Nonlinear Shallow Water Waves Research Articles

Formula omitted]-soliton solutions and localized structures for the [formula omitted]-dimensional Broer–Kaup–Kupershmidt system

In this paper, the ( 2 + 1 ) -dimensional Broer–Kaup–Kupershmidt system is investigated by means of Bäcklund transformation and the Hirota bilinear method. With symbolic computation, the N -soliton solutions with N + 1 arbitrary functions for that system are derived. Based on the derived solutions, some soliton structures, localized structures and special fission interaction behaviour are graphically studied, which might be helpful to understand the propagation processes for nonlinear and dispersive long waves in shallow water.

Real-Time Prediction of Tsunami Magnitudes in Osaka Bay, Japan, Using an Artificial Neural Network

This study examined the validity of using an artificial neural network (ANN) to predict tsunami water levels at several locations in Osaka Bay. The metropolitan areas of Osaka Bay have short warning times for tsunamis; a real-time tsunami forecast will allow for improved evacuation plans and will reduce the effect of these coastal disasters. Different tsunami conditions changing the relative strength of the asperities and background sources, such as fault displacement, fault length, fault width, fault slope, depth from sea bottom, and strike, were used for training the ANN; the data sets were generated by applying the nonlinear shallow water wave equations assuming different earthquake fault models. The linear activation function produced optimal results for the ANN output units, and the tangent-sigmoid function yielded good results for the ANN hidden layer units. The Levenberg-Marquardt method with Bayesian regulation was employed for the training of the ANN. Output from the trained ANN was the prelimina...

Modeling of inundation dynamics on Banda Aceh, Indonesia during the great Sumatra tsunamis December 26, 2004

The tsunami inundation flows on Banda Aceh, Indonesia reached 5 km inland during the December 26, 2004, event and devastated most of the houses, buildings, and infrastructure along the coast and killed more than 167,000 people. The overland flows from the northwest coast and the west coast collided at Lampisang village approximately 3.7 km from Ulee Lheue (northwest coast) and 6.8 km from Lhok Nga (west coast) as reported by survivors. Inundation modeling based on the nonlinear shallow-water wave equations reproduces the inundation pattern and demonstrates a colliding of the overland flows. The model suggests that wave characteristics on the northwest coast of Banda Aceh were different from those on the waves that impacted upon the west coast. The areas, which experienced higher inundation levels, did not always experience greatest overland flow speeds, and the damage areas mostly coincide with the flow speed distribution rather than the runup and inundation depth.

Rogue waves in nonlinear hyperbolic systems (shallow-water framework)**This invited article is one of a series published throughout 2011 in Nonlinearity on the topic of Rogue Waves.

The formation of rogue waves is studied in the framework of nonlinear hyperbolic systems with an application to nonlinear shallow-water waves. It is shown that the nonlinearity in the random Riemann (travelling) wave, which manifests in the steeping of the face-front of the wave, does not lead to extreme wave formation. At the same time, the strongly nonlinear Riemann wave cannot be described by the Gaussian statistics for all components of the wave field. It is shown that rogue waves can appear in nonlinear hyperbolic systems only in the result of nonlinear wave–wave or/and wave–bottom interaction. Two special cases of wave interaction with a vertical wall (interaction of two Riemann waves propagating in opposite directions) and wave transformation in the basin of variable depth are studied in detail. Open problems of the rogue wave occurrence in nonlinear hyperbolic systems are discussed.

EVOLUTION OF DAMAGED ARMOR LAYER PROFILE

A probabilistic hydrodynamic model for the wet and dry zone on a permeable structure is developed to predict irregular wave action on the structure above the still water level. The model is based on the time-averaged continuity and momentum equations for nonlinear shallow-water waves coupled with the exponential probability distribution of the water depth. The model predicts the cross-shore variations of the mean and standard deviation of the water depth and horizontal velocity. Damage progression of a stone armor layer is predicted by modifying a formula for bed load on beaches with input from the hydrodynamic model. The damage progression model is compared with three tests by 
 Melby and Kobayashi (1998) that lasted up to 28.5 hours. The model predicts the temporal progression of the eroded area quite well. The numerical model is very efficient and suited for a risk-based design of rubble mound structures.

416 集中系モデルによる非線形浅水波の解析

416 集中系モデルによる非線形浅水波の解析

Solitonic interactions and double-Wronskian-type solutions for a variable-coefficient variant Boussinesq model in the long gravity water waves

Under investigation in this paper is a variable-coefficient variant Boussinesq (vcvB) model for the nonlinear and dispersive long gravity waves in shallow water traveling in two horizontal directions with varying depth. Connection between the vcvB model and a variable-coefficient Ablowitz–Kaup–Newell–Segur system is revealed under certain constraints with the help of the symbolic computation. Multi-solitonic solutions in terms of the double Wronskian determinant for the vcvB model are derived. Interactions among the vcvB-solitons are discussed. A novel dynamic property is observed, i.e., the coexistence of elastic–inelastic-interactions.

Derivation and analysis of a new 2D Green–Naghdi system**Ce travail a bénéficié d'une aide de l'Agence Nationale de la Recherche portant la référence ANR-08-BLAN-0301-01.

We derive here a variant of the 2D Green–Naghdi equations that model the propagation of two-directional, nonlinear dispersive waves in shallow water. This new model has the same accuracy as the standard 2D Green–Naghdi equations. Its mathematical interest is that it allows a control of the rotational part of the (vertically averaged) horizontal velocity, which is not the case for the usual Green–Naghdi equations. Using this property, we show that the solution of these new equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition. Finally, we prove that the new Green–Naghdi equations conserve the almost irrotationality of the vertically averaged horizontal component of the velocity.

Viscous and inviscid regularizations in a class of evolutionary partial differential equations

We investigate solution properties of a class of evolutionary partial differential equations (PDEs) with viscous and inviscid regularization. An equation in this class of PDEs can be written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. A recently developed two-step iterative method (P.H. Chiu, L. Lee, T.W.H. Sheu, A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation, J. Comput. Phys. 228 (2009) 8034–8052) is employed to study this class of PDEs. The method is in principle superior for PDE’s in this class as it preserves their physical dispersive features. In particular, we focus on a Leray-type regularization (H.S. Bhat, R.C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci. 16 (2006) 615–638) of the Hopf equation proposed in alternative to the classical Burgers viscous term. We show that the regularization effects induced by the alternative model can be vastly different from those induced by Burgers viscosity depending on the smoothness of initial data in the limit of zero regularization. We validate our numerical scheme by comparison with a particle method which admits closed form solutions. Further effects of the interplay between the dispersive terms comprising the Leray-regularization are illustrated by solutions of equations in this class resulting from regularized Burgers equation by selective elimination of dispersive terms.

On the best quantity reconstructions for a well balanced finite volume method used to solve the shallow water wave equations with a wet/dry interface

Well balanced finite volume methods used to solve the shallow water wave equations are designed to preserve the steady state of a `lake at rest'. Unfortunately, for problems involving wet/dry interfaces, this steady state is not preserved unless the involved quantities are reconstructed with care. We test four reconstruction options: stage and momentum (where bed is fixed); stage and velocity (where bed is fixed); stage, water height, and velocity; and stage, bed, and velocity (with modification at wet/dry interfaces). Reconstructions based on stage, water height, and velocity are shown to preserve the steady state problem and accurately solve a representative unsteady state problem, whereas the other options lead to problems in both maintaining the steady solution and accurately solving non-steady problems. Our results indicate the appropriate choice of reconstruction variables for various situations when solving the shallow water wave equations using finite volume methods. References E. Audusse, F. Bouchut, M O. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing , 25(6):2050--2065, 2004. doi:10.1137/S1064827503431090 A. Kurganov, S. Noelle, and G. Petrova. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM Journal on Scientific Computing , 23(3):707--740, 2001. doi:10.1137/S1064827500373413 S. Noelle, N. Pankratz, G. Puppo, and J R. Natvig. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. Journal of Computational Physics , 213(2):474--499, 2006. doi:10.1016/j.jcp.2005.08.019 S. Roberts, O. Nielsen, D. Gray, and J. Sexton. ANUGA User Manual . Geoscience Australia, 2009. http://datamining.anu.edu.au/anuga W C. Thacker. Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics , 107:499--508, 1981. doi:10.1017/S0022112081001882

Fully nonlinear shallow water waves simulation using Green-Naghdi theory

Green-Naghdi (G-N) theory is a fully nonlinear theory for water waves. Some researchers call it a fully nonlinear Boussinesq model. Different degrees of complexity of G-N theory are distinguished by “levels” where the higher the level, the more complicated and presumably more accurate the theory is. In the research presented here a comparison was made between two different levels of G-N theory, specifically level II and level III G-N restricted theories. A linear analytical solution for level III G-N restricted theory was given. Waves on a planar beach and shoaling waves were both simulated with these two G-N theories. It was shown for the first time that level III G-N restricted theory can also be used to predict fluid velocity in shallow water. A level III G-N restricted theory is recommended instead of a level II G-N restricted theory when simulating fully nonlinear shallow water waves.

Wave Overtopping and Damage Progression of Stone Armor Layer

A probabilistic hydrodynamic model for the wet and dry zone on a permeable structure is developed to predict irregular wave action on the structure above the still water level. The model is based on the time-averaged continuity and momentum equations for nonlinear shallow-water waves coupled with the exponential probability distribution of the water depth. The model predicts the cross-shore variations of the mean and standard deviation of the water depth and horizontal velocity. The model is compared with four test series in which measurement was made of the wave overtopping rate and probability as well as the water depth, velocity, and discharge exceeded by 2% of incident 1,000 waves. The agreement is mostly within a factor of 2. Damage progression of a stone armor layer is predicted by modifying a formula for bed load on beaches with input from the hydrodynamic model. The damage progression model is compared with three tests that lasted up to 28.5 h. The model predicts the eroded area of the damaged armor layer well but overpredicts the deposited area because it does not account for discrete stone units deposited at a distance seaward of the toe of the damaged armor layer.

Gender and Age Factors in Tsunami Casualties

A tsunami casualty model is developed with criteria based on whether a person can remain standing within tsunami flow. A simplified model of a human body is made using anthropometric data. By applying the model to tsunami inundation predictions generated by analytic solution of fully nonlinear shallow-water wave theory, we demonstrate quantitatively that differences in gender and age fatalities are most pronounced when a tsunami is marginally significant; strong tsunamis wipe out population almost indiscriminately. This trend is consistent with some of the field observations. It is important to consider gender and age factors when a coastal community plans for tsunami preparedness. Because the development of the casualty model is based solely on physical mechanisms, the model results can be used to remove physical causes from field data, which then allows for further investigation of social and behavioral factors.

Acoustic Ducting, Reflection, Refraction, and Dispersion by Curved Nonlinear Internal Waves in Shallow Water

Nonlinear internal waves in shallow water have been shown to be effective ducts of acoustic energy, through theory, numerical modeling, and experiment. To date, most work on such ducting has concentrated on rectilinear internal wave ducts or those with very slight curvature. In this paper, we examine the acoustic effects of significant curvature of these internal waves. (By significant curvature, we mean lateral deviation of the internal wave duct by more than half the spacing between internal waves over an acoustic path, giving a transition from ducting to antiducting.) We develop basic analytical models of these effects, employ fully 3-D numerical models of sound propagation and scattering, and examine simultaneous acoustical and oceanographic data from the 2006 Shallow Water Experiment (SW06). It will be seen that the effects of curvature should be evident in the mode amplitudes and arrival angles, and that observations are consistent with curvature, though with some possible ambiguity with other scattering mechanisms.

A Lagrangian model for boundary layer growth and bed shear stress in the swash zone

A new model for the boundary layer development and associated skin friction coefficients and shear stress within the swash zone is presented. The model is developed within a Lagrangian reference frame, following fluid trajectories, and can be applied to both laminar flow and smooth turbulent flow. The model is based on the momentum integral approach for steady, flat-plate boundary layers, with appropriate modifications to account for the unsteady flow regime and flow history. The model results are consistent with previous measurements of bed shear stress and skin friction coefficients within the swash zone. These indicate strong temporal and spatial variation throughout the swash cycle, and a clear distinction between the uprush and backwash phase. This variation has been previously attributed the unsteady flow regime and flow history effects, both of which are accounted for in the new model. Fluid particle trajectories and velocity are computed using the non-linear shallow water wave equations and the boundary layer growth across the entire swash zone is estimated. Predictions of the bed shear stress and skin friction coefficients agree reasonably well with direct bed shear stress measurements reported by Barnes et al. (Barnes, M.P., O’Donaghue, T., Alsina, J.M., Baldock, T.E., 2009. Direct bed shear stress measurements in bore-driven swash. Coastal Engineering 56 (8), 853–867) and, for a given flow velocity, give stresses which are consistent with the bias toward uprush sediment transport which has consistently been observed in measurements. The data and modelling suggest that the backwash boundary layer is initially laminar, which results in the late development of significant bed shear during the backwash, with a transition to a turbulent boundary layer later in the backwash. A new conceptual model for the boundary layer structure at the leading edge of the swash is proposed, which accounts for both the no-slip condition at the bed and the moving wet–dry interface. However, further development of the Lagrangian Boundary Layer Model is required in order to include bore-generated turbulence and to account for variable roughness and mobile beds.

Modeling of nonlinear surface gravity waves in shallow water with regard to dispersion

This report considers the problem of the propagation of nonlinear surface gravity waves over shallow waters with regard to dispersion. By means of the semigraphical method proposed, the transformation of the profile of nonlinear surface gravity waves is described. The profiles of surface gravity waves are presented for the stages of crest sharpening and front steepening, as well as the wave train disintegration caused by dispersion.

Numerical modeling of wave evolution and runup in shallow water

Based on the Navier-Stokes (N-S) equations for viscous, incompressible fluid and the VOF method, 2-D and 3-D Numerical Wave Tanks (NWT) for nonlinear shallow water waves are built. The dynamic mesh technique is applied, which can save computational resources dramatically for the simulation of solitary wave propagating at a constant depth. Higher order approximation for cnoidal wave is employed to generate high quality waves. Shoaling and breaking of solitary waves over different slopes are simulated and analyzed systematically. Wave runup on structures is also investigated. The results agree very well with experimental data or analytical solutions.

Lagrangian Modeling of Weakly Nonlinear Nonhydrostatic Shallow Water Waves in Open Channels

A Lagrangian, nonhydrostatic, Boussinesq model for weakly nonlinear and weakly dispersive flow is presented. The model is an extension of the hydrostatic model—dynamic river model. The model uses a second-order, staggered grid, predictor-corrector scheme with a fractional step method for the computation of the nonhydrostatic pressure. Numerical results for solitary waves and undular bores are compared with Korteweg-de Vries analytical solutions and published numerical, laboratory, and theoretical results. The model reproduced well known features of solitary waves, such as wave speed, wave height, balance between nonlinear steepening and wave dispersion, nonlinear interactions, and phase shifting when waves interact. It is shown that the Lagrangian moving grid is dynamically adaptive in that it ensures a compression of the grid size under the wave to provide higher resolution in this region. Also the model successfully reproduced a train of undular waves (short waves) from a long wave such that the predict...

A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation

The paper presents an iterative algorithm for studying a nonlinear shallow-water wave equation. The equation is written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step iterative method that first solves the evolution equation by the implicit midpoint rule and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The midpoint rule is a symplectic time integrator for Hamiltonian systems, which may be a preferable method to solve the spatially discretized evolution equation. To give an assessment of the dispersion-preserving scheme, we provide a detailed analysis of the dispersive and dissipative errors of this algorithm. Via a variety of examples, we illustrate the efficiency and accuracy of the proposed scheme by examining the errors in different norms and providing the rates of convergence of the method. In addition, we provide several examples to demonstrate that the conserved quantities of the equation are well preserved by the implicit midpoint time integrator. Finally, we compare the accuracy, elapsed computing time, and spatial and temporal rates of convergence among the proposed method, a complete integrable particle method, and the local discontinuous Galerkin method.

A numerical solution for moving boundary shallow water flow above parabolic bottom topography

A numerical method has been applied to the nonlinear shallow water wave equations for unforced linear frictional flow above parabolic bottom topography and is found to be accurate. This solution involves moving shorelines. The motion decays over time. The numerical scheme used is adapted from the Selective Lumped Mass numerical scheme. The wetting and drying algorithm used in the numerical scheme is different to that in the Selective Lumped Mass scheme. The numerical scheme is finite element in space, using fixed triangular elements, finite difference in time and is explicit. The numerical solution compares well with an analytical solution. References Balzano, A., Evaluation of methods for numerical simulation of wetting and drying in shallow water flow models, Coastal Engineering, 34, 1998, 83--107. Goraya, S., A study of finite element tidal models, PhD. thesis, Swinburne University of Technology, Melbourne, Australia, 2001. Holdahl, R., Holden, H., and Lie,K-A., Unconditionally Stable Splitting Methods For the Shallow Water Equations, BIT, 39, 1998, 451--472. Kawahara, M., Hirano, H., and Tsubota, K., Selective lumping finite element method for shallow water flow, International Journal for Numerical Methods in Fluids, 2, 1982, 89--112. Lewis, C. H. III and Adams, W. M., Development of a tsunami-flooding model having versatile formation of moving boundary conditions,The Tsunami Society Monograph Series, 1983, No. 1, 128 pp. Parker, B. B., Frictional Effects on the Tidal Dynamics of a Shallow Estuary, PhD thesis, The John Hopkins University, 1984. Peterson P., Hauser J., Thacker W. C., Eppel D., An Error-Minimizing Algorithm for the Non-Linear Shallow-Water Wave Equations with Moving Boundaries. In Numerical Methods for Non-Linear Problems, editors C. Taylor, E. Hinton, D. R. J. Owen and E. Onate, 2, Pineridge Press, 1984, 826--836, http://www.cle.de/hpcc/publications/ Sampson, J., Easton, A., and Singh, M., Moving boundary shallow water flow in parabolic bottom topography, ANZIAM Journal, 47 (EMAC2005), C373-387, 2006, http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1050. Sampson, Joe, Easton, Alan and Singh, Manmohan, A New Moving Boundary Shallow Water Wave Numerical Model, Australian and New Zealand Industrial and Applied Mathematics Journal, 48 (CTAC2006), C605--C617, 2007, http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/78. Sampson, Joe, Easton, Alan and Singh, Manmohan, Moving boundary shallow water flow in a region with quadratic bathymetry,Australian and New Zealand Industrial and Applied Mathematics Journal, 49 (EMAC2007), C666--C680, 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/306. Thacker, W. C., Some exact solutions to the nonlinear shallow-water wave equations,J. Fluid. Mech., 107, 1981, 499--508. Vreugdenhil, C. B., Numerical Methods for Shallow-Water Flow, Kluwer Academic Publishers, 1998. Yoon S., B., and Cho J. H., Numerical simulation of Coastal Inundation over Discontinuous Topography, Water Engineering Research, 2(2), 2001, 75--87.