Nonlinear Shallow Water Research Articles

Tsunami run-up estimation based on a hybrid numerical flume and a parameterization of real topobathymetric profiles

Abstract. Tsunami run-up is a key value to determine when calculating and assessing the tsunami hazard in a tsunami-prone area. Run-up can be accurately calculated by means of numerical models, but these models require high-resolution topobathymetric data, which are not always available, and long computational times. These drawbacks restrict the application of these models to the assessment of small areas. As an alternative method, to address large areas empirical formulae are commonly applied to estimate run-up. These formulae are based on numerical or physical experiments on idealized geometries. In this paper, a new methodology is presented to calculate tsunami hazard at large scales. This methodology determines the tsunami flooding by using a coupled model that combines a nonlinear shallow water model (2D-H) and a volume-of-fluid model (RANS 2D-V) and applies the optimal numerical models in each phase of the tsunami generation–propagation–inundation process. The hybrid model has been widely applied to build a tsunami run-up database (TRD). The aim of this database is to form an interpolation domain with which to estimate the tsunami run-up of new scenarios without running a numerical simulation. The TRD was generated by simulating the propagation of parameterized tsunami waves on real non-scaled profiles. A database and hybrid numerical model were validated using real and synthetic scenarios. The new methodology provides feasible estimations of the tsunami run-up; engineers and scientists can use this methodology to address tsunami hazard at large scales.

Communication-aware adaptive Parareal with application to a nonlinear hyperbolic system of partial differential equations

In the strong scaling limit, the performance of conventional spatial domain decomposition techniques for the parallel solution of PDEs saturates. When sub-domains become small, halo-communication and other overhead come to dominate. A potential path beyond this scaling limit is to introduce domain-decomposition in time, with one such popular approach being the Parareal algorithm which has received a lot of attention due to its generality and potential scalability. Low efficiency, particularly on convection dominated problems, has however limited the adoption of the method. In this paper we demonstrate trough large-scale numerical experiments that it is possible not only to obtain time-parallel speedup on the non-linear shallow water wave equation, but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain-decomposition techniques alone. Two factors were essential in achieving this. First, for Parareal to converge on the hyperbolic problem we used an approximate Riemann solver as the preconditioner. This preconditioner introduces only dissipative errors with respect to the 3rd order accurate WENO-RK discretization used to solve the PDE system. The preconditoner is comparatively expensive and convergence is slow unless time-subdomains are short. We therefore introduce a new scheduler that we denote Communication Aware Adaptive Parareal (CAAP). CAAP increases obtainable speed-up by minimizing the time-subdomain length without making communication of time-subdomains too costly whilst also adaptively overlapping consecutive cycles of Parareal so to mitigate the impact of a relatively expensive coarse operator.

The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

We propose a new computational approach for embedded boundary simulations of hyperbolic systems and, in particular, the linear wave equations and the nonlinear shallow water equations. The proposed approach belongs to the class of surrogate/approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforced weakly, are appropriately modified to preserve optimal error convergence rates. This framework is applied here in the setting of a stabilized finite element method, even though other spatial discretization techniques could have been employed. Accuracy, stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations. Comparisons with standard weak boundary conditions imposed on body-fitted grids, which conform to the geometry of the computational domain boundaries, are also presented.

Experimental and Numerical Investigation of Solitary Wave Run-up Reduction

Rehman, K.; Park, K.-Y., and Cho, Y.-S., 2018. Experimental and Numerical Investigation of Solitary Wave Run-up Reduction. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 1111–1115. Coconut Creek (Florida), ISSN 0749-0208.Rising sea-levels and extreme wave events threaten coastal communities and stability of coastal regions. Accurate prediction of wave over-topping over coastal protection structures is challenging, but vital for effective hazard mitigation. Non-hydrostatic numerical modelling and laboratory experiments are used to assess magnitude of run-up over coastal protection structures under varying relative wave heights and structural features. The primary focus is the investigation of solitary wave impact with breakwaters, the consequent run-up and measures for its reduction. The experiments consisted of generating solitary waves in a 1.1 m high and 32.5 m long flume and observing its run-up for different heights of incident waves. A slope adjuster was used to vary the slope of a plywood plank for reproducing coastal features. Experimental observations were verified by proposing a numerical model based on non-linear shallow water equations (NLSWE) and solution is obtained by Godunov-type finite volume method. The NLSWE provide good approximation of shoaling, wave breaking, and wave reflection which arise due to wave overtopping in the swash and surf zones. The novel feature of the numerical model is the introduction of bed slope discretization technique – applicable on both structure and unstructured meshes- which offers well-balanced solution even for steep slopes encountered in case of breakwaters. Shock-capturing capabilities of Harten, Lax, and van Leer with contact wave restoration (HLLC) solver are utilized for accurate estimation of shocks and bore waves features during flow transitions. The proposed model gives excellentn agreement with experimental observations. The findings will further enhance the understanding of extreme wave propagation events over submerged coastal structures and related mitigation techniques.

Long Wave Run-up on Plane and “Non-Reflecting” Slopes

The long wave run-up on two types of slopes is investigated numerically within the framework of nonlinear shallow water theory using the CLAWPACK software. One of the slopes represents a plane slope widely used in the laboratory and numerical experiments; the second is the so-called “non-reflecting” slope (h ~ x4/3, where h is the basin depth and x is the distance from the shoreline). In the case of very low wave amplitudes when there is no wave breaking, the run-up height is greater on the non-reflecting beach than that on the plane slope. As the wave amplitude increases, the breaking effects have the stronger impact in the case of non-reflecting beach and the run-up height becomes smaller.

Process-based modelling of bank-breaking mechanisms of tidal sandbanks

Tidal sandbanks are large-scale dynamic bed forms observed in shallow shelf seas. Their plan view evolution may display a single bank breaking into two or more banks, for which two mechanisms have been proposed in the literature. However, as both were based on interpretation of observations, generic support from a process-based modelling perspective is lacking so far. Here we present a new idealised process-based model study into the transient evolution of tidal sandbanks. Key elements are the inclusion of nonlinear dynamics for topographies that vary in both horizontal directions, and the focus on long-term evolution (centuries and longer). As a further novelty, the hydrodynamic solution, satisfying the nonlinear shallow water equations including bottom friction and the Coriolis effect, is obtained from a truncated expansion in the ratio of maximum bank elevation (w.r.t.mean depth) and mean water depth. Bed evolution follows from the tidally averaged bed load sediment transport, enhanced by depth-dependent wind-wave stirring. From our model results, we identify two paths of evolution, leading to either bank-breaking or an S-shape. Which of these paths occurs depends on initial topography, with bank orientation and bank length as major control parameters. The breaking and S-shape obtained in our model results show resemblance with banks observed in the North Sea.

Three-Dimensional Weakly Nonlinear Shallow Water Waves Regime and its Traveling Wave Solutions

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

Numerical modelling of flow in Little Pigeon Bay due to the 2016 Kaikoura tsunami

The magnitude 7.8 Kaikoura earthquake on the north-eastern coast of South Island, New Zealand induced a tsunami, which affected many north-facing bays in Banks Peninsula. The V-shaped Little Pigeon Bay experienced pronounced inundation compared to adjacent bays, causing heavy damage to a cottage at the head of the bay. In order to investigate the characteristics of the tsunami flow in the bay, a numerical model based on nonlinear shallow water theory was established, using tidal levels from the neighbouring Sumner sea level gauge. Three incident wave angles were tested, ranging from due north to 30° east of north. The simulated maximum run-up heights were validated with surveyed results. For all the wave directions, a resonance with a period of 6.3 min was found from 50 to 100 min in the simulation. Based on the physical dimensions of Little Pigeon Bay, idealised models were set-up to analyse the influence of V-shaped embayment on maximum water level, maximum velocity and maximum momentum flux. Bays with sand bars of different crest heights were also investigated. A sand bar with a relatively low crest height (dw/d≥0.5) did not significantly reduce the maximum velocity and the maximum momentum flux.

A high‐order PIC method for advection‐dominated flow with application to shallow water waves

SummaryA hybrid Eulerian‐Lagrangian particle‐in‐cell–type numerical method is developed for the solution of advection‐dominated flow problems. Particular attention is given over to the high‐order transfer of flow properties from the particles to the grid. For smooth flows, the method presented is of formal high‐order accuracy in space. The method is applied to solve the nonlinear shallow water equations resulting in a new, and novel, shock capturing shallow water solver. The approach is able to simulate complex shallow water flows, which can contain an arbitrary number of discontinuities. Both trivial and nontrivial bottom topography is considered, and it is shown that the new scheme is inherently well balanced, exactly satisfying the ‐property. The scheme is verified against several one‐dimensional benchmark shallow water problems. These include cases that involve transcritical flow regimes, shock waves, and nontrivial bathymetry. In all the test cases presented, very good results are obtained.

A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids

Within the framework of the two-dimensional cell-centered Godunov-type finite volume (CCFV) method, this paper presents a novel multislope scheme on the basis of the monotone upstream scheme for conservation law (MUSCL) for numerically solving nonlinear shallow water equations on two-dimensional triangular grids. The Riemann states of the considered edge are calculated by an edge-based reconstructing procedure, where a limited scalar slope is employed to prevent potential numerical oscillations. The novel aspect of the new scheme is that it takes advantage of the geometrical characteristics of triangular grids in the reconstructing and limiting procedures, which effectively reduces the cost of computation and provides higher resolution and accuracy compared with classical MUSCL schemes. Seven tests are adopted to verify the scheme, and the results indicate that this scheme is efficient, accurate, robust, and high-resolution, and can be an ideal alternative for solving shallow water problems over uneven and frictional topography.

A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

This work extends a robust second-order Runge–Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green–Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modeling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model.

A well-balanced element-free Galerkin method for the nonlinear shallow water equations

In this paper, we consider the nonlinear shallow water equations over variable bottom topography in one dimension and propose a well-balanced element-free Galerkin method for solving this system. The proposed scheme has the features of being high-order accurate for general solutions and exactly preserving the still-water stationary solution. The main ingredient to achieve the well-balanced property is to use a special decomposition to the source term and discretize the source term as the flux term. Numerical tests are presented to illustrate the accuracy and validity of the proposed scheme.

Hydraulic jump and Bernoulli equation in nonlinear shallow water model

A shallow water model was applied to study the hydraulic jump and Bernoulli equation across the jump. On a flat terrain, when a supercritical flow plunges into a subcritical flow, discontinuity develops on velocity and Bernoulli function across the jump. The shock generated by the obstacle may propagate downstream and upstream. The latter reflected from the inflow boundary, moves downstream and leaves the domain. Before the reflected wave reaching the obstacle, the short-term integration (i.e., quasi-steady) simulations agree with Houghton and Kasahara’s results, which may have unphysical complex solutions. The quasi-steady flow is quickly disturbed by the reflected wave, finally, flow reaches steady and becomes critical without complex solutions. The results also indicate that Bernoulli function is discontinuous but the potential of mass flux remains constant across the jump. The latter can be used to predict velocity/height in a steady flow.

A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations

Abstract The high-order numerical solution of the non-linear shallow water equations is susceptible to Gibbs oscillations in the proximity of strong gradients. In this paper, we tackle this issue by presenting a shock capturing model based on the numerical residual of the solution. Via numerical tests, we demonstrate that the model removes the spurious oscillations in the proximity of strong wave fronts while preserving their strength. Furthermore, for coarse grids, it prevents energy from building up at small wave-numbers. When applied to the continuity equation to stabilize the water surface, the addition of the shock capturing scheme does not affect mass conservation. We found that our model improves the continuous and discontinuous Galerkin solutions alike in the proximity of sharp fronts propagating on wet surfaces. In the presence of wet/dry interfaces, however, the model needs to be enhanced with the addition of an inundation scheme which, however, we do not address in this paper.

Development of a Tsunami Inundation Analysis Model for Urban Areas Using a Porous Body Model

To evaluate tsunami hazards with strong locality in urban areas, this study developed a novel tsunami inundation model based on nonlinear shallow water wave equations and a porous body model (PBM). By applying a kinematic boundary condition that includes both porosity and surface permeability of the porous medium, the proposed model could accurately incorporate geometric effects such as the flow anisotropy caused by the distributions of buildings. The proposed PBM demonstrated as good accuracy for the inundation heights around buildings near the coastline as with a conventional three-dimensional simulation with high resolution. In addition, the model showed its capability to reproduce a tsunami’s essential behaviors in urban areas. In particular, the amplification effect of flow velocity along straight roads surrounded by buildings was reasonably reproduced. It can be expected that the present model can become a useful tool to accurately evaluate the tsunami risks in urban areas.

The shoreline problem for the one-dimensional shallow water and Green-Naghdi equations

The Green-Naghdi equations are a nonlinear dispersive perturbation of the nonlinear shallow water equations, more precise by one order of approximation. These equations are commonly used for the simulation of coastal flows, and in particular in regions where the water depth vanishes (the shoreline). The local well-posedness of the Green-Naghdi equations (and their justification as an asymptotic model for the water waves equations) has been extensively studied, but always under the assumption that the water depth is bounded from below by a positive constant. The aim of this article is to remove this assumption. The problem then becomes a free-boundary problem since the position of the shoreline is unknown and driven by the solution itself. For the (hyperbolic) nonlinear shallow water equation, this problem is very related to the vacuum problem for a compressible gas. The Green-Naghdi equation include additional nonlinear, dispersive and topography terms with a complex degenerate structure at the boundary. In particular, the degeneracy of the topography terms makes the problem loose its quasilinear structure and become fully nonlinear. Dispersive smoothing also degenerates and its behavior at the boundary can be described by an ODE with regular singularity. These issues require the development of new tools, some of which of independent interest such as the study of the mixed initial boundary value problem for dispersive perturbations of characteristic hyperbolic systems, elliptic regularization with respect to conormal derivatives, or general Hardy-type inequalities.

Asymptotic stability of a polar vortex perturbed by harmonic waves describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet

We examine stability of the vortex that represents one particular class of exact solution of a a nonlinear shallow water model describing atmospheric gravity waves circulating in an equatorial plane of a spherical planet. The mathematical model is represented by a two-dimensional free boundary Cauchy–Poisson problem on the nonstationary motion of a perfect uid around a solid circle with a sufficiently large radius so that the gravity is directed to the center of the circle. It is shown that the model admits two functionally independent nonlinear systems of shallow water equations. Two essential parameters that control stability of the vortex for both systems are identified. The order of their importance is analyzed and it is shown that one of the systems is more resistant to small perturbations and remains stable for larger range of these two parameters.

Wave Breaking in a Shallow Water Model

We prove wave breaking --- bounded solutions with unbounded derivatives --- in the nonlinear nonlocal equations which combine the dispersion relation of water waves and the nonlinear shallow water equations, and which generalize the Whitham equation to permit bidirectional wave propagation, provided that the slope of the initial data is sufficiently negative.

Advanced numerical models for the propagation of floods with high-sediment concentrations in mountain rivers

Rapid floods induced by extreme precipitation are common events in regions near the Andes mountain range. Growing urban development, combined with the changing climate and the influence of El Niño, have increased the exposure of the population in many regions of South America. Simulations of flash floods in these watersheds are very challenging, due to the complex morphology, the insufficient hydrometeorological data, and the uncertainty posed by the variability of sediment concentration. To address these issues, we develop a high-resolution numerical model of the non-linear shallow water equations, coupled with the mass conservation of sediment, and considering the density effects and changes of rheology in the momentum equation. Based on these simulations we develop a real-time early-warning system, by creating a surrogate model or meta-model from the simulations. Using a small set of parameters, we define storms for a wide range of meteorological conditions, and utilize the high-fidelity model results to create a database of flood propagation under different conditions. Through this second model we perform a sophisticated interpolation/regression, and approximate efficiently the flow depths and velocities. This is the first application of its kind in the Andes region, which can be used to improve the prediction of flood hazard in real conditions, employing low computational resources. We also create a framework to develop early warning systems, and to help decision makers and city planners in these mountain regions.

‘The Equator–Pole grid system’: an overset grid system for the sphere, with an optimal uniformity property

This article proposes a new grid system for the sphere, which consists of three orthogonal and almost uniform grids. The basic one is a latitude-longitude grid covering an annular band around the equator. For the rest of the sphere this grid is complemented by two grids covering the polar regions, and based on suitably modified stereographic coordinates. The rectangular structure of the grids makes highly efficient implementation on massively parallel computer systems possible. Numerical experiments with the new grid system are carried out for two advection examples, namely smooth deformational flow and rotation of the Cosine bell, and for the test problems 2, 3, and 6 from Williamson et al., concerning the non-linear shallow water equations. For problem 6, the Rossby–Haurwitz wave, we study conservation properties for mass and energy. The computational results compare favourably with results for other grids. Our focus is the definition of the grid system together with the connection between grids by overlapping and the use of centred interpolation formulas. The method of centred finite differences is used for the spatial discretisation of the differential equations, because it is well-known and easy to implement.