General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow

Equations which describe single phase fluid flow and transport through an elastic porous media are obtained by applying constitutive theory to a set of general multiphase mass, momentum, energy, and entropy equations. Linearization of these equations yields a set of equations solvable upon specification of the material coefficients which arise. Further restriction of the flow to small velocities proves that Darcy's law is a special case of the general momentum balance.     

Upwelling flow in Earth's mantle and sea-floor spreading

Upwelling flows in the Earth's mantle are accompanied by mass, momentum and energy transports from deep to upper layers. Those flows beneath the mid-ocean ridges give rise to sea-floor spreading. Mantle plumes, on the other hand, cause hot spots to be formed on the Earth's surface. Using the basic equations of fluid dynamics, temperature and velocity distributions in two-dimensional upwelling and cylindrical plumes can be obtained by an integral-relation method. Then the mass, momentum and energy transported to the lithosphere by these upwelling flows can readily be calculated. Based on those results we can more thoroughly discuss problems of plate dynamics, such as the driving mechanism of plate motion, the causes of formation of rift valleys over mid-ocean ridges, and the effect of mantle plumes on sea-floor spreading.     

Dynamic equations for steady spatially varied open channel flow: a critical review

A review of the dynamic equations governing steady spatially varied flow in open channels is presented. These equations are derived by employing either the momentum or the energy principle: the choice of the method employed is based on convenience. Nevertheless, the two approaches yield different results when applied to a particular flow situation. Recent researches have established that this anamoly is due to the omission of the influence of the lateral flow. The inconsistencies existing among the different forms of these equations and the rather incomplete nature of their derivation are discussed. It is believed that with the present state of knowledge, it is possible to obtain identical spatially varied flow profiles when the influencing parameters are properly evaluated whether one uses the momentum or the energy approach. The need for further study to provide a better understanding of this practically important phenomenen is established and potential research directions are defined. Recent contributions to the analysis of the hydraulic characteristics of the spatially varied flow phenomenon and the delineation of the spatially varied flow profiles into eight possible patterns are also presented.     

Analytical solutions of kinematic equations for erosion on a plane I. Rainfall of indefinite duration

Kinematic equations have been utilized in the past for modeling erosion from a sloping plane subject to rainfall. This paper, part I in a series of two, presents analytical solutions of these equations under the assumption of constant effective rainfall of indefinite duration. These solutions contain a function which is related to a generalization of Dawson's integral. Some pertinent properties of this function are briefly discussed. The solutions obtained by previous investigators constitute a part of the solutions developed here. In part II the solutions are derived for the case when the duration of the effective rainfall is finite.     

Wake effects on turbulent transport in the convective boundary layer

The effect of the downstream propagation of a wake on the transport of momentum, energy and scalars (such as humidity) in the convective boundary layer (CBL) is studied using a direct numerical simulation. The incompressible Navier–Stokes and energy equations are integrated under neutral and unstable thermal stratification conditions in a rotating coordinate frame with the Ekman layer approximation. Wake effects are introduced by modifying the mean velocity field as an initial condition on a converged turbulent Ekman layer flow. With this initial velocity distribution, the governing equations are integrated in time to determine how turbulent transport in the CBL is affected by the wake. Through the use of Taylor’s hypothesis, temporal evolution of the flow field in a doubly periodic computational domain is transformed into a spatial evolution. The results clearly indicate an increase in the scalar flux at the surface for the neutrally stratified case. An increase in wall scalar and heat flux is also noted for the CBL under unstable stratification, though the effects are diminished given the enhanced buoyant mixing associated with the hot wall.     

Earth's magnetic field modelling and earth's structure and evolution

The advantages of the approximation of the Earth's magnetic field by means of the field of the so-called natural magnetic sources are discussed. The shifting of these natural magnetic sources, determined for different epochs, is used to forecast the Earth's magnetic field and to draw conclusions about the motion of the corresponding part of the Earth. On the basis of the representation of the Earth's magnetic field from several past geological epochs as a field of one optimum dipole a new theory about the Earth's evolution is proposed.     

Compressibility,core dynamics and the subseismic wave equation

It is shown that in the dynamics of a deep fluid of planetary scale such as the Earth's core, compressibility, stratification and self-gravitation are all important as well as rotation. The existing proof of Cowling's theorem prohibiting non-stationary axisymmetric dynamos, and the application of the Proudman-Taylor theorem to core flows, both based on the assumption of solenoidal flow, need to be reconsidered. For sufficiently small (subacoustic) frequencies or reciprocal time scales, an approximation which neglects the effect of flow pressure on the density is valid. We call this the “subseismic approximation” and show that it leads to a new second-order partial differential equation in a single scalar variable describing the low frequency dynamical behaviour. The new “subseismic wave equation” allows a direct connection to be made between the various possible physical regimes of core structure and its dynamics.     

Applicability of some infiltration formulae to rangeland infiltrometer data

Three algebraic infiltration equations (Kostiakov's, Horton's and Philip's) were examined to determine which one would best fit infiltrometer data collected from a variety of mostly semi-arid rangeland plant communities from both Australia and the United States. Approximately 1,100 infiltrometer plots were included in the analysis. Results indicated that, in every instance, Horton's equation best fit the infiltrometer data. Variability of “point” measures of short-term infiltration rates were never satisfactorily accounted for by using either Kostiakov's or Philip's equation. Though Horton's equation provided a best fit to the overall infiltration data, R² values indicated a potential usefulness of this equation only under the certain conditions that were sampled in several rangeland plant communities in the Northern Territory, Australia. The equation could not be considered consistently useful under conditions sampled on rangelands in the United States.     

The method of calculation of the geophysical fields in piecewise homogeneous media

Recurrent algorithms for solving 3D direct problems on the distribution of geophysical fields in the piecewise-homogeneous media are built, based on methods of integral transforms, integral representations, and integral equations. The algorithms allow the recalculation of Green’s functions of one medium into Green’s function of another medium. The computational procedure is applicable to problems with equations of elliptic, parabolic, hyperbolic, and mixed types. The algorithms can be split and implemented on computing clusters and multiprocessor complexes. Inverse problems and algorithms of searching for boundaries of piecewise-homogeneous media as extremals of the Tikhonov regulating functional, approximated by the spline functions, are formulated.     

液核自由运动的变分方法

从流体静力平衡地球的弹性-引力运动动量方程出发,以角动 量方程控制弹性地幔、液态外核和弹性内核之间的相对转动,在球对称近似下考虑了地幔和 内核对液核边界扰动的形变响应,并以此作为液核边界运动的约束条件．根据地球简正模对 称性的一般特征,建立了自转、非黏性、椭球分层流体外核自由振荡运动的变分原理并给出 了相应的泛函．     

Dynamic programming and the principle of optimality: A systematic approach

The dynamic programming recursive procedure has provided an efficient method for solving a variety of sequential decision problems related to water resources systems. In many investigations Bellman's principle of optimality is used as a proof for the optimality of the dynamic programming solutions. In this paper the dynamic programming procedure is systematically studied so as to clarify the relationship between Bellman's principle of optimality and the optimality of the dynamic programming solutions.Our main result is that although the principle is valid, in order to use it as a proof for the optimality of the dynamic programming solution certain modeling requirements should be met.The mathematical model presented in this paper provides a convenient framework for the modeling and analysis of dynamic programming problems encountered by in water resources management studies.The results derived here resolve few of the fundamental questions raised in the literature regarding the validity of Bellman's principle of oplimality and the optimality of the dynamic programming solutions.     

Weak-contrast approximation of the elastic scattering matrix in anisotropic media

The plane-wave reflection and transmission coefficients at a plane interface between two anisotropic media constitute the elements of the elastic scattering matrix. For a 1-D anisotropic medium the eigenvector decomposition of the system matrix of the transformed elasto-dynamic equations is used to derive a general expression for the scattering matrix. Depending on the normalization of the eigenvectors, the expressions give scattering coefficients for amplitudes or for vertical energy flux.Computing the vertical slownesses and the corresponding polarizations, the eigenvector matrix and its inverse can be found. We give a simple formula for the inverse, regardless of the normalization of the eigenvectors. When the eigenvectors are normalized with respect to amplitudes of displacement (or velocity), the calculation of the scattering matrix for amplitudes is simplified.When the relative changes in all parameters are small, a weak-contrast approximation of the scattering matrix, based on the exactly determined polarization vectors in an average medium, is obtained. The same approximation is also derived directly from the transformed elasto-dynamic equations for a smooth vertically inhomogeneous medium, proving the consistency of the approximation.For monoclinic media, with the mirror symmetry plane parallel to the interface, the approximative scattering matrix is given in terms of analytic expressions for the non-normalized eigenvectors and vertical slownesses. For transversely isotropic media with a vertical axis of symmetry (VTI) and isotropic media, explicit solutions for the weak-contrast approximations of the scattering matrices have been obtained. The scattering matrix for amplitudes for isotropic media is well known. The scattering matrix for vertical energy flux may have applications in AVO analysis and inversion due to the reciprocity of the reflection coefficients for converted waves.Numerical examples for monoclinic and VTI media provide good agreement between the approximative and the exact reflection matrices. It is, however, expected that the approximations cannot be used when the symmetry properties of the two media are very different. This is because the approximation relies on a small relative contrast between the eigenvectors in the two media.Presented at the Workshop Meeting on Seismic Waves in Laterally Inhomogeneous Media, Castle of Trest, Czech Republic, May 22–27, 1995.     

Equations of state of close-packed phases of iron and their implications for the Earth's core

Theoretical equations of state for Fe at high pressures and temperatures are derived from the expression of the free energy written as a sum of the static energy, the harmonic, the anharmonic and the electronic contributions. All the calculations have been performed for the various crystal structures of Fe using different intermolecular potentials, and namely: Lennard-Jones, Morse and Rydberg functions. The available experimental data do not allow a definite choice between them thus leaving open the problem of the composition of the earth's core.     

Delayed yield. An exact quasi-three dimensional model for free-aquifers

Quasi-three-dimensional models have been quite successful in the numerical treatment of leaky aquifers. Similar models are not available for free aquifers, except for Boulton's theory which can be properly interpreted as such. In the present paper a quasi-three-dimensional model of free surface flows is developed for free aquifers and waves. The zero-order approximation of this model yields Boulton's theory of delayed yield, elucidating in this manner the nature of the latter theory. The model presented here possesses numerical and theoretical possibilities which will be explored more thoroughly in further work.     

Induction in laterally non-uniform conductors: Theory and numerical models

The study of electromagnetic induction in laterally non-uniform conductors is briefly reviewed. The two-dimensional perturbation problem is considered and the two polarization cases which arise from Maxwell's equations are discussed. Techniques for the solution of the equations for laterally non-uniform conductors are discussed with emphasis on numerical methods.     

Analytical approach for predicting local scour downstream of submerged sluice gate with an apron

A new analytical method was evaluated for predicting scour profile downstream of a submerged sluice gate with an apron. The differential equations between bed Shear stress and Scour profile Curvature(SSC model) were utilized to predict the scour profile in both temporal and equilibrium stages. A jet momentum flux was considered as an external source of erosion on a hypothetical particle ring as the boundary between the flow and sediment bed. The scour length and sediment resistance factor were t...     

CALCULATION OF GALVANIC EFFECTS BY MEANS OF THE METHOD OF SUB-AREAS*

A numerical method is proposed for solving the problem of steady current flow. The electrodynamic model is replaced by the equivalent stationary charge distribution obtained by Poisson's analysis, in which the surface integral equation for field intensity is reduced to a set of simultaneous linear algebraic equations by means of the method of sub-areas. The solution of the set allows the calculation of an approximation for the charge density distribution on the discontinuity surfaces of conductivity. The method is valid for complex conductivities, whereby the apparent phase shift of IP can be calculated from the complex potential or field intensity. The phase shift anomaly calculated as an application is very similar to the corresponding frequency effect anomaly. The method allows the calculation of the mise-à-la-masse effect as a solution to a potential problem, in which the primary current electrode is located within the body to be surveyed.     

风的垂直切变对滞弹性大气中静力适应过程的影响

本文基于描写滞弹性大气静力适应过程的线性方程组,从波动频散关系、气团运动规律和能量转换的角度出发,研究了水平基流及其垂直切变对该模式大气静力适应过程的影响.构造四种水平基流垂直分布模型进行比较,分别为常数型、线性切变型、反气旋切变型和气旋切变型,得到结论：（1）具有重力波性质的波动是滞弹性大气静力适应过程中扰动能量传播的方式,当垂直折射指数大于零时,基本气流及其垂直切变的存在,不仅改变了波动频率的大小,而且改变波动传播的方向;（2）在静力适应过程中气块的运动轨迹呈椭圆形,水平基流及其垂直切变影响椭圆的扁率,同时也影响扰动物理量之间的偏振关系;（3）水平基流的垂直切变是扰动能量和水平基流能量发生转换的媒介,当存在垂直向上的动量输送时,正的垂直风切变对应扰动能量的衰减,水平基流能量的增加,负的垂直风切变对应扰动能量的增加,水平基流能量的衰减;（4）不同的风的垂直切变模型,对静力适应过程的影响不同;对于水平基流呈反气旋切变型和气旋切变型,扰动发展的波动垂直结构为,急流轴上方波动等相位线自下而上向西倾斜,急流轴下方波动等相位线自下而上向东倾斜,反之亦然.     

The ellipticities of surfaces of equal density inside the earth

Values of the ellipticities ? of surfaces of constant density inside the earth and of Radau's parameter η have been re-calculated for several recent earth models. The effects on the calculated ? and η of changes in the assumed moment of inertia of the earth, of the assumed inner-core density, and of taking account of free earth oscillation data, are separately indicated. The results show that there is no need at present to re-calculate the ellipticity corrections to seismic travel times, though the need may arise later. For all recent earth models, η continues to be between zero and 0.6, so that Radau's approximation continues to be satisfactory.