A SF6 tracer study of horizontal mixing in Lake Constance

Horizontal mixing processes in the hypolimnion of the western part of Lake Constance are studied by measuring the dispersion of a sulfur hexafluoride (SF₆) tracer plume. Only 1 liter gaseous SF₆ (STP) was released at a central hypolimnic depth of 60 m in August 1990. Over a period of 3 months the horizontal dispersion of the tracer plume was measured by 19 surveys using a new, vertically integrating sampling device. The observed horizontal dispersion is marked by strong storm-induced stirring events. Nevertheless mean turbulent diffusion coefficients for the whole period can be computed. They rise about linear from 0.7 10⁵ cm²/s to 3.0 10⁵ cm²/s with the distance from the western end of the lake. For the hypolimnion of Überlingersee, a sill-separated basin in the western part of Lake Constance, a simple budget model gives an exchange time of 67 ± 6 days with the main basin (Obersee).     

Interaction of trace metals with natural particle surfaces: Comparison between adsorption experiments and field measurements

The distribution of the metal ions Zn and Pb between particulate and dissolved phase in river Glatt was studied by field measurements and compared with calculated simulations, using parameters obtained by adsorption experiments with natural suspended particulate material. Differences in distribution coefficients obtained from field data are observed in function of the sampling locations and of the composition of the particulate matter.Experiments in which metal ion solutions are titrated with a suspension of natural particles and analyzed by anodic stripping voltammetry, are interpreted in terms of binding capacities and conditional stability constants of Zn and Pb with the surface sites. Binding constants of a particular metal ion varied very little for all samples. We obtained mean values for the conditional average complex formation constants at pH 8 of: log^cond K _Pb = 9.44 ± 0.18 and log^cond K _Zn = 8.17 ± 0.20. At this pH, binding capacities of 5 10^–3 – 1.7 10^–2 mol/kg of particles were obtained for samples collected at different locations and times; organic material, iron and manganese oxides are considered to be the main components that control the adsorption to the particles.Distribution coefficients are calculated from the experimentally obtained binding capacities and conditional stability constants. Calculated distribution coefficients for Zn agree with those obtained from the field data and are not very sensitive to changes in the composition of the solution. Good agreement was obtained for lead as well; for some samples it was important to take two types of sites with different affinity into consideration.     

Accuracy of deflections of vertical determined from discrete solution of the geodetic boundary value problem

Summary The general problem of determining the figure of the earth leads to the solution of the geodetic boundary value problem. By its discrete approximation we obtain the discrete disturbing potential that maintains all properties of the original problem. Thus, the discrete approximation of the disturbing potential can be used in studying the behaviour of the earth's gravity field outside the disturbing masses. The deflections of the vertical are one of the quantities describing the behaviour of the earth's gravity field. A method for their computation from the discrete solution of the geodetic boundary value problem is put forth and estimates for its accuracy are given.     

曲面上的位场转换（二维和三维情况）

在文献[1]中我们利用复变函数论的方法只解决了二维的曲面上的位场转换问题.在本文中,我们利用单层位势理论将它化为一个奇异积分方程求解的问题,并且提出了用奇点分离法来解此奇异积分方程,这对二维和三维情况都适用.通过理论模型的计算,结果的精度是令人满意的.     

On the discrete outer gravity field

Summary In the present paper the gravity field of the earth in the neighbourhood of the local disturbing masses is studied. The object of the method presented consists of the approximation of the disturbing potentialT _h , which fulfils Laplace's equation outside disturbing masses, on the earth's surface the fundamental boundary value condition of gravity and in infinity it is to be regular by the approximation of the disturbing potential (or by the discrete disturbing potential)T _h , which fulfils the respective finite difference approximation of Laplace's equation and the boundary value conditions in infinity and on the earth's surface. It is also shown that the approximation of the disturbing potentialT _h has the same properties as the disturbing potentialT. The method under consideration will be derived quite generally without any hypothesis about the distribution of the mass between the earth's surface and the geoid. It commences from the gravity data related to the earth's surface only-from the given geodetic measurements.     

Acoustic viscoelastic modeling by frequency-domain boundary element method

Earth medium is not completely elastic, with its viscosity resulting in attenuation and dispersion of seismic waves. Most viscoelastic numerical simulations are based on the finite-difference and finite-element methods. Targeted at viscoelastic numerical modeling for multilayered media, the constant-Q acoustic wave equation is transformed into the corresponding wave integral representation with its Green’s function accounting for viscoelastic coefficients. An efficient alternative for full-waveform solution to the integral equation is proposed in this article by extending conventional frequency-domain boundary element methods to viscoelastic media. The viscoelastic boundary element method enjoys a distinct characteristic of the explicit use of boundary continuity conditions of displacement and traction, leading to a semi-analytical solution with sufficient accuracy for simulating the viscoelastic effect across irregular interfaces. Numerical experiments to study the viscoelastic absorption of different Q values demonstrate the accuracy and applicability of the method.     

The boundary value problem of Poisson’s equation with Stokes’ boundary condition

The following Poisson’s equation with the Stokes’ boundary condition is dealt with $$\left\{ \begin{gathered} \nabla ^2 T = - 4\pi Gp outside S, \hfill \\ \left. {\frac{{\partial T}}{{\partial h}} = \frac{1}{\gamma }\frac{{\partial y}}{{\partial h}}T} \right|_s = - \Delta g, \hfill \\ T = O\left( {r^{ - 3} } \right) at infinity, \hfill \\ \end{gathered} \right.$$ whereS is reference ellipsord. Under spherical approximation transformation, the ellipsoidal correction terms about the boundary condition, the equation and the density in the above BVP are respectively given. Therefore, the disturbing potentialT can he obtained if the magnitudes aboveO(ε⁴) are neglected.     

Integral Equations for Kinematic Dynamo Models

A new technique for the treatment of the kinematic dynamo problem is presented. The method is applicable when the dynamo is surrounded by a medium of finite conductivity and is based on a reformulation of the induction equation and boundary conditions at infinity into an integral equation. We show that the integral operator involved here is compact in the case of homogeneous conductivity, which is important for both mathematical and numerical treatment. A lower bound for the norm of then yields a necessary condition for the generation of magnetic fields by kinematic dynamos. Numerical results are presented for some simple ²-dynamo models. The far-field asymptotics for stationary and time-dependent field modes are discussed.     

地球外部扰动重力严密改化模型及分析检验

直接积分模型是计算地球外部扰动重力的主要数学工具,将全球积分模型改化为局域积分模型是实现地球外部重力场赋值的前提条件.相比表层积分模型和向上延拓积分模型,Stokes积分模型要求的输入信息种类最少,故在减轻数据保障压力方面具有比较明显的优势.在实施Stokes积分模型工程化应用过程中,我们发现扰动重力径向分量积分模型从...     

应用Bjerhammar方法确定GPS重力似大地水准面

GPS技术的发展提出了新的大地边值问题——GPS重力边值问题.本文将Bjerhammar方法应用于GPS重力问题的求解,并在给出理论公式的基础上,针对实际计算中虚拟场元的分布和求解、虚拟球半径的确定及奇异积分等问题提出了具体的解决方案.文中通过比例因子k在虚拟球半径和GPS重力数据密度间建立起联系,并推导出其近似值.在...     

Mass heterogeneities and convection in the earth's mantle inferred from gravity and core-mantle boundary irregularities

Mass heterogeneities in the earth's mantle are retrieved from the gravity data and the topography of the core-mantle boundary as well as the topography of the earth's surface. A mantle circulation induced by the heterogeneities is modelled by solving the Stokes problem for incompressible Newtonian fluid. The derived models of mantle motions correlate well with the plate tectonics and point at a close relation between the surface tectonic activity and the processes in the vicinity of the core-mantle boundary.     

Precipitation of electrons after geomagnetic substorms

Summary Precipitating electrons E_e>10 keV and E_e>30 keV and protons E_p=150 to 300 keV are analyzed using data obtained from the low-altitude polar-orbiting satellite Interkozmos-17 after the magnetic storm of December 2, 1977. On December 3, an isolated substrom occurred, and strong electron precipitation was observed in the midnigt sector, while the sequence of substorms of December 5 expanded its longitudinal extent from 22 MLT through the morning to 12 MLT. The different character of the precipitation, in the two cases mentioned, is discussed in relation to the changes of conditions for the generation of cyclotron instability as well as to the effect of particle injection. Proton precipitation occurred only in the night sector, most probably just inside the plasmapause.

---

¶rt; aau nau m E_e>10 u E_e>30 u nm E_p=150–300 n uu n uuma nmua n um m-17 n aum u 2. ¶rt;a 1977. 3. 12. nua uuaa u u nau m aumua m mo u. a n¶rt;amm 5. 12. nua auu ¶rt;m umaa, m m nau, a u m 22 MLT mu m ¶rt; 12 MLT. a aam nau nu¶rt; ¶rt; a ¶rt;am aa uu u ¶rt; uu um mumu u ¶rt;mu uuu amu. nau nm n¶rt;u m m u m a naana mu naa.

     

An integral equation for the homogeneous Neumann condition1

Modelling the theoretical response of several important geophysical systems involves the solution of Poisson's equation with homogeneous Neumann boundary conditions (i.e. a zero normal gradient) imposed over either open or closed surfaces. A simple integral equation solution to this problem is derived from first principles. It is applicable to both types of surface and in this respect represents an improvement on existing integral equation techniques. However, the present surface integral equation displays a strong singularity of order 1/R³ which requires an appropriate interpretation for its implementation. A comparison of some numerical results with analytical data taken from the literature demonstrates that the proposed integral equation technique is suitably robust, accurate and efficient for practical application in geophysical interpretation.     

A GALERKIN FINITE-ELEMENT PROGRAM FOR SIMULATING UNSATURATED FLOW IN POROUS MEDIA

Abstract. A fully documented Galerkin finite-element FORTRAN program is presented for solving the one-dimensional, transient flow equation in unsaturated porous media. Material balance error summaries are presented to demonstrate accuracy of the numerical scheme. Comparison of our simulated results with other existing numerical solutions using the Galerkin scheme provided excellent agreement.     

SH-wave diffraction by a semi-circular hill revisited: A null-field boundary integral equation method using degenerate kernels

Following the success of seismic analysis of a canyon [1], the problem of SH-wave diffraction by a semi-circular hill is revisited using the null-field boundary integral equation method (BIEM). To fully utilize the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-circular hill scattering problem, the problem is decomposed into two regions to produce circular boundaries using the technique of taking free body. One is the half-plane problem containing a semi-circular boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial full circular boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by a circular boundary. The degenerate kernel in the polar coordinates for two subdomains is utilized for the closed-form fundamental solution. The semi-analytical formulation along with matching boundary conditions yields six constraint equations. Instead of finding admissible wave expansion bases, our null-field BIEM approach in conjunction with degenerate kernels have five features over the conventional BIEM/BEM: (1) free from calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless and (5) well-posed system. All the numerical results are comparing well with the available results in the literature. It is interesting to find that a focusing phenomenon is also observed in this study.     

求解双相和黏弹性介质波传播方程的间断有限元方法及其波场模拟

间断有限元（Discontinuous Galerkin：DG）方法具有低数值频散、网格剖分灵活、能模拟地震波在复杂介质中传播等优点.因此,本文将一种新的DG方法推广到双相和黏弹性等复杂介质的地震波场模拟,发展了求解Biot弹性波方程和D'Alembert介质波动方程的DG方法.首先通过引入辅助变量将Biot双相介质弹性波方程和D'Alembert介质波动方程转化为关于时间-空间的一阶偏微分方程组,然后对该方程组进行DG空间离散,得到半离散化的常微分方程组.最后,对此常微分方程组,应用加权的Runge-Kutta格式进行时间推进计算.数值结果表明,DG方法可以有效地求解Biot双相介质弹性波方程和D'Alembert介质波动方程,并能很好地压制因离散求解波动方程而产生的数值频散,获得清晰的各种地震波震相.     

Boundary integral solutions of geoelectric potential problems for the perturbing body of extreme conductivity

Summary Theoretical formulae for boundary integral solutions of the geoelectric potential fields for a perfectly conducting perturbing 3D body, as well as for a perfectly insulating body are presented. It is shown that the boundary integral equation for the double-layer density, can be solved for all physically possible conductivities of the perturbing body.Dedicated to the Memory of K. P     

Expressions for the Global Gravimetric Moho Modeling in Spectral Domain

We apply a newly developed numerical method to improve the Moho geometry by the implementation of gravity data. This method utilizes expressions for the gravimetric forward and inverse modeling derived in a frequency domain. Methods for a spectral analysis and synthesis of the gravity field and crust density structures are applied in the gravimetric forward modeling of the consolidated crust-stripped gravity disturbances, which have a maximum correlation with the (a priori) Moho model. These gravity disturbances are obtained from the Earth’s gravity disturbances after applying the topographic and stripping gravity corrections of major known anomalous crust density structures; in the absence of a global mantle model, mantle density heterogeneities are disregarded. The isostatic scheme applied is based on a complete compensation of the crust relative to the upper mantle density. The functional relation is established between the (unknown) Moho depths and the complete crust-stripped isostatic gravity disturbances, which according to the adopted isostatic scheme have (theoretically) a minimum correlation with the Moho geometry. The system of observation equations, which describes the relation between spherical functions of the isostatic gravity field and the Moho geometry, is defined by means of a linearized Fredholm integral equation of the first kind. The Moho depths are determined based on solving the gravimetric inverse problem. The regularization is applied to stabilize the ill-posed solution. This numerical procedure is utilized to determine the Moho depths globally. The gravimetric result is presented and compared with the seismic Moho model. Our gravimetric result has a relatively good agreement with the CRUST2.0 Moho model by means of the RMS of differences (of 3.5 km). However, the gravimetric solution has a systematic bias. We explain this bias between the gravimetric and seismic Moho models by the unmodelled mantle heterogeneities and uncertainties in the CRUST2.0 global crustal model.     

Propagator matrices in elastic wave and vibration problems

The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order and ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients. In this paper we summarize some of the better known properties of propagators plus numerica methods for their computation. When the dispersion relation is somem ^th order minor of the integral matrix it is possible to deal withm ^th minor propagators so that the dispersion relation is a single element of them ^th minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided. Propagator equations forSH and forP-SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator equations forP-SV waves are given. Matrix polynomial approximations to the propagators, obtained from the method of mean coefficients by the Cayley-Hamilton theorem and the Lagrange-Sylvester, interpolation formula, are derived.     

Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media

The behavior of the mean equivalent hydraulic conductivity normal and parallel to stratification (K₁, and K₂, respectively) is studied here through Monte Carlo simulations of three-dimensional, steady-state flow in statistically anisotropic, bounded, and heterogeneous media. For water flow normal to stratification in strongly heterogeneous porous media (²_Y=3) the value of K₁ is not unique; it ranges from an arithmetic to a geometric, and finally, to a harmonic mean behavior depending on field dimensions, and medium anisotropy. For a fixed anisotropy ratio and variance of Y = ln K, the larger the distance, in the direction perpendicular to stratification, over which water flow takes place, the faster the rate at which, K_H, behavior is approached. However, even for large anisotropy ratios, harmonic mean behavior appears to be a good approximation only for aquifer thickness L₁ that is large enough to allow stratified flow to occur. For small aquifer thickness (L₁/₁<8, where ₁ is the integral scale normal to stratification) the limiting behavior, for large anisotropy ratios, appears to be, instead, that of two-dimensional flow, i.e., water flows primarily parallel to the planes of stratification. When the aquifer thickness is very small compared to the horizontal dimensions (and with relative similar integral scales in the three directions) a behavior resembling arithmetic mean conditions is exhibited, i.e., water flow takes place through heterogeneous, vertical, soil volumes. The geostatistical expressions of Desbarats (1992a) for upscaling hydraulic conductivity values were utilized and closed form empirical relations were developed for the main components of the upscaled hydraulic conductivity tensor.