On the runup parameterisation for reef-lined coasts
Introduction
Wave transformation along reef environments is more complex than on beaches due to the presence of lagoons, reef or ridge crests, abrupt slopes, large flats, and large roughness values. As offshore sea-swell propagates on reef-lined coasts waves break and the majority of the wave energy at these frequencies dissipates, and hence waves reform into a mixture of short- and long-period waves that propagate across the reef towards the shore, resulting in a bimodal spectrum on the reef flat (Lowe et al., 2005, Van Dongeren et al., 2013). This spectrum is typically a result of the breakpoint-forced momentum flux (Pomeroy et al., 2012) and non-linear wave–wave interactions (Nwogu and Demirbilek, 2010). Over certain reefs these longer period infragravity waves (25 s T 250 s) dominate (Pequignet et al., 2009, Cheriton et al., 2016). Recent studies have highlighted the importance of reef morphological controls on infragravity waves (Cheriton et al., 2021). In some cases resonance has been reported over the reef flat, where infragravity wave energy can be distributed around the natural frequency of the reef (Nwogu and Demirbilek, 2010, Torres-Freyermuth et al., 2012), amplifying wave runup at the coast and further complicating its prediction (Demirbilek et al., 2007, Pequignet et al., 2009).
Coral reefs offer natural coastal protection to many regions where they are present, dissipating up to 97% of incident wave energy (Ferrario et al., 2014). However, recent studies have reported that coral reefs are degrading (Eakin, 1996, Sheppard et al., 2005, Alvarez-Filip et al., 2009), caused by a combination of different factors including coastal development, marine pollution, ocean acidification and temperature increase, and overfishing, among others. This affects wave runup since it modifies the spatial gradient of wave dissipation, influencing incident swash and wave-induced setup (Franklin et al., 2018).
Runup is defined as the wave-induced height of discrete water-level maxima (Stockdon et al., 2006) dependent on both the time-averaged wave setup and swash. Coastal flooding often occurs when runup exceeds the height of the dune crest during extreme events (e.g., storms) (Sallenger, 2000, Park and Cox, 2016). The Hunt scaling parameter (Hunt, 1959a, Hunt, 1959b), where is the characteristic slope angle, and and are the deep-water wave height and wavelength, respectively, is typically used to characterise wave runup. For beaches, a range of definitions have been used for , including the foreshore slope around the still water shoreline, the slope at breakpoint, and the mean slope of the active part of the surf zone (Mayer and Kriebel, 1994, Park and Cox, 2016).
Runup parameterisations on beaches and coastal structures have been developed for wind waves (e.g., Hunt, 1959a, Hunt, 1959b, Holman, 1986, Van der Meer, 1988, Stockdon et al., 2006, Park and Cox, 2016) and tsunami waves (e.g., Kobayashi and Karjadi, 1994, Madsen and Schaeffer, 2010). da Silva et al. (2020) presented a comprehensive review of runup parameterisations. Table A.1 shows a summary of the runup models typically used in these environments (see Table A.2 for nomenclature), where the most widely used parameterisation is that of Stockdon et al. (2006). There is still much discussion on the influence of beach slope on the infragravity swash component of run-up (Holman and Sallenger, 1985, Raubenheimer et al., 2001, Hanslow and Nielsen, 1993, Passarella et al., 2018) with some authors finding swash to be independent from the beach slope (Ruessink et al., 1998, Senechal et al., 2011), while others have found that including the slope improves predictability (Cohn and Ruggiero, 2016). Moreover, scatter in the runup predictors with respect to data has been ascribed to aleatoric (Beuzen et al., 2019, Torres-Freyermuth et al., 2019, Rutten et al., 2021) and statistical (Fiedler et al., 2018) uncertainty.
In the case of coral reef-lined coasts, there are fewer studies that have focused on wave runup (Demirbilek et al., 2007, Buckley et al., 2015, Cheriton et al., 2016, Pearson et al., 2017, Lashley et al., 2018, Rueda et al., 2019). In a recent study on flood prediction, Stockdon’s parameterisation was assumed to be applicable to predict runup in reef environments (e.g., Beck et al., 2018). However, since runup on reef-lined coasts depends not only on the fore slope angle of the beach but also on the morphology of the reef, parameterisations traditionally used for beaches do not appear to be readily applicable (Franklin et al., 2018, Lashley et al., 2018, da Silva et al., 2020). To improve the coastal modelling of physical processes the parameterisation of small-scale processes needs to be improved (Fringer et al., 2019).
The most recent study regarding wave runup in reef environments, which takes into account certain aspects of reef geometry, is that by Pearson et al. (2017). These authors used a numerical model and probabilistic Bayesian network to estimate wave-induced flooding on reef-lined coasts, identifying water depth over the reef flat, incident wave conditions, and the width of the reef flat as the most important parameters to accurately predict flooding hazards in these environments. Beach slope and bed friction were found to be less important (Pearson et al., 2017). Similarly, a study performed by Rueda et al. (2019) involves the development of a metamodel to predict wave runup under a range of morphologic and forcing characteristics. The study employs an idealised reef profile and seven primary parameters (offshore water level, significant wave height, reef flat width, beach slope, wave steepness and seabed roughness). Their results provide an open-source code that serves as a tool for estimating runup estimations along reef-lined coasts, with a mean error of 30 cm (Rueda et al., 2019). More recently, Scott et al. (2020) presented a study using representative cluster profiles to represent the shoreline hydrodynamics of coral reef-lined coasts, and predicted runup with a mean error of 9.7–13.1%. This study included parameters used by Pearson et al. (2017), excluding roughness, and concludes that the representative cluster profiles (RCPs) would enhance predictive accuracy for the reef profiles most dissimilar to that used in BEWARE (Pearson et al., 2017). These are significant advances in the prediction of wave runup for fringing reefs, however, none of these studies present a readily applicable alternative runup formula that takes into account the morphologic and hydrodynamic parameters identified, making their application slightly less straightforward.
In recent years, interest in the use of Machine Learning for coastal and hydraulics applications has grown (e.g., Goldstein et al., 2013, Tinoco et al., 2015, Passarella et al., 2018, Goldstein et al., 2019, Beuzen et al., 2019, da Silva et al., 2020; among others). Machine learning is a subdiscipline of computer science concerned with the construction of computer programs that automatically improve with experience (Mitchell, 1997). The main aim of this approach is to develop predictors that can be generalised (Passarella et al., 2018). Passarella et al. (2018) employed this approach for swash parameterisation on beaches, finding an improvement in total swash prediction by developing a dimensional runup parameterisation.
This study aims to use the numerical model Simulating WAves till Shore (SWASH, Zijlema et al., 2011) and Machine Learning to develop and evaluate new formulae to predict runup associated with wave transformation over idealised reef-lined bathymetries, comparing traditional formulations with the results obtained using genetic programming. The outline of the paper is as follows. Section 2 describes the numerical model, model setup and simulated cases. The Machine Learning Method is described in Section 3, followed by the results in Section 4. Finally, the Discussion, focused on the comparison of the new formulations with those developed for beaches, and concluding remarks are presented (Sections 5 Discussion, 6 Conclusions).
Section snippets
Model description
In this study, a phase-resolving nonlinear non-hydrostatic free surface model, the SWASH model (http://swash.sourceforge.net, last access: 20 January 2017) developed at Delft University of Technology (Zijlema et al., 2011), is used. This model is suitable for simulating wave transformation due to nonlinear wave–wave interactions in the surf and swash regions since it solves the nonlinear shallow water equations and includes the non-hydrostatic pressure terms (Zijlema et al., 2011). The SWASH
Genetic programming for runup parameterisation
Machine learning (ML) is a sub discipline of computer science and a sector of artificial intelligence that applies algorithms systematically in order to synthesise relationships between information and data (Awad and Khanna, 2015), based on the notion that systems are able to identify patterns, learn from data, and make decisions. Thus, ML automates the construction of analytical models to find relationships between variables involved in processes of interest (Passarella et al., 2018). Various
Effect of reef geometry on runup
The effects of different geometry features on wave runup were tested using the numerical model SWASH. Model results show that increased with decreasing lagoon width, increasing forereef slope, and in the presence of a ridge or simplified box-shaped crest (Fig. 4a and b). The increase in water depth also results in an increase in runup and causes the effect of lagoon width to become less relevant.
The aforementioned pattern is similar for both the gentler and steeper slopes, however in the
Discussion
The primary mechanisms that control the water level at the coast are tides, storm surge, and wave runup (Buckley et al., 2016). The results presented here show that wave runup was significantly higher in the presence of a reef crest, which is due primarily to an increase in wave setup over the flat consistent with the findings of Yao et al. (2017). Furthermore, they also found that increasing crest width resulted in greater wave setup and identified crest submergence as the primary parameter
Conclusions
This study uses a non-linear wave transformation model and Machine Learning to investigate wave runup associated with wave transformation on reef-lined coasts. Different wave conditions were simulated over an idealised reef geometry at prototype scale. Findings show runup to be dominated by the contributions of wave setup and infragravity swash depending on the wave energy. The forereef slope, reef crest, reef flat width and water depth were varied to elucidate their role in runup. Modelling
CRediT authorship contribution statement
Gemma L. Franklin: Methodology, Writing – original draft, Writing – reviewing and editing, Visualization, Investigation, Formal analysis. Alec Torres-Freyermuth: Conceptualization, Writing – reviewing and editing, Supervision, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
Many thanks to Gonzalo Martín Ruiz for technical support and to Delft University of Technology for making the development of the SWASH model possible. Thanks also to the anonymous reviewers whose comments and suggestions helped to greatly improve the manuscript.
Funding
This work was supported by funding received by the first author through a postdoctoral scholarship awarded by the Programa de Becas Posdoctorales, Mexico and PAPIIT, Mexico IN102221 from DGAPA-UNAM, Mexico; the National Council of
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