A simple method for determining the vertical growth rate of vertical motion at the top of earth's outer core

A simple new method is described for extracting, from magnetic observations taken at Earth's surface, the vertical growth rate of vertical motion, ?u/?r, at special isolated points on the top surface of Earth's liquid core. The technique utilizes only the radial component of the frozen-flux induction equation and it requires information only on the radial magnetic field, B_r, its horizontal gradient, and its secular variations, ?B_r/?t, at the core-mantle boundary.     

How valid are dynamic models of subduction and convection when plate motions are prescribed?

A detailed comparison between fully dynamic and kinematic plate formulations has been made in models of mantle convection. Plate velocity is computed self-consistently from fully dynamic plate models with temperature- and stress-dependent viscosity and preexisting mobile faults. In fully dynamic models, the flow is driven solely by internal buoyancy, while in kinematic models the flow is driven by a combination of the prescribed surface velocity and internal buoyancy. Only a temperature-dependent viscosity, close to the effective viscosity determined from the fully dynamic models, is used in the kinematic models. The two types of models give very similar temperature structures and slab evolutionary histories when the effective viscosity and surface velocity are nearly identical. In kinematic plate models, the additional work introduced by the prescribed velocity boundary condition is apparently dissipated within the lithosphere and has little influence on the convection under the lithosphere. In models with periodic lateral boundary conditions, slabs sink into the lower mantle at an oblique angle and this contrasts with the vertical sinking which occurs with reflecting boundary conditions. Models show that we can simulate fully dynamic models with kinematic models under either periodic boundary conditions or reflecting boundary conditions.     

A three-dimensional,iterative mapping procedure for the implementation of an ionosphere-magnetosphere anisotropic Ohm’s law boundary condition in global magnetohydrodynamic simulations

The mathematical formulation of an iterative procedure for the numerical implementation of an ionosphere-magnetosphere (IM) anisotropic Ohm’s law boundary condition is presented. The procedure may be used in global magnetohydrodynamic (MHD) simulations of the magnetosphere. The basic form of the boundary condition is well known, but a well-defined, simple, explicit method for implementing it in an MHD code has not been presented previously. The boundary condition relates the ionospheric electric field to the magnetic field-aligned current density driven through the ionosphere by the magnetospheric convection electric field, which is orthogonal to the magnetic field B, and maps down into the ionosphere along equipotential magnetic field lines. The source of this electric field is the flow of the solar wind orthogonal to B. The electric field and current density in the ionosphere are connected through an anisotropic conductivity tensor which involves the Hall, Pedersen, and parallel conductivities. Only the height-integrated Hall and Pedersen conductivities (conductances) appear in the final form of the boundary condition, and are assumed to be known functions of position on the spherical surface R=R₁ representing the boundary between the ionosphere and magnetosphere. The implementation presented consists of an iterative mapping of the electrostatic potential , the gradient of which gives the electric field, and the field-aligned current density between the IM boundary at R=R₁ and the inner boundary of an MHD code which is taken to be at R₂>R₁. Given the field-aligned current density on R=R₂, as computed by the MHD simulation, it is mapped down to R=R₁ where it is used to compute by solving the equation that is the IM Ohm’s law boundary condition. Then is mapped out to R=R₂, where it is used to update the electric field and the component of velocity perpendicular to B. The updated electric field and perpendicular velocity serve as new boundary conditions for the MHD simulation which is then used to compute a new field-aligned current density. This process is iterated at each time step. The required Hall and Pedersen conductances may be determined by any method of choice, and may be specified anew at each time step. In this sense the coupling between the ionosphere and magnetosphere may be taken into account in a self-consistent manner.     

Beach water table fluctuations due to spring–neap tides: moving boundary effects

Tidal water table fluctuations in a coastal aquifer are driven by tides on a moving boundary that varies with the beach slope. One-dimensional models based on the Boussinesq equation are often used to analyse tidal signals in coastal aquifers. The moving boundary condition hinders analytical solutions to even the linearised Boussinesq equation. This paper presents a new perturbation approach to the problem that maintains the simplicity of the linearised one-dimensional Boussinesq model. Our method involves transforming the Boussinesq equation to an ADE (advection–diffusion equation) with an oscillating velocity. The perturbation method is applied to the propagation of spring–neap tides (a bichromatic tidal system with the fundamental frequencies ω₁andω₂) in the aquifer. The results demonstrate analytically, for the first time, that the moving boundary induces interactions between the two primary tidal oscillations, generating a slowly damped water table fluctuation of frequency ω₁−ω₂, i.e., the spring–neap tidal water table fluctuation. The analytical predictions are found to be consistent with recently published field observations.     

Boundary layer models of ocean floor spreading

Summary The equations of conservations of momentum and energy scaled with the characteristic values of the mantle indicate the presence of the upper boundary layer to produce the estimated rate of the ocean floor spreading by convection and the importance of the frictional heating. The depth of the upper boundary layer can be estimated from the balance of the viscous force with the horizontal pressure gradient at the sea floor. It is of the orders of 100 km and becomes deeper for the Pacific than for the Atlantic Ocean and also with frictional heating than without it. The frictional heating increases the surface heat flow of the heat conduction by ten to twenty percent for the Pacific Ocean but only by a few percent for the Atlantic Ocean. The similarity solutions are determined for the temperature and horizontal velocity in the upper boundary layer. These solutions are expressed in power series of the variabley x ^–n , wherex, y, andn are horizontal and vertical coordinates and numerical constant, respectively. Both temperature and horizontal velocity within the boundary layer are higher for the Pacific than for the Atlantic Ocean. When a larger viscosity is applied, it causes the increase of horizontal velocity below the surface because of the surface boundary conditions of the finite velocity and of vanishment of the velocity shear. The higher horizontal velocity generates higher temperature because it advects hotter material from the mid-ocean ridge site. The direct effect of frictional heating on the temperature distribution of the similarity solution is almost negligible, since the shear zone is deep and near the lower boundary of the upper boundary layer. In the similarity solution, the surface heat flow which is increased by the frictional heating is given as the boundary value. The effect of the frictional heating is important below the mid-ocean ridge.     

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.     

Shape factor in the drawdown solution for well testing of dual-porosity systems

One of the important parameters in existing commercial dual-porosity reservoir simulators is matrix–fracture shape factor, which is customarily obtained by assuming a constant pressure at the matrix–fracture boundary. In his work, Chang [1] and [2] addressed the impact of boundary conditions at the matrix–fracture interface and presented analytical solutions for the transient shape factor and showed that for a slab-shaped matrix block a constant pressure boundary condition leads to an asymptotic (long-time) shape factor of π²/L², and that a constant volumetric flux leads to an asymptotic shape factor of 12/L². In a recent paper [3], we reconfirmed Chang’s [1] and [2] results using a Laplace transform approach. In this study, we extend our previous analysis and use infinite-acting radial and linear dual-porosity models, where the boundary condition is chosen at the wellbore, as opposed to at the matrix boundary. The coupled equations for fracture and matrix are solved analytically, taking into account the transient exchange between matrix and fracture. The analytical solution that invokes the time dependency of fracture boundary condition under constant rate is then used to calculate the transient shape factors. It is shown that, for a well producing at constant rate from a naturally fractured reservoir, the appropriate value of stabilized shape factor is 12/L². This contrasts with the commonly used shape factor for a slab-shaped matrix block that is subject to a constant pressure boundary condition, which is π²/L². The errors in the matrix–fracture exchange term in a dual-porosity model associated with the use of a shape factor derived based on constant pressure boundary condition at the matrix boundary are then evaluated.     

Conditions governing the use of approximations for the Saint-Vénant equations for shallow surface water flow

The solutions of the Saint-Vénant equations are compared with those of the kinematic, diffusion and gravity wave approximations, for a range of constant Froudé and kinematic wave numbers, with two different lower boundary conditions: (1) critical flow; and (2) zero depth gradient. For each lower boundary condition, zones are defined in the F₀,k-field in which either kinematic, diffusion or gravity wave solutions may be used to approximate the full Saint-Vénant solutions.     

Scaling relationships for the near-fieldP-SV ground motion

A set of monoial scaling relations to parametrize several measures of strong motion (peak velocity, peak starting-phase acceleration, peak stopping acceleration) is proposed. Dynamic solutions are obtained for a 2-D (P-SV) stress-drop model of faulting, and ground motion from these calculations is used to calibrate the scaling relations. Geometrical spreading, radiation patterns, low frequency near-field radiation, and free surface response are analysed, and introduced as corrections. The calculational finite difference method is sound within its frequency range of validity, which is found to be about 0 to 5–6 Hz for the chosen grid steps. A strong difference is obtained between theP andSV motion scalings, mainly with the source rupture velocity. Also noted are significant differences between the starting and stopping accelerations due to different frequency content and the influence of low frequency near-field radiation. To test the estimated scaling relationships, some synthetic predictions of the kinematic parameters are made, with quite good agreement when compared with dynamic computations (errors within 30%). The results emphasize some features of the near-fieldP-SV radiation and allow a kinematic prediction for a simple and smooth source model, but show the limited reliability of such predictions, arising from the problem's complexity.     

Accuracy of kinematic wave and diffusion wave approximations for space-independent flows with lateral inflow neglected in the momentum equation

Error equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the momentum equation are derived under simplified conditions for space-independent flows. These equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow and channel roughness when the initial condition is non-vanishing; and it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error equations are found to be the Riccati equation. The structure of the error equations in the case when the momentum equation neglects lateral inflow is different from that when the lateral inflow is included.     

Analysis of hysteretic behaviour of a hillslope-storage kinematic wave model for subsurface flow

The objective of this work is to analyse the storage–flux hysteretic behaviour of a simplified model for subsurface flow processes. The subsurface flow dynamics is analysed by means of a model based on the kinematic wave assumptions and by using a width weighting/depth averaging scheme which allows to map the three-dimensional soil mantle into a one-dimensional profile. Continuity and a kinematic form of Darcy’s law lead to a hillslope-storage kinematic wave equation for subsurface flow, solvable with the method of characteristics. Adopting a second order polynomial function to describe the bedrock slope and an exponential function to describe the variation of the width of the hillslope with hillslope distance, we derive general solutions to the hillslope-storage kinematic wave equations, applicable to a wide range of hillslopes. These solutions provide a physical basis for deriving two geometric parameters α and ψ which define the hydrological similarity between hillslopes with respect to their characteristic response and hysteresis. The hysteresis η, quantified by the area of the hysteretic dimensionless loop, has been therefore computed for a range of values of parameters α and ψ. Slopes exhibit generally clockwise hysteretic loop in the flux-storage plot, with higher groundwater mean volume for given discharge on rising limb than at same discharge on falling limb. It has been found that hysteresis increases with decreasing α and ψ, i.e. with increasing convergence (for the shape) and concavity (for the profile), and vice versa. For relatively large values of α and ψ the hysteresis may take a complex pattern, with combination of clockwise to anticlockwise loop cycles. Application of the theory to three hillslopes in the Eastern Italian Alps provides an opportunity to examine how natural topographies are represented by the two hillslope hydrological similarity parameters.     

Accuracy of kinematic wave and diffusion wave approximations for space-independent flows on infiltrating surfaces with lateral inflow neglected in the momentum equation

Error equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the momentum equation are derived under simplified conditions for space-independent flows. These equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parameterized through a dimensionless parameter γ which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow, infiltration and channel roughness when the initial condition is non-vanishing; it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error equations are found to be the Riccati equation. The structure of the error equations in the case when the momentum equation neglects lateral inflow is different from that when the lateral inflow is included.     

The Green’s function of the harmonic horizontal force applied at the interior of the saturated half-space soil

By using integral transform methods, the Green’s functions of horizontal harmonic force applied at the interior of the saturated half-space soil are obtained in the paper. The general solutions of the Biot dynamic equations in frequency domain are established through the use of Hankel integral transforms technique. Utilizing the above-mentioned general solutions, and the boundary conditions of the surface of the half-space and the continuous conditions at the plane of the horizontal force, the solutions of the boundary value problem can be determined. By the numerical inverse Hankel transforms method, the Green’s functions of the harmonic horizontal force are obtainable. The degenerate case of the results deduced from this paper agrees well with the known results. Two numerical examples are given in the paper. Foundation item: State Natural Science Foundation (59879012) and Doctoral Foundation from State Education Commission (98024832).     

On the frozen flux velocity field at the surface of earth's core necessary to account for the poloidal main magnetic field and its secular variation

An expression for the inviscid horizontal velocity field at the surface of the Earth's core necessary to account for the poloidal main magnetic field and its secular variation seen at the Earth's surface is derived for an insulating mantle in the limit of infinite core conductivity. The starting point of derivation is Ohm's law rather than the magnetohydrodynamic induction equation. Maps of the resulting motion for epoch 1965.0 at different truncation levels are presented and discussed.     

Diffusive wave solutions for open channel flows with uniform and concentrated lateral inflow

The diffusive wave equation with inhomogeneous terms representing hydraulics with uniform or concentrated lateral inflow into a river is theoretically investigated in the current paper. All the solutions have been systematically expressed in a unified form in terms of response function or so called K-function. The integration of K-function obtained by using Laplace transform becomes S-function, which is examined in detail to improve the understanding of flood routing characters. The backwater effects usually resulting in the discharge reductions and water surface elevations upstream due to both the downstream boundary and lateral inflow are analyzed. With a pulse discharge in upstream boundary inflow, downstream boundary outflow and lateral inflow respectively, hydrographs of a channel are routed by using the S-functions. Moreover, the comparisons of hydrographs in infinite, semi-infinite and finite channels are pursued to exhibit the different backwater effects due to a concentrated lateral inflow for various channel types.     

Comparative analysis of dynamic coefficient β(T, n) curves obtained by different methods

The problem of curve construction for dynamic coefficient β(T, n) based on the integral parameter of ground motion (cumulative absolute velocity (CAV)) is described. Comparative analysis of the curves β(T, n), obtained by maximum values of ground motion acceleration and via cumulative absolute velocity application, is carried out. The characteristic properties of changes in the considered dynamic coefficient curves are revealed and investigated.     

A worldwide comparison of predicted S-wave delays from a three-dimensional upper mantle model with P-wave station corrections

This paper is a comparison, on a worldwide scale, between P-station corrections deduced from ISC residuals (Dziewonski and Anderson) and synthetic S-station corrections computed using a three-dimensional upper mantle model obtained from mantle wave data (Woodhouse and Dziewonski). The upper mantle S-velocity model is described by a spherical harmonic expansion up to degree 8; the P-station corrections are smoothed using a similar expansion, in order that the two data sets can be compared.Correlations between P-station corrections, δt_P, and synthetic S-station corrections, δt_S relevant to various depths of integration indicate that a station correction contains information about structures down to at least 670 km. For this depth of integration, the correlation coefficient of the two data sets is 0.59; the slope ‘a’ of the relation δt_S = aδt_P + b, obtained for the worldwide distribution of stations, is in good agreement with results of previous regional studies using direct readings of P and S arrival times (a = 3.61 ± 0.13).An analysis of regional variations of the relation δt_S = aδt_P + b is carried out on the basis of two published global tectonic patterns (Okal; Jodan). Results for oceanic regions are not reliable, due to the lack of data. On continental areas, a significant difference appears between mountains (a = 2.7 ± 0.3) and shields (a = 4.5 ± 0.4 for Okal's pattern, a = 5.7 ± 1.5 for Jordan's pattern). The largest a-value for shields rules out an explanation by partial melting, as proposed in previous studies. Thermal heterogeneities lead to low a-values; undulations of the lithosphere-asthenosphere boundary appear to be the most feasible explanation of the high slope beneath shields; they are also able to explain the range of variation of the station corrections; the lowest values of the station corrections correspond to a total vanishing of the low velocity zone beneath the oldest shields. For mountains, the mean values of the station corrections as well as the low a-value can be accounted for by a slight increase of Poisson's ratio together with a significant density increase.     

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.     

THEORETICAL APPROACH TO THE SOLUTION OF THE INFILTRATION PROBLEM

ABSTRACT

The one-dimensional transient downward entry of water in unsaturated soils is investigated theoretically. The mathematical equation describing the infiltration process is derived by combining Darcy's dynamic equation of motion with the continuity and thermodynamic state equations adjusted for the unsaturated flow conditions. The resulting equation together with the corresponding initial and boundary conditions constitues a mathematical initial boundary value problem requiring the solution of a nonlinear partial differential equation of the parabolic type. The volumetric water content is taken as the dependent variable and the time and the position along the vertical direction are taken as the independent variables. The governing equation is of such nature that a solution exists for t > 0 and is uniquely determined if two relationships are defined, together with the specified state of the system, at the initial time t = 0 and at the two boundaries. The two required relations are those of pressure versus permeability and pressure versus volumetric water content.

Since the partial differential equation has strong non-linear terms, a discrete solution is obtained by approximating the derivatives with finite-differences at discrete mesh points in the solution domain and integrated for the corresponding initial and boundary conditions. The use of an implicit difference scheme is employed in order to generate a system of simultaneous non-linear equations that has to be solved for each time increment. For n mesh points the two boundary conditions provide two equations and the repetition of the recurrence formula provides n—2 equations, the total being n equations for each time increment. The solution of the system is obtained by matrix inversion and particularly with a back-substitution technique. The FORTRAN statements used for obtaining the solution with an electronic digital computer (IBM 704) are presented together with the input data.

Analysis of the errors involved in the numerical solution is made and the stability and convergence of the solution of the approximate difference equation to that of the differential equation is investigated. The method applied is that of making a Fourier series expansion of a whole line of errors and then following the progress of the general term of the series expansion and also the behavior of each constituent harmonic. The errors (forming a continuous function of points in an abstract Banach space) are represented by vectors with the Fourier coefficients constituting a second Banach space. The amplification factor of the difference equation is shown to be always less than unity which guarantees the stability of the employed implicit recurrence scheme.

Experiments conducted on a vertical column packed uniformly with very fine sand, show a satisfactory agreement between the theoretically and experimentally obtained values. Many experimental results are shown in an attempt to explain the infiltration phenomenon with emphasis on the shape and movement of the wet front, and the effects of the degree of compaction, initial water content and deaired water on the infiltration rate.