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Spherical Coordinates (spherical + coordinate)
Selected AbstractsA practical grid-based method for tracking multiple refraction and reflection phases in three-dimensional heterogeneous mediaGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2006M. De Kool SUMMARY We present a practical grid-based method in 3-D spherical coordinates for computing multiple phases comprising any number of reflection and transmission branches in heterogeneous layered media. The new scheme is based on a multistage approach which treats each layer that the wave front enters as a separate computational domain. A finite-difference eikonal solver known as the fast-marching method (FMM) is reinitialized at each interface to track the evolving wave front as either a reflection back into the incident layer or a transmission through to the adjacent layer. Unlike the standard FMM, which only finds first arrivals, this multistage approach can track those later arriving phases explicitly caused by the presence of discontinuities. Notably, the method does not require an irregular mesh to be constructed in order to connect interface nodes to neighbouring velocity nodes which lie on a regular grid. To improve accuracy, local grid refinement is used in the neighbourhood of a source point where wave front curvature is high. The method also provides a way to trace reflections from an interface that are not the first arrival (e.g. the global PP phase). These are computed by initializing the multistage FMM from both the source and receiver, propagating the two wave fronts to the reflecting interface, and finding stationary points of the sum of the two traveltime fields on the reflecting interface. A series of examples are presented to test the efficiency, accuracy and robustness of the new scheme. As well as efficiently computing various global phases to an acceptable accuracy through the ak135 model, we also demonstrate the ability of the scheme to track complex crustal phases that may be encountered in coincident reflection, wide-angle reflection/refraction or local earthquake surveys. In one example, a variety of phases are computed in the presence of a realistic subduction zone, which includes several layer pinch-outs and a subducting slab. Our numerical tests show that the new scheme is a practical and robust alternative to conventional ray tracing for finding various phases in layered media at a variety of scales. [source] Localized spectral analysis on the sphereGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2005Mark A. Wieczorek SUMMARY It is often advantageous to investigate the relationship between two geophysical data sets in the spectral domain by calculating admittance and coherence functions. While there exist powerful Cartesian windowing techniques to estimate spatially localized (cross-)spectral properties, the inherent sphericity of planetary bodies sometimes necessitates an approach based in spherical coordinates. Direct localized spectral estimates on the sphere can be obtained by tapering, or multiplying the data by a suitable windowing function, and expanding the resultant field in spherical harmonics. The localization of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. Two kinds of axisymmetric windows are here constructed that are ideally suited to this purpose: bandlimited functions that maximize their spatial energy within a cap of angular radius ,0, and spacelimited functions that maximize their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem that is solved by an orthogonal family of data tapers, and the properties of these windows depend almost entirely upon the space,bandwidth product N0= (L+ 1) ,0/,. The first N0, 1 windows are near perfectly concentrated, and the best-concentrated window approaches a lower bound imposed by a spherical uncertainty principle. In order to make robust localized estimates of the admittance and coherence spectra between two fields on the sphere, we propose a method analogous to Cartesian multitaper spectral analysis that uses our optimally concentrated data tapers. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased for stochastic processes when the input spectrum is white and when averages are made over all possible realizations of the random variables. In physical situations, only one realization of such a process will be available, but in this case, a weighted average of the spectra obtained using multiple data tapers well approximates the expected spectrum. While developed primarily to solve problems in planetary science, our method has applications in all areas of science that investigate spatiospectral relationships between data fields defined on a sphere. [source] Semi-Lagrangian advection on a spherical geodesic gridINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2007Maria Francesca Carfora Abstract A simple and efficient numerical method for solving the advection equation on the spherical surface is presented. To overcome the well-known ,pole problem' related to the polar singularity of spherical coordinates, the space discretization is performed on a geodesic grid derived by a uniform triangulation of the sphere; the time discretization uses a semi-Lagrangian approach. These two choices, efficiently combined in a substepping procedure, allow us to easily determine the departure points of the characteristic lines, avoiding any computationally expensive tree-search. Moreover, suitable interpolation procedures on such geodesic grid are presented and compared. The performance of the method in terms of accuracy and efficiency is assessed on two standard test cases: solid-body rotation and a deformation flow. Copyright © 2007 John Wiley & Sons, Ltd. [source] A convergent three-level finite difference scheme for solving a dual-phase-lagging heat transport equation in spherical coordinatesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004Weizhong Dai Abstract Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three-level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60,71, 2004. [source] Designing herbicide formulation characteristics to maximize efficacy and minimize rice injury in paddy environmentsPEST MANAGEMENT SCIENCE (FORMERLY: PESTICIDE SCIENCE), Issue 6 2001Steven A Cryer Abstract Mathematical descriptors, coupled with experimental observations, are used to quantify differential uptake of an experimental herbicide in Japonica and Indica rice (Oryza sativa, non-target) and barnyardgrass (Echinochloa crus-galli, target). Partitioning, degradation, plant uptake and metabolism are described using mass-balance conservation equations in the form of kinetic approximations. Estimated environmental concentrations, governed by the pesticide formulation, are described using superimposed analytical solutions for the one-dimensional diffusion equation in spherical coordinates and by a finite difference representation of the two-dimensional diffusion equation in Cartesian coordinates. Formulation attributes from granules include active ingredient release rates, particle sizes, pesticide loading, and granule spacing. The diffusion model for pesticide transport is coupled with the compartment model to follow the fate and transport of a pesticide from its initial application location to various environmental matrices of interest. Formulation effects, partitioning and degradation in the various environmental matrices, differential plant uptake and metabolism, and dose-response information for plants are accounted for. This novel model provides a mechanism for selecting formulation delivery systems that optimize specific attributes (such as weed control or the therapeutic index) for risk-assessment procedures. In this report we describe how this methodology was used to explore the factors affecting herbicide efficacy and to define an optimal release rate for a granule formulation. © 2001 Society of Chemical Industry [source] Treatment of vector equations in deep-atmosphere, semi-Lagrangian models.THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 647 2010I: Momentum equation Abstract Application of the semi-Lagrangian method to the vector momentum equation in orthogonal curvilinear systems is considered, with emphasis on spherical and spheroidal coordinates for deep-atmosphere models (in which the shallow-atmosphere approximation is not made). In spherical coordinates, a certain rotation matrix allows vector components at semi-Lagrangian departure points to be expressed in the required component forms at the corresponding arrival points. This rotation matrix also allows the unit vector triad at the arrival point to be expressed in terms of the unit vector triad at the departure point. The resulting formalisation of the momentum equation applies to timesteps of arbitrary length, and in the limit of infinitesimal timestep it delivers the familiar Eulerian components. Extension of the rotation matrix technique to more general orthogonal curvilinear systems is remarkably straightforward, thus demonstrating the versatility of the semi-Lagrangian method. Explicit forms are given for systems having longitude as one coordinate and general orthogonal coordinates in the meridional plane, and in particular these forms are applicable to both confocal oblate spheroidal and similar oblate spheroidal systems. A specific factorisation of the rotation matrix in the spherical polar case involves three matrices, one of which represents rotation of the unit vector triad along a great circle arc; the non-Euclidean, shallow-atmosphere approximation of the total rotation matrix results when the great circle rotation matrix is replaced by the identity matrix (but the other two matrices involved in the factorisation are kept intact). This shallow-atmosphere rotation matrix agrees with that used by ECMWF and Météo-France. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd. [source] Unsteady exact solutions of the flow equations for three-dimensional spherical atmospheresTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 635 2008A. Staniforth Abstract Time-dependent, closed-form solutions of the 3D Euler equations describing motion relative to a uniformly rotating coordinate frame are derived. The spherical geopotential approximation is applied but not the shallow-atmosphere and hydrostatic approximations. The solutions correspond to cyclostrophically and hydrostatically balanced vortices that are steady in inertial space and whose symmetry axes do not coincide with the rotation axis of the coordinate frame. The inertial-frame flow velocities are readily transformed to a precisely spherical rotating coordinate system in which the 3D Euler equations contain centrifugal as well as Coriolis terms. In this form the solutions may be used to test numerical models formulated in spherical coordinates under the spherical geopotential approximation, so long as the centrifugal terms are explicitly included as forcing terms. The development is repeated for the hydrostatic and non-hydrostatic primitive equations (with the shallow-atmosphere approximation) and for the shallow-water equations. In the latter case, the required explicit centrifugal force may be provided by a zonally symmetric addition to the free surface height, with an identically equal orographic elevation to ensure conservation of mass. The solutions are then identical to the unsteady shallow-water solutions of Läuter et al. that inspired this study. ©Crown Copyright 2008. Reproduced with the permission of the Controller of HMSO and the Queen's Printer for Scotland. Published by John Wiley & Sons, Ltd. [source] Multipolar Ordering in Electro- and Magnetostatic Coupled NanosystemsCHEMPHYSCHEM, Issue 9 2008Elena Y. Vedmedenko Dr. habil. Abstract Electric and magnetic multipole moments and polarizabilities are important quantities in studies of intermolecular forces, non-linear optical phenomena, electrostatic, magnetostatic or gravitational potentials and electron scattering. The experimental determination of multipole moments is difficult and therefore the theoretical prediction of these quantities is important. Depending on purposes of the investigation several different definitions of multipole moments and multipole,multipole interactions are used in the literature. Because of this variety of methods it is often difficult to use published results and, therefore, even more new definitions appear. The first goal of this review is to give an overview of mathematical definitions of multipole expansion and relations between different formulations. The second aim is to present a general theoretical description of multipolar ordering on periodic two-dimensional lattices. After a historical introduction in the first part of this manuscript the static multipole expansion in cartesian and spherical coordinates as well as existing coordinate transformations are reviewed. On the basis of the presented mathematical description multipole moments of several symmetric charge distributions are summarized. Next, the established numerical approach for the calculation of multipolar ground states, namely Monte Carlo simulations, are reviewed. Special emphasis is put on the review of ground states in multipolar systems consisting of moments of odd or even order. The last section is devoted to the magnetization reversal in dense packed nanomagnetic arrays, where the magnetic multipole,multipole interactions play an important role. Comparison between the theory and recent experimental results is given. [source] | |||||||||||||