多次各向异性散射模式对S波能量密度包络曲线的影响

在多次各向异性散射理论的基础上,本文重新推导了方向性散射系数的球函数展开式.引入特征时间的概念,来定义震源处初始地震波脉冲宽度,并在地震波能量密度积分方程中引入任意给定频率的初始脉冲能量谱密度的解析表达.通过离散波数方法求解了修正的地震波能量密度积分方程.基于积分方程的数值解,研究了不同散射模式对S波能量密度包络曲线的影响.计算结果表明:随着震源距的增加,在S波到时之后,多次各向异性散射模式与多次各向同性散射模式合成的能量密度包络差异逐渐增大.其中通过多次前散射模式,我们可以得到不同震源距的尾波能量密度包络的同一衰减趋势,以及S波能量密度包络随着震源距的增加而出现的展宽现象.最后,利用美国内华达州Wells地震余震的台站记录验证了多次前散射模式的实用性与有效性.     

Rossby waves and cascade phenomena

Nonlinear triad interactions in the Navier-Stokes equation describing quasi-geostrophic thermal convection in a spherical layer are considered. The analysis provides an insight into the nature of kinetic energy transfer across the spectrum and allows us to assess the degree of nonlocality in the energy transfer. The distinguishing feature of the spherical problem is the emergence of the Rossby waves, which cause nonlocal redistribution of energy in the system. The results of this study are compared with their counterparts for the plane model.     

On Solving the Forward Problem of Gravimetry in Curvilinear and Cartesian Coordinates: Krasovskii’s Ellipsoid and Plane Modeling

Correcting the effects of the sphericity of the Earth in the results of the interpretation of gravimetric data is a topical issue in modern gravimetry. Estimating the error of the gravity field calculations due to the replacement of the spherical Earth model by the plane model is an important part of this problem. In this paper, a method is proposed for transforming the plane density models into spherical ones and vice versa. Algorithms for calculating the vertical component of gravity field for both model types are presented. For two extensive plane models of the Earth’s density, their transformation into spherical models is carried out and the resulting gravity fields are compared. The relative root mean square residuals between the fields calculated with this replacement are at most 5%.     

Comparison of methods for modelling the behaviour of bubbles produced by marine seismic airguns

Three models for the dynamics of seismic airgun‐generated bubbles and their associated far‐field signals are developed and compared with geophysical data. The first model of an airgun‐generated bubble uses a spherical approximation, the second is an approximate Lagrangian model which allows for small deformations from a spherical shape, whilst the final model is an axisymmetric boundary‐integral method which permits the bubble to evolve into highly non‐spherical geometries. The boundary‐integral method also allows both geometric interference and strong dynamic interactions in multi‐bubble studies. When comparing the spherical model to experimental data there are three apparent, significant differences: the magnitude of the primary pressure peak, which is greater in the model; the subsequent decay of the pressure peaks and motion – the experimental data demonstrating greater decay and a slower rise rate; and the frequency of oscillation, which is slower in the experimental data. It is believed that the first discrepancy is due to the initial stages of expansion where the compressed air is forced to sparge through the airgun ports. The other differences indicate that there is some other energy‐loss mechanism which is not accounted for in the spherical bubble model. Non‐spherical bubble behaviour is investigated through the use of two different deformable many‐bubble codes and their predictions are compared with the spherical model and experimental data. The Lagrangian model predicts the formation of a buoyancy‐driven liquid jet on the first collapse of a typical airgun bubble; however, the model breaks down when the bubble becomes significantly deformed, due to a low‐order spherical‐harmonic approximation for the potential. The axisymmetric boundary‐integral code models the jet shape accurately and it is found that these bubbles evolve to toroidal geometries when the jet impacts on the opposite surface of the bubble. This highly non‐spherical behaviour is readily observed on high‐speed films of airgun bubbles, and is one key source of energy loss; it damps the pulsations of the bubble and slows its rise speed. Inter‐bubble interactions are investigated using the two deformable bubble models, and the predictions are compared to field data. It was found that as the bubbles approach each other, their periods of oscillation increase in accordance with observations, and jets are formed in the direction of motion upon collapse.     

An efficient and adaptive approach for modeling gravity effects in spherical coordinates

The need to obtain more reliable Earth structures has been the impetus for conducting joint inversions of disparate geophysical datasets. For seismic arrival time tomography, joint inversion of arrival time and gravity data has become an important way to investigate velocity structure of the crust and upper mantle. However, the absence of an efficient approach for modeling gravity effects in spherical coordinates limits the joint tomographic analysis to only local scales. In order to extend the joint tomographic inversion into spherical coordinates, and enable it to be feasible for regional studies, we develop an efficient and adaptive approach for modeling gravity effects in spherical coordinates based on the longitudinal/latitudinal grid spacing. The complete gravity effects of spherical prisms, including gravitational potential, gravity vector and tensor gradients, are calculated by numerical integration of the Gauss–Legendre quadrature (GLQ). To ensure the efficiency of the gravity modeling, spherical prisms are recursively subdivided into smaller units according to their distances to the observation point. This approach is compatible with the parameterization of regional arrival time tomography for large areas, in which both the near- and far-field effects of the Earth's curvature cannot be ignored. Therefore, this approach can be implemented into the joint tomographic inversion of arrival time and gravity data conveniently. As practical applications, the complete gravity effects of a single anomalous density body have been calculated, and the gravity anomalies of two tomographic models in the Taiwan region have also been obtained using empirical relationships between P-wave velocity and density.     

Local Ionospheric Modeling Using the Localized Global Ionospheric Map and Terrestrial GPS

global ionosphere maps are generated on a daily basis at CODE using data from about 200 GPS/GLONASS sites of the IGS and other institutions. The vertical total electron content is modeled in a solargeomagnetic reference frame using a spherical harmonics expansion up to degree and order 15. The spherical Slepian basis is a set of bandlimited functions which have the majority of their energy concentrated by optimization inside an arbitrarily defined region, yet remain orthogonal within the spatial region of interest. Hence, they are suitable for decomposing the spherical harmonic models into the portions that have significant strength only in the selected areas. In this study, the converted spherical harmonics to the Slepian bases were updated by the terrestrial GPS observations by use of the least-squares estimation with weighted parameters for local ionospheric modeling. Validations show that the approach adopted in this study is highly capable of yielding reliable results.     

Estimation of the gravitational potential energy of the earth based on different density models

The estimation of the Earth’s gravitational potential energy E was obtained for different density distributions and rests on the expression E = − (W_min + ΔW) derived from the conventional relationship for E. The first component W_min expresses minimum amount of the work W and the second component ΔW represents a deviation from W_min interpreted in terms of Dirichlet’s integral applied on the internal potential. Relationships between the internal potential and E were developed for continuous and piecewise continuous density distributions. The global 3D density model inside an ellipsoid of revolution was chosen as a combined solution of the 3D continuous distribution and the reference PREM radial piecewise continuous profile. All the estimates of E were obtained for the spherical Earth since the estimated (from error propagation rule) accuracy σ_E of the energy E is at least two orders greater than the ellipsoidal reduction and the contribution of lateral density inhomogeneities of the 3D global density model. The energy E contained in the 2nd degree Stokes coefficients was determined. A good agreement between E = E_Gauss derived from Gaussian distribution and other E, in particular for E = E_PREM based on the PREM piecewise continuous density model and E-estimates derived from simplest Legendre-Laplace, Roche, Bullard and Gauss models separated into core and mantle only, suggests the Gaussian distribution as a basic radial model when information about density jumps is absent or incomplete.     

多点冲击荷载作用下K6型单层球面网壳结构失效机理分析

为研究单层球面网壳结构在多点冲击荷载作用下的失效机理,在ANSYS/LS-DYNA中建立了60m跨度的Kiewitt 6型单层球面网壳结构的有限元模型,将整个网壳结构沿肋杆分成6个不同扇区,沿环杆分成5个不同环区,分别对网壳结构在同一环区和同一扇区受到多点冲击时的动力响应及冲击全过程能量传递的规律进行了分析研究。基于网壳结构失效变形特点,定义了多点冲击荷载作用下单层球面网壳结构的4类失效模式;根据整个冲击过程中能量的传递特点,揭示了单层球面网壳结构每类失效模式所对应的失效机理。     

Relativistic expansion of a magnetized fluid

We study semi-analytical time-dependent solutions of the relativistic MHD equations for the fields and the fluid emerging from a spherical source. We assume uniform expansion of the field and the fluid and a polytropic relation between the density and the pressure of the fluid. The expansion velocity is small near the base but approaches the speed of light at the light sphere where the flux terminates. We find self-consistent solutions for the density and the magnetic flux. The details of the solution depend on the ratio of the toroidal and the poloidal magnetic field, the ratio of the energy carried by the fluid and the electromagnetic field and the maximum velocity it reaches.     

Comparison of two approaches for considering laterally varying density in topographic effect on satellite gravity gradiometric data

The satellite gravity gradiometric data are influenced by laterally varying density in topographic masses, while in most of studies a constant density for the masses was considered. This assumption causes an error in estimating the topographic effect. This paper theoretically and numerically investigates the methods of Sjöberg as well as Novák and Grafarend to consider the laterally varying density for topographic masses in formulation of topographic potential in spherical harmonics.     

A model of thermal convection in the major planets with a strongly density-stratified atmospheric layer

Abstract

As an extension of a model by Busse (1983a), a two-layer model of thermal convection in the self-gravitating rotating spherical fluid is considered. The upper layer with arbitrary vertical distributions of density and potential temperature representing the atmospheric layer of major planets is imposed on the spherical Boussinesq fluid. The Prandtl number P and the ratio of the mass of the upper layer to that of the lower layer are used as small expansion parameters. The modification of the critical Rayleigh number by imposing the upper layer are clearly separated into two parts, proportional to (1) the mass of the upper layer and to (2) an integral representing a measure of convective instability of the upper layer. Some implications for atmospheric dynamics of the major planets are also presented.     

Gravitational attraction and potential of spherical shell with radially dependent density

Solutions to the direct problem in gravimetric interpretation are well-known for wide class of source bodies with constant density contrast. On the other hand, sources with non-uniform density can lead to relatively complicated formalisms. This is probably why analytical solutions for this type of sources are rather rare although utilization of these bodies can sometimes be very effective in gravity modeling. I demonstrate an analytical solution to that problem for a spherical shell with radial polynomial density distribution, and illustrate this result when applied to a special case of 5th degree polynomial. As a practical example, attraction of the normal atmosphere is calculated.     

Planetary structures in general relativity

Summary For a static single-zone planet assuming that 1) the pressure and density are connected by the equationP =K _Q ^1+1/n–s, 2) a complete spherical symmetry is preserved and the system is in hydrostatic equilibrium; in section II, the expressions for the field equations have been obtained in suitable dimensionless forms. In section III the solution of the field equations forn=0 (which represents a homogeneous liquid) has been given in explicit form; for other prescribed value ofn=2/5 it has been pointed out that solutions must be performed by numerical integrations. Expressions for the mass-radius relation, the ratio of central to average density, the total energy, the proper energy and gravitational potential energy, which give some informations about the internal structure of the planet, have also been given in this section. Section IV discusses the velocity of sound at the centre of the planet. A few concluding remarks regarding the structure of the planet have been given in section V.     

Turning surface behaviour for internal waves subject to general gravitational fields

Abstract

It is shown that the turning surfaces associated with internal waves in a uniformly rotating, density stratified, Boussinesq fluid in the presence of an arbitrary gravitational field are regular points for the governing eigenvalue differential equation. The results are illustrated for two particular examples that have geophysical and astrophysical significance, namely radially directed spherical gravity, and the gravitational field in a rapidly rotating cylinder.     

Analytical wavefront curvature correction to plane‐wave reflection coefficients for a weak‐contrast interface

Most amplitude versus offset (AVO) analysis and inversion techniques are based on the Zoeppritz equations for plane‐wave reflection coefficients or their approximations. Real seismic surveys use localized sources that produce spherical waves, rather than plane waves. In the far‐field, the AVO response for a spherical wave reflected from a plane interface can be well approximated by a plane‐wave response. However this approximation breaks down in the vicinity of the critical angle. Conventional AVO analysis ignores this problem and always utilizes the plane‐wave response. This approach is sufficiently accurate as long as the angles of incidence are much smaller than the critical angle. Such moderate angles are more than sufficient for the standard estimation of the AVO intercept and gradient. However, when independent estimation of the formation density is required, it may be important to use large incidence angles close to the critical angle, where spherical wave effects become important. For the amplitude of a spherical wave reflected from a plane fluid‐fluid interface, an analytical approximation is known, which provides a correction to the plane‐wave reflection coefficients for all angles. For the amplitude of a spherical wave reflected from a solid/solid interface, we propose a formula that combines this analytical approximation with the linearized plane‐wave AVO equation. The proposed approximation shows reasonable agreement with numerical simulations for a range of frequencies. Using this solution, we constructed a two‐layer three‐parameter least‐squares inversion algorithm. Application of this algorithm to synthetic data for a single plane interface shows an improvement compared to the use of plane‐wave reflection coefficients.     

Forward computation of the magnetic field of a 3D body with arbitrary boundary and continually varying magnetization

The forward computation of the gravitational and magnetic fields due to a 3D body with an arbitrary boundary and continually varying density or magnetization is an important problem in gravitational and magnetic prospecting. In order to solve the inverse problem for the arbitrary components of the gravitational and magnetic anomalies due to an arbitrary 3D body under complex conditions, including an uneven observation surface, the existence of background anomalies and very little or no a priori information, we used a spherical coordinate system to systematically investigate forward methods for such anomalies and developed a series of universal spherical harmonic expansions of gravitational and magnetic fields. For the case of a 3D body with an arbitrary boundary and continually varying magnetization, we have also given the surface integral expressions for the common spherical harmonic coefficients in the expansion of the magnetic field due to the body, and a very precise numerical integral algorithm to calculate them. Thus a simple and effective method of solving the forward problem for magnetic fields due to 3D bodies of this kind has been found, and in this way a foundation is laid for solving the inverse problem of these magnetic fields. In addition, by replacing the parameters and unit vectors in the spherical harmonic expansion of a magnetic field by gravitational parameters and a downward unit vector, we have also derived a forward method for the gravitational field (similar to that for the magnetic case) of a 3D body with an arbitrary boundary and continually varying density.     

全球地表热流的产生与分布

全球地表热流是反映地球内部热与动力学过程的一种主要能流.本文在三维球坐标框架下,就几个不同的粘度模型分别研究地幔内部密度异常(基于全球地震层析结果)以及板块运动激发的地幔流动的热效应及其对于观测地表热流产生和分布特征的贡献.由于地幔动力系统具有较高的Pe数,可以期望由板块运动激发的地幔流动将强烈地扰动地幔内部初始传导状态下的温度场以及地表热的热流分布.结果表明,与地幔内部密度异常产生的热效应相比,运动的板块及其激发的地幔流动在全球地表观测热流的产生和分布特征上起着更为重要的作用.观测到的大洋中脊处的高热流在很大程度上可以归因于板块激发的地幔流动的热效应.计算的平均温度剖面较好地揭示了岩石圈和D″层的温度特征,即温度随深度的剧烈变化,这与我们目前通过其他手段对岩石圈和D″层的温度结构了解是一致的.一个下地幔粘度比上地幔高出30倍的粘度结构(文中使用的粘度模型2)较之其余模型的拟合程度似乎更好.     

Saturated,collisional flows of spheres over an inclined,erodible bed between vertical sidewalls

We consider a steady, uniform, dense, fully-saturated, gravity-driven, inclined flow of water and identical spherical grains over an erodible bed between parallel, vertical sidewalls. The grains are inelastic and the energy lost in their interaction is also influenced by the fluid viscosity. We use an extension of kinetic theory for dense flows and employ approximate integrations of the momentum and energy balances for the grains in order to obtain analytical expressions for the depth of flow and for the volume flow rate of the mixture as functions of inclination and average concentration. We also predict the range of slopes for which dense, fully-saturated flows are possible. The predictions are in reasonable agreement with already published experimental results.     

全球地表热流的产生与分布

全球地表热流是反映地球内部热与动力学过程的一种主要能流.本文在三维球坐标框架下，就几个不同的粘度模型分别研究地幔内部密度异常(基于全球地震层析结果)以及板块运动激发的地幔流动的热效应及其对于观测地表热流产生和分布特征的贡献.由于地幔动力系统具有较高的Pe数，可以期望由板块运动激发的地幔流动将强烈地扰动地幔内部初始传导状态下的温度场以及地表热的热流分布.结果表明，与地幔内部密度异常产生的热效应相比，运动的板块及其激发的地幔流动在全球地表观测热流的产生和分布特征上起着更为重要的作用.观测到的大洋中脊处的高热流在很大程度上可以归因于板块激发的地幔流动的热效应.计算的平均温度剖面较好地揭示了岩石圈和D″层的温度特征，即温度随深度的剧烈变化，这与我们目前通过其他手段对岩石圈和D″层的温度结构了解是一致的.一个下地幔粘度比上地幔高出30倍的粘度结构(文中使用的粘度模型2)较之其余模型的拟合程度似乎更好.