Influence of bottom frictional effects in sill regions upon lee wave generation and implications for internal mixing

A cross-sectional nonhydrostatic model using idealized sill topography is used to examine the influence of bottom friction upon unsteady lee wave generation and flow in the region of a sill. The implications of changes in shear and lee wave intensity in terms of local mixing are also considered. Motion is induced by a barotropic tidal flow which produces a hydraulic transition, associated with which are convective overturning cells, wave breaking, and unsteady lee waves that give rise to mixing on the lee side of the sill. Calculations show that, as bottom friction is increased, current profiles on the shallow sill crest develop a highly sheared bottom boundary layer. This enhanced current shear changes the downwelling of isotherms downstream of the sill with an associated increase in the hydraulic transition, wave breaking, and convective mixing in the upper part of the water column. Both short and longer time calculations with wide and narrow sills for a number of sill depths and buoyancy frequencies confirm that increasing bottom friction modifies the flow and unsteady lee wave distribution on the downstream side of a sill. Associated with this increase in bottom friction coefficient, there is increased mixing in the upper part of the water column with an associated decrease in the vertical temperature gradient. However, this increase in mixing and decrease in temperature gradient in the upper part of the water column is very different from the conventional change in near-bed temperature gradient produced by increased bottom mixing that occurs in shallow sea regions as the bottom drag coefficient is increased.     

Effect of water depth and the bottom boundary layer upon internal wave generation over abrupt topography

The role of water depth and bottom boundary layer turbulence upon lee-wave generation in sill regions is examined. Their effect upon vertical mixing is also considered. Calculations are performed using a non-hydrostatic model in cross-section form with a specified tidal forcing. Initial calculations in deeper water and a sill height such that the sill top is well removed from the surrounding bed region showed that downstream lee-wave generation and associated mixing increased as bottom friction coefficient k increased. This was associated with an increase in current shear across the sill. However, for a given k, increasing vertical eddy viscosity A _v reduced vertical shear in the across sill velocity, leading to a reduction in lee-wave amplitude and associated mixing. Subsequent calculations using shallower water showed that for a given k and A _v, lee-wave generation was reduced due to the shallower water depth and changes in the bottom boundary layer. However, in this case (unlike in the deepwater case), there is an appreciable bottom current. This gives rise to bottom mixing which in shallow water extends to mid-depth and enhances the mid-water mixing that is found on the lee side of the sill. Final calculations with deeper water but small sill height showed that lee waves could propagate over the sill, thereby reducing their contribution to mixing. In this case, bottom mixing was the major source of mixing which was mainly confined to the near bed region, with little mid-water mixing.     

The effects of large and small-scale topography upon internal waves and implications for tidally induced mixing in sill regions

A free surface non-hydrostatic model in a cross-sectional form, namely, two-dimensional, in the vertical is used to examine the role of larger-scale topography, namely, sill width, and smaller scale topography, namely, ripples on the sill upon internal wave generation and mixing in sill regions. The present work is set in the context of earlier work and the wider literature in order to emphasise the problems of simulating mixing in hydrographic models. Highlights from previous calculations and references to the literature for detail, together with new results presented here with smooth and “ripple” topography, are used to show that an idealised cross-sectional model can reproduce the dominant features found in observations at the Loch Etive sill. Calculations show that on both the short and long time scales, the presence of small-scale “ripple” topography influence the mixing and associated Richardson number distribution in the sill region. Subsequent calculations in which the position and form of the small-scale sill topography is varied show for the first time that it is the small-scale topography near the sill crest that is particularly important in enhancing mid-water mixing on the lee side of the sill. Both short-term and longer-term calculations with a reduced sill width and associated time series show that as the sill width is reduced, the non-linear response of the system increases. In addition, Richardson number plots show that the region of critical Richardson number, and hence enhanced mixing, increases with time and a reduction in sill width. Calculations in which buoyancy frequency N varies through the vertical show that buoyancy frequency close to the top of the sill is primarily controlling mixing rather than its mean value. Hence, a Froude number based on sill depth and local N is the critical parameter rather than one based on total depth and mean N.     

Effects of seismic wave propagation upon buried pipelines

The behaviour of long straight buried pipelines subjected to seismic wave propagation is investigated. Well-known relationships for determining upper bounds for the axial strain and curvature in the pipeline as well as relationships for relative displacement and rotation at the pipeline joints are discussed. The assumption that the seismic excitation can be modelled as a travelling wave having a shape which remains unchanged as it traverses the pipeline is examined in detail. It is shown that this assumption is unconservative when the effective propagation velocity of the seismic waves with respect to the pipeline is such that the actual time lag (separation distance between points divided by effective propagation velocity) is less than a ‘cross-over’ time lag. Cross-over time lags for 22 pairs of ground displacements recorded during the 1971 San Fernando Earthquake are presented in this paper. Finally, methods for estimating the propagation speed of the seismic waves along or with respect to the pipeline are discussed.     

On the influence of stratification and tidal forcing upon mixing in sill regions

A cross-sectional non-hydrostatic model with idealized topography was used to examine the processes influencing tidal mixing in the region of sills. Initial calculations with appropriate parameters for the sill at the entrance to Loch Etive showed that the model could reproduce the main features of the observed mixing in the region. In particular, the hydraulic jump in the sill region was reproduced, as was an intense mid-water jet that was observed to separate from the lee side of the sill. Shear instabilities associated with the jet appeared to be a source of mixing within the thermocline. In addition, internal lee waves were generated on the lee side of the sill, with the observed amplification because of trapping during the flood stage. Their magnitude and hence the mixing increased with increasing Froude number (F _r). In the case of vertically varying buoyancy frequency, its value near the sill top determined the F _r number, with its value below influencing internal waves magnitude at depth. At high F _r values particularly with strong currents, short waves and overturning occurred.     

A viscous-spring transmitting boundary for cylindrical wave propagation in saturated poroelastic media

Based on the u–U formulation of Biot equation and the assumption of zero permeability coefficient, a viscous-spring transmitting boundary which is frequency independent is derived to simulate the cylindrical elastic wave propagation in unbounded saturated porous media. By this viscous-spring boundary the effective stress and pore fluid pressure on the truncated boundary of the numerical model are replaced by a set of spring, dashpot and mass elements, and its simplified form is also given. A u–U formulation FEA program is compiled and the proposed transmitting boundaries are incorporated therein. Numerical examples show that the proposed viscous-spring boundary and its simplified form can provide accurate results for cylindrical elastic wave propagation problems with low or intermediate values of permeability or frequency content. For general two dimensional wave propagation problems, spuriously reflected waves can be greatly suppressed and acceptable accuracy can still be achieved by placing the simplified boundary at relatively large distance from the wave source.     

Non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions

A three-dimensional non-linear, non-hydrostatic model in cross-sectional form is used to determine the factors influencing the relative importance of the linear, non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions due to tidal forcing. The importance of the free surface elevation term is also considered. Idealised topography representing the sill at the entrance to Loch Etive, the site of a recent measurement programme, is used. Calculations show that the non-linear terms in the energy flux become increasingly important as the sill Froude Number (F _s) increases and the sill aspect ratio is increased. The vertical profile of the stratification, in particular its value close to the sill crest where internal waves are generated, has a significant influence on unsteady lee wave and mixed tidal–lee wave generation and the non-linear contribution to the energy flux. Calculations show that as F _s increases, the energy flux due to the non-linear and non-hydrostatic terms increases more rapidly than the linear term. The importance of the non-linear terms in the energy flux also increases as the sill aspect ratio is increased. Increasing the buoyancy frequency reduces the contribution of the non-hydrostatic and non-linear terms to the total energy flux. Also, as the buoyancy frequency is increased, this reduces unsteady lee wave and mixed tidal–lee wave generation. In essence, these calculations show that the energy flux due to the non-hydrostatic and non-linear terms is appreciable in sill regions.     

多人工波速优化透射边界在谱元法地震波动模拟中的应用

将作者最近发展的多人工波速优化透射边界(记为ca j-MTF)应用于高精度谱元法的地震波动模拟中,并与经典的廖氏透射(MTF)边界、完美匹配层(PML)边界、黏弹性边界以及一阶旁轴近似边界进行了比较分析.结果显示:①ca j-MTF边界与MTF边界在形式上非常接近,它继承了后者公式简单、易于实现、精度可控、计算量低以及...     

A modified domain reduction method for numerical simulation of wave propagation in localized regions

A modified domain reduction method(MDRM) that introduces damping terms to the original DRM is presented in this paper. To verify the proposed MDRM and compare the computational accuracy of these two methods, a numerical test is designed. The numerical results of the MDRM and DRM are compared using an extended meshed model. The results show that the MDRM significantly improved the computational accuracy of the DRM. Then, the MDRM is compared with two existing conventional methods, namely Liao's transmitting boundary and viscous-spring boundary with Liu's method. The MDRM shows its great advancement in computational accuracy, stability and range of applications. This paper also discusses the influence of boundary location on computational accuracy. It can be concluded that smaller models tend to have larger errors. By introducing two dimensionless parameters, φ_1 and φ_2, the rational distance between the observation point and the MDRM boundary is suggested. When φ_1 2 or φ_213, the relative PGA error can be limited to 5%. In practice, the appropriate model size can be chosen based on these two parameters to achieve desired computational accuracy.     

复杂场地对地震波传播的影响

本文基于两相介质动力学方程组，利用显式集中质量有限元结合透射人工边界，研究了复杂介质情况下盆地和凸起地形对地震波传播的影响。研究表明，控制波在盆地传播的主要因素有沉积层和基岩的剪切波速比、沉积层的厚度、饱和度。欠饱和土的基本共振频率小于单相土和饱和土的基本共振频率。凸起地形对地震动有放大作用，但是在山脚附近有抑制作用。饱水土的地形效应要比单相土的地形效应更加明显。山间峡谷的放大作用和两山间的距离有密切的关系。     

A local time-domain transmitting boundary for simulating cylindrical elastic wave propagation in infinite media

Finite element simulation of the time-dependent wave propagation in infinite media requires enforcing the transmitting boundary to replace the truncated far-field infinite domain so as to model the effect of the wave radiation towards infinity. This paper proposed a novel local time-domain transmitting boundary for simulating the cylindrical elastic wave radiation problem. This boundary is a mechanical model consisting of the spring, dashpot and mass elements, with the auxiliary degrees of freedom introduced, which is dynamically stable and easily implemented into the commercial finite element codes. Numerical analysis of the cylindrical elastic wave radiation problem indicates that the proposed transmitting boundaries with the order N=3 for cylindrical P and SV waves and with the order N=4 for cylindrical SH wave have very high accuracy, even when the artificial boundary at wave source. The proposed transmitting boundary with order N=0 can be applied approximately to the general two-dimensional infinite elastic wave problems that contain the more complex outgoing wave fields at artificial boundary than the cylindrical waves. The plane-strain Lamb problem is analyzed with the acceptable engineering accuracy achieved. On the other hand, the proposed transmitting boundary with higher order can be a tool to localize the temporal convolution that appears in an exact time-domain transmitting boundary for the general infinite wave problems. This potential applicability is mentioned.     

复杂介质地震波传播模拟中边界元法与有限差分法的比较研究

地震波场数值模拟是理论地球物理学和勘探地球物理学的重要研究手段.在众多数值模拟方法中边界元法和有限差分法是两种典型的地震波传播模拟计算方法.边界元法是一种半解析-半数值的边界型方法,它显式地利用边界连续条件,沿着地层边界进行离散,具有降维、高精度和自动满足远场辐射条件的优点;有限差分法是一种典型的基于微分的区域型方法,它隐式地使用边界连续条件,以空间网格形式进行离散和数值逼近,具有高效、实用和容易数值实现的优点.本文以一个半圆形均匀Valley模型和两个非均匀断裂/断层模型为例,从计算精度、计算效率、频散特性以及适用性等方面对这两种方法进行了比较研究.数值计算结果表明:边界元法可以精确地几何描述有内部断点、断面的复杂构造,能够精确地模拟内部不规则界面之间波的反射/传播;有限差分法不能以足够的精度描述几何断点和内部不规则边界.边界元法在高频时计算量大于有限差分法,有限差分法则需要更小的网格间距以压制数值频散.因此,在处理内部非均质和高频计算时,有限差分法更有效;在处理内部不规则边界、断点、大尺度等问题时,边界元法比有限差分法更有优势.     

A semi-analytical artificial boundary for time-dependent elastic wave propagation in two-dimensional homogeneous half space

This paper presents a time-dependent semi-analytical artificial boundary for numerically simulating elastic wave propagation problems in a two-dimensional homogeneous half space. A polygonal boundary is considered in the half space to truncate the semi-infinite domain, with an appropriate boundary condition imposed. Using the concept of the scaled boundary finite element method, the wave equation of the truncated semi-infinite domain is represented by the partial differential equation of non-constant coefficients. The resulting partial differential equation has only one spatial coordinate variable and time variable. Through introducing a few auxiliary functions at the truncated boundary, the resulting partial differential equations are further transformed into linear time-dependent equations. This allows an artificial boundary to be derived from the time-dependent equations. The proposed artificial boundary is local in time, global at the truncated boundary and semi-analytical in the finite element sense. Compared with the scaled boundary finite element method, the main advantage in using the proposed artificial boundary is that the requirement for solving a matrix form of Lyapunov equation to obtain the unit-impulse response matrix is avoided, so that computer efforts are significantly reduced. The related numerical results from some typical examples have demonstrated that the proposed artificial boundary is of high accuracy in dealing with time-dependent elastic wave propagation in two-dimensional homogeneous semi-infinite domains.     

A high-order time-domain transmitting boundary for cylindrical wave propagation problems in unbounded saturated poroelastic media

Based on the u–p formulation of Biot equation with an assumption of zero permeability coefficient, a high-order transmitting boundary is derived for cylindrical elastic wave propagation in infinite saturated porous media. By this transmitting boundary the total stresses on the truncated boundaries of a numerical model, such as a finite element model, are replaced by a set of spring, dashpot and mass elements, with some additionally introduced auxiliary degrees of freedom. The transmitting boundaries are incorporated into the DIANA SWANDYNE II program and an unconditionally stable implicit time integration algorithm is adopted. Despite the assumption made in the derivation of the transmitting boundary, numerical examples show that it can provide highly accurate results for cylindrical elastic wave propagation problems in infinite saturated porous medium in case the u–p formulation is applicable. Although the direct applications of the proposed transmitting boundary to general two dimensional wave problems in infinite saturated porous media are not highly accurate, acceptable accuracy can still be achieved by placing the transmitting boundary at relatively large distance from the wave source.     

Combined paraxial-consistent boundary conditions finite element model for simulating wave propagation in elastic half-space media

In this paper, a finite element model of a soil island is coupled to both a consistent transmitting boundary and a paraxial boundary, which are then used to model the propagation of waves in semi-infinite elastic layered media. The formulation is carried out in the frequency domain while assuming plane strain conditions. It is known that a discrete model of this type, while providing excellent results for a wide range of physical parameters in the context of a half-space problem, may deteriorate rapidly at low frequencies of excitation. This is so because at low frequencies the various waves in the model eventually attain characteristic wavelengths which exceed the distance of the bottom boundary, which then causes that boundary to fail. Also, the paraxial boundaries themselves break down at very low frequencies. In this paper, this difficulty is overcome and the model׳s performance is improved upon dramatically by incorporating an artificial buffer layer sandwiched between the bottom of the soil medium and the underlying elastic half-space. Applications dealing with rigid foundations resting on homogenous or layered half-space media are shown to exhibit significant improvement. Following extensive simulations, clear guidelines are provided on the performance of the coupled model and an interpretation is given on the engineering significance of the findings. Finally, clear recommendations are provided for the practical use of the proposed modelling strategy.     

The discrete wave number formulation of boundary integral equations and boundary element methods: A review with applications to the simulation of seismic wave propagation in complex geological structures

We review the application of the discrete wave number method to problems of scattering of seismic waves formulated in terms of boundary integral equation and boundary element methods. The approach is based on the representation of the diffracting surfaces and interfaces of the medium by surface distributions of sources or by boundary source elements, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green's functions are evaluated by the discrete wave number method, and the boundary conditions yield a linear system of equations. The inversion of this system allows the calculation of the full wave field in the medium. We investigate the accuracy of the method and we present applications to the simulation of surface seismic surveys, to the diffraction of elastic waves by fractures, to regional crustal wave propagation and to topographic scattering.     

Effect of surface wave propagation in a four-layered oceanic crust model

Dispersion of Rayleigh type surface wave propagation has been discussed in four-layered oceanic crust. It includes a sandy layer over a crystalline elastic half-space and over it there are two more layers—on the top inhomogeneous liquid layer and under it a liquid-saturated porous layer. Frequency equation is obtained in the form of determinant. The effects of the width of different layers as well as the inhomogeneity of liquid layer, sandiness of sandy layer on surface waves are depicted and shown graphically by considering all possible case of the particular model. Some special cases have been deduced, few special cases give the dispersion equation of Scholte wave and Stoneley wave, some of which have already been discussed elsewhere.     

Least square approach to simulate wave propagation in irregular profiles using the indirect boundary element method

In earthquake engineering and seismology it is of interest to know the surface motion at a given site due to the incoming and scattered seismic waves by surface geology. This can be formulated in terms of diffraction of elastic waves and then the indirect boundary element method (IBEM) for dynamic elasticity is used. It is based on the explicit construction of diffracted waves at the boundaries from which they radiate. This provides the analyst with insight on the physics of diffraction. The IBEM has been applied to study the amplification of elastic waves in irregular soil profiles. From the strong or weak satisfaction of boundary conditions and a simple analytical discretization scheme a linear system of equations for the boundary sources is obtained. Here, we explore the use of a weak discretization strategy with more collocation points than force densities. The least squares enforcement of boundary conditions leads to a system with reduced number of unknowns. This approach naturally allows one to use both coarser and finer boundary discretizations for smooth and rapidly varying profiles, respectively. A well studied semicircular canyon under incident P or SV in-plane waves is used to calibrate this method. Several benefits are obtained using mixed meshing that leads to the least squares condensation of the IBEM.