Some aspects of vortices in rotating stratified fluids

Summary Ideas concerning the overturning of unstably stratified, rotating fluids are explored using potential vorticity.A set of equations governing axi-symmetric flow in a quasi-Boussinesq system are found based on the gradient wind approximation, and a transformation analogous to that developed byHoskins [6] is used.The time-development of a linear, thermally unstable vortex under the action of Ekman pumping is studied with these equations. The changing radial scale during amplification of the vortex is well represented.Finally, some exact steady vortex states for stably stratified fluids are found and their possible relevance to atmospheric vortices is discussed.     

周期径向荷载作用下圆弧格构拱动力稳定性分析

基于结构弹性稳定理论,根据能量法推导出以位移为基本未知量的圆弧格构拱总势能,从Hamilton原理出发,建立了考虑剪切变形的圆弧格构拱的动力稳定微分方程.利用Galerkin方法将其转化为二阶常微分Mathieu型参数共振方程,求得周期解所包围的动力不稳定区域,探讨了圆弧格构拱发生参数共振的动力稳定性问题,分析了缀条面...     

Unsteady boundary layer flows induced by a wavy plate

Summary Unsteady boundary layer flows generated in an incompressible, homogeneous, nonrotating viscous fluid bounded by a rigid wavy plate are studied theoretically. The Laplace transform method is employed to obtain exact solutions of the unsteady boundary layer equations in a wavy plate configuration. The structures of the unsteady velocity distribution and the associated boundary layers are determined explicitly and several particular solutions are recovered as special cases of this analysis. The physical interpretation of the mathematical results are examined.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

The role of hand calculations in ground water flow modeling

Most ground water modeling courses focus on the use of computer models and pay little or no attention to traditional analytic solutions to ground water flow problems. This shift in education seems logical. Why waste time to learn about the method of images, or why study analytic solutions to one-dimensional or radial flow problems? Computer models solve much more realistic problems and offer sophisticated graphical output, such as contour plots of potentiometric levels and ground water path lines. However, analytic solutions to elementary ground water flow problems do have something to offer over computer models: insight. For instance, an analytic one-dimensional or radial flow solution, in terms of a mathematical expression, may reveal which parameters affect the success of calibrating a computer model and what to expect when changing parameter values. Similarly, solutions for periodic forcing of one-dimensional or radial flow systems have resulted in a simple decision criterion to assess whether or not transient flow modeling is needed. Basic water balance calculations may offer a useful check on computer-generated capture zones for wellhead protection or aquifer remediation. An easily calculated "characteristic leakage length" provides critical insight into surface water and ground water interactions and flow in multi-aquifer systems. The list goes on. Familiarity with elementary analytic solutions and the capability of performing some simple hand calculations can promote appropriate (computer) modeling techniques, avoids unnecessary complexity, improves reliability, and is likely to save time and money. Training in basic hand calculations should be an important part of the curriculum of ground water modeling courses.     

Approximate Absorption Corrections for Complete SH Seismograms

An approximate absorption correction is proposed that can be applied retroactively to the response of a perfectly elastic medium, not decomposed into elementary waves, for example in the finite-difference method. As the duration of excitation decreases, the approximate correction approaches the exact, and becomes quite exact for impulsive excitation. The correction makes use of the perfectly elastic response and dissipation operator, but their relation is not simply convolutory. The method is restricted to a spatially constant Q with an arbitrary power-law frequency dependence. The advantage of the approximate solutions, as compared to the exact, in which the absorption is included in the equation of motion, is in a considerable saving of computer time if several absorption models (or several Q values) are repeatedly applied to the same structure. The simplest computations are for Q linearly dependent of the frequency, where the absorption correction reduces to an exponential time windowing.     

Approximate absorption corrections for complete SH seismograms

Summary An approximate absorption correction is proposed that can be applied retroactively to the response of a perfectly elastic medium, not decomposed into elementary waves, for example in the finite-difference method. As the duration of excitation decreases, the approximate correction approaches the exact, and becomes quite exact for impulsive excitation. The correction makes use of the perfectly elastic response and dissipation operator, but their relation is not simply convolutory. The method is restricted to a spatially constant Q with an arbitrary power-law frequency dependence. The advantage of the approximate solutions, as compared to the exact, in which the absorption is included in the equation of motion, is in a considerable saving of computer time if several absorption models (or several Q values) are repeatedly applied to the same structure. The simplest computations are for Q linearly dependent of the frequency, where the absorption correction reduces to an exponential time windowing.     

Travelling wave convection in a rotating layer

Abstract

Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.     

基础平动与转动结构随机地震响应分析的复模态法

本文对基础平动与转动结构随机地震响应问题进行了系统研究。首先建立了结构运动方程，然后针对体系运动方程为非经典阻尼的情况．用复模态法解耦获得了体系位移、速度和加速度随机地震响应的解析解，最后通过一个例子来说明考虑结构一地基相互作用的必要性。本文方法可推广到非平稳地震响应分析。     

Point vortex dynamics for coupled surface/interior QG and propagating heton clusters in models for ocean convection

Abstract

The dynamic behavior of baroclinic point vortices in two-layer quasi-geostrophic flow provides a compact model for studying the transport of heat in a variety of geophysical flows including recent heton models for open ocean convection as a response to spatially localized intense surface cooling. In such heton models, the exchange of heat with the region external to the compact cooling region reaches a statistical equilibrium through the propagation of tilted heton clusters. Such tilted heton clusters are aggregates of cyclonic vortices in the upper layer and anti-cyclonic vortices in the lower layer which collectively propagate almost as an elementary tilted heton pair even though the individual vortices undergo shifts in their relative locations. One main result in this paper is a mathematical theorem demonstrating the existence of large families of long-lived propagating heton clusters for the two-layer model in a fashion compatible to a remarkable degree with the earlier numerical simulations. Two-layer quasi-geostrophic flow is an idealization of coupled surface/interior quasi-geostrophic flow. The second family of results in this paper involves the systematic development of Hamiltonian point vortex dynamics for coupled surface/interior QG with an emphasis on propagating solutions that transport heat. These are novel vortex systems of mixed species where surface heat particles interact with quasi-geostrophic point vortices. The variety of elementary two-vortex exact solutions that transport heat include two surface heat particles of opposite strength, tilted pairs of a surface heat particle coupled to an interior vortex of opposite strength and two interior tilted vortices of opposite strength at different depths. The propagation speeds of the tilted elementary hetons in the coupled surface/interior QG model are compared and contrasted with those in the simpler two-layer heton models. Finally, mathematical theorems are presented for the existence of large families of propagating long-lived tilted heton clusters for point vortex solutions in coupled surface/interior QG flow.     

非对称型强飓风中的准平衡流特征分析

本文在论述飓风发生发展生命史过程中平衡、准平衡和非平衡态动力学特征的基础上,应用PV-ω方法,对具有非轴对称和长时间强度维持特征的飓风Bonnie（1998）进行了反演诊断分析,结果表明：基于非线性平衡模式的平衡流能够描述飓风水平涡旋场的基本特征,而加入准平衡ω方程得到的准平衡流能反映边界层入流、高层出流、眼墙区的剧烈倾斜上升运动和眼心区域的下沉运动.准平衡流描述了具有较长生命史组织化过程的强对流系统,而与其相联系的辐散运动与涡度同量级,证明了飓风准平衡流场具有涡散运动共存的特征,但在边界层顶的入流急流区和高层出流区仍存在高度非平衡态的超梯度流.利用反演的准平衡流场分析发现,当由环境风场低层到高层存在顺切变时,飓风内中尺度对流带移动方向的左侧,有利于强对流单体的发展和新对流单体的形成,右侧则相反,同时强气旋式旋转流场的作用,使得对流单体形成后随基本气流传播至对流减弱区,造成了飓风非对称结构的形成和维持.     

Three-dimensional time-harmonic Green's functions of saturated soil under buried loading

Three-dimensional time-harmonic response of a poroelastic half space subjected to an arbitrary buried loading is investigated. The analysis starts with the field equations in cylindrical coordinates based on Biot's general theory of poroelasticity. General solutions for the displacements are first derived using the Fourier expansions and Hankel integral transform with respect to the circumferential and radial coordinates, respectively. The transformed-domain solutions are obtained in explicit form. The physical-domain displacements and stress components are then obtained numerically by inverse integral transform. Comparisons illustrating the accuracy of the developed approach are made with existing solutions for an elastic half space, which is reduced directly from the general solution developed in the paper. Numerical results are presented for the displacements of a saturated soil subjected to a horizontal internal excitation.     

On the evolution and stability of interfacial ripples on and off the critical frequency

Abstract

Under consideration are interfaces between two media of different densities and which arise from the interaction between the Mth and Nth harmonics of the motion where 1 ≤ N < M. By means of the method of multiple scales in both space and time a pair of nonlinear coupled partial differential equations is derived which model the progression of the interface. The equations contain a detuning parameter [sgrave] which allow imperfections in the resonance to be taken into account. Stokes-type sinusoidal solutions to the equations were sought. It was found that solutions exist for all values of the interaction ratio M/N. In some situations interfaces exist at both exact and near resonance; while in others they are destroyed by amplifications in the detuning. In yet others, a quantity of detuning is actually necessary for the profiles to exist. In all cases, even when the parameters are fixed, a very large class of interface profiles is possible. Finally, the stability of the profiles is studied. It is found that some are quite stable, even to perturbations with wavenumbers close to the main flow.     

预条件方程组及其应用

建立了与一般Toeplitz方程组ANX=B所对应的ω循环型预条件方程组PN［ω］Xω=B.通过采用不同的准则构造预条件矩阵,可以得到不同的预条件方程组,计算出合理的ω值.理论分析和实际计算证明了该方法所得到的近似计算结果优于普通Fourier变换方法(ω=1)的分析结果.     

High-order finite volume WENO schemes for the shallow water equations with dry states

The shallow water equations are used to model flows in rivers and coastal areas, and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. These equations have still water steady state solutions in which the flux gradients are balanced by the source term. It is desirable to develop numerical methods which preserve exactly these steady state solutions. Another main difficulty usually arising from the simulation of dam breaks and flood waves flows is the appearance of dry areas where no water is present. If no special attention is paid, standard numerical methods may fail near dry/wet front and produce non-physical negative water height. A high-order accurate finite volume weighted essentially non-oscillatory (WENO) scheme is proposed in this paper to address these difficulties and to provide an efficient and robust method for solving the shallow water equations. A simple, easy-to-implement positivity-preserving limiter is introduced. One- and two-dimensional numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.     

Some exact solutions of unsteady boundary layer equations-I

Summary Unsteady boundary layer flows generated in a homogeneous, non-rotating viscous fluid are considered. The method of Laplace transform is used to obtain exact solutions of the unsteady boundary layer equations in a more general situation. The structures of the unsteady velocity field and the associated boundary layers are determined. Several particular solutions are recovered as special cases of the present general theory. The physical implications of the mathematical results are investigated.     

Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations

Shallow water equations with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. An important difficulty arising in these simulations is the appearance of dry areas where no water is present, as standard numerical methods may fail in the presence of these areas. These equations also have still water steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. In this paper we propose a high order discontinuous Galerkin method which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. A simple positivity-preserving limiter, valid under suitable CFL condition, will be introduced in one dimension and then extended to two dimensions with rectangular meshes. Numerical tests are performed to verify the positivity-preserving property, well-balanced property, high order accuracy, and good resolution for smooth and discontinuous solutions.     

岩石圈下层塑性流动网带的剪切热效应

基于“网状塑性流动”大陆动力学模型和中国大陆大地热流的分布特征,文中针对网带黏性剪切热效应的平衡态,通过热力学能量方程的求解,导出网带温度异常方程等关系式,并通过与网带热流正异常的拟合分析,确定了相关的参数值。研究表明,塑性流动网带黏性剪切热的产生为温度的正异常提供了热源,并通过热传导扩展网带的影响范围,形成相应的热异常网带。作为对中东亚塑性流动网络系统平均状况的估计,网带的热异常宽度与地震带所显示的网带视宽度一致,约为65km;剪切网带的宽度为44km,约为网带热异常宽度的23;网带中线处的温度正异常约为210K,剪切网带边界处的温度正异常约为67K,至热异常带边缘处降低为零,其起伏形态与网带大地热流正异常“凸峰”相一致。文中为岩石圈下层塑性流动网络的存在及其对大地热流正异常分布的控制提供了理论依据     

Comparison of Ray Methods with the Exact Solution in the 1-D Anisotropic “Simplified Twisted Crystal” Model

The exact analytical solution for the plane S-wave, propagating along the axis of spirality in the simple 1-D anisotropic simplified twisted crystal model, is compared with four different approximate ray-theory solutions. The four different ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. The comparison is carried out numerically, by evaluating both the exact analytical solution and the analytical solutions of the equations of the four ray methods. The comparison simultaneously demonstrates the limits of applicability of the isotropic and anisotropic ray theories, and the superior accuracy of the coupling ray theory over a broad frequency range. The comparison also shows the possible inaccuracy due to the quasi-isotropic perturbation of travel times in the equations of the coupling ray theory. The coupling ray theory thus should definitely be preferred to the isotropic and anisotropic ray theories, but the quasi-isotropic perturbation of travel times should be avoided. Although the simplified twisted crystal model is designed for testing purposes and has no direct relation to geological structures, the wave-propagation phenomena important in the comparison are similar to those in the models of the geological structures.In additional numerical tests, the exact analytical solution is numerically compared with the finite-difference numerical results, and the analytical solutions of the equations of different ray methods are compared with the corresponding numerical results of 3-D ray-tracing programs developed by the authors of the paper.