Nonlinear Shallow Water Research Articles

Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations

This paper considers the dynamical behavior of solutions near explicit self-similar singularity for a class of nonlinear shallow water models including the Dullin–Gottwald–Holm equation, the Korteweg–de Vires equation, the Camassa–Holm equation, the dispersive rod equation and the Benjamin–Bona–Mahony equation. We first show there are explicit self-similar solutions for those five nonlinear shallow water equations, then we find that those explicit self-similar solutions for the Dullin–Gottwald–Holm equation, the Camassa–Holm equation and the dispersive rod equation are asymptotic stable, but for the Korteweg–de Vries equation and the Benjamin–Bona–Mahony equation being unstable.

Multi-directional focused wave group interactions with a plane beach

Numerical simulations and laboratory measurements are presented of multi-directional focused wave groups interacting with a plane beach. The numerical model is a two-dimensional-horizontal (2DH) hybrid flow solver, governed by a Boussinesq equation set with enhanced dispersion characteristics pre-breaking, and the nonlinear shallow water equations post-breaking. Waves are introduced into the model via an in-built multi-element piston wavemaker, allowing for complete reproduction of laboratory experiments. A wetting and drying algorithm models shoreline movement in both cross-shore and longshore directions. Predicted free surface motions of the multi-directional focused wave groups are in good agreement with wave gauge data from laboratory experiments previously carried out at the UK Coastal Research Facility (UKCRF) using a linear paddle wave generator. Both phase decomposition into Stokes-like harmonic components and wavelets provide insight into nonlinear interactions as the wave groups propagate up the beach. Free second-order error waves in a multi-directional wave group are smaller than for the corresponding uni-directional case, and spread laterally around the incoming wave group. Of the free error waves generated by linear paddle signals, only the low-frequency second-order error wave affects extreme run-up on the beach. By applying a second-order correction to the paddle signals used to generate a multi-directional wave group, it is shown that, whereas the long error wave causes a significant increase in the maximum run-up, the impact is not as severe as for the uni-directional analogue. Shoaling and run-up of the bound long waves in a spread sea are studied. Examination of the transverse structure of these subharmonic components reveals that sideways spreading in the inner surf zone contributes to reduced run-up in directionally spread groups.

Tsunami Simulations of the Sulawesi Mw 7.5 Event: Comparison of Seismic Sources Issued from a Tsunami Warning Context Versus Post-Event Finite Source

The 28 September 2018 Sulawesi earthquake generated a much larger tsunami than expected from its Mw = 7.5 magnitude and from its dominant strike-slip mechanism. Within a few minutes after the earthquake, the tsunami devastated the seafront of Palu bay, destroying houses and infrastructures over a few hundred meters. Coastal subsidence and slumping at various locations around the bay were also observed. There is debate in the scientific community as to whether submarine landslides and shore collapses contributed to the generation of strong and destructive waves locally. The objective of this study is threefold: first, to determine whether standard seismic inversions could predict the source in the context of tsunami early warning; second, to define a new seismic source built from optical image correlation and based on the geological and tectonic context; third, to assess whether the earthquake alone is able to generate up to 9-m wave heights at the coast. Numerical simulations of the tsunami propagation are performed for different seismic dislocation sources. Nonlinear shallow water equations are solved by a finite-difference method in grids with 200-m and 10-m resolutions. The early CMT focal solutions calculated by seismological institutes show dominant strike-slip mechanisms with a homogenous slip distribution. These sources produce maximum tsunami heights of 40-cm on the coast of Palu city. Two heterogeneous sources are tested and compared: the USGS “finite fault” model calculated from seismic inversion and a new “hybrid” source inferred from different techniques. The latter is based on a segmented fault in agreement with the geological context and built from both from seismic parameters of a CMT solution and the observed horizontal ground displacements. This source produces water wave heights of 4 to 5-m in the Palu bay. The observed inundation heights and distances are reproduced satisfactorily by the model at Pantoloan and at the southwestern tip of Palu bay. However, the “hybrid” source is unable to reproduce the largest 8 to 12-m water heights as reported from field surveys. Thus, even though this “hybrid” source produces most of the reported tsunami energy, we cannot exclude that the numerous coastal collapses observed in Palu bay contributed to increase the local tsunami run-up.

A Comparison of the Explicit and Implicit Hybridizable Discontinuous Galerkin Methods for Nonlinear Shallow Water Equations

An explicit implementation of the hybridizable discontinuous Galerkin (HDG) method for solving the nonlinear shallow water equations is presented. We follow the common construction of the implicit HDG for nonlinear conservation laws, and then explain the differences between the explicit formulation and the implicit version. For the implicit implementation, we use the approximate traces of the conserved variables ( $${\widehat{\varvec{q}}}$$ ) to express the numerical fluxes in each element. Next, we impose the conservation of the numerical fluxes via a global system of equations. Using the Newton–Raphson method, this global system can be solely expressed in terms of the increments of the approximate traces in each iteration. For the explicit method, having $${\varvec{q}}_h$$ at each time level, we first obtain $${\widehat{\varvec{q}}}_h$$ such that the conservation of the numerical flux is satisfied. This will result in a nonlinear system of equations which is local to each edge of the mesh skeleton. Having the solution ( $${\varvec{q}}_h$$ , $$\widehat{{\varvec{q}}}_h$$ ) for the previous time step, we use the Runge–Kutta time integration method to obtain $${\varvec{q}}_h$$ in the next time step. Hence, the introduced explicit technique is based on local operations over the faces and elements of the mesh. Using different numerical examples, we show the optimal convergence of the solution of the explicit and implicit approach in $$L^2$$ norm. Finally, through numerical experiments, we discuss the advantages of the implicit and explicit techniques from the computational cost point of view.

Numerical study of tsunami wave run-up and land inundation on coastal vegetated beaches

Vegetation exerts a significant damping effect on tsunami wave run-up on coastal beaches, thus effectively mitigating the tsunami hazard. A depth-integrated two-dimensional numerical model (HydroSed2D, Liu et al., 2008; Liu et al., 2010) is developed to investigate tsunami wave run-up and land inundation on coastal beaches covered with Pandanus odoratissimus (P. odoratissimus). The present model is based on a finite volume Roe-type scheme, that solves the non-linear shallow water equations with the capacity of treating the wet or dry boundary at the wave front. The momentum equations in this model are modified by adding a drag force term, thus considering the resistance effects of vegetation on tsunami waves. The accuracy of the numerical scheme and the vegetation drag force are validated by three experimental cases of dam-break flow propagation in a dry channel, solitary wave propagation in a vegetated flume, and tsunami run-up over an uneven bed. Subsequently, a numerical model is applied to simulate tsunami run-up and land inundation on actual-scale vegetated beaches and a series of sensitive analyses are conducted by comparing numerical results. The obtained numerical results suggest that P. odoratissimus can effectively attenuate tsunami run-up and land inundation distance on coastal beaches, and a higher attenuation rate for tsunami wave can be achieved by increasing both vegetation width and vegetation density. The tsunami wave height is also an important factor that impacts the tsunami wave run-up and land inundation on vegetated beaches.

Numerical modelling of landslide-tsunami propagation in a wide range of idealised water body geometries

Large landslide-tsunamis are caused by mass movements such as landslides or rock falls impacting into a water body. Research of these phenomena is essentially based on the two idealised water body geometries (i) wave flume (2D, laterally confined wave propagation) and (ii) wave basin (3D, unconfined wave propagation). The wave height in 2D and 3D differs by over one order of magnitude in the far field. Further, the wave characteristics in intermediate geometries are currently not well understood. This article focuses on numerical landslide-tsunami propagation in the far field to quantify the effect of the water body geometry. The hydrodynamic numerical model SWASH, based on the non-hydrostatic non-linear shallow water equations, was used to simulate approximate linear, Stokes, cnoidal and solitary waves in 6 different idealised water body geometries. This includes 2D, 3D as well as intermediate geometries consisting of “channels” with diverging side walls. The wavefront length was found to be an excellent parameter to correlate the wave decay along the slide axis in all these geometries in agreement with Green's law and with diffraction theory in 3D. Semi-theoretical equations to predict the wave magnitude of the idealised waves at any desired point of the water bodies are also presented. Further, simulations of experimental landslide-tsunami time series were performed in 2D to quantify the effect of frequency dispersion. This process may be negligible for solitary- and cnoidal-like waves for initial landslide-tsunami hazard assessment but becomes more important for Stokes-like waves in deeper water. The findings herein significantly improve the reliability of initial landslide-tsunami hazard assessment in water body geometries between 2D and 3D, as demonstrated with the 2014 landslide-tsunami event in Lake Askja.

Non-linear shallow water flow modelling over topography with depth-averaged potential equations

Hunter Rouse was the father of the Fluid Mechanics of Open Channel Flow. In this review article the fundamental advances on the non-hydrostatic modeling of shallow open channel flow over topography, since the publication of his 1938 book, are discussed. Flows over uneven topography, like at open channel controls, occurs in a short length, thus rendering ideal fluid flow theory an adequate mathematical tool. This paper emphasis is on depth-averaged modeling of these flows using Boussinesq equations, given the significant advances basically since the 80’s. The one-dimensional steady flow model from Picard iteration is reviewed in detail, including energy and momentum formulations, flow at a weir crest, and on a slope. A new numerical scheme is developed for the solution of transcritical steady weir flow, showing excellent match with experiments. A new derivation for the two-dimensional depth-averaged unsteady ideal fluid model is presented based on a continuum mechanics formulation. The result is the Serre–Green–Naghdi equations (SGNE). It is demonstrated that the extended Bernoulli equation derived from Picard iteration is an integral form of the SGNE for 1D steady flow. A robust MUSCL-Hancock finite volume model is developed to solve the unsteady 1D SGNE, showing excellent agreement with the steady transcritical flow solutions previously constructed. The accuracy of the MUSCL-Hancock solver is critically assessed using a higher-order solver for the SGNE.

Numerical modelling of coastal inundation from Cascadia Subduction Zone tsunamis and implications for coastal communities on western Vancouver Island, Canada

In the present study, numerical simulations were conducted to estimate the spatio-temporal characteristics of tsunami inundation for municipalities on Vancouver Island, Canada, as a result of various potential Cascadia Subduction Zone earthquake deterministic scenarios. By varying the earthquake magnitude and associated slip distance, the influence of these parameters on the tsunami wave propagation, inundation, and the possible damage on infrastructure and buildings was investigated. A numerical tsunami inundation model based on nonlinear shallow water equations was constructed using publically available bathymetric and topographic data and applied to three study areas: the City of Port Alberni, the District of Ucluelet, and the District of Tofino. The numerical results obtained in this study show that the maximum tsunami inundation depth and spatial extent of inundation are sensitive to the earthquake magnitude, whereas the tsunami arrival time is not. For the worst-case earthquake scenario investigated (MW 9.3), all of the three study areas were extensively inundated. Results of tsunami wave amplitude and overland inundation depth, flow velocity, hydrodynamic force, and depth–velocity product are analysed in detail to assess the impacts of the tsunami on inland buildings. The potential damage was estimated with previously proposed fragility curves for wood, reinforced concrete, and steel frame buildings. In conjunction with a site reconnaissance visit by the authors for better understanding the general characteristics of these areas, model results suggest that significant damage to buildings would occur, especially to wooden constructions, with considerable risk of loss of human life.

Effect of kinematic fault rupture process on tsunami propagation

The scope of this study is to investigate the effect of the kinematic rupture process on tsunami propagation and corresponding tsunami hazards in coastal regions. It is significant for the tsunageneric earthquakes of long fault length and subsequent long duration of the fault rupture in the strike direction. Most previous studies on the tsunami hazard assessments of the Manila Trench did not consider the effect of the rupture process of huge plates. The movements of the seabed near the earthquake source are calculated by Okada model. The free surface elevation triggered by the seafloor motion is modeled by the nonlinear shallow water equations. The 2004 Indian Ocean tsunami and the 2011 Japan Tohoku tsunami are used to validate the numerical model by comparing with measurement data. The numerical experiments of the tsunami propagation for the cases of different source rupture velocities in uniform depth and continental shelf are carried out to understand the mechanism of the changes of the tsunami arrival time and the tsunami amplitude due to the kinematic rupture process. Stimulated by the long duration of plate motion of 2004 Indian Ocean tsunami, we propose several typical patterns of the rupture process of the long fault of the Manila Trench and analyze quantitatively on the role of the uncertain rupture process of tectonics source in tsunami propagation in the South China Sea. Our results have highlighted that the slow rupture velocity will remarkable affect the tsunami arrival time and maximum amplitude in the South China Sea, comparing to the instantaneous rupture. It is suggested that the rupture process is essential for assessments of the tsunami hazards generated by submarine mass failure with extreme long earthquake rupture source.

Non-hydrostatic modeling of drag, inertia and porous effects in wave propagation over dense vegetation fields

A new wave-vegetation model is implemented in an open-source code, SWASH (Simulating WAves till SHore). The governing equations are the nonlinear shallow water equations, including non-hydrostatic pressure. Besides the commonly considered drag force induced by vertical vegetation cylinders, drag force induced by horizontal vegetation cylinders in complex mangrove root systems, as well as porosity and inertia effects, are included in the vegetation model, providing a logical supplement to the existing models. The vegetation model is tested against lab measurements and existing models. Good model performance is found in simulating wave height distribution and maximum water level in vegetation fields. The relevance of including the additional effects is demonstrated by illustrative model runs. We show that the difference between vertical and horizontal vegetation cylinders in wave dissipation is larger when exposed to shorter waves, because in these wave conditions the vertical component of orbital velocity is more prominent. Both porosity and inertia effects are more pronounced with higher vegetation density. Porosity effects can cause wave reflection and lead to reduced wave height in and behind vegetation fields, while inertia force leads to negative energy dissipation that reduces the wave-damping capacity of vegetation. Overall, the inclusion of both effects leads to greater wave reduction compared to common modeling practice that ignores these effects, but the maximum water level is increased due to porosity. With good model performance and extended functions, the new vegetation model in SWASH code is a solid advancement toward refined simulation of wave propagation over vegetation fields.

Development and Validation of a Tsunami Numerical Model with the Polygonally Nested Grid System and its MPI-Parallelization for Real-Time Tsunami Inundation Forecast on a Regional Scale

We have developed a new numerical model suitable for rapid and wide-area estimation of tsunami inundation and damage. The model is based on the world-renowned TUNAMI code solving the two-dimensional nonlinear shallow water equations, and enables one-stop simulation of the initial tsunami distribution based on a fault model, tsunami propagation and inundation, and damage estimation. It extends the configuration of the grid system from conventional rectangular regions to polygonal regions so that deployment of high-resolution grids can be confined to the coastal lowland, resulting in remarkably improved efficiency in computation and better precision. For the purpose of real-time implementation of tsunami inundation simulation using a high-performance computing infrastructure, vectorization and MPI parallelization have also been conducted. Moreover, the model was verified and validated through several benchmark problems that the National Tsunami Hazard Mitigation Program, organized by federal agencies and states in the U.S., developed as the quality standards for simulating and assessing tsunami hazard and risk. The newly-developed model is named “Real-time Tsunami inundation (RTi) model,” and its computational performance was examined using the SX-ACE, a vector supercomputer installed at Tohoku University. The results show that it requires only 128 cores of the SX-ACE for implementing six-hour tsunami inundation simulation with a 10-meter grid resolution within 10 minutes for the 700 km long coastline of Kochi Prefecture, Japan. This means that the RTi model is over 10 times more efficient as the conventional tsunami model with the rectangular domains, and it can be inferred that 2,451 cores of the SX-ACE are the overall computational resources needed for real-time tsunami inundation forecast on the whole coastal regions along the Nankai Trough subduction zone, corresponding to the computational performance of 170 Tflop/s. The resources required are equivalent to 24% of all the SX-ACE resources at Tohoku University, indicating the feasibility of real-time tsunami inundation forecast on a regional scale by using the RTi model. Since the Disaster Information System operated by the Cabinet Office of the Japanese Government adopted a function of tsunami damage estimation using the aforementioned numerical model, at the end of this paper, a brief overview of the subsystem for rapidly estimating tsunami damage on a regional scale is described.

Generation mechanism of large later phases of the 2011 Tohoku-oki tsunami causing damages in Hakodate, Hokkaido, Japan

The 2011 Tohoku-oki earthquake generated a large tsunami that caused catastrophic damage along the Pacific coast of Japan. The major portion of the damage along the Pacific coast of Tohoku in Japan was mainly caused by the first few cycles of tsunami waves. However, the largest phase of the tsunami arriving surprisingly late in Hakodate in Hokkaido, Japan; that is, approximately 9 h after the origin time of the earthquake. It is important to understand the generation mechanism of this large later phase. The tsunami was numerically computed by solving both linear shallow water equations and non-linear shallow water equations with moving boundary conditions throughout the computational area. The later tsunami phases observed on southern Hokkaido can be much better explained by tsunami waveforms computed by solving the non-linear equations than by those computed by solving the linear equations. This suggests that the later tsunami waves arrived at the Hokkaido coast after propagating along the Pacific coast of the Tohoku region with repeated inundations far inland or reflecting from the coast of Tohoku after the inundation. The spectral analysis of the observed waveform at Hakodate tide gauge shows that the later tsunami that arrived between 7.5 and 9.5 h after the earthquake mainly contains a period of 45–50 min. The normal modes of Hakodate Bay were also computed to obtain the eigen periods, eigenfunctions, and spatial distribution of water heights. The computed tsunami height distributions near Hakodate and the fundamental mode of Hakodate Bay indicate that the large later phases are mainly caused by the resonance of the bay, which has a period of approximately 50 min. The results also indicate that the tsunami wave heights near the Hakodate port area, the most populated area in Hakodate, are the largest in the bay because of the resonance of the fundamental mode of the bay. The results of this study suggest that large future tsunamis might excite the fundamental mode of Hakodate Bay and cause large later phases near the Hakodate port.

Coupling Conditions for Water Waves at Forks

We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this analysis justifies the well-known Stoker interface conditions, so that the coupling does not depend on the angle of the fork. We also find this in the numerical solution. Large amplitude solutions in a symmetric fork also tend to follow Stoker’s relations, due to the symmetry constraint. For non symmetric forks, 2D effects dominate so that it is necessary to understand the flow inside the fork. However, even then, conservation laws give some insight in the dynamics.

Momentum Conservative Scheme for Simulating Wave Runup and Underwater Landslide

This paper focuses on the numerical modelling and simulation of tsunami waves triggered by an underwater landslide. The equation of motion for water waves is represented by the Nonlinear Shallow Water Equations (NSWE). Meanwhile, the motion of underwater landslide is modeled by incorporating a term for bottom motion into the NSWE. The model is solved numerically by using a finite volume method with a momentum conservative staggered grid scheme that is proposed by Stelling & Duinmeijer 2003 [12]. Here, we modify the scheme for the implementation of bottom motion. The accuracy of the implementation for representing wave runup and rundown is shown by performing the runup of harmonic wave as proposed by Carrier & Greenspan 1958 [2], and also solitary wave runup of Synolakis, 1986 [14], for both breaking and non-breaking cases. For the underwater landslide, result of the simulation is compared with simulation using the Boundary Integral Equation Model (BIEM) that is performed by Lynett and Liu, 2002 [9].

Shock capturing staggered grid scheme for simulating dam-break flow and runup

This paper focuses on a numerical implementation for simulating complex flow phenomena such as shock wave and wave runup. We use the Nonlinear Shallow Water Equations (NSWE) for representing the flow. The wave model is implemented by using finite volume with momentum conservative staggered grid scheme. A relatively simple leap-frog scheme, upwind method, and special treatment in advection term are used in the numerical implementation. To test the numerical implementation, we reconstruct a physical experiment that was proposed by Aureli et al. 2000, i.e. a dam-break flow that creates shock wave and wave runup. Results of numerical simulation shows a good agreement with experimental data of.

Dissipation of Solitary Wave Due To Mangrove Forest: A Numerical Study by Using Non-Dispersive Wave Model

In this paper, we study a dissipation of solitary wave due to mangrove forest by using numerical simulation. Here, the solitary wave is chosen to represent tsunami wave form. To simulate the wave dynamic, we use the non-dispersive Nonlinear Shallow Water Equations (NSWE). The model is implemented numerically by using finite volume method in a momentum conservative staggered grid. By using the proposed numerical scheme, the numerical code is able to simulate solitary wave breaking phenomenon. Wave dissipation due to mangrove forest is modelled as bottom roughness with an approximate value of manning roughness, which is derived from the classical Morisson’s formula. To test the modelled dissipation by mangrove forest, we reconstruct a physical experiment in hydrodynamic laboratory where a solitary wave propagates above a sloping bottom, which has a parameterized mangrove in the shallower part. Two cases are performed to test the performance of the numerical implementation, i.e. the non-breaking and breaking solitary waves. Results of simulation agree quite well with the measurement data. The results of simulation are also analyzed quantitatively by calculating errors as well as correlation with the measurement data. Moreover, to investigate effects of wave steepness on solitary wave, to the reduction of wave energy, we perform numerical investigation. Various solitary waves with different wave steepness are simulated to see their effects on amplitude and energy reduction due to mangrove forest.

Differential invariants of Camassa–Holm equation

Under investigation in this work is a nonlinear shallow water equation, Camassa–Holm equation, which is very important in fluid dynamics, nonlinear dynamics and physical application. The new equivariant moving frames method is implemented to obtain a finite generating set of differential invariants, recurrence relations, and syzygies among the generating differential invariants, for Lie symmetry group of Camassa–Holm equation. This method is very efficient, only using the infinitesimal determining equations and choice of cross-section normalization. The results are useful support for describing the invariant properties and trend of physical, oceanic or atmospheric motion.

An Application of Semi-empirical Physical Model of Tsunami-Bore Pressure on Buildings

Characteristic patterns of tsunami wave pressure on buildings is divided into three types, depending on its vertical profiles and time, which is observed after the tsunami impacted the buildings. The first one is the impulsive pressure, which is observed just after the tsunami impacted the buildings. The second one is the bore pressure, which is observed after the impulsive pressures. The third one is the quasi-steady pressure, which is observed after the bores go away from the buildings. In this study, based on characteristics of bore pressure observed in a hydraulic experiment, a semi-empirical physical model of bore pressure is developed by applying a turbulent bore theory. Also, we present an application method of the semi-empirical physical model to evaluations of bore pressure with usage of numerical results of inundation simulations of two-dimensional nonlinear shallow water equation models. Furthermore, we apply the semi-empirical physical model to evaluations of pressure acting on buildings in an inundation area by carrying out numerical simulations of tsunami inundation.

Run-up of long waves generated by bottom-tilting wave maker

ABSTRACTUsing the recently developed bottom-tilting wave maker, run-ups of very long waves are investigated on an adjustable plane beach. Both leading-elevation waves and leading-depression waves are considered, in which the experimental results are compared with the numerical solution of the nonlinear shallow water equations. A surf similarity parameter based on the steepness of the accelerating phase of the wave is suggested. It is shown that both the maximum run-up height and the theoretical breaking criterion can be described in terms of the surf similarity parameter, which is consistent with the results for the solitary waves.

Multi-layer non-hydrostatic free surface modelling using the discontinuous Galerkin method

A multi-layer non-hydrostatic version of the unstructured mesh, discontinuous Galerkin finite element based coastal ocean model, Thetis, is developed. This is accomplished using the PDE solver framework, Firedrake, which is used to automatically produce the code for the discretised model equations in a rapid and efficient manner. The motivation for this work is a need to accurately simulate dispersive nearshore free surface processes.In order to resolve both frequency dispersion and non-linear effects accurately, additional non-hydrostatic terms are included in the layer-integrated hydrostatic equations, producing a form similar to the layered non-linear shallow water equations, but with extra vertical velocities at the layer interfaces. An implementation process is adopted to easily handle the inter-layer connection, i.e. the governing equations are transformed into a depth-integrated system through the introduction of depth-averaged variables.The model is verified and validated through comparisons against several idealised and experimentally-based test cases. All the comparisons demonstrate good agreement, showing that the developed non-hydrostatic model has excellent capabilities in representing coastal wave phenomena including shoaling, refraction and diffraction of dispersive short waves, as well as propagation, run-up and inundation of non-linear tsunami waves.