Contamination of a well in a uniform background flow

In this paper we describe the transport of pollution in groundwater in the neighbourhood of a well in a uniform background flow. We compute the rate at which contaminated particles reach the well as a function of the place of the source of pollution. The motion of a particle in a dispersive flow is seen as a random walk process. The Fokker-Planck equation for the random motion of a particle is transformed using the complex potential for the advective flow field. The resulting equation is solved asymptotically after a stretching transformation. Finally, the analytical solution is compared with results from Monte Carlo simulations with the random walk model. The method can be extended to arbitrary flow fields. Then by a numerical coordinate transformation the analytical results can still be employed.     

Motion of wetting fronts moving into partially pre-wet soil

We study the motion of wetting fronts for vertical infiltration problems as modeled by Richards’ equation. Parlange and others have shown that wetting fronts in infiltration flows can be described by traveling wave solutions. If the soil layer is not initially dry, but has an initial distribution of water content then the motion of the wetting front will change due to the interaction of the infiltrating flow with the pre-existing soil conditions. Using traveling wave profiles, we construct simple approximate solutions of initial-boundary value problems for Richards’ equation that accurately describe the position and moisture distribution of the wetting front. We show that the influences of surface boundary conditions and initial conditions produce shifts to the position of the wetting front. The shifts can be calculated by examining the cumulative infiltration, and are validated numerically for several problems for Richards’ equation and the linear advection–diffusion equation.     

A model of a homogeneous isotropic turbulent flow and its application for the simulation of cloud drop tracks

Abstract

A model of a homogeneous isotropic turbulent flow is presented. The model provides different realizations of the random velocity field component with given correlation latitudinal and lateral functions and a spatial structure which obeys the Kolmogorov theory of homogeneous and isotropic turbulence. For the generation of the turbulent flow the structural function of the flow in the form suggested by Batchelor (Monin and Yaglom, 1975) was used. This function describes the spectrum of turbulence both in the viscous and inertial ranges. The isotropy and homogeneity of the velocity field of the model are demonstrated.

The model is aimed at simulating the ‘‘fine'’ features of drop's (aerosol particles') motion, such as the deviations of drops’ velocity from the velocity of the flow, detailed structures of drops’ tracks, related to drops’ (particles') inertia. The model is intended also for the purpose of studying cloud drops’ and aerosol particles’ motion and their diffusional spreading utilizing the Monte Carlo methods.

Some examples of drop tracks for drops of different size are presented. Drops’ tracks are very sophisticated, so that the relative position of drops falling initially from the same point can vary drastically. In some cases drops’ tracks diverge very quickly, in other cases all drops move within a turbulent eddy along nearly the same closed tracks, but with different speed. The concentration of drop tracks along isolated paths is found in spite of the existence of a large number of velocity harmonics. It is shown that drops (aerosol particles) tend to leave some areas of the turbulent flow apparently due to their inertia. These effects can possibly contribute to inhomogeneity of drops’ concentration in clouds at different spatial scales.     

On the stationary charge distribution on aerosol particles in a bipolar ionic atmosphere

Summary By the «limiting sphere» method the combination coefficients for gaseous ions and aerosol particles were calculated, allowing for the jump in ion concentration at the surface of the particles. Hence the stationary charge distribution on aerosol particles in a symmetrical bipolar ionic atmosphere was determined. The use of the Boltzmann equation for this purpose proposed by some authors is theoretically wrong asthis equation applies to equilibrium rather than to stationary states. In practice, the Boltzmann equation can be used for particles with radius 3·10^–5 cm (under atmospheric pressure). Within this range the image forces and the jump in ion concentration may be neglected. The conditions of the applicability of the steady diffusion equations to the theory of the stationary charge distribution in aerosols are discussed.     

The instability of barotropic circular vortices

Abstract

The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this linear instability analysis is not valid very far into the finite amplitude range, and that a barotropic vortex, whose fastest growing perturbation is internal, is vertically fragmented by the instability.     

Determination of the size distribution of polydisperse submicron aerosols by a response matrix method

Summary An analytical method for determining the size distribution of polydisperse submicron aerosols are described. It is shown that we can determine approximately the size distribution function analyzingZ(Q) (dependence of the aerosol concentration at the exit of a diffusion battery upon the flow rate) by a iterative technique using response matrixes. Numerical experiments were carried out to evaluate the practical application and limitations of this technique, and the technique was found to be useful itZ(Q) may be determined within the errors of about ±5 percents. Practical measurements of atmospheric submicron aerosols were made on the university campus and some examples of the size distributions of aerosols determined by this method were shown.     

The dependence of response spectrum on the tectonic ambient shear stress field

Introduction The acceleration response spectrum and peak ground acceleration are the necessary and im-portant parameters in earthquake-resistant design at present. They are still active research field. With the increase of digital high accurate strong motion observation data, especially the earth-quakes of Loma Prieta (M=7.0) in 1989; Landers (M=7.3) in 1992; Big Bear (M=6.4) in 1994 and Northridge (M=6.7) in 1994 in USA; Kozani (M=6.6) earthquake and afteshocks in 1995 in Greece; Dinar…     

STOCHASTIC THEORY OF THE DISTRIBUTION OF SEDIMENT PARTICLES SUSPENDED IN A TURBULENT FLOW

The random motion of sediment particles suspended in a turbulent flow is studied by means of stochastic process. Results of analysis of particle's frequency response to the random force exerted on the particle due to fluid turbulence suggest that only the lower part of the whole frequency range of the eddy motion will govern the motion of the particle. The mean values of particle velocity and displacement in the vertical direc- tion are calculated and it is found that particle velocity vp- can be decomposed into a mean motion and a velocity fluctuation vp- , where is equal to the fall velocity in tranquil fluid. An Ito^ random differential equation for particle dis- placement Yp is developed, from which a Fokker-Planck equation for the probability density function p(y,t) is derived on the basis of the theory of Markov process, where y denotes the vertical coordinate. The vertical distribution of the particle is thus interrelated to the random motion of the particle. The an effect that a particle will be subject to in the neighborhood or the bottom boundary is taken into consideration and a corresponding Fokker-Planck equation is developed. Analytical solution of the Fok- ker-Planck equation including the lift force effect shows that probability density p(y,t) for the particle displacement has a maximum value at y = H where the perpen- dicular component of the lift force balances the particle gravity. This theoretical result agrees with experimental observations as reported in literature.     

A linear perturbation analysis of magnetopause motion in the Newton-Busemann limit

The response of the magnetopause surface to time-varying solar wind dynamic pressure is examined. We argue that to a first approximation the magnetopause surface may be considered as analogous to an elastic membrane. Upon displacement from equilibrium resulting from a change in applied external pressure, it moves to a new equilibrium under the equation of motion of a forced, damped, simple harmonic oscillator. We derive this equation of motion by linearising for small perturbations the momentum equation for flow past a nonrigid ellipsoidal body in the Newton-Busemann limit. Though our approach is only an approximation to the real dynamics of the magnetopause boundary, it serves to demonstrate the importance of inertia in the system response. It allows us to estimate the natural eigenperiod of magnetopause oscillation as typically around 7 min, the precise value depending on solar wind conditions. However, the magnetopause eigenoscillation is furthermore found to be strongly damped, regardless of solar wind conditions. One consequence of these properties is that short-period fluctuations in the solar wind dynamic pressure elicit a suppressed magnetospheric response. We outline other theoretical expectations by which our model may be tested against observation, and discuss the implications of our findings for current interpretations of spacecraft observations made in the dynamic magnetopause environment.     

Existence of the solutions and the attractors for the large-scale atmospheric equations

The atmosphere is a kind of fluid surrounding therotating earth, and its state can be described by thevelocity vector v, the temperature T, the density ρ, andthe pressure p at each point. Its evolution process isessentially governed by the Navier-Stockes equationand the temperature equation. Due to the specialty andthe complexity of the atmospheric problem, althoughsome modified and simplified work have made onthese fluid mechanics and temperature equations, theyare still a very complex forc…     

An externally forced barotropic circulation model for the New York Bight

The sea level and the barotropic, frictional circulation response for the New York Bight are used to demonstrate the effects of external sea-level forcing, bathymetry, and variable friction. The governing equation is the steady, integrated vorticity equation and is computed by finite differencing over a curvilinear grid conforming to the 10- and 100-m isobaths and extending for 250 km alongshore. The boundary conditions are based on the hypothesis that the dynamics of the shelf are driven by the external sea-level gradient and the coastal no-flux condition; and consequently the conditions at the lateral boundaries are dependent thereon. Therefore, the external sea-level slope must be independently specified, and the lateral boundary conditions must be dependently generated. The diabathic component of the external sea slope forces the calm wind circulation by its effect on the transport through the upstream boundary; and the parabathic component has also an important modifying effect by forcing a shelf convergent transport. The parabathic sea slope at the coast is independent of its offshore value, being instead a direct product of the coastal boundary condition.The bottom friction is expressed as related to the sea level through a bottom length parameter and a veer angle, both of which are taken to increase shoreward. An additional bottom stress component, related to the surface stress, is determined for bottom depths less than the Ekman depth. Such bottom stress variability produces significant alterations in the nearshore flow field, over the constant bottom stress formulation, by reducing it and causing it to veer downgradient and downwind in the nearshore.The model is forced by different wind directions and the results are discussed. The circulations generally conform to the observed mean flow patterns, but with several smaller-scale features. The strong bathymetric feature of the Hudson Shelf Valley causes a polarized up- and downvalley flow for winds with an eastward or westward component, respectively. Under mean westerly winds, there is a divergence in the shelf valley flow at about the 60-m isobath. The Apex gyre existing off the western tip of Long Island becomes more extensive for winds changing from northeast to southwest. Mean flow reversals (to the northeast) occur off both Long Island and New Jersey for wind directions changing counterclockwise from northwest to southeast and from west to east, respectively. Southeastward transport over the outer New Jersey shelf tends to be enhanced by wind and external sea-level conditions; and the transport over the New Jersey midshelf, particularly in the lee of the shelf valley, tends to be weak and variable also under these mean conditions.     

Development and testing of a physically based,three-dimensional model of surface and subsurface hydrology

We present a numerical, catchment-scale model that solves flow equations of surface and subsurface flow in a three-dimensional domain. Surface flow is described by the two-dimensional parabolic approximation of the St. Venant equation, using Manning’s equation of motion; subsurface flow is described by the three-dimensional Richards’ equation for the unsaturated zone and by three-dimensional Darcy’s law for the saturated zone, using an integrated finite difference formulation. The hydrological component is a dynamic link library implemented within a comprehensive model which simulates surface energy, radiation budget, snow melt, potential evapotranspiration, plant development and plant water uptake. We tested the model by comparing distributed and integrated three-dimensional simulated and observed perched water depth (PWD), stream flow data, and soil water contents for a small catchment. Additional tests were performed for the snow melting algorithm as well as the different hydrological processes involved. The model successfully described the water balance and its components as evidenced by good agreement between measured and modelled data.     

Modeling the initiation of sediment motion under a wide range of flow conditions using a Geno-Mamdani Fuzzy Inference System method

The current study introduces a novel approach to estimate the incipient motion of sediments under a wide range of flow regimes by developing a fuzzy model with a fuzzy-band that refers to a transition from weak motion to general motion of sediment. The partial sediment entrainment is defined by fuzzy sets considering the uncertainty related to the individual ratio of inertia to viscous forces which is the definition of shear Reynolds number. In the current study, the Mamdani Fuzzy Inference System (Mamdani FIS) is used to develop a comprehensive fuzzy model of the incipient motion of sediment. The Mamdani FIS has a shortcoming regarding the training of the fuzzy model. To estimate the dimensionless shear stress, a new method is developed by combining a genetic algorithm with the fuzzy approach which is named the Geno-Mamdani Fuzzy Inference System (GMFIS) method. The performance of the GMFIS model is evaluated using experimental data by considering root mean square error (RMSE), Nash-Sutcliffe coefficient of efficiency (CE), degree of robustness (D_r), and concordance coefficient (CC) as evaluation criteria. The GMFIS model performed very well based on the RMSE, CE, D_r, and CC values and satisfactorily represented the three types of incipient motion. Finally, a new range of fuzzy, dimensionless, critical shear stress values is established in all flow conditions from weak to general sediment entrainment.     

Scattering of elastic waves by a general anisotropic basin. Part 2: a 3D model

Scattering of plane harmonic waves by a three‐dimensional basin of arbitrary shape embedded within elastic half‐space is investigated by using an indirect boundary integral equation approach. The materials of the basin and the half‐space are assumed to be the most general anisotropic, homogeneous, linearly elastic solids without any material symmetry (i.e. triclinic). The unknown scattered waves are expressed in terms of three‐dimensional triclinic time harmonic full‐space Green's functions. The results have been tested by comparing the surface response of semi spherical isotropic and transversely isotropic basins for which the numerical solutions are available. Surface displacements are presented for a semicircular basin subjected to a vertical incident plane harmonic pseudo‐P‐, SV‐, or SH‐wave. These results are compared with the motion obtained for the corresponding equivalent isotropic models. The results show that presence of the basin may cause significant amplification of ground motion when compared to the free‐field displacements. The peak amplitude of the predominant component of surface motion is smaller for the anisotropic basin than for the corresponding isotropic one. Anisotropic response may be asymmetric even for symmetric geometry and incidence. Anisotropic surface displacement generally includes all three components of motion which may not be the case for the isotropic results. Furthermore, anisotropic response strongly depends upon the nature of the incident wave, degree of material anisotropy and the azimuthal orientation of the observation station. These results clearly demonstrate the importance of anisotropy in amplification of surface ground motion. Copyright © 2003 John Wiley & Sons, Ltd.     

On the propagation of pressure-patterns: Part I

Summary The propagation speed of sinoidal troughs and wedges in a steady state flow is determined from consideration of the mass transport due to the bodily motion of the system. Fundamental propositions are established regarding the mutual motion of wind-, pressure-, temperature-, and density-fields.It is found that in a frictionless barotropic general flow, all perturbations are propagated with the same speed—the speed of the general current. In a baroclinic general flow a perturbation will only be propagated without dispersion if it has a specific (sinoidal) horizontal structure.When a sinoidal perturbation is embedded in a baroclinic general flow-field, it will be propagated as though by a barotropic flow with the sameeffective speed. The effective speed can be computed when the vertical structure of the perturbation and of the mean flow are known.It is frequently assumed that the speed of mean flow at some particular level (500 mb is often assumed) gives the «steering» of the surface perturbation by a baroclinic general flow, that is to say, a baroclinic flow steers a perturbation with the speed of an equivalent barotropic field. The present paper provides a rational basis for the concept of an equivalent barotropic flow, but it is to be remembered that the «steering level» does not depend uniquely on the vertical structure of the mean flow-field, but varies from perturbation to perturbation, being lower for shallow perturbations than for (vertically) deep ones.     

On a simplified radiative-conductive heat transfer equation

Summary A simplified equation purported to represent the joint influence of radiative and turbulent transfers of heat in the atmosphere is derived by dividing the absorption spectrum of terrestrial radiation into strongly and weakly absorbed regions, classified according to the local scale of variation or to the local heating rate, and introducing two mean absorption coefficients for these two groups of regions. Assurning the validity of theK-theory of turbulent diffusion of heat, it is found that the temperature of the atmosphere is governed by a sixth-order partial differential equation in the heightz. This equation can be simplified to the second order if the mean absorption coefficient of the strongly absorbed regions is much larger while that of the weakly absorbed regions is much smaller than the local scale variation, and the influence of the former is equivalent to an added diffusion while that of the latter is a newtonian cooling, and these two influences are present simultaneously. The values of the two coefficients and their dependencies on the concentration of the absorbing material have been obtained. The equation has been applied to the problem of the thermal interaction between the atmosphere and the underlying earth, as related to the diurnal heat wave, and it was found that the temperature changes within the first few hundred meters from the earth can be predicted accurately by the model when the partition of the two groups is adjusted to the heating rate and the eddy transfer coefficient is allowed to increase very rapidly within the lowest 10 m.     

Transport and sedimentation of particles in a fluid flow

A simple two-dimensional random walk model is developed for the motion of a particle in a fluid flow. Some earlier results for the persistent injection of particles into the flow are extended, and the distribution of the maximum number of particles in suspension over the period (0,t) is derived.     

A multi‐azimuth seismic refraction study for a horizontal transverse isotropic medium: physical modelling results

To investigate the characteristics of the anisotropic stratum, a multi‐azimuth seismic refraction technique is proposed in this study since the travel time anomaly of the refraction wave induced by this anisotropic stratum will be large for a far offset receiver. To simplify the problem, a two‐layer (isotropy–horizontal transverse isotropy) model is considered. A new travel time equation of the refracted P‐wave propagation in this two‐layer model is derived, which is the function of the phase and group velocities of the horizontal transverse isotropic stratum. In addition, the measured refraction wave velocity in the physical model experiment is the group velocity. The isotropic intercept time equation of a refraction wave can be directly used to estimate the thickness of the top (isotropic) layer of the two‐layer model because the contrast between the phase and group velocities of the horizontal transverse isotropic medium is seldom greater than 10% in the Earth. If the contrast between the phase and group velocities of an anisotropic medium is small, the approximated travel time equation of a refraction wave is obtained. This equation is only dependent on the group velocity of the horizontal transverse isotropic stratum. The elastic constants A₁₁, A₁₃, and A₃₃ and the Thomsen anisotropic parameter ε of the horizontal transverse isotropic stratum can be estimated using this multi‐azimuth seismic refraction technique. Furthermore, under a condition of weak anisotropy, the Thomsen anisotropic parameter δ of the horizontal transverse isotropic stratum can be estimated by this technique as well.     

Measurement of lower-stratospheric aerosol by a balloon-borne photometer

Height distribution of the stratospheric aerosol extinction coefficient was measured in the altitude range 10 to 20 km by a balloon-borne multi-color sunphotometer in May 1978. It is demonstrated that detailed structures of the distribution of stratospheric aerosol can be remotely measured by the solar occultation method as well as by lidar andin situ particle counter observations. In the aerosol layer appearing at 18 km altitude the extinction coefficient at 800–1000 nm wavelength reached to 3×10^–7 m^–1, which was reasonable compared with lidar observations. Wavelength dependence of the aerosol optical depth was crudely estimated to be proportional to ^–1.5.     

An active mass damper designed using ARX models of a building structure

This study proposes a new design method for an active mass damper (AMD) that is based on auto‐regressive exogenous models of a building structure. The proposed method uses the results of system identification in the field of active structural control. The uncontrolled structure is identified as auto‐regressive exogenous models via measurements under earthquake excitation and forced vibration. These models are linked with an equation of motion for the AMD to introduce a state equation and output equation for the AMD–structure interaction system in the discrete‐time space; the equations apply modern control theories to the AMD design. In the numerical applications of a 10‐degree‐of‐freedom building structure, linear quadratic regulator control is used to understand the fundamental characteristics of the proposed design procedure. The feedback control law requires the AMD's acceleration, velocity and stroke; the structure's acceleration; and the ground acceleration as vibration measurements. The numerical examples confirm the high applicability and control effectiveness of the proposed method. One remarkable advantage of the proposed method is that an equation of motion for the structure becomes unnecessary for designing controllers. Copyright © 2016 John Wiley & Sons, Ltd.