Non-hydrostatic Wave Model Research Articles

Reconstruction of the 2002 tsunami at Stromboli using the non-hydrostatic WAVE model (NHWAVE)

Abstract In December 2002, two landslides along the Sciara del Fuoco at Stromboli triggered large tsunami waves that caused significant damage on the coast of the island up to an elevation of about 10 m above sea-level. In this work, we report in detail the items and the methods used to reconstruct the 2002 tsunami at Stromboli highlighting their strengths and limits. In particular, we describe: (1) the non-hydrostatic WAVE model used to simulate the triggering landslide, the wave propagation and the inundation/runup on the land; (2) the data and methods used to generate the topo-bathymetric computational grid; and (3) the field data acquired on Stromboli after the 2002 tsunami used as ground truth for checking the simulation outputs. Our results show that the most severe damage on the coast of Stromboli could have been caused by the interaction of successive waves triggered by the same landslide. In addition, we also describe the influence that the bathymetry had on the waves propagation and interaction.

Tidal modulation of infragravity wave dynamics on a reflective barred beach

Nearshore dynamics of infragravity (IG) waves on a barred beach with a steep foreshore (~1:10) is discussed. One-day nearshore wave measurements were conducted under a low-energetic offshore wave condition (wave height: ~0.5 m). Notably, IG waves were modulated in antiphase with the tidal water level. The observed data suggested that IG waves significantly developed over the shoaling zone as bound waves at low tide, whereas the breakpoint-forcing excited the dynamic wave setup in the swash zone at high tide. The tidal modulation mechanisms were investigated using a one-dimensional nonhydrostatic wave model, SWASH, which successfully reproduced the observed wave fields over tidal cycles. The analyses of energy transfer processes based on the model results revealed that the sandbar played a key role in modulating IG waves. Incident short waves did not break over the sandbar during high tides and excited bound IG waves dominantly on the shoreward of the sandbar. Subsequently, the IG waves lost approximately 40% of their energy even on the steep foreshore due to the breakpoint-forcing by the wave groups which were unbroken over the sandbar. In contrast, decreased water depth during low tides intensified energy transfer to IG waves on the seaward of the sandbar, and short wave attenuation over the sandbar resulted in nearly full reflection of IG waves from the shore. This study highlights the strong variability of IG wave dynamics on a barred beach profile.

Numerical investigation of wave attenuation by coupled flexible vegetation dynamic model and XBeach wave model

Quantitative investigation of wave attenuation by flexible vegetation is an increasingly prominent area in wave mechanics. Given that the existing XBeach non-hydrostatic wave model can only simulate wave attenuation by rigid vegetation, a mechanical model for simulating submerged flexible vegetation dynamics driven by waves and a coupling simulation method between the flexible vegetation dynamic model and the XBeach non-hydrostatic wave model were proposed. To assess the performance of the proposed coupled model, a wave flume experiment was conducted. This study demonstrates the validity of the new flexible vegetation dynamic model in simulating forces and motions of a single submerged flexible stem. The validation between modeled results and experimental data shown that the coupled model is capable of reproducing the wave attenuation in the presence of flexible vegetation. The above results further indicate that the coupling implementation is valid and reliable. Besides, this study further confirms that flexible vegetation is able to attenuate wave height, and the elasticity modulus of flexible vegetation plays a vital role in wave attenuation when all the other variables are invariant. This study can broaden the application of the XBeach model and provide technical support for the engineering practice of coastal nature-based solutions.

Numerical Study on Wave Attenuation of Tsunami-Like Wave by Emergent Rigid Vegetation

The extreme surges and waves generated in tsunamis can cause devastating damages to coastal infrastructures and threaten the intactness of coastal communities. After the 2004 Indian Ocean tsunami, extensive physical experiments and numerical simulations have been conducted to understand the wave attenuation of tsunami waves due to coastal forests. Nearly all prior works used solitary waves as the tsunami wave model, but the spatial-temporal scales of realistic tsunamis differ drastically from that of solitary waves in both wave period and wavelength. More recent work has questioned the applicability of solitary waves and been looking towards more realistic tsunami wave models. Therefore, aiming to achieve more realistic and accurate results, this study will use a parameterized tsunami-like wave based on wave observations during the 2011 Japan tsunami to study the wave attenuation of a tsunami wave by emergent rigid vegetation. This study uses a high-resolution numerical wave tank based on the non-hydrostatic wave model (NHWAVE). This work examines effects of prominent factors, such as wave height, water depth, vegetation density and width, on the wave attenuation efficiency of emergent rigid vegetation. Results indicate that the vegetation patch can dissipate a considerable amount of the total wave energy of the tsunami-like wave. However, the tsunami-like wave has a higher total wave energy, but also a lower wave energy dissipation rate. Results show that using a solitary instead of a tsunami-like wave profile can overestimate the wave attenuation efficiency of the coastal forest.

Investigation of wave-driven currents and statistical moments of irregular waves over a one dimensional horizontal fringing reef

Effects of significant wave height and seaward depth on the distribution of the wave-induced set-up, wave-driven currents and the statistical moments (kurtosis and skewness) of the horizontal velocities and surface elevations of irregular waves propagating over a fringing reef are investigated using a fully non-hydrostatic wave model. It is found that wave-induced set-up and wave-driven currents obviously increase with increasing significant wave height and seaward depth. Besides, as the seaward depth is relatively shallow, the local maximum skewness of horizontal fluid velocities and surface elevations are achieved nearly at the same location. However, the location of the maximum kurtosis of the horizontal fluid velocities is on the inshore side of the flat compared to that of the surface elevations. Additionally, as a seaward depth increases, variations of skewness and kurtosis of horizontal velocities are more significant compared to surface elevations, and the maximum kurtosis and skewness of the horizontal velocities located downstream compared to those of the surface elevations.

A two-layer non-hydrostatic landslide model for tsunami generation on irregular bathymetry. 2. Numerical discretization and model validation

We describe a numerical treatment for a two-layer coupled model developed for the investigation of submarine landslides and resulting tsunami generation over irregular bathymetry. The landslide model is formulated in a Cartesian coordinate system oriented with the still water level in order to facilitate coupling between water and ground motions. Motions in the upper water layer are simulated using the 3D non-hydrostatic wave model NHWAVE. Governing equations for the lower-layer landslide are integrated over the slide thickness, and include options for choosing either Newtonian viscous or granular rheology closures. The lower layer equations retain the effects of non-hydrostatic pressure, allowing the model to simulate motions over arbitrary and locally steep bathymetry. The model equations are solved using a Runge–Kutta scheme for time integration and a Godunov-type finite volume scheme for spatial derivatives, with non-conservative terms formulated using a finite difference scheme. The resulting model is verified in comparison to two laboratory experiments involving granular slide motion, and compared to observations for a field event during the 1964 Alaska earthquake.

A two-layer non-hydrostatic landslide model for tsunami generation on irregular bathymetry. 1. Theoretical basis

We describe a two-layer, coupled model for landslide motion over irregular bathymetry and resulting water column motion and tsunami generation. The three-dimensional non-hydrostatic wave model NHWAVE (Ma et al., 2012) is applied as the upper-layer model to simulate landslide-generated tsunami waves. Here, we focus on the derivation of governing equations for the lower-layer model, where the landslide is described as either a viscous mud flow or a saturated granular debris flow. The governing equations are depth-integrated in a Cartesian coordinate system referenced to the still water surface. Vertical acceleration of the slide is taken into account by including non-hydrostatic pressure effects within the slide. The model is validated in comparison to analytic results of Mangeney et al. (2000) and the slope-oriented numerical approach of Ma et al. (2015). The numerical approach and further examples are described in the companion paper Zhang et al. (2021).

On the accuracy of internal wave generation method in a non-hydrostatic wave model to generate and absorb dispersive and directional waves

The weakly reflective wave generation is a wave generation and absorption method in phase-resolving models, based on the assumption that the waves propagating towards the wave generation boundary are small amplitude shallow water waves with direction perpendicular to the boundary. This assumption makes the method weakly reflective for dispersive and directional waves. The internal wave generation method was proposed by Vasarmidis et al. (2019b) as an alternative, for the non-hydrostatic wave model, SWASH, to avoid reflections. In this study, a comparison is made between the performance of the new internal wave generation method and the weakly reflective wave generation method. It is shown that using the internal wave generation leads to a significantly more accurate prediction of the resulting wave field in case of waves reflected back to the numerical boundary. Additionally, the internal wave generation method is extended to short-crested waves and SWASH is validated for the first time with experimental data for the cases of wave propagation over a shoal and wave diffraction around a wall. The proposed extended internal wave generation method increases the capability of SWASH towards the study of wave propagation of highly dispersive short-crested waves in coastal environments with minimal reflection from the boundaries.

Efficient two-layer non-hydrostatic wave model with accurate dispersive behaviour

A 2-layer non-hydrostatic model with improved dispersive behaviour is presented. Due to the assumption of a constant non-hydrostatic pressure distribution in the lower layer, the dispersive behaviour is improved without much additional computational time. A comparison with linear wave theory showed that this 2-layer model gives a better result for the dispersion relation and shoaling of waves in intermediate water. This means that the 2-layer model is applicable in shallow and intermediate water depths (up to relative depths kh equals 4), whereas the 1-layer model is only applicable in shallow water depths (kh smaller than 1). Three laboratory experiments, including a fringing reef and a barred beach, were used to validate the presented mode for different hydrodynamic conditions. Based on these results, it can be concluded that the 2-layer model can be applied to accurately simulate the bulk wave height and spectral properties. The low frequency wave height, the setup and in particular the second order statistics contain more scatter, but the model accurately captured the general trend. Furthermore, the model showed good results for complex bathymetries in shallow to intermediate water.

Development of Two-Dimensional Non-Hydrostatic Wave Model Based on Central-Upwind Scheme

In this study, a two-dimensional depth-integrated non-hydrostatic wave model is developed. The model solves the governing equations with hydrostatic and non-hydrostatic pressure separately. The velocities under hydrostatic pressure conditions are firstly obtained and then modified using the biconjugate gradient stabilized method. The hydrostatic front approximation (HFA) method is used to deal with the wave breaking issue, and after the wave breaks, the non-hydrostatic model is transformed into the hydrostatic shallow water model, where the non-hydrostatic pressure and vertical velocity are set to zero. Several analytical solutions and laboratory experiments are used to verify the accuracy and robustness of the developed model. In general, the numerical simulations are in good agreement with the theoretical or experimental results, which indicates that the model is able to simulate large-scale wave motions in practical engineering applications.

Experiences with SWASH on modelling wave propagation over vegetation: comparisons with lab and field data

The vegetation capacity to protect the coasts from wave action is becoming more important and attractive due to ongoing sea level rise and increasing storminess. In addition, it is a quite environmentally friendly way. Quantifying the vegetation effect in wave propagation will be relevant for coastal management. A non-hydrostatic wave model based on the nonlinear shallow water equations, SWASH, offers opportunities to quantify the wave dissipation effect in vegetation fields. However, limited applications of SWASH addressing this subject can be found in the literature and therefore it is important to enhance the existing knowledge on the model behaviour. In this research, in order to understand the characteristics of the SWASH model further, the model is applied to reproduce the significant wave height (Hs) evolution over vegetation fields measured in flume experiments and in field campaign. Overall, SWASH performed very well in reproducing the Hs evolution measured both in the laboratory and in the field. In the case of flume data, the statistical scores MBE, RMSE and MRE, showed that the SWASH performance clearly improved when increasing the number of vertical layers assumed in the simulations. In the case of field data, considering a vegetation factor (Vf ) between 0.1 and 0.5, that represents the overall effect of scarcely known numerical vegetation parameters, led to a fairly good SWASH performance in modelling the Hs evolution over vegetation.

Reconstruction and Analysis of Freak Waves Generated From Unidirectional Random Waves

Abstract The generation of two freak waves in a broadband and a narrowband random series registered in the experiments of Li, J. X., Li, P. F., and Liu, S. X. (2013, “Observations of Freak Waves in Random Wave Field in 2D Experimental Wave Flume,” China Ocean Eng., 27(5), pp. 659–670) is precisely reconstructed using a fully non-hydrostatic water wave model. The simulation results indicate that even when the background spectral bandwidths are different, the evolution processes of the two freak waves are similar. Both freak waves emerge quickly during the transition from normal states to extreme events. The freak waves can persist over a long distance, i.e., approximately 5 peak wavelengths. The reconstructed time series in both the backward and forward locations at which the freak waves were recorded reveal that the largest freak wave crests were not captured in the experiment. The freak waves gradually emerged from an intense wave group. The waves developed quickly during the transition from a normal state to an extreme event. Very deep troughs were also formed in the evolution process. The two freak waves were actually generated via different spectral bandwidth processes, but the generation mechanisms of the rogue waves were similar. By analyzing the time series of the freak wave groups, the formation of the freak waves is found to result from the combined effect of the dispersive focusing, the third-order resonant wave interactions, and the higher harmonics.

Numerical Simulation of Wave Overtopping on Breakwater with an Armor Layer of Accropode Using SWASH Model

In this paper, a new method for predicting wave overtopping discharges of Accropode armored breakwaters using the non-hydrostatic wave model Simulating WAves till SHore (SWASH) is presented. The apparent friction coefficient concept is proposed to allow the bottom shear stress term calculated in the momentum equation to reasonably represent the effect of comprehensive energy dissipation caused by the roughness and seepage during the wave overtopping process. A large number of wave overtopping cases are simulated with a calibrated SWASH model to determine the values of equivalent roughness coefficients so that the apparent friction coefficients can be estimated to achieve the conditions with good agreement between numerical overtopping discharges and those from the EurOtop neural network model. The relative crest freeboard and the wave steepness are found to be the two main factors affecting the equivalent roughness coefficient. A derived empirical formula for the estimation of an equivalent roughness coefficient is presented. The simulated overtopping discharges by the SWASH model using the values of the equivalent roughness coefficient estimated from the empirical formula are compared with the physical model test results. It is found that the mean error rate from the present model predictions is 0.24, which is slightly better than the mean error rate of 0.26 from the EurOtop neural network model.

Laboratory and Numerical Experiments on Stem Waves due to Random Waves

Shin, C.H.; Yoon, S.B., and Jho, M.H., 2019. Laboratory and numerical experiments on stem waves due to random waves. In: Lee, J.L.; Yoon, J.-S.; Cho, W.C.; Muin, M., and Lee, J. (eds.), The 3rd International Water Safety Symposium. Journal of Coastal Research, Special Issue No. 91, pp. 111-115. Coconut Creek (Florida), ISSN 0749-0208.In this study, precisely-controlled laboratory experiments were conducted to investigate the characteristics of stem waves developed by the incidence of unidirectional random waves. The measured data were compared with numerical simulation results and analytical solutions. Numerical simulation using the non-hydrostatic wave model, SWASH, were performed. The followings were found: for the case of small amplitude waves no stem wave is developed, and the analytical solution agrees well with the measured data. However, for the finite amplitude waves the analytical solution gives no stem wave, while the measured data show clearly stem waves along the wall. On the other hand, the numerical results are found to agree well with the measured data. The width and normalized wave height of stem waves were found to be wider and smaller, respectively, as the nonlinear property of the incident waves increased. This tendency is the same as that generated by monochromatic waves. However, the normalized significant wave height distribution normal to the wall showed that, unlike in the case of monochromatic waves, no standing wave pattern is observed.

Sediment Transport Processes during Barrier Island Inundation under Variations in Cross-Shore Geometry and Hydrodynamic Forcing

Inundation of barrier islands can cause severe morphological changes, from the break-up of islands to sediment accretion. The response will depend on island geometry and hydrodynamic forcing. To explore this dependence, the non-hydrostatic wave model SWASH was used to investigate the relative importance of bedload transport processes, such as transport by mean flow, short- (0.05–1 Hz) and infragravity (0.005–0.05 Hz) waves during barrier island inundation for different island configurations and hydrodynamic conditions. The boundary conditions for the model are based on field observations on a Dutch barrier island. Model results indicate that waves dominate the sediment transport processes from outer surfzone until landwards of the island crest, either by transporting sediment directly or by providing sediment stirring for the mean flow transport. Transport by short waves was continuously landwards directed, while infragravity wave and mean flow transport was seaward or landward directed. Landward of the crest, sediment transport was mostly dominated by the mean flow. It was forced by the water level gradient, which determined the mean flow transport direction and magnitude in the inner surfzone and on the island top. Simulations suggest that short wave and mean flow transport are generally larger on steeper slopes, since wave energy dissipation is less and mean flow velocities are higher. The slope of the island top and the width of the island foremost affect the mean flow transport, while variations in inundation depth will additionally affect transport by short-wave acceleration skewness.

Internal Wave Generation in a Non-Hydrostatic Wave Model

In this work, internal wave generation techniques are developed in an open source non-hydrostatic wave model (Simulating WAves till SHore, SWASH) for accurate generation of regular and irregular long-crested waves. Two different internal wave generation techniques are examined: a source term addition method where additional surface elevation is added to the calculated surface elevation in a specific location in the domain and a spatially distributed source function where a spatially distributed mass is added in the continuity equation. These internal wave generation techniques in combination with numerical wave absorbing sponge layers are proposed as an alternative to the weakly reflective wave generation boundary to avoid re-reflections in case of dispersive and directional waves. The implemented techniques are validated against analytical solutions and experimental data including water surface elevations, orbital velocities, frequency spectra and wave heights. The numerical results show a very good agreement with the analytical solution and the experimental data indicating that SWASH with the addition of the proposed internal wave generation technique can be used to study coastal areas and wave energy converter (WEC) farms even under highly dispersive and directional waves without any spurious reflection from the wave generator.

A New Multilayer Nonhydrostatic Formulation for Surface Water Waves

Wang, G.; Liang, Q.; Zheng, J., and Wan, P., 2019. A new multilayer nonhydrostatic formulation for surface water waves. Journal of Coastal Research, 35(3), 693–710. Coconut Creek (Florida), ISSN 0749-0208.This work presents a new multilayer nonhydrostatic formulation for surface water waves. The new governing equations define velocities and pressure at an arbitrary location of a vertical layer and only contain spatial derivatives of maximum second order. Stoke-type Fourier and shoaling analyses are carried out to scrutinize the mathematical properties of the new formulation, subsequently optimizing the representative interface and the location to define variables in each layer to improve model accuracy. Following the analysis, the one-layer model exhibits accurate linear and nonlinear characteristics up to kd = π, demonstrating similar solution accuracy to the existing second-order Boussinesq-type models. The two-layer model with optimized coefficients can maintain its linear and nonlinear accuracy up to kd = 4π, which boasts of better solution accuracy a larger application range than most existing fourth-order Boussinesq model and two-layer Boussinesq models. The three-layer model presents accurate linear and nonlinear characteristics up to kd = 10π, effectively removing any shallow water limitation. The current multilayer nonhydrostatic water wave model does not predefine the vertical flow structures, and more accurate vertical velocity distributions can be obtained by considering the velocity profiles in coefficient optimization.

Efficient Wave Modeling Using Nonhydrostatic Pressure Distribution and Free Surface Tracking on Fixed Grids

For the estimation of wave loads on offshore structures, relevant extreme wave events need to be identified. In order to achieve this, long-term wave simulations of relatively large scales need to be performed. Computational fluid dynamics (CFD) based numerical wave tanks with an interface capturing two-phase flow approach typically require too large computational resources. In this paper, a three-dimensional (3D) nonhydrostatic wave model is presented, which solves the Navier–Stokes equations and employs an interface tracking method based on the continuity of the horizontal velocities along the vertical water column. With this approach, relatively fewer cells are needed in the vicinity of the air–water interface compared to CFD-based numerical wave tanks. The numerical model solves the governing equations on a rectilinear grid, which allows for the employment of high-order finite differences. The capabilities of the new wave model are presented by comparing the wave propagation in the tank with the CFD approach in a two-dimensional (2D) simulation. Further, a 3D simulation is carried out to determine the wave forces on a vertical cylinder. The calculated wave forces using the new approach are compared to those obtained using the CFD approach and experimental data. It is seen that the new approach provides a similar accuracy to that from the CFD approach while providing a large reduction in the time taken for the simulation. The gain is calculated to be about 4.5 for the 2D simulation and about 7.1 for the 3D simulation.

Non-hydrostatic numerical modelling of nearshore wave transformation over shore-oblique sandbars

Observations from high resolution bathymetric surveys on the coast of North Carolina indicate areas with shore-parallel depth contours and other areas with shore-oblique sandbars. These shore-oblique bars are large-scale morphological features that extend up to 1000 m from shore, and previous studies have shown that they are spatially correlated with higher shoreline change rates and induce erosional hotspots. To investigate the influence the oblique sandbar morphology has on nearshore wave transformation and energy distribution, a non-hydrostatic wave model is used with the bathymetry corresponding to two areas with shore-parallel depth contours and two areas with shore-oblique sandbars. The model is applied with a horizontal resolution of 5 m and a boundary condition representative of the directional spectrum of a typical southeast swell event. An idealized approach with relatively short alongshore model domains and cyclic lateral boundary conditions is adopted due to limited availability of high resolution bathymetry, however this approach elucidates the influence of shore-oblique sandbars on the nearshore hydrodynamics. The numerical results show that higher alongshore variability in depth due to the shore-oblique bars drive large alongshore changes in cross-shore wave energy flux and cause spatial variability in dissipation in the nearshore region. The simulations indicate that alongshore variations in wave energy caused by the shore oblique bars results in a 50–75% reduction in wave setup and a 25–40% decrease in alongshore velocity averaged across the surf zone for the conditions investigated. The results suggest that wave-driven hydrodynamics can influence nearshore sediment dynamics to create erosional zones compared to areas with shore-parallel bathymetric contours, since the morphology of shore-oblique sandbars are an important control on wave energy that impacts the beach.

Numerical Evaluation of Design Rules for Non-Breaking Wave Loads on Vertical Walls

This paper describes a numerical evaluation of design rules for the determination of wave loads of non-breaking waves on vertical structures. Design guidelines were proposed by Sainflou (1928) and Goda (1974). These guidelines use geometric parameters of the structure, an incident wave height and a wave period. In practice (cf. CERC, 1984), a Rayleigh distribution of individual wave heights is assumed to determine the design wave height in an irregular wave field. Their reliability and range of applicability are poorly known, especially when the incident wave condition consists of a mixed sea state, like a local wind sea and a low-frequency (swell) component. To validate the above described design methods, we applied the non-hydrostatic numerical wave model SWASH to simulate wave loading on a rigid vertical wall for single and mixed sea states. In addition, we compared the results with linear wave theory and the spectral response approach using transfer functions based on linear wave theory.