Solitary Wave Run-up Research Articles

Modeling wave dynamics with coastal vegetation using a smoothed particle hydrodynamics porous flow model

Coastal vegetation serves as an effective nature-based solution for protecting shorelines from erosion and storm surges. A detailed understanding of the dynamic interactions between waves and vegetation is crucial for quantifying the mitigation potential of coastal vegetation during extreme climatic events. This study investigates wave dynamic interactions with coastal vegetation by developing a numerical model using the weakly compressible Smoothed Particle Hydrodynamics (WCSPH) method. The model simulates the effects of vegetation by incorporating vegetation porosity, drag, and inertia forces into the momentum equations. By integrating porosity information at specific background points, the SPH interpolation method effectively treats the interfaces between fluid and vegetation regions, allowing versatility in modeling various vegetation layouts without explicitly simulating their solid structures, which is computationally prohibitive. The numerical model is validated using laboratory-based measurements and analytical models for both submerged and emergent scenarios involving cylindrical and strip-type vegetation. The model is then utilized to investigate the effects of vegetation as both a natural line of defense and a retrofitting solution in front of a sea dike, to quantify the performance of nature-based solutions for enhancing climate resilience in natural and hybrid coastal protection infrastructures. The results reveal that canopy density significantly impacts solitary wave run-up and energy dissipation. For the configurations tested in this study, high-density vegetation reduces normalized wave run-up by 52% compared to simulations without vegetation, while low-density vegetation achieves a 17% reduction. These findings demonstrate the potential of vegetation, particularly at higher densities, to mitigate solitary wave energy and reduce run-up hazards. Furthermore, simulation results indicate that using vegetation as a retrofitting solution effectively mitigates overtopping rates, achieving a 37% reduction with minimal forest density (∅ = 0.023). These results underscore the effectiveness of nature-based interventions in enhancing coastal protection and resilience.

2DH Numerical Study of Solitary Wave Processes around an Idealized Reef-Fringed Island

In order to better understand the role of coral reefs around an isolated island in mitigating tsunami hazards, this study performed a horizontally two-dimensional (2DH) numerical study of tsunami-like solitary wave propagation and run-up around an idealized reef-fringed island. The shock-capturing Boussinesq wave model, the FUNWAVE-TVD is used in the present study and well-validated with existing experimental data for its robustness in predicting 2DH solitary wave processes around an island. Based on the validated model, the typical solitary propagation process around the reef-fringed island and the effects of morphological and hydrodynamic parameters on the maximum run-up heights were systematically investigated. It is found that coral reefs can effectively reduce maximum run-up heights around an isolated island. The reef flat’s water depth, reef flat width, and reef surface roughness are the main factors affecting maximum run-up heights around an island, while the fore-reef slope has little impact. For the idealized reef-fringed island in this study, sea-level rise will cause coral reefs to lose their protective capability on the lee side, and the presence of coral reefs may even enhance tsunami hazards around an island when the reef flat width is very narrow or coral bleaching happens.

Tsunami Squares: Leapfrog scheme implementation and benchmark study on wave–shore interaction of solitary waves

Impulse waves are generated by rapid subaerial mass movements including landslides, avalanches and glacier break-offs, which pose a potential risk to public facilities and residents along the shore of natural lakes or engineered reservoirs. Therefore, the prediction and assessment of impulse waves are of considerable importance to practical engineering. Tsunami Squares, as a meshless numerical method based on a hybrid Eulerian–Lagrangian algorithm, have focused on the simulation of landslide-generated impulse waves. An updated numerical scheme referred to as Tsunami Squares Leapfrog, was developed which contains a new smooth function able to achieve space and time convergence tests as well as the Leapfrog time integration method enabling second-order accuracy. The updated scheme shows improved performance due to a lower wave decay rate per unit propagation distance compared to the original implementation of Tsunami Squares. A systematic benchmark testing of the updated scheme was conducted by simulating the run-up, reflection and overland flow of solitary waves along a slope for various initial wave amplitudes, water depths and slope angles. For run-up, the updated scheme shows good performance when the initial relative wave amplitude is smaller than 0.4. Otherwise, the model tends to underestimate the run-up height for mild slopes, while an overestimation is observed for steeper slopes. With respect to overland flow, the prediction error of the maximum flow height can be limited to ± 50% within a 90% confidence interval. However, the prediction of the front propagation velocity can only be controlled to ± 100% within a 90% confidence interval. Furthermore, a sensitivity analysis of the dynamic friction coefficient of water was performed and a suggested range from 0.01 to 0.1 was given for reference.

Energy Transfer and Reverse Flow Characteristics in the Interaction Process between Non-Breaking Solitary Wave and a Steep Seawall: A Case Study

This study utilized a two-dimensional numerical viscous wave tank to simulate the run-up and run-down processes of non-breaking solitary waves on a steep seawall. The research aimed to investigate the transformation between wave potential energy and kinetic energy, the evolution mechanisms of the wave and flow fields, and the correlation between the dynamic pressure gradient and the reverse flow near the sloping bed. The numerical model results were consistent with laboratory measurements of free surface elevations and flow velocity profiles, demonstrating the accuracy of the numerical model. This study focused on a solitary wave with a wave-height-to-water-depth ratio of 0.15, propagating on a representative seawall with a steep slope of 1:3 along the western coast of Taiwan. The simulation results indicate that the maximum run-up height occurs when the potential energy is at its highest. Undertow is caused by the adverse pressure gradient within the flow field, and the dynamic pressure on the sloping bed is directly proportional to the free surface elevation. Therefore, by observing the spatial changes in the free surface elevation, we can indirectly determine the occurrence time of undertow.

The solitary wave run-up on sloped beaches protected by submerged rigid breakwaters

The reduction of wave run-up induced by solitary waves at sloping beaches, by means of submerged barriers has been investigated. To this purpose, 198 laboratory tests were undertaken in the wave channel of the University of Calabria, by adopting three beach slopes, three water depths and square and rectangular rigid type-barriers. To make comparisons on the performance of the barriers in reducing the run-up, 99 tests were carried out considering the same experimental conditions, without the barriers. Two approaches were used to deduce the resulting solitary wave run-up in the presence of the barriers starting from the knowledge of wave parameters without them. The first approach uses the wave amplitude, whereas the second, more physically based, uses the wave energy, depending on the horizontal velocity and the wave pressure. Based on the calculation of the transmission coefficients using the dimensional analysis, the transmitted wave amplitude and energy after the barriers were deduced. Using these quantities, the recent semi-empirical formulas deduced by Casella et al. (2022) in the absence of the barriers were extended to determine the solitary wave run-up in their presence.

Performance Evaluation of DualSPHysics and COMCOT Programs through Numerical Testing for Simulating Tsunami Propagation and Overtopping on Seawalls

A tsunami is a destructive wave that can cause massive damage to coastal infrastructure. One of the disaster mitigation measures that can be chosen is the construction of coastal protection infrastructure, such as a seawall. Seawalls play a crucial role in protecting coastal areas as they can reduce wave energy and minimize the impact of tsunami-induced damage. However, during the 2011 Japan tsunami, the seawall built in Taro city failed as it proved to be less effective in handling the tsunami waves. The research conducted aims to model the propagation and overtopping of tsunami waves on the seawall, initially carried out through physical laboratory experiments, and later transformed into numerical test models using the Smoothed Particle Hydrodynamics (SPH) method and the Cornell Multigrid Coupled Tsunami (COMCOT) model. In this study, the parameters being compared include wave height, wave propagation, and overtopping on the seawall under two scenarios : the run-up of solitary waves on a shore without a seawall and the overtopping condition with a seawall using the Solitary Wave generation type, following the experiments conducted by Huang et al en 2022. Several parameters in the laboratory case study should be considered to expand our understanding of the systematically discussed tsunami propagation and overtopping processes and evaluate the capabilities of the SPH and COMCOT models.

Numerical Simulation on Reduced Runup Height of Solitary Wave by Fixed Submerged and Floating Rectangular Obstacles

The wave runup height is one of the most important parameters for affecting the design of coastal structures such as dikes, revetments, and breakwaters. In this study, SWASH (Zijlema et al., 2011), a non-hydrostatic pressure numerical model, was used to analyze the effect of reducing The wave runup height of solitary waves by submerged and floating rectangular obstacles. It was confirmed that the SWASH model reproduces the propagation, breaking, and runup of solitary waves quite well. In addition, it was confirmed that the wave deformation of the solitary wave by submerged and floating rectangular obstacles was well reproduced. Finally, we conducted an examination of the effect of reducing the runup height of submerged and floating rectangular obstacles. Reduced runup heights are calculated and the characteristics of runup height reduction according to the dimensions of the obstacle were analyzed. The energy attenuation effect of the floating obstacle is greater than the submerged obstacle, and it is shown to be more effective in reducing the runup height.

Large-Amplitude Internal Wave Transformation into Shallow Water

Abstract Large-amplitude internal solitary wave (ISW) shoaling, breaking, and run-up was tracked continuously by a dense and rapidly sampling array spanning depths from 500 m to shore near Dongsha Atoll in the South China Sea. Incident ISW amplitudes ranged between 78 and 146 m with propagation speeds between 1.40 and 2.38 m s−1. The ratio between wave amplitude and a critical amplitude A0 controlled breaking type and was related to wave speed cp and depth. Fissioning ISWs generated larger trailing elevation waves when the thermocline was deep and evolved into onshore propagating bores in depths near 100 m. Collapsing ISWs contained significant mixing and little upslope bore propagation. Bores contained significant onshore near-bottom kinetic and potential energy flux and significant offshore rundown and relaxation phases before and after the bore front passage, respectively. Bores on the shallow forereef drove bottom temperature variation in excess of 10°C and near-bottom cross-shore currents in excess of 0.4 m s−1. Bores decelerated upslope, consistent with upslope two-layer gravity current theory, though run-up extent Xr was offshore of the predicted gravity current location. Background stratification affected the bore run-up, with Xr farther offshore when the Korteweg–de Vries nonlinearity coefficient α was negative. Fronts associated with the shoaling local internal tide, but equal in magnitude to the soliton-generated bores, were observed onshore of 20-m depth.

A comprehensive analysis of solitary wave run-up at sloping beaches using an extended experimental dataset

In this work, the phenomenon of wave run-up induced by solitary waves in breaking and nonbreaking conditions has been investigated. Despite several studies have been carried out to study this kind of physical process, a robust determination of solitary wave run-up on plane beaches over a wide range of wave conditions and beach slopes is still lacking. In this context, 99 laboratory tests were undertaken in the wave channel of the University of Calabria by adopting three beach slopes and three water depths. The experimental set-up was made up of four wave gauges, a pressure transducer, a velocimeter and an adjustable waterproof ramp to reproduce the plane sloping beach. Using graduated markers along the beach and a video camera, the run-up was deduced. The present experiments cover a range not extensively analyzed in literature, and specifically related to wave run-up comparable or greater than the water depth. The laboratory run-up, velocity and pressure data were validated using the IH-2VOF numerical model. Two different approaches, both for breaking and nonbreaking conditions, were used to obtain semi-analytical formulas for the wave run-up. The first approach is conventional, being based on wave non-linearity and beach slope, while the second is based on the wave energy.

Numerical Simulation of Solitary Waves Propagating on Stepped Slopes Beaches

The objective of the current paper is to study the propagating and breaking of solitary waves on stepped slopes beaches, to simulate the shoaling and breaking, specifically the location of breaking point Xb, and solitary wave height at breaking Hb of solitary waves on the different stepped slopes. Ansys Fluent is used to implement the simulation, a two-dimensional volume of fluid (VOF) which is based on the Reynolds-Averaged Navier–Stokes (RANS) equations and the k–ε turbulence closure solver. The obtained results were firstly validated with existing empirical formulas for solitary wave run-up on the slope without stepped structure and are compared with the experimental and numerical results. The numerical computation has been carried out for several, configurations of beach slopes with tan ß= 1:15, 1:20, 1:25, wave height H0= 0.04, 0.06, 0.08m, water depth h0= 0.15, 0.2, 0.25m, and step height Sh= 0.025, 0.05, 0.075m. A set of numerical simulations were implemented to analyze shoaling and breaking of solitary waves, wave reflection, wave transmission, and wave run-up with various parameters wave heights, water depth, beach slopes, and Sh step height.

Overtopping volume of impulse waves in glacier lakes: Experimental and numerical investigation using rigid dams

Glacier lake outburst floods (GLOFs) that result from the breaching of moraine dams are typical disasters in high mountain regions of the Tibetan Plateau, which endanger vital infrastructures such as the Sichuan-Tibet Railway and Highway. Breaching can be caused by tsunami-type waves, resulting from glaciers calving and falling into impounded lakes, which overtop and destroy their natural boundaries. In this work, we develop an equation for the overtopping volume based on the simplified geometry of fluids running up an incline. Specifically, we propose that overtopping can be viewed in terms of the run-up of solitary waves impacting a structure from which relationships with the dam and wave properties can be established. This equation is tested and verified against physical experiments and numerical simulations of waves overtopping rigid, impermeable triangular structures which serve as idealized natural dams. The proposed equation is found to adequately account for the first-order dependencies of the overtopping volume on the different dam and wave properties. We extend the applicability of this analysis to erodible dams made up of glacier tills, wherein we show the basic dependence of the overtopping volume on the erodibility of the dams and flood scales.

Effects of the bed roughness and beach slope on the non-breaking solitary wave runup height

In this study, a series of laboratory experiments were carried out to investigate the effects of bed roughness and beach slope on the non-breaking solitary wave runup height. Experimental measurements reveal that the bottom friction plays different roles in the solitary wave runup process for different beach slopes. Under the same incident wave condition, the maximum runup height on a smooth steel bed was found to be larger than that on a rough sandpaper bed over a 10° beach slope, whereas the same runup heights were observed for the smooth and rough beds over a 45° beach slope. By introducing the runup similarity parameter, quantitative relationships between the maximum runup height and the bed roughness, as well as the beach slope, were analyzed. On the basis of Synolakis (1987), a new empirical model for estimating the non-breaking solitary wave induced runup height was proposed, after incorporating a slope-related enhancement coefficient and a friction-related reduction coefficient. It is found that both the enhancement coefficient (being always larger than unity) and the reduction coefficient (being always smaller than unity) are large under the small relative incident wave height and steep beach slope condition. Influence of the bed roughness is more significant under the small runup similarity parameter condition, under which the runup height monotonically decreases with the increase of the runup similarity parameter for the small bed roughness, whereas it first increases then decreases with the increase of the runup similarity parameter for the large bed roughness. Nevertheless, influence of the bed roughness could be neglected for the large runup similarity parameter condition. These results are consistent with experimental observations. Finally, the present model was validated using a large amount of existing solitary wave runup data with different bed frictions and beach slopes. The good agreement between the model predictions and measured data indicates the reliability and accuracy of the present model.

Modeling of sediment transport in rapidly-varying flow for coastal morphological changes caused by tsunamis

Tsunamis can cause significant coastal erosion and harbor sedimentation that exacerbate the concomitant flood hazards and hamper recovery efforts. Coupling of the non-hydrostatic model NEOWAVE and the sediment transport model STM provides a tool to understand and predict these morphological changes. The non-hydrostatic model can describe flow fields associated with tsunami generation, wave dispersion as well as shock-related and separation-driven coastal processes. The sediment transport module includes non-equilibrium states under rapidly-varying flows with a variable exchange rate between bed and suspended loads. A previous flume experiment of solitary wave runup on a sandy beach provides measurements for a systematic evaluation of sediment transport driven by shock-related processes. The extensive impacts at Rikuzentakata, Iwate, Japan and Crescent City Harbor, California, USA from the 2011 Tohoku tsunami provide pertinent case studies for model benchmarking. We utilize a self-consistent fault-slip model to define the tsunami source mechanism and field survey data to determine the characteristic grain sizes and morphological changes. The near-field modeling at Rikuzentakata gives reasonable fits with observed large-scale erosion and sedimentation associated with transition of the incoming wave into a surge and formation of a hydraulic jump in the receding flow. The non-hydrostatic module becomes instrumental in resolving tsunami waves at the far-field shore of Crescent City. The results show good agreement with local tide-gauge records as well as observed scour around coastal structures and deposition in basins resulting from separation-driven processes. While the erosion patterns in the laboratory and field cases can be explained by suspended sediment transport in the receding flow, bed load transport can be a dominant mechanism in sediment laden flows and scour around coastal structures.

Parallel-Computing Two-Way Grid-Nested Storm Surge Model with a Moving Boundary Scheme and Case Study of the 2013 Super Typhoon Haiyan

This study presents a numerical tool for calculating storm surges from offshore, nearshore, and coastal regions using the finite-difference method, two-way grid-nesting function in time and space, and a moving boundary scheme without any numerical filter adopted. The validation of the solitary wave runup on a circular island showed the perfect matches between the model results and measurements for the free surface elevations and runup heights. After the benchmark problem validation, the 2013 Super Typhoon Haiyan event was selected to showcase the storm surge calculations with coastal inundation and flood depths in Tacloban. The catastrophic storm surges of about 8 m and wider, storm-induced inundation due to the Super Typhoon Haiyan were found in the Tacloban Airport, corresponding to the findings from the field survey. In addition, the anti-clockwise, storm-induced currents were explored inside of Cancabato Bay. Moreover, the effect of the nonlinear advection terms with the fixed and moving shoreline and the parallel efficiency were investigated. By presenting a storm surge model for calculating storm surges, inundation areas, and flood depths with the model validation and case study, this study hopes to provide a convenient and efficient numerical tool for forecasting and disaster assessment under a potential severe tropical storm with climate change.

Runup of breaking solitary waves on rough uniform slopes

An experiment consisting of 122 tests was conducted for breaking solitary waves on rough and smooth slopes. The constant water depth and the incident wave height were varied to cover a wide range of solitary wave nonlinearity, i.e., from 0.039 to 0.458. Free surface elevations, velocities and maximum runup heights were measured for four slope surfaces of different roughness. A surf similarity parameter proposed previously for smooth slopes was used to analyze the runup data on rough slopes. A reduction factor was introduced to compare the runup heights of breaking solitary waves on rough and smooth slopes and to unify the runup formula for smooth and rough slopes. The fitted reduction factor for each of the rough surfaces was less than unity (smooth slope) and in the range of 0.48–0.88. Existing runup formulas are used to confirm the accuracy and applicability of runup height predictions. The measured free surface elevations and velocities were analyzed to quantify the roughness effect on solitary wave runup. Laboratory observations reveal that the slope roughness noticeably influences the hydrodynamics in the surf and swash zones.

Study of beach permeability's influence on solitary wave runup with ISPH method

Coastal protections, such as dykes, seawalls, breakwaters and natural beaches can often be considered as porous structures. The accurate prediction of wave motion around porous structures is necessary for the effective design of durable coastal protections. Smoothed particle hydrodynamics (SPH) is a meshless particle-based method suitable for the simulations of violent free-surface flows and their interaction with porous structures. In this paper, an incompressible SPH (ISPH) model is applied to the simulations of solitary wave runup on permeable slopes. The apparent density concept is introduced to allow the smooth particles’ volume to change when shifting between the pure-fluid region and porous region. The present simulations consider both the triangular beaches with uniform permeability and the solid beaches with overlying porous layers. The study focuses on the influence of permeability on the maximum wave runup heights. New runup laws are proposed, which offer guidelines for the design of porous coastal protections.

MPM simulation of solitary wave run-up on permeable boundaries

Waves attenuate rapidly as they propagate through porous media due to significant energy dissipation. The ability of permeable armour layers to absorb wave energy is therefore of great interest to the researchers and engineers tasked with the construction of structures that defend vulnerable coastlines from the wave attack. The goal of this research is to determine the effectiveness of vertical and sloped permeable barriers in minimising the wave run-up. Traditional methods for ascertaining the efficacy of protective barriers have used small-scale physical models. However, these are expensive and have been shown to suffer from the scaling problems, therefore numerical methods are gaining popularity. This paper investigates the effect of modifying the mean grain size of a permeable barrier on the run-up response to a solitary wave, using the Material Point Method (MPM), which is capable of handling large deformation problems within a Lagrangian framework, with a background mesh facilitating the solution of the governing equations and allowing for simple imposition of the boundary conditions. A double-point MPM is adopted, with two sets of material points representing the solid and liquid phases respectively, to accurately model situations where the fluid moves through the solid skeleton, such as in the case of wave run-up on porous structures. The multi-phase version of the MPM package Anura3D (www.anura3d.com) is used in the study, with a focus on the influence of changing the mean grain size of composition particles of a porous structure on solitary wave run-ups, on both vertical and sloped permeable boundaries. It has been shown that with an increase in the mean grain size, and therefore the permeability, the overall wave run-up height can be significantly reduced. The proposed study could contribute to a better understanding on the wave run-up reduction on porous structures, and provide useful design guidelines to the coastal defences.

Boussinesq modeling for the runup of solitary-like waves on a slope and circular island

This paper presents numerical simulations for the runup of solitary-like waves on a sloping beach and circular island. The solitary-like wave is constructed by introducing an elongation parameter to the wavenumber of the solitary wave. All simulations are performed with a Boussinesq model solving the fully nonlinear and weakly dispersive Boussinesq equations. The numerical model is validated by comparing the computed runup of solitary waves on sloping beach and circular island to the experimental runup data. The propagation of the solitary-like wave in a constant flume and the runup on a sloping plane beach and the circular island are simulated. Initial surface elevation of a solitary-like wave is provided as the initial condition. The waveform of a solitary-like wave with different values of the elongation parameter shows that undular bores and a train of solitary waves could appear in offshore shallow regions. Runup of solitary-like waves on the sloping beach and circular island shares the same decay trends but different powers. Abnormal rear runup only exceeds the front runup for some cases of given nonlinearity parameters and relative wavelength and the waterline diameter of a circular island. The computed ratio of rear-runup/front-runup is less than 1.3.

Stagnant peaked free surface released at a sloping beach

A stagnant free-surface flow is an instantaneous flow field of pure acceleration with zero velocity and a deformed surface. There exists a potential-flow acceleration field. With zero velocity and the acceleration field given, there is a limiting free-surface position which possesses one peak at its point of highest elevation. By complex analysis, it can be shown that the surface peak has a right angle. We elaborate on an elementary model of two-dimensional stagnant free-surface flow with a peak. Our model may serve to describe a situation of maximal single-wave run-up with a given energy at a uniformly sloping beach. The highest possible run-up of an incoming solitary wave corresponds to zero kinetic energy. It encompasses an idealized situation where the kinetic wave energy is converted into potential energy in a water mass piling up along the slope to become stagnant at one single moment. Multipoles with singularities outside the fluid domain may give rise to a smooth and gradual deceleration needed for a non-breaking run-up process. A pair of dipoles with an orientation perpendicular to a given slope represents the stagnant acceleration fields with the highest surface peak spatially concentrated along the slope. Thereby, a one-parameter family of surface shapes is constituted, only dependent on the slope angle. The initial flow field, the initial free surface, the initial isobars and the geometric parameters are all calculated for different slope angles.

Assessment of Numerical Methods for Plunging Breaking Wave Predictions

This study evaluates the capability of Navier–Stokes solvers in predicting forward and backward plunging breaking, including assessment of the effect of grid resolution, turbulence model, and VoF, CLSVoF interface models on predictions. For this purpose, 2D simulations are performed for four test cases: dam break, solitary wave run up on a slope, flow over a submerged bump, and solitary wave over a submerged rectangular obstacle. Plunging wave breaking involves high wave crest, plunger formation, and splash up, followed by second plunger, and chaotic water motions. Coarser grids reasonably predict the wave breaking features, but finer grids are required for accurate prediction of the splash up events. However, instabilities are triggered at the air–water interface (primarily for the air flow) on very fine grids, which induces surface peel-off or kinks and roll-up of the plunger tips. Reynolds averaged Navier–Stokes (RANS) turbulence models result in high eddy-viscosity in the air–water region which decays the fluid momentum and adversely affects the predictions. Both VoF and CLSVoF methods predict the large-scale plunging breaking characteristics well; however, they vary in the prediction of the finer details. The CLSVoF solver predicts the splash-up event and secondary plunger better than the VoF solver; however, the latter predicts the plunger shape better than the former for the solitary wave run-up on a slope case.