Tsunami Wave Run-up Research Articles

Efficient numerical scheme based on the method of lines for the shallow water equations

In this paper, a nonlinear shallow-water model of tsunami wave propagation at different points along a coastline of an ocean has been numerically simulated using method of lines. The simulation is carried out for various coastal slopes and the ocean depths. The effects of the coast slope and sea depth on the tsunami wave run-up height and velocity are illustrated. The accuracy of the mathematical model is verified by solving a classical test problem with known analytic solution. The computed run-up height and velocity show satisfactory agreement with the tsunami wave physics.

Tsunami Wave Run-up on a Vertical Wall in Tidal Environment

We solve analytically a nonlinear problem of shallow water theory for the tsunami wave run-up on a vertical wall in tidal environment. Shown that the tide can be considered static in the process of tsunami wave run-up. In this approximation, it is possible to obtain the exact solution for the run-up height as a function of the incident wave height. This allows us to investigate the tide influence on the run-up characteristics.

A numerical study of tsunami wave impact and run-up on coastal cliffs using a CIP-based model

Abstract. There is a general lack of understanding of tsunami wave interaction with complex geographies, especially the process of inundation. Numerical simulations are performed to understand the effects of several factors on tsunami wave impact and run-up in the presence of gentle submarine slopes and coastal cliffs, using an in-house code, a constrained interpolation profile (CIP)-based model. The model employs a high-order finite difference method, the CIP method, as the flow solver; utilizes a VOF-type method, the tangent of hyperbola for interface capturing/slope weighting (THINC/SW) scheme, to capture the free surface; and treats the solid boundary by an immersed boundary method. A series of incident waves are arranged to interact with varying coastal geographies. Numerical results are compared with experimental data and good agreement is obtained. The influences of gentle submarine slope, coastal cliff and incident wave height are discussed. It is found that the tsunami amplification factor varying with incident wave is affected by gradient of cliff slope, and the critical value is about 45°. The run-up on a toe-erosion cliff is smaller than that on a normal cliff. The run-up is also related to the length of a gentle submarine slope with a critical value of about 2.292 m in the present model for most cases. The impact pressure on the cliff is extremely large and concentrated, and the backflow effect is non-negligible. Results of our work are highly precise and helpful in inverting tsunami source and forecasting disaster.

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.

Wavelet Galerkin scheme for solving nonlinear dispersive shallow water waves: Application in bore propagation and breaking

An accurate and fast numerical method is developed to investigate the nonlinear (linear) shallow water wave propagation over flat (depth-varying) topography in one space dimension within an irrotational and inviscid flow. As physical model, we use a dispersive Boussinesq-type (BT) system for small-amplitude long waves with weak transverse variation. The problem is discretised in space using a wavelet-Galerkin method based on one-periodic Daubechies scaling functions. Assuming periodic boundary conditions, the discretised operators in spatial domain are circulant and skew-symmetric. These characteristics of discretised differential operators allow us to incorporate the Fast Fourier Transformation (FFT) in the matrix operations which results in a substantial improvement in the computational efficiency and accuracy of the numerical solver compared with the conventional finite difference or finite volume methods. We use a four-stage Runge–Kutta method to temporally discretise the governed spatially discretised differential equations. Several comparative test cases are conducted to validate the performance and efficiency of the proposed wavelet-Galerkin scheme for the BT model over flat beds relative to some existing analytical solutions and numerical results from a second-order finite difference method. We examine the numerical results of the BT system to investigate the two-way propagation of waves for some large L2-norm profiles of the initial free-surface elevation. We also assess the ability of the proposed method to predict the evolution and breaking of undular bores over a flat bed with a simple kinematic criterion. Moreover, we study (tsunami) wave runup and propagation by incorporating the effects of depth-varying topography in a simplified BT system in order to check the applicability of the approach to capture the interactions between the bathymetric features and the wet cells.

Modeling tsunami propagation and the emplacement of thumbprint terrain in an early Mars ocean

AbstractThe identification of lobate debris deposits in Arabia Terra, along the proposed paleoshoreline of a former northern ocean, has renewed questions about the existence and stability of ocean‐sized body of water in the early geologic history of Mars. The potential occurrence of impact‐generated tsunamis in a northern ocean was investigated by comparing the geomorphologic characteristics of the Martian deposits with the predictions of well‐validated terrestrial models (scaled to Mars) of tsunami wave height, propagation direction, runup elevation, and distance for three potential sea levels. Our modeling suggests several potential impact craters ~30–50 km in diameter as the source of the tsunami events. Within the complex topography of flat‐floored valleys and plateaus along the dichotomy boundary, the interference of the multiple reflected and refracted waves that are observed in the simulation may explain the origin of the arcuate pattern that characterizes the thumbprint terrain.

Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation

We consider the Cauchy problem with spatially localized initial data for the twodimensional wave equation degenerating on the boundary of the domain. This problem arises, in particular, in the theory of tsunami wave run-up on a shallow beach. Earlier, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi developed a method for constructing asymptotic solutions of this problem. The method is based on a modified Maslov canonical operator and on characteristics (trajectories) unbounded in the momentum variables; such characteristics are nonstandard from the viewpoint of the theory of partial differential equations. In a neighborhood of the velocity degeneration line, which is a caustic of a special form, the canonical operator is defined via the Hankel transform, which arises when applying Fock’s quantization procedure to the canonical transformation regularizing the above-mentioned nonstandard characteristics in a neighborhood of the velocity degeneration line (the boundary of the domain). It is shown in the present paper that the restriction of the asymptotic solutions to the boundary is determined by the standard canonical operator, which simplifies the asymptotic formulas for the solution on the boundary dramatically; for initial perturbations of special form, the solutions can be expressed via simple algebraic functions.

Numerical simulation of the impacts of reflected tsunami waves on Pulo Raya Island during the 2004 Indian Ocean tsunami

There are numerous small populated islands near western Sumatra in Indonesia. The Weh, Banyak, Mentawai, and Enggano islands are home to many Indonesians. These small islands have been found to be effective at reducing tsunami wave energy on several occasions. At the same time, they are situated around Sumatras active subduction zone and are often among the areas most affected by tsunamisas in the case of Pulo Raya Island, in the western Aceh Jaya district about 800 m from Sumatra. Pulo Raya was devastated by the 2004 Indian Ocean tsunami. This study investigates the hydrodynamic processes of reflected tsunami waves and their impact on Pulo Raya, using a Cornell Multi-grid COupled Tsunami (COMCOT) model to simulate the tsunami wave runup and subsequent hydrodynamic processes. The simulation confirms eyewitness accounts that it was not the initial runup, but the reflected waves that devastated the worst-hit areas of the island. As a result, we can recommend that governments and communities on small islands need to anticipate the impacts of reflected waves following a tsunami, especially at lee side of of the island that was perceived to be a safer place by the island community before the 2004 Indian Ocean tsunami.

Multi-method observation and analysis of a tsunami caused by glacier calving

Abstract. Glacier calving can cause violent tsunami waves which, upon landfall, can cause severe destruction. Here we present data acquired during a calving event from Eqip Sermia, an ocean-terminating glacier in west Greenland. During an exceptionally well-documented event, the collapse of 9 × 105 m3 ice from a 200 m high ice cliff caused a tsunami wave of 50 m height, traveling at a speed of 25–33 m s−1. This wave was filmed from a tour boat at 800 m distance from the calving face, and simultaneously measured with a terrestrial radar interferometer and a tide gauge. Tsunami wave run-up height on the steep opposite shore at a distance of 4 km was 10–15 m, destroying infrastructure and eroding old vegetation. These observations indicate that such high tsunami waves are a recent phenomenon in the history of this glacier. Analysis of the data shows that only moderately bigger tsunami waves are to be expected in the future, even under rather extreme scenarios.

Validation of Boussinesq–Green–Naghdi modeling for surf zone hydrodynamics

2D modeling for surf zone phenomenons are validated in present work using the rotational Boussinesq–Green–Naghdi model. Three benchmark test cases are simulated: tsunami wave runup a conical island; tsunami wave runup complex shelf; and rip current and wave setup over sand bars. The computed results are compared to the experimental data including the free surface deformation and depth-averaged velocities. The simulated 2D cases fundamentally validate the model׳s ability in predicting wave transformation, wave breaking, wave runup and the velocity field for complex hydrodynamic conditions and give the basis for moving on to more complex applications. The lack of irrotationality would strongly contribute to the depth-varying velocity profile of rotational modeling, which has been partially proven in 1D undertow test (Zhang et al., 2014a). Unfortunately, few 2D experiments with rotational vortex data measured could be found due to the difficulty of recording the vortex characteristics. Future work would be the model application to much more complex geophysical and engineering problems, where the lack of any irrotational constrain is expected to excel.

LONG WAVE PROPAGATION, SHOALING AND RUN-UP IN NEARSHORE AREAS

This paper discusses the possibility to study propagation, shoaling and run-up of these waves over a slope in a 300- meter long large wave flume (GWK), Hannover. For this purpose long bell-shaped solitary waves (elongated solitons) of different amplitude and the same period of 30 s are generated. Experimental data of long wave propagation in the flume are compared with numerical simulations performed within the fully nonlinear potential flow theory and KdV equations. Shoaling and run-up of waves on different mild slopes is studied hypothetically using nonlinear shallow water theory. Conclusions about the feasibility of using large scale experimental facility (GWK) to study tsunami wave propagation and run-up are made.

Effective coastal boundary conditions for tsunami wave run-up over sloping bathymetry

Abstract. An effective boundary condition (EBC) is introduced as a novel technique for predicting tsunami wave run-up along the coast, and offshore wave reflections. Numerical modeling of tsunami propagation in the coastal zone has been a daunting task, since high accuracy is needed to capture aspects of wave propagation in the shallower areas. For example, there are complicated interactions between incoming and reflected waves due to the bathymetry and intrinsically nonlinear phenomena of wave propagation. If a fixed wall boundary condition is used at a certain shallow depth contour, the reflection properties can be unrealistic. To alleviate this, we explore a so-called effective boundary condition, developed here in one spatial dimension. From the deep ocean to a seaward boundary, i.e., in the simulation area, we model wave propagation numerically over real bathymetry using either the linear dispersive variational Boussinesq or the shallow water equations. We measure the incoming wave at this seaward boundary, and model the wave dynamics towards the shoreline analytically, based on nonlinear shallow water theory over bathymetry with a constant slope. We calculate the run-up heights at the shore and the reflection caused by the slope. The reflected wave is then influxed back into the simulation area using the EBC. The coupling between the numerical and analytic dynamics in the two areas is handled using variational principles, which leads to (approximate) conservation of the overall energy in both areas. We verify our approach in a series of numerical test cases of increasing complexity, including a case akin to tsunami propagation to the coastline at Aceh, Sumatra, Indonesia.

Advanced numerical modelling of tsunami wave propagation, transformation and run-up

This paper presents a series of numerical models tests performed to assess the ability of the OpenFOAM® modelling system to represent the physics that stands behind the propagation of tsunami waves in coastal regions and in particular their transformation (by means of shoaling, refraction, reflection) over uneven bathymetries and their interaction with the shoreline (run-up, run-down, breaking). A series of tests is selected from the literature to explore the accuracy, applicability and limitations of this particular computational fluid dynamics model solving the three-dimensional Navier–Stokes equations for incompressible multiphase fluids. The range of scenarios tested includes experimental work on wave propagation over shoals, solitary wave run-up and the Monai valley tsunami physical model test recreating the Hokkaido–Nansei–Oki 1993 tsunami that struck Okushiri Island, Japan. This allowed direct comparison with experimental data from physical model tests as well as with other established two-dimensional – that is, depth integrated – and three-dimensional numerical models.

Three-dimensional tsunami runup simulation for the port of Koborinai on the Sanriku coast of Japan

Kim, D.C., Kim, K.O., Pelinovsky, E., Didenkulova, I. and Choi, B.H., 2013. Three-dimensional tsunami runup simulation for the port of Koborinai on the Sanriku coast of Japan.The huge tsunami generated by the earthquake that occurred off the Pacific coast of Japan at 14:46 JST (05:46 UTC) on Friday, 11 March 2011 produced a maximum runup of 40 m on the east coast of Japan. The earthquake triggered extremely destructive tsunami waves up to 37.9 m in height that struck Japan minutes after the quake. Koborinai is a tiny fishery port located north of Miyako City in the Iwate Prefecture. A survey team from the University of Tokyo Earthquake Research Institute (ERI) found high water marks and other evidence of a gigantic wave at the port of Koborinai. The port is on low land sandwiched between two mountains. A joint survey team from ERI-Sungkyunkwan University (SKKU) and Korea Ocean Research & Development Institute (KORDI) visited again and surveyed the site in detail. The 3D Princeton Ocean Model was applied to describe the propagation and runup of the tsunami on the Japanese coast. The numerical simulation results obtained were in satisfactory agreement with observations made in the general area, except those made in many v-shaped valleys along the northern Iwate coast. The extremely high runup of tsunami waves at the port at Koborinai were successfully reproduced by numerical simulation through stepwise refinement of the spatial scale using multi-nesting and consideration of the vertical acceleration of flow along steep slopes using a CFD model to solve the Reynolds-averaged Navier–Stokes (RANS) equations. The velocity field was also computed, and the simulation results show that the water flow that climbed the coast possessed a strong vertical velocity component.

On sediment extent and runup of tsunami waves

Tsunami sediments play an important role in tsunami hazard assessments. Recent tsunami events and their associated tsunami deposits have produced a wealth of sediment data. Surveys of recent tsunami events observed that maximum inland tsunami inundation exceeds the maximum inland extent of sand deposition. A driving question of tsunami sedimentology is whether the difference between maximum tsunami inundation and inland extent of sand can be employed as a measure of inundation for past events for which only the extent of the sand is known. Attempts to infer the causative flow characteristics from these sediments often suffer from uncertainties in the data and from restrictive assumptions made in inversion models. The latter demonstrates that the interaction between the tsunami flow and the movable bed is not fully understood. In our contribution, we employ numerical simulations of tsunami propagation and inundation with MOST and a simple model for deposition. In our simulations, we vary the onshore slope, the initial amplitude, which can be interpreted as offshore wave amplitude, and the grain size. While the initial amplitude and onshore slope influence the difference between maximum inundation and sand extent, the grain size has little effect. The link between the onshore slope, initial amplitude and the difference between the maximum tsunami inundation and inland sediment extent can be employed as a first-order quantitative tool to estimate tsunami flow characteristics and maximum inundation from tsunami deposits. We anticipate these results will improve tsunami hazard assessments and narrow the gap between field observations and numerical simulations.

Laboratory Investigation on Tsunami Wave Runup

The 2D laboratory experiments were performed to investigate tsunami wave runup on the combined sand beach. The N-wave was generated in three different water depths. The water surface elevations, maximum elevation of runup and snapshots of wave uprush and back wash were measured. The theoretical analysis of runup was presented. The results showed that uprush water wave had a decelerate process. The maximum elevation of runup R depends on incident wave height H and R is linear relationship with H plus water depth h.

Bed stress assessment under solitary wave run-up

Understanding and forecasting tsunami wave run-up is very important in mitigating tsunami hazards. The bed stress under wave motion governs viscous wave damping and sediment transport processes, which change coastal morphology. One of the most common methods used for simulation is the shallow water equation (SWE) model, often used with a Manning-style approach for modeling bottom friction. Boundary-layer approaches provide better information regarding bed stress, particularly since they are also valid for nonsteady flows. In this study, a simulation of wave run-up is carried out by simultaneous coupling of the SWE model with the k-ω model. The k-ω model is used near the flow boundary at the bottom, only for assessing the boundary layer shear stress. Free stream velocity and calculations of the free surface evolution are obtained from the SWE model. Both this method, and the conventional method, are applied to the canonical problem of a solitary wave propagating over a constant depth and then up a sloping beach (Synolakis, 1987). The new method is found to increase the computational accuracy and physical realism compared to the conventional Manning method. Comparison of bed shear stresses shows that the new method is able to accommodate the effect of deceleration, which leads to sign changes and a phase shift between the free stream velocity and the bed stress. Furthermore, it is found that during the run-up and run-down process, bed stress in the direction of leaving the shoreline is more dominant.

Numerical Modeling of Tsunami Wave Run-Up and Effects on Sediment Scour around a Cylindrical Pier

AbstractThe linked two-dimensional hydrodynamic and sediment-scour models have been enhanced to simulate tsunami wave run-up on a sloping beach to determine its effects on sediment scour around a cylinder pier located in the wave breaking and run-up zone. To resolve the steep wavefront of the tsunami bore, the kinetic flux vector splitting scheme was adopted to solve the hydrodynamic model equations in the unstructured triangular mesh. The models have been validated by comparing model simulations with experimental data. The results indicate that the model predictions of water surface elevations and velocity match well with the measured data. The error for peak flood elevation ranges from 0.01 to 0.11 m, and the maximum error for the peak velocity is 6%. The model simulations adequately characterize the tsunami wave propagations and transformations as the wave approaches the beach from offshore, especially for the sharp tsunami front before it breaks and the tsunami bore runs up in the beach slope. The mod...

The role of bottom friction in models of nonbreaking tsunami wave runup on the shore

The role of bottom friction in the runup of nonbreaking long waves on the shore is analyzed. The case of the normal incidence of monochromatic waves is considered. The relief of the model region consists of an even horizontal bottom area conjugated with a flat slope. The energy dissipation is estimated as the work of bottom friction forces over the wave field obtained using the known analytical solution based on the Carrier-Greenspan transforms. Estimates of energy losses for waves whose periods are typical for tsunami waves have been obtained. The energy dissipation is shown to be not concentrated in the shore line area as a rule. The question about the practicability of using partially reflecting boundary conditions on the coast to take into account the bottom friction in large-scale models of tsunami propagation is considered.

Draw-down and run-up of tsunami waves on sloping beaches

The dynamics of waves and their interaction with a beach depends on whether the leading wave component is elevated or depressed. These differences are explained in this paper using a hydraulic model and the principle of conservation of impulse. Laboratory experiments of depression waves, conducted using a novel wavemaker, are compared with model predictions. Over a sloping beach, these waves have a nearly constant V-shaped depression trailed by a growing Λ-shaped positive wave. The shoreline recedes over a significant distance, caused by shoreward water being drawn into the V-shaped depression. When the trailing Λ-shaped positive wave breaks, an energetic hydraulic bore develops and moves up the beach. The hydraulic model leads to general formulae for wave slopes, draw-down and run-up. The run-up of negative waves can be larger or smaller than that of positive waves, depending on the wave amplitude and beach parameters. The predictions are compared with results from photographs of depression waves taken during the 2004 Sumatra tsunami. Similar phenomena occurred in Japan in 2011. By incorporating up/down amplitude data in new tsunami warning systems, the properties of tsunamis on beaches could be estimated in real time using the present work, thus improving emergency response strategies. In future, the damage associated with tsunami waves, depending on coastal parameters, could increase with rising sea levels, erosion and destruction of coral reefs, and the loss of Arctic sea-ice.