Nonlinear Shallow Water Research Articles

Reef influence quantification in light of the 1771 Meiwa tsunami

While interactions between regular wave driven flooding and reefs have been widely studied due to climate change pressure, the effects of reefs on tsunami flooding have less been investigated. From studies of historical events, reefs can behave as buffers or as amplifiers of inundation, depending upon the location. Interactions between reefs and tsunamis have generally been analyzed with idealized models, and there have been only few studies of specific reefs and their characteristics. Using numerical NonLinear Shallow Water models, this study characterizes the influence of the Southeast Ishigaki Island reef during the 1771 tsunami that hit the Yaeyama Islands. In this work, we modified reef topography in silico and then, measured the impact of these changes using a new parameter, the Reef Impact Factor (RIF). First, a reference model was built, simulating the real event with an accurate reef representation and using run-up data to calibrate bottom friction. This calibration highlights the difficulty of representing reef friction with a homogeneous coefficient. Second, a model without a reef was compared to the reference model. The impact of reef removal varies considerably along the coastline and maximum wave heights at the shore were strongly affected, with a 12.5% increase on average. Overall, this suggests a protective role of the reef along most of the coast. However, at local scale, channels that break the continuity of the front reef, increased wave heights by up to 40% on the proximate coast, revealing their strong focusing influence. Finally, changes in tide level, which regulates reef depth, were investigated, showing a global positive correlation between sea level and maximum wave height at the coast. However, the impact of the reef depth appeared weak compared to the impact of incident wave parameters. This study contributes to a global effort to understand tsunami-reef interactions in a non-idealized framework, suggesting a Reef Impact Factor for inter-reef/study comparisons. Moreover, vulnerable and exposed coasts were identified at Ishigaki Island, which may help to improve inundation forecasting, resulting in more appropriate management of these vulnerable sections of the coast.

Wave Height Interval Distribution Method Based on Nonlinear Shallow Water Wave Function

Li, X.; Zhao, Y., and Jin, H., 2020. Wave height interval distribution method based on nonlinear shallow water wave function. In: Yang, Y.; Mi, C.; Zhao, L., and Lam, S. (eds.), Global Topics and New Trends in Coastal Research: Port, Coastal and Ocean Engineering. Journal of Coastal Research, Special Issue No. 103, pp. 843–846. Coconut Creek (Florida), ISSN 0749-0208.In order to improve the ability to analyze and judge the interval distribution of wave height, it is necessary to estimate the interval distribution of wave height. An interval distribution estimation method of wave height based on nonlinear shallow water wave function is proposed. A statistical time series analysis model for estimating the interval distribution of sea wave height is constructed, large data mining and feature extraction for estimating the interval distribution of sea wave height are carried out by adopting a large data feature detection method, ordered clustering of statistical feature sequences of the interval distribution of sea wave height is carried out based on a nonlinear shallow water wave function estimation idea, information clustering and attribute merging in the process of estimating the interval distribution of sea wave height are carried out by combining a fuzzy C-means clustering analysis method. The feature quantity of association rules of statistical time series of wave height interval distribution is extracted, and the association rule set of statistical time series of wave height interval distribution is analyzed by using principal component analysis method, so as to realize accurate estimation of wave height interval distribution quantity in weighted Markov chain. The simulation results show that the method has high accuracy in estimating the interval distribution of wave height and improves the quantitative analysis capability of the interval distribution of wave height.

Numerical study on a possible cause of the ‘strange tide’ in the coastal area of Jiangsu Province, China

There are many eyewitness reports on unexpected water level change called ‘strange tide’ around the area of the radial sand ridges off the coast of Jiangsu Province, China, which has caused several fatal accidents in history. This study aims to prove whether a fast-moving atmospheric pressure disturbance can be a possible cause of the ‘strange tide.’ Based on the nonlinear shallow water model, a large number of scenarios for waves induced by a fast-moving pressure disturbance are numerically simulated and the maximum of the wave amplification ratio in each scenario is discussed. In each scenario, we assume an idealized pressure disturbance passing through the concerned area. The heading angle and translational speed of atmospheric disturbances are considered as variables. The results show that a fast-moving pressure disturbance can induce large wave level fluctuation in this area. In some special situations, the water level rise can be amplified more than 40 times than the water level change caused by a pressure disturbance with equivalent intensity under static equilibrium state. Both the heading angle and translational speed of atmospheric disturbances play a very important role in the occurrence of dangerous situation. The most dangerous combination of the two parameters is discussed, and the disaster prone locations in this area are identified considering the historical accidents and the numerical experiments.

Combined Hybridizable Discontinuous Galerkin (HDG) and Runge-Kutta Discontinuous Galerkin (RK-DG) formulations for Green-Naghdi equations on unstructured meshes

In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms, which globally account for the non-hydrostatic effects through the introduction of auxiliary coupling variables. High-order discrete approximations of the main flow variables are obtained with a RK-DG method, while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second-order linear elliptic sub-problems with high-order HDG formulations. The combined use of hybrid unknowns and local post-processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches. The proposed formulation is then extended to a more complex family of three parameters enhanced Green-Naghdi equations. The resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies, showing good convergence properties and very good agreements with several sets of experimental data.

Invariant Subspace Classification and Exact Explicit Solutions to a Class of Nonlinear Wave Equation

In this paper, the invariant subspace classification of a class of nonlinear shallow water wave equation is given, then some exact explicit solutions to the nonlinear equation are provided by using the invariant subspace method. This method is a dynamical system method by nature, for its key step is to transform a nonlinear partial differential equation (PDE) into ordinary differential equation (ODE) systems, then by solving the ODE systems, the exact solutions to the nonlinear PDE are obtained.

Piston-Driven Numerical Wave Tank Based on WENO Solver of Well-Balanced Shallow Water Equations

A numerical wave tank equipped with a piston type wave-maker is presented for long-duration simulations of long waves in shallow water. Both wave maker and tank are modelled using the nonlinear shallow water equations, with motions of the numerical piston paddle accomplished via a linear mapping technique. Three approaches are used to increase computational efficiency and accuracy. First, the model satisfies the exact conservation property (C-property), a stepping stone towards properly balancing each term in the governing equation. Second, a high-order weighted essentially non-oscillatory (WENO) method is used to reduce accumulation of truncation error. Third, a cut-off algorithm is implemented to handle contaminated digits arising from round-off error. If not treated, such errors could prevent a numerical scheme from satisfying the exact C-property in long-duration simulations. Extensive numerical tests are performed to examine the well-balanced property, high order accuracy, and shock-capturing ability of the present scheme. Correct implementation of the wave paddle generator is verified by comparing numerical predictions against analytical solutions of sinusoidal, solitary, and cnoidal waves. In all cases, the model gives satisfactory results for small-amplitude, low frequency waves. Error analysis is used to investigate model limitations and derive a user criterion for long wave generation by the model.

Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography. In Chan (J Comput Phys 362:346–374, 2018), a SBP-like differentiation operator was introduced to construct the discretely entropy conservative DG methods. We extend this idea to the SWEs and establish an entropy stable scheme by adding additional dissipative terms. Careful approximation of the source term is included to ensure the well-balanced property of the resulting method. A simple positivity-preserving limiter, compatible with the entropy stable property, is included to guarantee the non-negative water heights during the computation. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

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.

Extreme Inundation Statistics on a Composite Beach

The runup of initial Gaussian narrow-banded and wide-banded wave fields and its statistical characteristics are investigated using direct numerical simulations, based on the nonlinear shallow water equations. The bathymetry consists of the section of a constant depth, which is matched with the beach of constant slope. To address different levels of nonlinearity, time series with five different significant wave heights are considered. The selected wave parameters allow for also seeing the effects of wave breaking on wave statistics. The total physical time of each simulated time-series is 1000 h (~360,000 wave periods). The statistics of calculated wave runup heights are discussed with respect to the wave nonlinearity, wave breaking and the bandwidth of the incoming wave field. The conditional Weibull distribution is suggested as a model for the description of extreme runup heights and the assessment of extreme inundations.

Effects of earthquake spatial slip correlation on variability of tsunami potential energy and intensities

Variability characterization of tsunami generation is quintessential for proper hazard estimation. For this purpose we isolate the variability which stems solely from earthquake spatial source complexity, by simulating tsunami inundation in the near-field with a simplified digital elevation model, using nonlinear shallow water equations. For earthquake rupture, we prescribe slip to have a log-normal probability distribution function and von Kármán correlation between each subfault pair, which we assume decreases with increasing euclidean distance between them. From the generated near-field inundation time-series, emanating from several thousand synthetic slip realizations across a magnitude 9 earthquake, we extract several tsunami intensity measures at the coast. Results show that all considered tsunami intensity measures and potential energy variability increase with increasing spatial slip correlations. Finally, we show that larger spatial slip correlations produce higher tsunami intensity measure exceedance probabilities within the near-field, which highlights the need to quantify the uncertainty of earthquake spatial slip correlation.

Tsunami inundation hazard across Japan

Japan faces the world's highest tsunami hazard and risk due to its tectonic environment, high population density and exposure concentration along its coastlines. It is therefore of paramount interest to quantify and differentiate absolute and relative tsunami hazard and risk on a countrywide scale. We quantify tsunami hazard in terms of inundation depth for the entire Japanese coastline with a probabilistic tsunami hazard assessment utilizing the earthquake source information of the 2017 Japanese seismic hazard model. We simulate a stochastic event set for offshore earthquake sources and generate, for each single source with MW ≥ 7.5, non-uniform finite-fault slip models. We calculate elastic seafloor and landmass deformations that serve as initial conditions to high-resolution numerical modelling of tsunami wave propagation and coastal inundations by solving the non-linear shallow water equation. Variable land surface roughness based on land cover data is used to simulate accurate hydraulics of coastal inundation.We differentiate tsunami hazard by inundation depth hazard curves and inundation depth return period maps aggregated to city ward polygons as meaningful administrative boundaries. We find mega-thrust events on the subduction interfaces constituting the largest hazard. However, events down to MW = 7.5 can contribute to substantial hazard in several regions along the coast. In particular for city wards within the Tokyo bay area, we find that earthquakes occurring on the Sagami trough and not the largest mega-thrust events on the Nankai trough, contribute to the highest inundation hazard. Our results also illustrate that tsunami hazard on the western Japanese coast is not negligible.

Goal-oriented error estimation and mesh adaptation for shallow water modelling

This study presents a novel goal-oriented error estimate for the nonlinear shallow water equations solved using a mixed discontinuous/continuous Galerkin approach. This error estimator takes account of the discontinuities in the discrete solution and is used to drive two metric-based mesh adaptation algorithms: one which yields isotropic meshes and another which yields anisotropic meshes. An implementation of these goal-oriented mesh adaptation algorithms is described, including a method for approximating the adjoint error term which arises in the error estimate. Results are presented for simulations of two model tidal farm configurations computed using the Thetis coastal ocean model (Kärnä et al. in Geosci Model Dev 11(11):4359–4382, 2018). Convergence analysis indicates that meshes resulting from the goal-oriented adaptation strategies permit accurate QoI estimation using fewer computational resources than uniform refinement.

Tsunamiopasnost' Krymskogo poberezh'ya Chernogo morya i Kerchenskogo proliva pri katastroficheskih tsunamigennyh zemletryaseniyah, blizkih po lokalizatsii k istoricheskomu YAltinskomu zemletryaseniyu 12 sentyabrya 1927 goda

Purpose. The main purpose of the work is to assess a possible tsunami hazard for the region of location of the Crimean Bridge constructed across the Kerch Strait, since up to present, numerical simulations of the tsunamigenic earthquakes in the Black Sea have never taken the Kerch Strait region (except for the water area at the strait entrance) into consideration as a possible object for a tsunami hazard. Methods and Results. To assess a tsunami hazard for the Kerch Strait and, particularly, for the region of location of the constructed Crimean Bridge, the present paper considers the historical catastrophic tsunamigenic earthquake in the southeastern part of the Crimea Peninsula on September 12, 1927 as well as possible strong earthquakes, the source locations and magnitudes of which can be close to those of the above-mentioned one. The available data on localization and intensity of the earthquake on September 12, 1927 allowed to model the source of such an earthquake. Besides, within the framework of the nonlinear shallow water equations, simulated were the tsunami generation and the tsunami waves’ propagation from the single- and two-block seismic sources in the Black Sea along the Crimea Peninsula, at the Kerch Strait entrance and in its water area. The seismic sources of close localization were similarly simulated for two possible tsunamigenic earthquakes in the southeastern part of the Crimea Peninsula. Conclusions. For all the considered scenarios, generation and propagation of the tsunami waves over the water area regions under study were numerically simulated, the histograms of distribution of the wave run-up maximum heights along the Crimea Peninsula and the Kerch Strait coasts were constructed. It is shown that for all the scenarios, in the region of the Crimean Bridge western pillars the tsunami wave heights do not exceed 0.3–0.5 m, whereas in the region of its eastern pillars, possible wave heights are within the range 0.6–1.95 m. The results represented in the paper are compared with the available data on the wave heights in a number of settlements of the Crimea Peninsula coast and at the Kerch Strait entrance, obtained by the other authors.

M-lump, high-order breather solutions and interaction dynamics of a generalized $$(2 + 1)$$-dimensional nonlinear wave equation

In this paper, a generalized $$(2+1)$$-dimensional nonlinear wave equation is obtained by extending the generalized $$(2+1)$$-dimensional Hirota bilinear equation into a more generalized form. The obtained new equation is useful in describing nonlinear wave phenomena in nonlinear optics, shallow water and oceanography. Based on the bilinear method, the N-soliton solutions of the generalized $$(2+1)$$-dimensional nonlinear wave equation are obtained. M-lump solutions are investigated by applying the long wave limit to the N-soliton solutions. The propagation orbits and velocities of the M-lump wave are analyzed. The high-order breather waves are obtained by establishing the complex conjugate relations in the parameters of the N-solitons. Furthermore, the interaction hybrid solutions are constructed, which contain hybrid solutions composed of breathers, solitons and lumps. The dynamical behaviors of the hybrid solutions are systematically analyzed via numerical simulations. The obtained results will enrich the study of theory of the nonlinear localized waves.

Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part

In this paper, we discuss the construction of a class of implicit-explicit (IMEX) methods for systems of ordinary differential equations which their right hand side can be split into two parts; nonstiff or mildly stiff part and stiff part. The proposed methods treat the non-stiff part by an explicit second derivative diagonally implicit multistage integration method (SDIMSIM) and the stiff part by an implicit diagonally implicit multistage integration method (DIMSIM). The explicit part of these methods has strong stability preserving (SSP) property and the implicit part is A- and L-stable. We will construct methods with p=q=r=s and p=q+1=r=s up to order four with large SSP coefficients with respect to the large region of absolute stability, assuming that the implicit part of the method has Runge–Kutta stability (RKS) property together with A- and L-stability. These methods are tested on the linear advection-diffusion, advection-reaction and nonlinear shallow water equations, and the numerical results are presented conforming the efficiency and order of constructed methods.

Wave breaking for periodic solutions of a nonlinear shallow water equation

ABSTRACT We consider a nonlinear shallow water equation. An inequality condition for wave breaking has been established for the Cauchy problem with periodic domain. Numerical computation confirms the result, as breaking happens for waves with a chosen initial profile satisfying the theoretical condition for wave breaking.

Lattice Boltzmann method for shallow water flow with wave radiation stress

The protection of the marine environment is of growing concern, and the development of hydrodynamic models is required to study ocean processes. In this work, a two-dimensional hydrodynamic lattice Boltzmann model was developed to study wave–current coupling problems that combine wave radiation, bed shear stress and wind shear stress. The model is based on nonlinear shallow water equations and two different forms are developed to model wave radiation stress. The first form uses an additional external force term, including a weight factor in the lattice Boltzmann equation at the streaming step. The second form modifies a local equilibrium distribution function at the collision step. Von Neumann stability method is used to study the two schemes. This model was verified by three classic cases: waves diffracted by a straight vertical wall, waves diffracted by two breakwaters, and waves in a circular channel. The numerical results demonstrate the validity of the proposed model, indicating that it can accurately describe the characteristics of water flow with interactions between waves and currents.

Wave Fields and Nearshore Currents in the Coastal Region Opposite San Mauro Cilento (Italy)

In this paper in order to simulate nearshore currents in computational domains representing the complex morphology of real coastal regions we use a model based on a contravariant integral form of the fully nonlinear Boussinesq equations (FNBE). The contravariant integral form, in which Christoffel symbols are absent, of the continuity equation does not contain the dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations (NSWE). The capacity of the proposed model to correctly simulate the wave train propagation on a highly distorted grid is verified against test case present in the literature. The simulation of wave fields and nearshore currents in the coastal region, opposite San Mauro Cilento (Italy) in presence of a system of T-head groins, is numerically reproduced by using the proposed model.

Well-posedness of a highly nonlinear shallow water equation on the circle

We present a comprehensive introduction and overview of a recently derived model equation for waves of large amplitude in the context of shallow water waves and provide a literature review of all the available studies on this equation. Furthermore, we establish a novel result concerning the local well-posedness of the corresponding Cauchy problem for space-periodic solutions with initial data from the Sobolev space Hs on the circle for s>3∕2.

Numerical Simulation of the Sea Bottom Modifications Behind a T-head Groin

In this paper, we simulate the sea bottom modifications produced by the presence of a T-head groin. We present a simulation model of sea bottom modifications composed of two sub-models: a two-dimensional phase-resolving model that simulate the variation of the fluid dynamic variables inside the wave; a second sub-model to simulate the sea bottom modifications, in which the suspended sediment concentration is calculated by the wave-averaged advection-diffusion equation. The fluid motion equation and the concentration equation are expressed in a new contravariant formulation. The velocity fields from deep water up to just seaward of the surf-zone are simulated by a new integral contravariant form of the Fully Nonlinear Boussinesq Equations. The new integral form of the proposed continuity equation does not contain the dispersive term. The Nonlinear Shallow Water Equations, expressed in an integral contravariant form, are solved in order to simulate the breaking wave propagation. The momentum equation, integrated over the turbulent boundary layer, is solved to calculate the near-bed instantaneous flow velocity and the intra-wave hydrodynamic quantities. Starting from the contravariant formulation of the advection–diffusion equation for the suspended sediment concentration, it is possible to calculate the sea bottom modification. The advective sediment transport terms in the advection-diffusion equation are formulated according to a quasi-three-dimensional approach