Non-hydrostatic Wave Research Articles

On the kinetics of internal gravity waves beyond the hydrostatic regime

We present a new derivation of the kinetic equation for weak, non-hydrostatic internal gravity wave turbulence. The equation is equivalent to the one obtained by Caillol & Zeitlin (Dyn. Atmos. Oceans, vol. 32, issue 2, 2000, pp. 81–112), but it takes a canonical form. We show that it conserves the energy without involving the resonance condition in frequency, and look for the isotropic part of the steady, scale-invariant solutions. We provide a parametrization of the resonant manifold of non-hydrostatic internal gravity wave triadic interactions. This allows us to simplify the collision integral, and to evaluate the transfer coefficients of all triadic interactions. In the hydrostatic limit, our equation is equivalent to the Hamiltonian description of Lvov & Tabak (Phys. Rev. Lett., vol. 87, issue 16, 2001, 168501).

Computational study on complex wave hydrodynamics of multidirectional extreme waves at fringing reef

Due to the special topography of coral reefs, most of the energy of incoming waves can be reduced through the wave breakings at the reef edge, thus playing a vital role in protecting the coastal regions behind coral reefs. In severe weather conditions, wind-generated waves tend to propagate in multiple directions in real ocean environment. The abrupt multidirectional wave-focusing mechanisms may result in the formation of huge waves in both deep and shallow waters, threatening the safety of coastal infrastructures and well-being of coastal communities at the reef islands. Therefore, it is crucial to thoroughly examine the complex hydrodynamics of multidirectional extreme waves at the fringing reefs. Previous studies on reef wave hydrodynamics predominantly focus on unidirectional waves. Only a small number of studies have considered the wave hydrodynamics of fringing reefs under multidirectional waves, let alone multidirectional extreme waves. To address the knowledge gap in previous studies, this study analyzes the complex hydrodynamic processes of multidirectional extreme waves at the fringing reef by applying a nonhydrostatic numerical wave model (NHWAVE). Influences of several key factors are analyzed, such as significant wave height, peak wave period, water depth, incident wave angle, and reef topography. It is expected that the research findings of this study will contribute to a deeper understanding of the hydrodynamics of multidirectional extreme waves at the fringing reef.

A multi-faceted methodology for calibration of coastal vegetation drag coefficient

The accurate prediction of wave height attenuation due to vegetation is crucial for designing effective and efficient natural and nature-based solutions for flood mitigation, shoreline protection, and coastal ecosystem preservation. Central to these predictions is the estimation of the vegetation drag coefficient (Cd). The present study undertakes a comprehensive evaluation of three distinct methodologies for estimating the drag coefficient: traditional manual calibration, calibration using a novel application of state-of-the-art metaheuristic optimization algorithms, and the integration of an empirical bulk drag coefficient formula (Tanino and Nepf, 2008) into the XBeach non-hydrostatic wave model. These methodologies were tested using a series of existing laboratory experiments involving nearshore vegetation on a sloping beach. A key innovation of the study is the first application of metaheuristic optimization algorithms for calibrating the drag coefficient, which enables efficient automated searches to identify optimal values aligning with measurements. We found that the optimization algorithms rapidly converge to precise drag coefficients, enhancing accuracy and overcoming limitations in manual calibration which can be laborious and inconsistent. While the integrated empirical formula also demonstrates reasonable performance, the optimization approach exemplifies the potential of computational techniques to transform traditional practices of model calibration. Comparing these strategies provides a framework to determine effective methodology based on constraints in determining the vegetation drag coefficient.

Numerical study on hydrodynamic impacts of 3D excavation pit on tsunami-like wave at fringing reef

In recent years, dredging reef sand and aggregate as building materials for infrastructures has become a common practice in many atoll islands, which can significantly reshape the wave hydrodynamic environment at the fringing reef. Hence, it becomes necessary to evaluate the hydrodynamic impacts of excavation pits at fringing reef. However, previous research adopted a simplified two-dimensional (2D) excavation pit model, which ignores the influences of 3D geometric characteristics of excavation pit. This study performs a three-dimensional (3D) numerical study on the hydrodynamic impacts of 3D excavation pit on tsunami-like wave at fringing reef by using a nonhydrostatic wave model (NHWAVE). Impacts of several main factors, such as water depth, wave height, pit location, geometric dimensions of pit, the spacing between two serially connected pits and the spacing between two parallel connected pits are carefully discussed. Research results reveal that 3D excavation pit can significantly reshape the wave hydrodynamic environment at the fringing reef, especially the local wave height at the pit. In addition to wave breaking at the reef edge, the 3D excavation pit can further damp out a portion of energy of the breaking surge bore, which can reduce the maximum value of wave runup height at centerline of computational domain. However, maximum value of wave runup height near the two sides of pit are slightly increased to some extent.

Numerical study on performance of submerged permeable breakwater under impacts of multi-directional focused wave groups

Multi-directional waves focusing is considered a crucial method for generating extreme waves, posing a significant threat to coastal infrastructure and personal property. In recent years, there has been widespread construction of permeable submerged breakwaters as a preventive measure against extreme wave damage. Therefore, this study aims to assess the attenuation of multi-directional focused waves by both permeable and impermeable submerged breakwaters based on nonhydrostatic numerical wave model (NHWAVE). Effects of several key parameters are carefully examined. According to research findings, permeable submerged breakwaters offer significant benefits in reducing the wave amplitude, flow velocity, water elevation, and energy flux of multi-directional focused wave groups, while also effectively inhibiting the transfer of wave energy to higher frequencies. Additionally, we observe alternating positive and negative vortices occur at the boundary of the permeable submerged breakwater. The findings of this study are expected to advance the investigation into the hydrodynamic characteristics of submerged permeable breakwaters subjected to focused waves.

Nonhydrostatic internal waves in the presence of mean currents and rotation

Nonhydrostatic internal waves in the presence of mean currents and rotation

A study of the non-linear properties and wave generation of the multi-layer non-hydrostatic wave model SWASH

Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model’s boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

Numerical investigation of infragravity wave hydrodynamics at fringing reef with a permeable layer

In tropical and subtropical coastal regions, coral reefs are abundant and play a vital role in maintaining ecosystem balance. Additionally, they effectively dissipate a significant amount of wave energy that propagates from the open sea towards the coastline, providing coastal areas with protection against wave impacts. Consequently, numerous scholars have conducted extensive research to investigate the hydrodynamic characteristics of wave propagation and transformation over coral reef topography. However, previous studies have often simplified the bottom boundary of coral reefs as impermeable layers, neglecting the fact that the coral reef consists of a permeable canopy structure in the actual marine environment. To fill the knowledge gap of previous research, this study is based on the Non-Hydrostatic Wave Model Solver (NHWAVE) to numerically simulate the propagation process of random waves over permeable coral fringing reefs. The study takes a comprehensive approach by considering the influences of various factors on the hydrodynamic characteristics of random waves over the fringing reef, including incident wave height, reef-flat water depth, peak wave period, permeable layer thickness, permeable layer porosity, and median diameter of the permeable layer. This paper focuses primarily on analyzing the variations in sea-swell wave height (HSS), infragravity wave height (HIG), and mean water level (MWL) along the fringing reef, while conducting a comparative analysis between fringing reefs with a permeable layer and impermeable fringing reefs. The findings reveal that the presence of a permeable layer reduces the shallow water deformation of waves on the fore-reef slope and mitigates the wave-breaking phenomenon near the reef edge, thereby significantly reducing the sea-swell wave height (HSS), infragravity wave height (HIG), and wave setup on the reef-flat. Furthermore, the permeable layer's existence also leads to a decrease in the maximum wave run-up height on the back-reef slope. The research findings of this study can further enhance our understanding of the hydrodynamic characteristics of infragravity waves over fringing reefs, which is significant for studying the impact of random waves on coastal areas and understanding the protective mechanisms of coral reefs in coastal regions.

Rising sea levels and the increase of shoreline wave energy at American Samoa

American Samoa is experiencing rapid relative sea level rise due to increases in global sea level and significant post-2009 earthquake land subsidence, endangering homes and critical infrastructure. Wave and water-level observations collected over a fringing reef at Faga‘itua Bay, American Samoa, in 2017 reveal depth-limited shoreline sea-swell wave heights over the range of conditions sampled. Using field data to calibrate a one-dimensional, phase-resolving nonhydrostatic wave model (SWASH), we examine the influence of water level on wave heights over the reef for a range of current and future sea levels. Assuming a fixed reef bathymetry, model results predict rising sea levels will escalate nearshore extreme water levels that are dominated by an increase in nearshore sea-swell wave heights. Model results provide insight into how and at what reef depths rising sea levels reduce reef capacity to dissipate wave energy, compounding shoreline threats. This study aims to bring increased attention to the immediate threats to American Samoa’s way of life, and to demonstrate the utility of SWASH for extrapolating wave transformation to future sea level.

Laboratory and non-hydrostatic modelling of focused wave group evolution over fringing reef

This paper presents physical experiments and numerical simulations to study the propagation of focused waves group across hypothetical fringing reef profiles. A wave flume is 69 m long and 1.0 m deep, and the reef cross section is made up of a reef face, a reef flat and a vertical wall. A reef crest of 0.085 m is optionally constructed on the outside to replicate the reef crown. By focusing wave trains of the JONSWAP or constant wave amplitude spectrum, the transient wave group is generated on the reef slope. Free surface elevations and flow velocity are measured over time along the flume's centreline. The focused wave process and the development of higher harmonics as a result of the nonlinear interaction over the reef face are clearly visible in the wavelet and FFT analyses of the observed free surface elevation. Low frequency wave is increasing on the reef flat while these short-period wave motions are primarily absorbed by rapid breaking on reef edge and crest. On the flat, it is discovered that reef crest has the effect of reducing short-period wave motion and increasing long-period wave motion. A numerical multi-layer non-hydrostatic wave model is employed and its ability to describe the propagation of focused wave groups over fringing reef profiles is assessed.

Submarine landslide-induced tsunami generation in Makassar strait, Indonesia using Non-hydrostatic wave model

Tsunami modelling of potential landslide-induced tsunami in Makassar Strait is carried out to quantify possible damage to the nearby cities. The numerical model is used to represent the wave generation by using NHWAVE model. The simulations consist of a series of scenarios based on distinct size of the landslide volume. Four landslides with volume 5, 8, 70, and 200 km3 are used as tsunami sources in the initiation stage. The sources are evenly distributed in the strait addressing different landslide location. Maximum wave heights of 1.5 m are found in the area between Palu and Bangkir from case 1 and around Talok from case 2 simulations. The empirical run-up calculation of 7.5 m is estimated at the land for the presented wave height. The value significantly elevates the case 3 and 4 proportional to the volume values. The waves impact more than half of coastline with maximum value found in the Sulawesi side. Interestingly, wide and narrow shelf next to Kalimantan Island plays an important role in reducing the tsunami hazard level.

Influence of Excavation Pits on the Wave Hydrodynamics of Fringing Reefs under Regular Waves

Dredging materials from reef flats have become an important source of sand and aggregates for meeting the infrastructure needs of coral-lined shores in subtropical and tropical regions, especially for low-lying atoll islands. Dredging at the reef flats can generate artificial excavation pits, which not only have profound influences on coral ecological stability but also deeply affect the hydrodynamic characteristics of coral reefs. To deepen the understanding of the influence of excavation on the wave hydrodynamics of fringing reefs, the wave propagation, wave transformation, wave setup, and wave runup processes of regular waves on fringing reefs with artificial pits have been systematically analyzed using a non-hydrostatic numerical wave solver (NHWAVE). The effects of some significant factors have been carefully investigated. According to the study findings, the existence of artificial pits can result in a slight decrease in the wave height around the artificial pit. The time-mean maximum of wave runup height at the backreef slope can be reduced to some extent when the artificial pit is present. When placed close to the reef edge, the artificial pit can have noticeable effects on the hydrodynamic characteristics of fringing reefs, particularly the wave setup along the reef flat. It is hoped that the study findings can provide further reference for evaluation of the influences of artificial pits on the wave hydrodynamics of fringing reefs.

Pollard's exact solution for nonhydrostatic geophysical internal waves with underlying currents in f-plane approximation

In the paper, we present Pollard's exact solution for nonhydrostatic geophysical internal waves with underlying currents in f-plane approximation. We construct the exact solution of nonhydrostatic geophysical internal waves in the presence of underlying currents, extending Pollard's solution to admit underlying horizontal, meridional, and vertical currents at mid-latitudes. Furthermore, we show the underlying meridional and vertical currents are constant. At the equator, the vertical current term vanishes and the meridional current term may vary both zonally and vertically.

Non-Hydrostatic Modelling of Coastal Flooding in Port Environments

Understanding key flooding processes such as wave overtopping and overflow (i.e., water flows over a structure when the crest level of the structure is lower than the water level in front) is crucial for coastal management and coastal safety assessment. In port and harbour environments, waves are not only perpendicular to the coastal structure but also very oblique, with wavefronts almost perpendicular to the main infrastructures in the harbour docks. Propagation and wave–structure interaction of such perpendicular and (very) oblique waves need to be appropriately modelled to estimate wave overtopping properly. Overflow can also be critical for estimating flooding behind any coastal defence. In this study, such oblique and parallel waves (i.e., main wave direction is parallel to the structures) are modelled in a non-hydrostatic wave model and validated with physical model tests in the literature. On top, overflow is also modelled and validated using an existing empirical formula. The model gives convincing behaviours on the wave overtopping and overflow.

Numerical simulating wave propagation over solid obstructions using a non-hydrostatic free surface flow model combined immersed boundary method

The vertical transformed σ-coordinate adopted by a family of non-hydrostatic free surface water flow models is highly efficient in capturing the free surface and uneven bottom. Nevertheless, it is difficult to simulate flows around complex solid obstructions. In this paper, the immersed boundary method (IBM) is developed and implemented in a non-hydrostatic wave model to promote the ability in simulating wave interactions with solid structures. In the IBM spirit, a virtual force term is introduced into the momentum equations to mimic the natural physical boundary conditions. The virtual force terms are added on the named ghost cells neighboring the solid boundary to enforce the no-slip velocity condition on the solid wall. An efficient and robust method is promoted to identify the virtual solid boundary and the ghost cells. A set of test cases are implemented to verify and validate the proposed model in modeling water wave interaction with solid obstructions. The free surface flow model coupled with IBM is well verified, accurate and efficient in modeling wave-structure interaction and is expected to promote the model in engineering applications.

Numerical Study on Hydrodynamics of Submerged Permeable Breakwater under Impacts of Focused Wave Groups Using a Nonhydrostatic Wave Model

Extreme waves, called rogue waves or freak waves, usually occur unexpectedly and with very large wave heights. In recent years, extreme waves were reported not only in deep ocean waters but also in shallow waters, which threaten the safety and intactness of the coastal regions. To prevent the coastal infrastructures and communities from the devastating power of extreme surges and waves, many coastal defense structures were built along the coastline, i.e., submerged permeable breakwaters. However, the number of studies on the hydrodynamic characteristics of a submerged permeable breakwater under the impact of extreme waves is relatively few. In addition, wave focusing has been widely used to generate extreme waves in the past few decades. Hence, as a necessary supplement to the previous research work, the hydrodynamic performance of a submerged permeable breakwater under the impacts of focused wave groups was numerically studied by using a nonhydrostatic numerical wave model (NHWAVE). The influences of several main factors, such as the incident significant wave height, water depth, wave peak period, porosity of the breakwater (n), and the side slope angle of the breakwater, were considered. It is expected that the results of this study will further strengthen the research on the hydrodynamic characteristics of a submerged permeable breakwater under extreme wave conditions.

Modelling wave attenuation through submerged vegetation canopies using a subgrid canopy flow model

Coastal canopies formed by aquatic vegetation can significantly modify nearshore wave processes, which has motivated numerous laboratory and field studies investigating how wave attenuation occurs for a wide range of canopies as a function of wave conditions and canopy properties. To predict wave dissipation by canopies, numerous formulations have been derived to describe canopy drag through use of empirical drag coefficients. In many cases, drag coefficients are used as a tuning parameter to account for complex processes that govern how wave-driven flows interact with individual plants within canopies. This approach, however, can greatly limit the predictive capabilities of both empirical and numerical models. In this study, we extend a non-hydrostatic (phase-resolving) depth-integrated wave model (XBeach non-hydrostatic) with an efficient and robust subgrid canopy flow model to account for the important properties of submerged canopy flows that govern canopy drag forces and wave dissipation. A combination of wave heights, velocities and direct canopy force measurements, as well as data obtained from literature, were used to validate the model, showing the model has good skill in capturing the governing flow dynamics for both rigid and flexible vegetation canopies. By incorporating the canopy flow dynamics, the extended model allows for application of well-established drag coefficients for individual canopy elements to accurately predict wave attenuation over submerged vegetation canopies, rather than relying on empirical formulations or calibration. With the current extension, the model remains computationally efficient, allowing for use on coastal (∼km) scales. • A canopy flow subgrid model was implemented in the open-source XBeach model. • The model was verified using observed wave heights, velocities and canopy forces. • It can accurately predict wave evolution over rigid and flexible submerged canopies. • The model is robust and computationally efficient for use in large scale applications. • Not accounting for canopy flow can lead to substantial errors in wave predictions.

A high accuracy/resolution spectral element/Fourier–Galerkin method for the simulation of shoaling non-linear internal waves and turbulence in long domains with variable bathymetry

A high-order hybrid continuous-Galerkin numerical method, designed for the simulation of non-linear, non-hydrostatic internal waves and turbulence in long computational domains with complex bathymetry, is presented. The spatial discretization in the non-periodic wave-propagating directions, utilizes the nodal spectral element method. Such a high-order element-based discretization allows the highly accurate representation of complex domain geometry along with the flexibility of concentrating resolution in areas of interest. Under the assumption of the normal-to-isobath propagation of non-linear internal waves, a third periodic direction is incorporated via a Fourier–Galerkin discretization. The distinct non-hydrostatic nature of non-linear internal waves and, any instabilities and turbulence therein, necessitates the numerically challenging solution of the pressure Poisson problem. A defining feature of this work is the application of a domain decomposition approach, combined with block-Jacobi/deflation-based preconditioning to the pressure Poisson problem. Such a combined approach is particularly suitable for the long high aspect-ratio complex domains of interest and enables the efficient high-accuracy reproduction of the non-hydrostatic dynamics of non-linear internal waves. Implementation details are also described in the context of the stability of the solver and its parallelization strategy. A series of benchmarks of increasing complexity demonstrate the robustness of the flow solver. The benchmarks culminate with the three-dimensional simulation of a convectively breaking mode-one non-linear internal wave over a realistic South-China-Sea bathymetric transect and background current/stratification profiles.

Performance Evaluation of Seawalls in Mitigating a Real-World Tsunami Wave Using a Nonhydrostatic Numerical Wave Model

In the past few decades, huge surges and waves generated during tsunami events have caused devastating destruction to both onshore and offshore infrastructures, seriously threatening the safety and intactness of coastal communities around the world. As one of the most important coastal protection infrastructures, a seawall can effectively reduce the impact intensity of surges and waves, thus playing a vital role in protecting coastal regions. Most previous studies have systematically investigated the hydrodynamic characteristics of seawalls under the action of regular and irregular waves. Meanwhile, solitary wave models are often used as the wave model for tsunamis. However, vast hydrodynamic differences exist between solitary wave models and real-world tsunamis in terms of both wave profile and wave period. Hence, our understanding of the performance of seawalls in mitigating the damaging power of real-world tsunamis is still insufficient. Hence, it is of great significance to systematically study the performance of seawalls in mitigating the huge surges and waves generated during real-world tsunamis. In comparison to previous research, where the wave profiles of solitary waves were used as tsunami wave models, a parameterized tsunami-like wave based on the recorded wave profile of the 2011 Japan tsunami is applied in this study to evaluate the performance of different seawalls in mitigating tsunami surges and waves by using a nonhydrostatic numerical wave model (NHWAVE). The effects of the prominent factors, such as incident wave height, still water depth, beach slope, and the side slope of the seawall, on the hydrodynamics of the seawall are systematically discussed. It is believed that the research findings drawn from this study can further broaden our understanding of the performance of seawalls in mitigating tsunamis.

Mountain Waves Produced by a Stratified Shear Flow with a Boundary Layer. Part III: Trapped Lee Waves and Horizontal Momentum Transport

Abstract The boundary layer theory for nonhydrostatic mountain waves presented in Part II is extended to include upward-propagating gravity waves and trapped lee waves. To do so, the background wind with constant shear used in Part II is smoothly curved and becomes constant above a “boundary layer” height d, which is much larger than the inner layer scale δ. As in Part II, the pressure drag stays well predicted by a gravity wave drag when the surface Richardson number J > 1 and by a form drag due to nonseparated sheltering when J < 1. As in Part II also, the sign of the Reynolds stress is predominantly positive in the near-neutral case (J < 1) and negative in the stable case (J > 1) but situations characterized by positive and negative Reynolds stress now combine when J ∼ 1. In the latter case, and even when dissipation produces positive stress in the lower part of the inner layer, a property we associated with nonseparated sheltering in Part II, negative stresses are quite systematically found aloft. These negative stresses are due to upward-propagating waves and trapped lee waves, the first being associated with negative vertical flux of pseudomomentum aloft the inner layer, the second to negative horizontal flux of pseudomomentum downstream the obstacle. These results suggest that the significance of mountain waves for the large-scale flow is more substantial than expected and when compared to the form drag due to nonseparated sheltering.