximum value of h r in Table 2 was set to 6 m. Moreover, the roughness of the fringing reef n 2 in Table 1 was allowed to vary from 0.09 to 0.02. The value and range of n 2 was set based on values used in previous studies (Gelfenbaum et al., 2011), where 0.0962 represented a highly rough reef, 0.05 represented a normal and healthy reef and 0.02 represented an essentially dead and smooth reef. Besides, n 1 was fixed as 0.02 (sand) for the horizontal bottom in front of the fore-reef slope and n 3 was fixed as 0.04 (cobbles) for the back-reef beach in all groups. Fig. 8 shows computed R ss , R IG and R 2% with the variation of the water level over the reef flat for Group A. As seen in Fig. 8, R ss increases notably with the increase of the water depth over the reef flat as greater water depth means smaller reflection and also smaller bottom friction. However, the variations of R IG and R 2% are quite different, in which case R IG and R 2% decrease with the increase of the water depth over the reef flat. Fig. 9 shows computed mean water level near the shoreline (at the toe of the back-reef beach) with the variation of the water level over the reef flat for Group A. It can be seen mean water level near the shoreline decreases obviously with the increase of the water depth over the reef flat, which is consistent with previous studies (Gourlay, 1996;Nwogu and Demirbilek, 2010) about wave setup over the reef flat. Their experimental data revealed that the setup over the reef is controlled by the relative submergence h r /H s and decreases significantly as h r /H s increases. The variation of R 2% in Fig. 8 should be contributed from both R IG and mean water level. The decrease of R IG in Fig. 8 is unexpected since larger water depth should mean smaller bottom friction and we think this may be related to the generation of infragravity waves. Péquignet et al. (2009) found that the unusually large wave setup over the reef flat could alter the dynamics of wave transformation, driving the modulation of breaking swell waves near the reef edge more significantly, so that the generation of in-fragravity waves was enhanced. Therefore in this study, when h r is small, large wave setup occurs and long wave components may be generated more easily. As h r increases continuously, setup becomes smaller and generation of the long wave components should be weaker. In this view, sea level rise seems to be able to reduce irregular wave run-up as both R IG and mean water level decrease; however, it should be noted that R 2% is the vertical distance calculated from the still water level and the net increase of the water depth induced by sea level rise is not included in the variation of R 2% . Therefore, instead of R 2% , Fig. 10 replotted the variation of R 2% * , which is the 2% highest run-up height calculated from the reef flat. As seen in Fig. 10, although R IG and mean water level decrease with sea level rise, R 2% * still increases significantly, which indicates the reduction of R IG and mean water level cannot offset the net increase of the water depth and larger inundation will still occur due to the net increase of the water depth induced by sea level rise. Fig. 11 shows computed R ss , R IG and R 2% with the variation of the roughness of the fringing reef for Group B. As seen in Fig. 11, R ss increases slightly with the decrease of n 2 while R IG and R 2% increase more evidently with the de-  crease of n 2 . Fig. 12 shows computed mean water level near the shoreline with the variation of the roughness of the fringing reef for Group B. The mean water level is demonstrated to be insensitive to the change of reef surface roughness. Results presented in Fig. 11 and Fig. 12 clearly indicate that the coral bleaching will have a negative impact on the safety of reef-fringed coasts mainly due to the enhanced low-frequency infragravity oscillations. Swash zone hydrodynamics in reef environment may become increasingly dominated by low-frequency wave motions with the projected change in the frictional characteristics of coral reefs. Moreover, we have to point out that coral reefs with dense coastal vegetation may impose wave energy dissipation not only by bottom friction but also drag forces induced by vegetation. It is reported that during the sea temperature rise, corals will expel the vegetation living in their tissues (Brown, 1997). However, the Manning coefficient used in the present model only represents small-scale irregularities of the reef surface without considering the drag force and therefore it is likely that for coral reefs with dense coastal vegetation, the negative impact of coral bleaching may be more serious than that demonstrated in this study. Fig. 13 shows computed R ss , R IG and R 2% with the vari-ation of the significant incident wave height for Group C. As seen in Fig. 13, R ss increases slightly while R IG and R 2% increase significantly with the increase of H s . As presented in Ning et al. (2019a), solitary wave with larger incident wave height breaks more violently with larger dissipation and the increase of the run-up height with the increase of incident wave height can be mitigated by the stronger wave breaking while in this study this mitigation is not observed for irregular waves. Instead, R IG increases almost linearly with the increase of incident wave height. Fig. 14 shows computed mean water level near the shoreline with the variation of the significant incident wave height for Group C. Mean water level also increases significantly with the increase of incident wave height, which is consistent with the relationship between setup and relative submergence as mentioned above. Large setup should also contribute to the generation of infragravity waves so that R IG increases almost linearly with the increase of incident wave height as seen in Fig. 13. The results presented in Fig. 13 and Fig.14 clearly indicate that the increase of extreme wave height will have a serious negative impact on the safety of reeffringed coasts due to the enhanced low-frequency infragravity oscillations and elevated mean water level.     13. Computed R ss , R IG and R 2% with the variation of the significant incident wave height for Group C.

Conclusions
In this paper, the fully nonlinear Boussinesq wave model with shock-capturing capability, FUNWAVE-TVD, was utilized to simulate irregular wave evolution and run-up over fringing reefs. Experimental data of irregular wave transformation and run-up over a reef-beach system were used to verify the present model. Laboratory experiments were designed with smaller relative submergence for a more realistic account of surf zone processes in a reef environment during extreme wave events. The model performed reasonably well with a proper grid size for irregular wave evolution over the reef flat and run-up over the back-reef slope. Infragravity wave heights over the reef flat were underestimated by the present model, but the model skill for infragravity wave height prediction in this study is demonstrated to be better than that in previous numerical studies with larger relative submergences. The validated model was utilized to further investigate the impacts of climate changerelated sea level rise, the degrading of the reef surface roughness and increase of the extreme wave height on irregular wave run-up over the back-reef beach. Numerical experiment results indicate although infragravity waves and mean water level are reduced, sea level rise will still cause larger inundations over reef-fringed coasts due to the net increase of the water depth. Coral bleaching will have a negative impact on the safety of reef-fringed coasts mainly due to the enhanced low-frequency infragravity oscillations. The increase of extreme wave height will have a serious negative impact on the safety of reef-fringed coasts mainly due to the enhanced low-frequency infragravity oscillations and elevated mean water level. The results presented in this paper clearly demonstrate the impact of climate change on the safety of reef-fringed coasts and the influencing mechanism of each climate change-related effect on the run-up, which may be useful for coastal engineers and managers to take future measures for fringing-reef coasts in the face of different climate change-related effects.