Propagation of oblique waves over small bottom undulation in an ice-covered two-layer fluid

Scattering of oblique incident waves by small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated within the framework of linearized water wave theory. The ice-cover is being modeled as an elastic plate of very small thickness. There exist two modes of time-harmonic waves–one with lower wave number propagating along the ice-cover (ice-cover mode) and the other with higher wave number along the interface (interfacial mode). A perturbation analysis is employed to solve the corresponding boundary value problem governed by modified Helmholtz equation and thereby evaluating the reflection and transmission coefficients approximately up to first order for both modes. A patch of sinusoidal ripples, having two different wave numbers over two consecutive stretches, is considered as an example and the related coefficients are determined. It is observed that when the wave is incident on the ice-cover surface we always find energy transfer to the interface, but for interfacial incident waves there are parameter ranges for which no energy transfer to the ice-cover surface is possible. Also it is observed that for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. These results are demonstrated in graphical form. From the derived results, the solutions for problems with free surface can be obtained as particular cases.     

Oblique wave scattering by undulating porous bottom in a two-layer fluid: Fourier transform approach

The problem involving scattering of oblique waves by small undulation on the porous ocean bed in a two-layer fluid is investigated within the framework of linearised theory of water waves where the upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different wave numbers (modes): wave with lower wave number propagates along the free surface whilst that with higher wave number propagates along the interface. When an oblique incident wave of a particular mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in conjunction with Fourier transform technique is used to derive the first-order corrections of velocity potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both modes. One special type of undulating bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the transformation of water wave energy between the two modes. Comparisons between the present results with those in the literature are made for particular cases and the agreements are found to be satisfactory. In addition, energy identity, an important relation in the study of water wave theory, is derived with the help of the Green’s integral theorem.     

均匀介质背景中三维异常体的面波波场响应及其动力学特征分析

除介质各向异性之外,地球内部介质的横向非均匀性也是控制面波速度变化的重要因素.本文基于振型耦合和多重散射的地震波传播理论,数值模拟并分析了在具有均匀介质背景的三维异常体——上地幔横向非均匀介质中传播时,地震面波的振幅与偏振等动力学响应参数;其中分别模拟了不同周期入射、不同角度入射和不同尺度非均匀介质模型等多种情形下面波波场,并对横向非均匀性诱导的面波偏振异常进行了分析.结果表明,相对于长周期面波而言,短周期面波的振幅和偏振方向受横向非均匀性的影响更大,特别是偏振方向对地球结构的非均匀性更为敏感;切向分量存在横向非均性引起的Rayleigh与Love面波耦合现象;异常体边界处表现出强的面波波场响应.     

径向分层TI孔隙介质井孔中激发的模式波的数值研究

横向各向同性介质是地层中普遍存在的一种各向异性介质.本文对径向分层TI孔隙介质包围井孔中激发的斯通利波和弯曲波的传播特性进行了理论计算,发现模式波在低频时更多的是反应原状地层的信息,而随着频率的增加侵入带参数逐渐起控制作用;Biot理论描述的地层衰减比速度更容易受井壁附近地层参数的影响.利用灵敏度曲线定量研究了不同频率下地层各个参数对相速度和衰减系数的贡献大小,主要结果显示模式波的衰减受水平渗透率影响明显,而垂直渗透率的变化对模式波几乎无影响;斯通利波对水平向传播的横波速度比弯曲波的灵敏度高.从单极子和偶极子声源在井孔中激发的全波波形也可发现,声波测井仪器较宽的声源频带和合适的源距设置有利于对不同径向深度上的地层声学参数进行成像.     

含孔隙、裂隙致密介质中多极子声波的传播特征

非常规油气藏（如致密性地层及蕴藏油气的页岩地层）的重要特征是低孔、低渗,但裂隙或裂缝比较发育.为满足非常规勘探的需求,本文将孔、裂隙介质弹性波传播理论应用于多极子声波测井的井孔声场模拟,重点研究了致密介质中裂隙发育时多极子声波的传播机理以及衰减特征.井孔声场的数值计算结果表明裂隙的存在明显改变了弹性波和井孔模式波的频散、衰减和激发强度,尤其是井壁临界折射纵波的激发谱的峰值随着频率的增加逐渐降低,这与应用经典的Biot理论下的计算结果相反,且裂隙的存在也使得饱含水和饱含气时临界折射纵波激发强度的差异变大.井孔模式波的衰减与地层横波衰减和井壁流体交换有关,井壁开孔边界下致密地层裂隙发育还使得井孔斯通利波和艾里相附近的弯曲波对孔隙流体的敏感性增强,在井壁闭孔边界条件下引起井孔模式波衰减的主要因素是裂隙引起的地层横波衰减造成的,且在截止频率附近弯曲波的衰减与地层的横波衰减一致.数值计算结果为解释非常规油气地层的声学响应特征提供了参考.     

Characteristics of coastal trapped waves along the northern coast of the South China Sea during year?1990

The structures and evolution of the coastal-trapped waves (CTW) along the northern coast of the South China Sea (SCS) in the year?1990 are studied using observed hourly sea level records collected from four sites around the northern SCS and a three-dimensional numerical model with realistic bathymetry and wind forcing. Analysis of the yearlong records of the observed sea level data indicates that the sea level variations are highly correlated between the stations and the sea level variability propagates southwestward along the coast. The sea level signals traveling from northeast to southwest along the coast with a propagation speed of 5.5–17.9?m?s^?1 during both the typhoon season and the winter month show the characteristics of a CTW. The wave speed is faster between stations Shanwei and Zhapo than that between Xiamen and Shanwei. Sea level variations during both typhoon season and winter month are reasonably well represented by the numerical model. The model runs focused on the wave signals related to typhoons and winter storm show that the CTW propagating southwestward along the coast can be reinforced or decreased by the local wind forcing during its propagation and there are apparent differences in the propagation characteristics between the waves along the mainland and those traveling around Hainan Island. The abrupt change of the shelf width and coastline around Leizhou Peninsula and Hainan Island are responsible for strong scattering of CTWs from one mode into higher modes. The alongshore velocities across different transects associated with CTW are investigated to examine the vertical structures of the waves. The alongshore velocity structures at transects during different events are related to the combined effect of stratification and shelf profile, which can be estimated using the Burger number. The empirical orthogonal function analysis of alongshore velocity and nodal lines of the mode structure suggest mode two CTWs in transect S2 during typhoon season and mode 1 CTWs during winter. Sensitivity model experiments are also performed to demonstrate the effects of local wind and topography on the wave propagation.     

起伏地表下基于改进BISQ模型双相介质中曲线网格有限差分法波场模拟

本文以基于改进BISQ模型的二维双相各向同性介质一阶速度-应力方程为基础,推导出了曲线坐标系下对应的方程,然后采用低频散、低耗散的同位网格MacCormack有限差分法来离散方程,并采用紧致的单边MacCormack差分格式结合牵引力镜像法来施加自由地表边界条件,实现了地震波场数值模拟.曲线网格有限差分法采用贴体网格来描述自由表面,地表的网格线紧贴地形,避免了台阶近似造成的数值散射.数值模拟结果表明,在双相介质起伏自由地表和分界面处,各类波型复杂的反射透射规律可以清晰展现,曲线网格有限差分法可以精确地解决地震波在含起伏地表的双相各向同性介质中的传播问题.     

Sensitivity of seismic waves to structure

We study how the perturbations of a generally heterogeneous isotropic or anisotropic structure manifest themselves in the wavefield, and which perturbations can be detected within a limited aperture and a limited frequency band. A short-duration broad-band incident wavefield with a smooth frequency spectrum is considered. In-finitesimally small perturbations of elastic moduli and density are decomposed into Gabor functions. The wavefield scattered by the perturbations is then composed of waves scattered by the individual Gabor functions. The scattered waves are estimated using the first-order Born approximation with the paraxial ray approximation. For each incident wave, each Gabor function generates at most 5 scattered waves, propagating in specific directions and having specific polarisations. A Gabor function corresponding to a low wavenumber may generate a single broad-band unconverted wave scattered in forward or narrow-angle directions. A Gabor function corresponding to a high wavenumber usually generates 0 to 5 narrow-band Gaussian packets scattered in wide angles, but may also occasionally generate a narrow-band P to S or S to P converted Gaussian packet scattered in a forward direction, or a broad-band S to P (and even S to S in a strongly anisotropic background) converted wave scattered in wide angles. In this paper, we concentrate on the Gaussian packets caused by narrow-band scattering. For a particular source, each Gaussian packet scattered by a Gabor function at a given spatial location is sensitive to just a single linear combination of 22 values of the elastic moduli and density corresponding to the Gabor function. This information about the Gabor function is lost if the scattered wave does not fall into the aperture covered by the receivers and into the legible frequency band.     

利用随钻正交偶极子声波测井评价地层各向异性的数值研究

在随钻测井条件下,由于钻铤占据了井孔内的大部分空间,充液井孔中沿着井轴方向传播的模式波的特性与电缆测井非常不同.本文建立了随钻正交偶极子测井声学模型,采用三维有限差分方法模拟了偶极子声源在随钻条件下各向异性地层井孔内激发的声场,研究了地层的声学各向异性在随钻正交偶极子声波测井中的响应特征.数值模拟结果表明,在随钻测井条件下,对于井轴同TI地层对称轴垂直的情况,弯曲波分裂现象仍然存在,通过正交偶极子测量方式和合适的反演算法能够准确有效地确定地层的快横波方位角,可以考虑采用同正演理论相结合的反演算法来获得地层的快、慢横波速度及声学各向异性信息;对于井轴同介质对称轴呈一定夹角的TI地层井孔,情况变得非常复杂,不同井斜倾角下弯曲波的速度的变化趋势并非同对应的地层横波速度的变化趋势完全一致,不过在一定的频段内,地层横波速度仍然是弯曲波的最主要控制因素.对于本文研究的模型,当井轴同介质对称轴的夹角大于大于60°时,此时获得的弯曲波的各向异性值基本能够反映对应角度下地层横波速度的各向异性信息.     

Seismic response of lined tunnels in the half-plane with surface topography

In this work, we examine the seismic response of multiple tunnels reinforced with liners and buried within the elastic homogeneous half-plane in the presence of surface relief. The seismic waves are upward propagating, time-harmonic, horizontally polarized shear (SH) waves. More specifically, we examine: (a) the scattered wave fields along the free surface and inside the half-plane with the embedded tunnels; (b) the dynamic stress concentration factors that develop at the soil-liner interfaces; (c) the stresses and displacements that develop inside the tunnel liners. We use a sub-structuring technique that is based on the direct boundary element method to model each constituent part of the problem separately. Then, assembly of the full problem is accomplished through the imposition of compatibility and equilibrium conditions at all interfaces. Next, a detailed verification study is carried out based on comparisons against available analytical and/or numerical results for a series of test examples. Subsequently, detailed numerical simulations are conducted and the results of these parametric studies reveal the influence of the following key parameters on the soil-tunnel system response: (a) the shape of the free-surface relief; (b) the depth of placement of the tunnels and their separation distance; (c) the SH-wavelength to tunnel diameter ratio; (d) the elastic properties of the tunnel lining rings and (e) the dynamic interaction effects between the free-surface relief and the tunnels.     

Scattering and attenuation of elastic waves in random media

In seismic exploration, elastic waves are sent to investigate subsurface geology. However, the transmission and interpretation of the elastic wave propagation is complicated by various factors. One major reason is that the earth can be a very complex medium. Nevertheless, in this paper, we model some terrestrial material as an elastic medium consisting of randomly distributed inclusions with a considerable concentration. The waves incident on such an inhomogeneous medium undergo multiple scattering due to the presence of inclusions. Consequently, the wave energy is redistributed thereby reducing the amplitude of the coherent wave.The coherent or average wave is assumed to be propagating in a homogeneous continuum characterized by a bulk complex wavenumber. This wavenumber depends on the frequency of the probing waves; and on the physical properties and the concentration of discrete scatterers, causing the effective medium to be dispersive. With the help of multiple scattering theory, we are able to analytically predict the attenuation of the transmitted wave intensity as well as the dispersion of the phase velocity. These two sets of data are valuable to the study of the inverse scattering problems in seismology. Some numerical results are presented and also compared, if possible, with experimental measurements.     

On the propagation characteristics of tunnel surface‐waves for seismic prediction

To increase the safety and efficiency of tunnel constructions, online seismic exploration ahead of a tunnel has become a valuable tool. One recent successful forward looking approach is based on the excitation and registration of tunnel surface‐waves. For further development and for finding optimal acquisition geometries it is important to study the propagation characteristics of tunnel surface‐waves. 3D seismic finite difference modelling and analytic solutions of the wave equation in cylindrical coordinates reveal that at higher frequencies, i.e., if the tunnel‐diameter is significantly larger than the wavelength of surface‐waves, these surface‐waves can be regarded as Rayleigh‐waves confined to the tunnel wall and following helical paths along the tunnel axis. For lower frequencies, i.e., when the tunnel surface‐wavelength approaches the tunnel‐diameter, the propagation characteristics of these surface‐waves are similar to S‐waves. We define the surface‐wave wavelength‐to‐tunnel diameter ratio w to be a gauge for separating Rayleigh‐ from S‐wave excitation. For w > 1.2 tunnel surface‐waves behave like S‐waves, i.e. their velocity approaches the S‐wave velocity and the particle motion is linear and perpendicular to the ray direction. For w < 0.6 they behave like Rayleigh‐waves, i.e., their velocity approaches the Rayleigh‐wave velocity and they exhibit elliptical particle motion. For 0.6 < w < 1.2 a mixture of both types is observed. Field data from the Gotthard Base Tunnel (Switzerland) show both types of tunnel surface‐waves and S‐waves propagating along the tunnel.     

Seismic migration and absorbing boundaries with a one-way wave system for heterogeneous media1

A first-order one-way wave system has been created based on characteristic analysis of the acoustic wave system and optimization of the dispersion relation. We demonstrate that this system is equivalent to a third-order scalar partial-differential equation which, for a homogeneous medium, reduces to a form similar to the 45° paraxial wave equation. This system describes accurately waves propagating in a 2D heterogeneous medium at angles up to 75°. The one-way wave system representing downgoing waves is used for a modified reverse time migration method. As a wavefield extrapolator in migration, the downgoing wave system propagates the reflection events backwards to their reflectors without scattering at the discontinuities in the velocity model. Hence, images with amplitudes proportional to reflectivity can be obtained from this migration technique. We present examples of the application of the new migration method to synthetic seismic data where P-P reflections P-SV converted waves are present. Absorbing boundaries, useful in the generation of synthetic seismograms, have been constructed by using the one-way wave system. These boundaries absorb effectively waves impinging over a wide range of angles of incidence.     

Mesospheric planetary wave effects on global PMC variability inferred from AIM–CIPS and TIMED–SABER for the northern summer 2007 PMC season

Polar Mesospheric Cloud (PMC) observations from the Cloud Imaging and Particle Size (CIPS) instrument on the Aeronomy of Ice in the Mesosphere (AIM) spacecraft are used to investigate the role of planetary wave activity on global PMC variability in the summer polar mesosphere during the 2007 Northern hemisphere season. This is coupled with an analysis of contemporaneous measurements of atmospheric temperature by the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument onboard the Thermosphere–Ionosphere–Mesosphere–Energetics and Dynamics (TIMED) spacecraft to characterize the importance of temperature as a dominant forcing mechanism of the dynamical state of the summer polar mesosphere. The study confirms results from a recent study using PMC data from the Student Nitric Oxide Explorer (SNOE) and temperature data from SABER, such that planetary wave activity is present in both PMCs and mesospheric temperature and that are strongly coherent and anti-correlated. The dominant wave present in the polar summer mesosphere in both PMCs and temperature is the 5-day wavenumber 1 Rossby normal mode. The maximum amplitude of the variation of the 5-day wave in temperature is small at 3 K but has a significant effect on PMC albedo. The phase relationship between PMC and temperature is variable between 150° and 180° out of phase, with PMC albedo reaching a maximum ～10 h before the minimum in temperature. We have identified two additional waves, the westward propagating 2-day wavenumber 2 (2DW2) and the eastward propagating 2-day wavenumber 1 (2DE1) are both present in PMC and temperature variability in the 2007 NH season. The 2DW2 wave is consistent with a Rossby normal mode excited by the instability in the zonal mean zonal wind. However, the source of the 2DE1 wave could be a nonlinear interaction of the 2DW2 with the migrating diurnal tide. This is the first time these two wave features have been detected in coincident PMC and temperature measurements. Analysis of the zonal variation of PMC occurrence and temperature shows they are also anti-correlated and supporting the conclusion that temperature is an important forcing mechanism in zonal variability.     

Ray path of head waves with irregular interfaces

Head waves are usually considered to be the refracted waves propagating along flat interfaces with an underlying higher velocity. However, the path that the rays travel along in media with irregular interfaces is not clear. Here we study the problem by simulation using a new approach of the spectral-element method with some overlapped elements (SEMO) that can accurately evaluate waves traveling along an irregular interface. Consequently, the head waves are separated from interface waves by a time window. Thus, their energy and arrival time changes can be analyzed independently. These analyses demonstrate that, contrary to the case for head waves propagating along a flat interface, there are two mechanisms for head waves traveling along an irregular interface: a refraction mechanism and transmission mechanism. That is, the head waves may be refracted waves propagating along the interface or transmitted waves induced by the waves propagating in the higher-velocity media. Such knowledge will be helpful in constructing a more accurate inversion method, such as head wave travel-time tomography, and in obtaining a more accurate model of subsurface structure which is very important for understanding the formation mechanism of some special areas, such as the Tibetan Plateau.     

Oblique wave diffraction by a flexible floating structure in the presence of a submerged flexible structure

Expansion formulae associated with the interaction of oblique surface gravity waves with a floating flexible plate in the presence of a submerged horizontal flexible structure are derived using Green’s integral theorem in water of finite and infinite water depths. The associated Green’s functions are derived using the fundamental solution associated with the reduced wave equation. The integral forms of the Green’s functions and the velocity potentials are advantageous over the eigenfunction expansion method in situation when the roots of the dispersion relation coalesce. As an application of the expansion formulae, diffraction of oblique waves by a finite floating elastic plate in the presence of a submerged horizontal flexible membrane is investigated in water of finite depth. The accuracy of the numerical computation is demonstrated by analysing the convergence of the complex amplitude of the reflected waves and the energy relation. Effect of the submerged membrane on the diffraction of surface waves is studied by analysing the reflection and transmission coefficients for various parametric values. Further, the derivation of long wave equation under shallow water approximation is derived in a direct manner in the appendix. The concept and methodology can be easily extended to deal with acoustic wave interaction with flexible structures and related problems of mathematical physics and engineering.     

Seismic wavefield modelling in two-phase media including undulating topography with the modified Biot/squirt model by a curvilinear-grid finite difference method

In this paper, we deduced the corresponding first-order velocity–stress equation for curvilinear coordinates from the first-order velocity–stress equation based on the modified Biot/squirt model for a two-dimensional two-phase medium. The equations are then numerically solved by an optimized high-order non-staggered finite difference scheme, that is, the dispersion relation preserving/optimization MacCormack scheme. To implement undulating free-surface topography, we derive an analytical relationship between the derivatives of the particle velocity components and use the compact finite-difference scheme plus a traction-image method. In the undulating free surface and the undulating subsurface interface of two-phase medium, the complex reflected wave and transmitted wave can be clearly recognized in the numerical simulation results. The simulation results show that the curvilinear-grid finite-difference method, which uses a body-conforming grid to describe the undulating surface, can accurately reduce the numerical scattering effect of seismic wave propagation caused by the use of ladder-shaped grid to fit the surfaces when undulating topography is present in a two-phase isotropic medium.     

Onset of an energy cascade and nonperiodic behaviour in the nonlinear propagation of MHD waves in the solar atmosphere

Abstract

We study the nonlinear stability of MHD waves propagating in a two-dimensional, compressible, highly magnetized, viscous plasma. These waves are driven by a weak, shear body force which could be imposed by large scale internal fluctuations present in the solar atmosphere.

The effects of anisotropic viscosity (leading to a cubic damping) and of the nonlinear coupling of the Alfven and the magnetoacoustic waves are analysed using Galerkin and multiple-scale analysis: the MHD equations are reduced to a set of nonlinear ordinary differential equations which is then suitably truncated to give a model dynamical system, representing the interaction of two complex Galerkin modes.

For propagation oblique to the background magnetic field, analytical integration shows that the low-wavenumber mode is physically unstable. For propagation parallel to the background magnetic field the high-wavenumber wave can undergo saddlenode bifurcations, in way that is similar to the van der Pol oscillator; these bifurcations lead to the appearance of a hysteresis cycle.

A numerical integration of the dynamical system shows that a sequence of Hopf bifurcations takes place as the Reynolds number is increased, up to the onset of nonperiodic behaviour. It also shows that energy can be transferred from the low- wavenumber to the high-wavenumber mode.     

The influence of background winds and attenuation on the propagation of atmospheric gravity waves

The influence of background winds and energy attenuation on the propagation of atmospheric gravity waves is numerically analyzed. The gravity waves, both in the internal and ducted forms, are included through employing ray-tracing method and full-wave solution method. Background winds with different directions may cause ray paths of internal gravity waves to be horizontally prolonged, vertically steepened, reflected or critically coupled, all of which change the accumulation of energy attenuation along ray paths. Only the penetrating waves propagating against winds can easily reach the ionospheric height with less energy attenuation. The propagation status of gravity waves with different periods and phase speeds is classified into the cut-off region, the reflected region and the propagating region. All the three regions are influenced significantly by winds. The area of the reflected region reduces when gravity waves propagate in the same direction of winds and expands when propagating against wind. In propagating region, the horizontal attenuation distances of gravity waves increase and the arrival heights decrease when winds blow in the same direction of gravity waves, while the attenuation distances decrease and the arrival heights increase when gravity waves propagate against winds. The results for ducted gravity waves show that the influence of winds on waves of lower atmospheric modes is not noticeable for they propagate mainly under mesosphere, where the wind field is relatively weak. However, strong winds at thermospheric height lead to considerable changes of dispersion relation and attenuation distance of upper atmospheric modes. Winds against the wave propagating direction support long-distance propagation of G mode, while the attenuation distances decrease when winds blow in the same direction of the wave. The distribution of TIDs observed by HF Doppler array at Wuhan is compared with the simulation of internal gravity waves. The observation of TIDs shows agreement with our numerical calculations.     

各向异性介质qP波传播描述II:分离纯模式标量波

在各向异性地震波场中,qP波与qS波常常是耦合在一起的.多分量地震数据处理中一个关键环节就是波型分离(即模式解耦),以纵波成分为主的常规单分量地震数据的成像则需要合理描述标量qP波的传播算子.本文作者曾构建了在运动学上同弹性波动方程等价,动力学上突出标量qP波的伪纯模式波动方程.为了彻底消除qS波残余,本文根据波矢量与qP波偏振矢量之间的偏差,提出从伪纯模式波场提取纯模式标量qP波的方法.数值分析展示了投影偏差算子在波数域和空间域的特征.基于不同复杂程度理论模型的试验结果表明,联合"伪纯模式传播算子"与"投影偏差校正"可为各向异性介质分离模式波场传播过程提供一种简便的描述工具.