Momentum Conservative Scheme for Simulating Wave Runup and Underwater Landslide

An algorithm is presented for numerical modelling of impulse water waves generated by submarine landslides moving along irregular bottom profiles. A spatially nonuniform submarine landslide moving on a spatially nonuniform slope is modelled by a “quasi-deformed” rigid body. In addition, a simplified model is studied for the particular case of landslide and bottom profiles dependent on one spatial coordinate only. Both models are used for the comparative analysis of numerical and experimental data for submarine rigid landslides moving along a plane slope. The simplified model is applied to analyse the dependences of wave characteristics on various parameters for submarine landslides moving along a sea bottom slope with monotonically increasing depth and in a bounded reservoir. The full model is used to study the landslide trajectories and the wave patterns for the model submarine landslide of a spatially irregular shape moving in the model reservoir with a spatially irregular bottom. All numerical computations are performed using the nonlinear shallow water equations with moving bottom and the time-stepping algorithm provided by a predictor–corrector scheme on adaptive grids.

In this work, we study about nonlinear surface wave generated by submarine landslides. The landslide is modeled as a rigid block sliding down along a sloping bottom. We implement a numerical scheme based on the staggered finite volume method to simulate surface wave. We enhance the conservative scheme for nonlinear shallow water equation (SWE) to include this bottom motion. Our numerical result show a good agreement with previous results of the nonlinear boundary integral equation model (BIEM) by Lynett and Liu and central-upwind scheme by Kurganov and Petrova.

Numerical study of periodic long wave run-up on a rigid vegetation sloping beach

Tsunamis, wave triggered by underwater earthquakes or volcanic eruptions, can achieve significant run-up heights upon reaching shore. Run-up refers to the maximum vertical distance a tsunami wave reaches above the normal sea level. This study employs a numerical model to simulate the run-up wave at Canti Beach, South Lampung Regency, during the 2018 Sunda Strait tsunami. The method approximates solutions to the shallow water model consisting of mass and momentum conservation equations using the finite difference method in staggered grid grids and incorporates the upwind method for nonlinear terms. Bathymetry data from GEBCO was projected in two dimensions using the haversine formula. The numerical scheme includes a wet-dry procedure for simulating run-up waves. Results indicate that waves with a 60-second period and 0.09-meter amplitude create a 40.0195-meter inundation area, although this amplitude is significantly lower than the observed data from the 2018 Sunda Strait tsunami. Additionally, a simulation with a 0.1-meter amplitude results in a 1.8299-meter run-up height, closely matching the observed data. This study demonstrates that nonlinear shallow water equations can effectively estimate run-up height and inundation area at Canti Beach.

Numerical study of vegetation damping effects on solitary wave run-up using the nonlinear shallow water equations

In this research, we investigate how submarine landslides are able to produce extremely damaging tsunami waves and estimate the maximum height of the waves in question using a mathematical model. The model is based on the non-linear shallow water equations, which are solved numerically using the finite volume method on a staggered grid. We also derive a new analytical solution in order to obtain a description of surface waves generated by a submarine landslide. To validate the numerical model, several benchmark tests are conducted by comparing the results from the numerical scheme to those obtained from the analytical solution, experimental data, or other numerical results. After validation, the scheme is implemented on the real topography of Palu Bay to simulate the generation of tsunami waves by submarine landslides. The simulations provide excellent results. Accurate estimates of the maximum tsunami wave amplitude are produced by the numerical scheme, confirmed by data from records of the 2018 Palu tsunami.

Irregular wave propagation with a 2DH Boussinesq-type model and an unstructured finite volume scheme

Landslide-generated tsunami waves (LGTWs) are gravity waves generated by impulsive impacts of the submarine or subaerial landslides into a water body. Large-scale LGTWs are extreme natural hazards with devastating and fatal consequences on infrastructures and communities. The study of LGTW processes, e.g. slides initiation (or triggering), motion, interaction with water and air, impulsive wave generation, propagation, run-up, overtopping, and inundation, is a multidisciplinary challenge and is an essential task to acquire an informed assessment and to manage the LGTW risks. Landslide-generated waves and the interactions between landslide and water body are the topics of this thematic issue of Landslides. The objective of this issue is to increase awareness and to extend knowledge, about this important theme, among the Landslides readers, both practitioners and researchers. Laboratory experiments are an indispensable component of obtaining a comprehensive understanding of the LGTW event complexities. Alessandro Romano, Marcello Di Risio, Giorgio Bellotti, Matteo Molfetta, Leonardo Damiani and Paolo De Girolamo provide a dataset of three-dimensional experiments for subaerial LGTWs triggered by slide at the edge of a conical island. The high spatial resolution of the dataset can be used for the benchmarking of analytical and numerical models. A series of experimental data in a wave basin with flexible mesh-packed slides is presented by Frederic Evers and Willi Hager. They examine the existing prediction methods for wave height in comparison with the test data. Modelling the interactions of the landslide and water body is a demanding and an elegant task and it is required for an intuitive understanding of the related phenomena and the risk assessment of LGTWs. There are two review articles in this issue with the focuses on the analytical and numerical modelling of LGTWs. An extensive review of the conceptual, mathematical and numerical aspects of the landslide-generated waves modelling for both submarine and subaerial is presented by Saeedeh Yavari-Ramsheh and Behzad Ataie-Ashtiani. They also scrutinise the shortcomings and the future challenges for the numerical modelling of LGTWs. Emiliano Renzi and Paolo Sammarco provide an overview of analytical models for the dynamics of landslide tsunami generation and propagation along beaches and around islands. The analytical modelling is in the framework of shallow-water wave theory. They also suggest directions for the future research of analytical modelling in this field. The application of numerical modelling in association with field measurements and investigations of recent events and case studies, as well as data analysis of historic events are key sources for an enhanced insight into the landslide-generated wave impacts. James Kirby, Fengyan Shi, Dmitry Nicolsky and Shubhra Misra present the numerical simulations for tsunami waves generated by the 27 April 1975 Kitimat, British Columbia submarine landslide. Slide motion is modelled using two approaches by assuming a solid sliding mass or a highly viscous Newtonian fluid lower layer. They show both approaches can resemble the observations. Kyoji Sassa, Khang Dang, Hideaki Yanagisawa and Bin He introduce a new model for the numerical simulation of the landslide-induced tsunami. The underwater landslide initiation and motion, as well as tsunami wave due to the landslide, are simulated. They applied the model to the largest landslide-induced tsunami disaster occurred in Japan and their numerical simulations successfully reproduce the historical tsunami heights, recorded for the 1792 Unzen-Mayuyama event. Sylfest Glimsdal, Jean-Sebastien L’Heureux, Carl B. Harbitz and Finn Lovholt present the results of the post-tsunami field survey and the numerical modelling of the 29th January 2014 submarine landslide at Statland, Norway. By comparing the simulated tsunami run-up for different scenarios with observations, they reconstruct the landslide dynamics, tsunami generation and wave runup. James Goff and James Terry discuss tsunamigenic slope failures as a possible source of the tsunami hazard for Pacific Island Countries and Territories. They suggest a range of indicators such as geological, oral tradition and archaeological components to identify evidence of past tsunamis. Jersain Gomez, Moises Berezowsky, Alejandro Lara and Elizabeth Gonzalez present a study of the water waves generated due a possible semi-submerged landslide in La Yesca Reservoir, Mexico. A physical model of the reservoir and numerical model based on shallow water equations are employed. The Guest Editor hopes that the thematic issue encourages further researches and publications in Landslides on various aspects of LGTW hazards including; warning procedures, understanding of related phenomena, and mitigation measures. I gratefully acknowledge the international experts who reviewed the manuscripts and in particular the support of Editor-in-Chief Professor Kyoji Sassa that made the publication of this thematic issue possible.

Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations

The Lower St. Lawrence Estuary (LSLE) is a 230 km long by 50 km wide trough with a broad, flat floor with maximum water depths of 400 m and “shelves” that sit in water depths of < 60 m. It is partly filled with thick glaciomarine and post-glacial sediments and lies within close proximity to the Charlevoix Seismic Zone, the most seismically active region of eastern Canada. The purpose of this paper is to present the modelled tsunami effects of two submarine landslides from the LSLE. A regional seafloor mapping project revealed several submarine landslides on the slopes and channel floor of the LSLE. The tsunamigenic effects of two instability features in the area were investigated. The features chosen were: (1) a blocky submarine landslide that covers an area of ~3 km2, a run-out distance of 1.2 km and maximum slab thickness of 20 m; and (2) a lateral spread feature with a 4 km long headwall escarpment and a maximum slab thickness of 10.5 m, which may be a candidate for a future landslide. Using a numerical wave tank, the nonlinear shallow water equations were solved for motions induced by the submarine instability features. The equations are solved numerically by the finite volume method, and the code is able to model accurately tsunami runup and drawdown.

The National Capital Integrated Coastal Development (NCICD) is a construction megaproject around Jakarta Bay targeted environmental revitalization and flood mitigation. One of the main projects of NCICD is to develop the Jakarta’s outer sea dike to prevent future disasters triggered by the increase of the sea level around Jakarta Bay. In this paper, we aim to assess and optimize the design of Jakarta’s outer sea dike by investigating the wave run-up phenomenon, which is measured as the maximum vertical extent of wave uprush on a structure above the still water level. We used the Non-linear Shallow Water Equations (NSWE) as our mathematical model to simulate this phenomenon. The NSWE model was solved numerically using the finite volume method on a staggered grid with a wet-dry procedure to obtain accurate wave run-up height. To validate our numerical scheme, we conducted benchmark tests against a publicized experimental dataset, resulting in a good agreement between the numerical and experimental data, which confirms the robustness and accuracy of our model. We then simulate the wave run-up over three different sea dike profiles: single slope, single berm, and single berm with rocks. Our study shows that among the cases we investigated, the single berm with rocks is the most effective design of the sea dike as even small-sized rock units can significantly reduce the wave run-up height.

The shallow water wave’s equation represents rapid unsteady flow frequently attended by shock waves. For shock phenomena, the influence of bottom friction may be assumed marginal, as the bottom width where shock arises is relatively very thin compared to the scale of the flow domain. However, the energy loss across the shock is significant. This energy loss is attributed to the internal stresses within the very thin infinitesimal shock interface. For practical computation, the contribution of the internal friction may be incorporated in the wall friction, in other words the internal stresses can be represented as Manning frictional resistance. Frictions either wall friction, surface friction, or internal friction between fluid particles are the sources or sinks of momentum. Strong simplification of modeling of the free surface shallow flows is necessary for the computer simulation. The material on the basis of shallow water models is essential, even considering a numerical method of any kind, similar to most of the shock-capturing numerical methods on the utilisation of local Riemann problem solution, both for the exact or approximate. However the role of the Riemann problem is wider. The Riemann problem can be useful in theoretical studies of simple shalow water models; it can also be used in conjunction with other numerical solution. This research deals with shock-capturing, finite volume numerical methods, particular devoted to the details of numerical methods of the shock-capturing type. Some hypothetical tests are modeled as a shallow water wave equation, which therefore can be cast as Riemann Problem, solved by utilizing the Godunov’s type solution. Finite volume methods of the Godunov type are used for the purpose of solving numerically the time-dependent, non-linear shallow water equations. Key words : shallow water, homogeneous, shock, sources, sinks, Riemann problem, finite volume, shock-capturing, Godunov’s type.

Equations of a landslide motion over an uneven underwater slope subject to gravity and buoyancy forces, water friction and resistance are presented. A simulation of surface waves generated by a landslide in a bounded reservoir with a parabolic bottom profile has been performed within the nonlinear shallow water equations, and the results of that simulation are given. The influence of the parameters of the motion equation on the maximal splashing size is studied numerically.

A combination of landslide and tsunami simulation model is presented to simulate landslide-induced tsunami. The submarine landslide and the tsunami are analyzed together using the nonlinear shallow water equations and the extended Boussinesq equations in order to predict the inter-connected phenomena. Numerical results show the initiation and movement of landslide and tsunami. The test cases show a good agreement between numerical results and analytical solutions. The model can be developed further to improve its accuracy and convenience for users.

Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations