海啸波作用下泥沙运动——Ⅲ. 数学模型的建立与验证

基于Boussinesq方程耦合泥沙运动和地形演变模型,建立海啸作用下泥沙运动数学模型。地形演变模型采用WENO差分格式,并将WENO差分格式与Lax-Wendroff格式和FTBS格式进行对比分析。运用Synolakis、Kobayashi和Young的实验数据分别对水动力模块和地形演变模块进行验证,数值模拟结果与实验数据吻合良好,模型能够很好地模拟海啸波的传播、破碎、上爬、回落过程以及岸滩的冲淤变化过程,该数学模型能够运用到海啸作用下的岸滩演变研究和预测中。     

Potential Tsunami Hazard Associated with the Kerepehi Fault,Firth of Thames,New Zealand

Geophysical data have identified four submarine segments of the Kerepehi Fault, roughly bisecting a back-arc rift (Hauraki Rift). These segments have been traced through the shallow waters of the Firth of Thames, which lies at the southern end of the Hauraki Gulf, New Zealand. No historical or paleotsunami data are available to assess the tsunami hazard of these fault segments.Analysis of the fault geometry, combined with paleoseismic data for three further terrestrial segments of the Fault, suggest Most Credible Earthquake (MCE) moment magnitudes of 6.5–7.1. Due to the presence of thick deposits of soft sediment, and thesemi-confined nature of the Firth, the MCE events are considered capable of generating tsunami or tsunami-like waves. Two numerical models (finite element and finite difference), and an empirical method proposed by Abe (1995), were used to predict maximum tsunami wave heights. The numerical models also modelled the tsunami propagation.The MCE events were found not to represent a major threat to the large metropolitan centre of Auckland City (New Zealand's largest population centre). However, the waves were a threat to small coastal communities around the Firth, including the township of Thames, and 35,000 ha of low-lying land along the southern shores of the Firth of Thames.The Abe method was found to provide a quick and useful method of assessing the regional tsunami height. However, for sources in water depths < 25 m the Abe method predicted heights 2–4 times larger than the numerical models. Since the numerical models were not intended for simulating tsunami generation in such shallow water, the Abe results are probably a good guide to the maximum wave heights.     

A transient finite element analysis of linear viscoelastic material model

Since the attenulation of propagating waves through soil/rock is related to the localized material properties as well as the strain developed, the commonly used Rayleigh-type damping model and its variations are not suitable for dynamic finite element analysis of such materials. A linear viscoelastic material model based on the concept of the relaxation spectrum is manipualted in place of the damping model in this paper. The method proposed by Day and Minster¹¹ to transform the convolutional form of the stress–strain relationship to a set of differential operators using the Pade approximant method is generalized to non-scalar waves and implemented for transient finite element analyses. A time-marching scheme is also proposed to incorporate the resultant differential operators into the governing equation of motion. The accuracy related to the Pade approximant method and the time-marching scheme is investigated by critically analysing some scalar wave propagation problems. The proposed technique is further verified using two one-dimensional stress wave propagation problems and a two-dimensional transient propagating wave through an unbounded linear viscoelastic medium. Some encouraging results have been obtained using the proposed technique and guidelines for using this technique are also presented. Comparisons of analytical solutions obtained by Fourier synthesis and numerical results have been provided.     

Comparison of linear and nonlinear shallow wave water equations applied to tsunami waves over the China Sea

This paper discusses the applications of linear and nonlinear shallow water wave equations in practical tsunami simulations. We verify which hydrodynamic theory would be most appropriate for different ocean depths. The linear and nonlinear shallow water wave equations in describing tsunami wave propagation are compared for the China Sea. There is a critical zone between 400 and 500 m depth for employing linear and nonlinear models. Furthermore, the bottom frictional term exerts a noticeable influence on the propagation of the nonlinear waves in shallow water. We also apply different models based on these characteristics for forecasting potential seismogenic tsunamis along the Chinese coast. Our results indicate that tsunami waves can be modeled with linear theory with enough accuracy in South China Sea, but the nonlinear terms should not be neglected in the eastern China Sea region.     

Analysis of one-dimesional wave propagation in a fluid-saturated finite soil column

Biot's dynamic equations of motion for one-dimensional wave propagation in a fluid-saturated linear elastic isotropic soil are solved using Laplace transformation followed by numerical inversion and the results compared with a direct finite element formulation. A soil column of finite dimension subjected to velocity boundary conditions is analysed, allowing for reflection of waves from boundaries. Comparison of time histories at given points along the column shows that the finite element solution gives good agreement with the Laplace transform solution for low as well as high drag.     

Visualization of tsunami waves with Amira package

In recent years numerical investigations of tsunami wave propagation have been spurred by the magnitude 9.3 earthquake along the Andaman–Sumatra fault in December, 2004. Visualization of tsunami waves being modeled can yield a much better physical understanding about the manner of wave propagation over realistic seafloor bathymetries. In this paper we will review the basic physics of tsunami wave propagation and illustrate how these waves can be visualized with the Amira visualization package. We have employed both the linear and nonlinear versions of the shallow-water wave equation. We will give various examples illustrating how the files can be loaded by Amira, how the wave-heights of the tsunami waves can be portrayed and viewed with illumination from light sources and how movies can be used to facilitate physical understanding and give important information in the initial stages of wave generation from interaction with the ambient geological surroundings. We will show examples of tsunami waves being modeled in the South China Sea, Yellow Sea and southwest Pacific Ocean near the Solomon Islands. Visualization should be a part of any training program for teaching the public about the potential danger arising from tsunami waves. We propose that interactive visualization with a web-portal would be useful for understanding more complex tsunami wave behavior from solving the 3-D Navier–Stokes equation in the near field.     

Characteristics of Tsunamis Propagating over Oceanic Ridges: Numerical Simulation of the 1996 Irian Jaya Earthquake Tsunami

The 1996 Irian Jaya earthquake tsunami was simulated by using the numerical model based on the linear long wave theory including Coriolis force in the spherical coordinate system. The numerical modeling result at Chichijima is in good agreement with the observed tide gauge data. The distinctive oscillation at Chichijima can be interpreted as the formation of boundary waves, so called ridge waves that are excited on the South-Honshu ridge. The mechanism of tsunami propagation trapped on an oceanic ridge is analyzed with the simple ridge model. The result explains the characteristics of ridge waves excited on theSouth-Honshu ridge.     

The 1987–88 Alaskan Bight tsunamis: Deep ocean data and model comparisons

Excellent deep ocean records have been obtained of two tsunamis recently generated in the Alaskan Bight on 30 November 1987 and 6 March 1988, providing the best available data set to date for comparison with tsunami generation/propagation models. Simulations have been performed with SWAN, a nonlinear shallow water numerical model, using source terms estimated by a seafloor deformation model based on the rectangular fault plane formalism. The tsunami waveform obtained from the model is quite sensitive to the specific source assumed. Significant differences were found between the computations and observations of the 30 November 1987 tsunami, suggesting inadequate knowledge of the source characteristics. Fair agreement was found between the data and the model for the first few waves of the 6 March 1988 tsunami. Model estimates of the seismic moment and total slip along the fault plane are also in fair agreement with those derived from the published Harvard centroid solution for the 6 March 1988 event, implying that the computed seafloor deformation does bear some similarity to the actual source.     

探地雷达有限元数值模拟及衰减特性探讨

从Maxwell方程组出发,推导了带衰减项与不带衰减项的探地雷达(GPR)有限元波动方程。利用matlab平台编译了二维GPR有限元正演模拟程序。通过对典型模型例子的模拟分析,给出了GPR电磁波衰减比的概念及其计算公式,加深理解了GPR电磁波在不同介质中的传播特性。在GPR正演模拟中,必须考虑衰减项的吸收作用,得到的模拟结果更符合GPR电磁波在地下介质中的传播规律,更有助于理解和掌握地下介质中探地雷达波的传播规律,对探地雷达资料的地质解释具有更好地指导意义。     

基于波浪理论的水库地质灾害涌浪数值分析方法

有别于国内现行广泛应用Navie-Stokes方程进行地质灾害涌浪的数值模拟技术,本文采用波浪理论对地质灾害涌浪波进行了分析。地质灾害涌浪波是非周期性波,并且有强烈的非线性,介于中等水波至浅水波之间;可用浅水波模型和Boussinesq模型进行数学描述。本文采用有限差分法的Boussinesq模型,以三峡库区龚家方崩滑体涌浪为例,模拟了涌浪波的传播和爬高问题。该模型能够计算形成涌浪瞬时河面、河面最大波高图、最大流速矢量图、最大爬坡和预警分布图。模拟计算结果与调查结果吻合非常好。这说明基于波浪理论的地质灾害涌浪分析方法精度较高,为涌浪的预测研究提供了一种新的研究方法。     

应用二阶完全非线性Boussinesq方程模拟破碎波浪

建立了基于高阶Boussinesq水波方程的一维波浪破碎数值模型。基于一组具有二阶完全非线性特征的Boussinesq水波方程,建立了交错网格下的高精度差分格式,推导了适用于该组方程的永形波解析解,其和松弛造波技术相结合实现了数值波浪水槽中(强)非线性波浪的无反射入射。通过模拟封闭容器内水体晃动问题对数值格式进行了验证,通过模拟孤立波在斜坡海岸上的浅化过程说明了将方程从弱非线性发展到完全非线性的必要性。采用涡粘方法处理波浪破碎,利用物理模型实验数据,分析了模型中各波浪破碎参数对数值结果的影响并对参数进行了率定。应用该模型对规则波在斜坡海岸上的传播、变浅以及破碎过程进行了数值模拟研究,数值结果同实验数据吻合良好,验证了模型的有效性。     

ISPH numerical simulation of tsunami generation by submarine landslides

The present investigation focused on the numerical simulation of the gravity currents of non-Newtonian fluids by means of the incompressible smoothed particle hydrodynamics (ISPH) approach. Firstly, to solve the rheological properties of the non-Newtonian Bingham model by ISPH method, the multi-viscodensity approach has been introduced. Then, this methodology has been used to simulate the annular viscometer and landslide deformation test cases. Through simulating these test cases, the viscoplastic behavior of the non-Newtonian viscosity and propagation of tsunami waves due to underwater landslide movement have been observed. Numerical results were in good agreement with the theoretical and experimental studies and showed that this methodology can be used to investigate precisely the generation and propagation of tsunami surface waves.     

Fully nonlinear Boussinesq equations for long wave propagation and run-up in sloping channels with parabolic cross sections

A general framework for derivation of long wave equations in narrow channels, and their transformation to Lagrangian coordinates is briefly established. Then, fully nonlinear Boussinesq equations are derived for channels of parabolic cross sections. The simplified version with normal nonlinearity is compared with corresponding models from the literature, and propagation properties are discussed. A Lagrangian run-up model is adapted to the fully nonlinear set. This model is tested by means of controlled residues and by a well-controlled comparison to exact analytic solutions from the literature. Then, run-up of solitary waves in simple geometries is simulated and compared to a semi-analytic solution that is derived for propagation and run-up in a composite channel. The dispersive model retains the higher run-up height in a parabolic channels, as reported in the recent literature for NLSW solutions, as compared to a rectangular channel.     

Computational dynamic homogenization for the analysis of dispersive waves in layered rock masses with periodic fractures

The analysis of the wave propagation in layered rocks masses with periodic fractures is tackled via a two-scale approach in order to consider shape and size of the rock inhomogeneities. To match the displacement fields at the two scales, an approximation of the micro-displacement field is assumed that depends on the first and second gradients of the macro-displacement through micro-fluctuation displacement functions obtained by the finite element solution of cell problems derived by the classical asymptotic homogenization. The resulting equations of motion of the equivalent continuum at the macro-scale result to be not local in space, thus a dispersive wave propagation is obtained from the model. The simplifying hypotheses assumed in the multi-scale kinematics limit the validity of the model to the first dispersive branch in the frequency spectrum corresponding to the lowest modes.Although the homogenization procedure is developed to study the macro-scale wave propagation in rock masses with bounded domain, the reliability of the proposed method has been evaluated in the examples by considering unbounded rock masses and by comparing the dispersion curves provided by the rigorous process of Floquet–Bloch with those obtained by the method presented. The accuracy of the method is analyzed for compressional and shear waves propagating in the intact-layered rocks along the orthotropic axes. Therefore, the influence of crack density in the layered rock mass has been analyzed. Vertical cracks have been considered, periodically located in the stiffer layer, and two different crack densities have been analyzed, which are differentiated in the crack spacing. A good agreement is obtained in case of compressional waves travelling along the layering direction and in case of both shear and compressional waves normal to the layering. The comparison between two crack systems with different spacing has shown this aspect to have a remarkable effect on waves travelling along the direction of layering, and limited in the case of waves propagating normal to the layers.The equivalent continuous model obtained through the dynamic homogenization technique here presented may be applied to the computational analysis of non-stationary wave propagation in rock masses of finite size, also consisting of sub-domains with different macro-mechanical characteristics. This avoids the use of computational models represented at the scale of the heterogeneities, which may be too burdensome or even unfeasible.     

Modelling December 2004 Indian Ocean tsunami: a coastal study

December 2004 tsunami in the Indian Ocean region has been simulated using MIKE-21 HD model. The vertical displacement of the seabed is incorporated into the numerical simulation by using time-varying bathymetry data. In the open ocean, sea surface height from altimeter observation has been used to validate the model results. To the west of the rupture zone, the crest is observed to precede the trough of the tsunami waves while to the east, trough preceded the crest. The model performance along the coastal region has been validated using de-tided sea levels from tide gauge measurements at Tuticorin, Chennai, Vishakapattanam, and Paradip ports along the east coast of India. Unique coastal characteristics of the tsunami waves, wave height, and wave celerity are reasonably simulated by the numerical model. Spectral analysis of tide gauge observations and corresponding model results has been done, and the distribution of frequency peaks from the analysis of gauge observations and the model results is observed to have a reasonable comparison. Low-frequency waves, contributed from the coastally trapped edge waves, are found to dominate both the tide gauge observations and the model results. The subsequent increase in the tsunami wave height observed at Chennai, Vishakapattanam, and Paradip has been explained on the basis of coastally trapped edge waves. From the validation studies using altimeter data and tide gauge data, it is observed that the model can be used effectively to simulate the tsunami wave height in the offshore as well as in the coastal region with satisfying performance.     

Absorption characteristics of a Stoneley wave in multilayered media and the logging method for estimating permeability

The aim of the study is to increase the reliability of estimation of the acoustic parameters of a layered permeable formation using new software, based on two-velocity continuum equations for 2D numerical simulation of acoustic waves propagating in a borehole. The key method is the method of finite differences as applied to solving equations of the continuum filtration theory. The dissipative characteristics of Stoneley waves have been identified for the case of their propagation in a layered formation. These aspects should be taken into account when using Stoneley waves to estimate the permeability of geologic media.     

Pacific Basin tsunami hazards associated with mass flows in the Aleutian arc of Alaska

We analyze mass-flow tsunami generation for selected areas within the Aleutian arc of Alaska using results from numerical simulation of hypothetical but plausible mass-flow sources such as submarine landslides and volcanic debris avalanches. The Aleutian arc consists of a chain of volcanic mountains, volcanic islands, and submarine canyons, surrounded by a low-relief continental shelf above about 1000–2000 m water depth. Parts of the arc are fragmented into a series of fault-bounded blocks, tens to hundreds of kilometers in length, and separated from one another by distinctive fault-controlled canyons that are roughly normal to the arc axis. The canyons are natural regions for the accumulation and conveyance of sediment derived from glacial and volcanic processes. The volcanic islands in the region include a number of historically active volcanoes and some possess geological evidence for large-scale sector collapse into the sea. Large scale mass-flow deposits have not been mapped on the seafloor south of the Aleutian Islands, in part because most of the area has never been examined at the resolution required to identify such features, and in part because of the complex nature of erosional and depositional processes. Extensive submarine landslide deposits and debris flows are known on the north side of the arc and are common in similar settings elsewhere and thus they likely exist on the trench slope south of the Aleutian Islands. Because the Aleutian arc is surrounded by deep, open ocean, mass flows of unconsolidated debris that originate either as submarine landslides or as volcanic debris avalanches entering the sea may be potential tsunami sources.To test this hypothesis we present a series of numerical simulations of submarine mass-flow initiated tsunamis from eight different source areas. We consider four submarine mass flows originating in submarine canyons and four flows that evolve from submarine landslides on the trench slope. The flows have lengths that range from 40 to 80 km, maximum thicknesses of 400–800 m, and maximum widths of 10–40 km. We also evaluate tsunami generation by volcanic debris avalanches associated with flank collapse, at four locations (Makushin, Cleveland, Seguam and Yunaska SW volcanoes), which represent large to moderate sized events in this region. We calculate tsunami sources using the numerical model TOPICS and simulate wave propagation across the Pacific using a spherical Boussinesq model, which is a modified version of the public domain code FUNWAVE. Our numerical simulations indicate that geologically plausible mass flows originating in the North Pacific near the Aleutian Islands can indeed generate large local tsunamis as well as large transoceanic tsunamis. These waves may be several meters in elevation at distal locations, such as Japan, Hawaii, and along the North and South American coastlines where they would constitute significant hazards.     

Finite element modeling of wave propagation problems in multilayered soils resting on a rigid base

A new finite element scheme is proposed, in this paper, for solving two-dimensional wave propagation problems in multilayered soils resting on a rigid base. The multilayered soils are treated as multiple horizontal layers of lateral infinite extension in geometry. Since these horizontal layers can be truncated by two artificially truncated vertical boundaries, two high-order artificial boundary conditions are applied for propagating the incoming waves from the interior domain into the far field of the system. Both the semi-analytical method and the truncated boundary migration procedure are used to derive the high-order artificial boundary conditions, which are comprised of a physically meaningful dashpot and a generalized energy absorber. The main advantage of using the proposed finite element scheme is that the derived artificial boundary condition can be straightforwardly implemented in the finite element analysis, without violating the band/sparse structure of the conventional finite element equation. The related numerical examples have demonstrated that the proposed finite element scheme is of high accuracy in dealing with wave propagation problems in multiple horizontal layers.     

Insights on the small tsunami from January 28, 2020, Caribbean Sea MW7.7 earthquake by numerical simulation and spectral analysis

A major left-lateral strike-slip Mw7.7 earthquake occurred in the vicinity of the Caribbean Sea on January 28, 2020. As a result, a small-scale tsunami was generated. The properties of the seismogenic source were described using observational data gathered for the earthquake and tsunami, as well as information on the regional tectonic setting. The tsunami was simulated with the COMCOT model and Okada’s dislocation model from finite fault solutions for M_W7.7 Caribbean Sea earthquakes published by the United States Geological Survey. The simulation results were compared to tide gauge records to validate whether the seafloor’s vertical displacement generated by the strike-slip fault caused a small-scale tsunami. We conducted a spectral analysis of the tsunami to better understand the characteristics of tsunami records. The tsunami simulation results showed that the co-seismic vertical displacement caused by a strike-slip M_W7.7 earthquake could have contributed to the small-scale tsunami, but the anomalously large high-frequency tsunami waves recorded by the George Town tide gauge 11 min after the earthquake were unrelated to the earthquake-generated tsunami. According to the spectrum analysis, the predominant period of noticeable high-frequency tsunami waves recorded by the George Town tide gauge occurred only two minutes after the earthquake. This indicates that the source of the small-scale tsunami was close to the George Town station and the possible tsunami source was 150 km away from George Town station. These facts suggest that a submarine landslide was caused by the strike-slip earthquake. The comprehensive analysis showed that the small-scale tsunami was not caused solely by co-seismic seafloor deformation from the strike-slip event but that an earthquake-triggered submarine landslide was the primary cause. Therefore, the combined impact of two sources led to the small-scale tsunami.

     

Pseudo-plane strain analysis of wave propagation problems arising from detonations of explosives in cylindrical boreholes

Wave propagation problems, such as blasting for excavation of a new tunnel oriented perpendicular to an existing tunnel, are truly three dimensional in nature. Dynamic finite element analysis with three-dimensional elements is, however, very expensive. The cheaper and simpler alternative would be to model the problem approximately in two dimensions. This paper shows that dynamic finite element analysis of such problems using conventional two-dimensional plane strain elements produces responses which are erroneously excessive. This is accredited to the inability to the inability of the two-dimensional elements to correctly model the rapid attenuation of the amplitudes of the outward propagating waves. To overcome this problem, a pseudo-plane strain concept is introduced and has been found to be a viable alternative. Numerical results are presented to demonstrate the application of the concept.