Smoothing, filtering, and boundary effects
TL;DR: A general procedure is outlined for the construction of an ‘ideal’ low-pass filter, a filter that removes the shortest resolvable wave component but restores all other wave components as close to their original amplitudes without amplifying or changing the phase of any wave component.
Abstract: Numerical integrations of finite-difference analogs of systems of nonlinear partial differential equations, such as those arising in atmospheric dynamics, are subject to computational instability from a variety of causes. One type of instability is produced by a spurious, nonlinear growth of high-frequency components that may be introduced by roundoff, truncation, and observational error. This type of instability, first discussed by N. A. Phillips, can be suppressed by a suitable choice of finite-difference method or by the use of a filter that selectively damps the high-frequency components. Though much effort is being devoted to the development of stable finite-difference procedures, and considerable success has been achieved, all such methods involve high-frequency smoothing either implicitly or explicitly. It is therefore important that the effects of such filtering be fully understood. Filtering and smoothing operators are developed for use in conjunction with the numerical integration of nonlinear systems and for other purposes. The general procedure is demonstrated for simple one-dimensional operators and the properties of such operators are thoroughly explored. The development is then expanded to allow for compound operators designed to suit some particular requirement and further extended to more than one dimension. Both real and complex operators are discussed. Reverse smoothers or wave amplifiers are introduced, and some of the problems associated with their use are discussed. A general procedure is outlined for the construction of an ‘ideal’ low-pass filter; that is, a filter that removes the shortest resolvable wave component (the 2-grid-interval wave) but restores all other wave components as close as is desired to their original amplitudes without amplifying or changing the phase of any wave component. Finally, the effects, sometimes disastrous, of finite domains on the properties of the smoothing operators are explored for a variety of common boundary assumptions.
Citations
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01 Jan 1995
3,954 citations
Book•
01 Nov 2002TL;DR: A comprehensive text and reference work on numerical weather prediction, first published in 2002, covers not only methods for numerical modeling, but also the important related areas of data assimilation and predictability.
Abstract: This comprehensive text and reference work on numerical weather prediction, first published in 2002, covers not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of the subject, equations of motion and their approximations, a modern and clear description of numerical methods, and the determination of initial conditions using weather observations (an important science known as data assimilation). Finally, this book provides a clear discussion of the problems of predictability and chaos in dynamical systems and how they can be applied to atmospheric and oceanic systems. Professors and students in meteorology, atmospheric science, oceanography, hydrology and environmental science will find much to interest them in this book, which can also form the basis of one or more graduate-level courses.
2,240 citations
TL;DR: This monograph is an outstanding monograph on current research on skewelliptical models and its generalizations and does an excellent job presenting the depth of methodological research as well as the breath of application regimes.
Abstract: (2005). Atmospheric Modeling, Data Assimilation, and Predictability. Technometrics: Vol. 47, No. 4, pp. 521-521.
1,580 citations
Cites methods from "Smoothing, filtering, and boundary ..."
...(1977) combined the use of an energy-conserving fourth order model with a sixteenth order filter (similar to the eighth power of the horizontal Laplacian (Shapiro, 1970))....
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TL;DR: In this paper, a benchmark calculation is proposed for evaluating the dynamical cores of atmospheric general circulation models (GCMs) independently of the physical parameterizations, focusing on the long-term statistical properties of a fully developed general circulation; thus, it is particularly appropriate for comparing the dynamics used in climate models.
Abstract: A benchmark calculation is proposed for evaluating the dynamical cores of atmospheric general circulation models (GCMs) independently of the physical parameterizations. The test focuses on the long-term statistical properties of a fully developed general circulation; thus, it is particularly appropriate for intercomparing the dynamics used in climate models. To illustrate the use of this benchmark, two very different atmospheric dynamical cores--one spectral, one finite difference--are compared. It is found that the long-term statistics produced by the two models are very similar. Selected results from these calculations are presented to initiate the intercomparison.
958 citations
References
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TL;DR: In this article, a wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws, and the best ones are determined, i.e., those which have the smallest truncation error and in which the discontinuities are confined to a narrow band of 2-3 meshpoints.
Abstract: : In this paper a wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws. Among these schemes we determine the best ones, i.e., those which have the smallest truncation error and in which the discontinuities are confined to a narrow band of 2-3 meshpoints. These schemes are tested for stability and are found to be stable under a mild strengthening of the Courant-Friedrichs-Levy criterion. Test calculations of one dimensional flows of compressible fluids with shocks, rarefaction waves and contact discontinuities show excellent agreement with exact solutions. In particular, when Lagrange coordinates are used, there is no smearing of interfaces. The additional terms introduced into the difference scheme for the purpose of keeping the shock transition narrow are similar to, although not identical with, the artificial viscosity terms, and the like of them introduced by Richtmyer and von Neumann and elaborated by other workers in this field.
2,408 citations
Book•
14 Jul 2012
TL;DR: This account attempts to provide and relate the necessary ideas and techniques in reasonable detail to develop the insight necessary to plan both the acquisition of adequate data and sound procedures for its reduction to meaningful estimates.
Abstract: The measurement of power spectra is a problem of steadily increasing importance which appears to some to be primarily a problem in statistical estimation. Others may see it as a problem of instrumentation, recording and analysis which vitally involves the ideas of transmission theory. Actually, ideas and techniques from both fields are needed. When they are combined, they provide a basis for developing the insight necessary (i) to plan both the acquisition of adequate data and sound procedures for its reduction to meaningful estimates and (ii) to interpret these estimates correctly and usefully. This account attempts to provide and relate the necessary ideas and techniques in reasonable detail — Part I of this article appeared in the January, 1958 issue of THE BELL SYSTEM TECHNICAL JOURNAL.
1,353 citations
TL;DR: In this article, it was shown that the derived form of the finite difference Jacobian can prevent nonlinear computational instability and thereby permit long-term numerical integrations, which is not the case in finite difference analogues of the equation of motion for two-dimensional incompressible flow.
1,328 citations
TL;DR: In this paper, a two-level quasi-geostrophic model is used to forecast a long-period numerical forecast of the atmosphere, starting with an atmosphere in relative rest, including a jet and zonal surface westerlies in middle latitudes.
Abstract: A long-period numerical forecast is made with a two-level quasi-geostrophic model, starting with an atmosphere in relative rest. Both friction and non-adiabatic effects are included in the equations, the latter as a linear function of latitude. Principal empirical elements in the experiment are the intensity of the heating, the value of the vertical stability, and the type of frictional dissipation. The flow patterns which develop are quite realistic, including a jet and zonal surface westerlies in middle latitudes, and the growth of a large disturbance. The associated energy transformations are investigated, and demonstrate the important role of the disturbance in the development of the zonal currents. The meridional circulation is also studied, together with its contribution to the zonal momentum budgets of the lower and upper halves of the atmosphere. Truncation errors eventually put an end to the forecast by producing a large fictitious increase in energy.
546 citations
TL;DR: In this article, the conservation and stability properties of the spatial differencing methods devised by Arakawa are investigated by means of spectral analysis of the stream function into finite Fourier modes.
Abstract: The satisfactory numerical solution of the equations of fluid dynamics applicable to atmospheric and oceanic problems characteristically requires a high degree of computational stability and accurate conservation of certain statistical moments. Methods for satisfying these requirements are described for various systems of equations typical of low. Mach number fluid dynamics systems, and are investigated in detail as applied to the two-dimensional, inertial-plane equation for conservation of vorticity in a frictionless non-divergent fluid. The conservation and stability properties of the spatial differencing methods devised by A. Arakawa are investigated by means of spectral analysis of the stream function into finite Fourier modes. Any of two classes of linear and quadratie conserving schemes are shown to eliminate the non-linear instability discussed by Phillips, although the “aliasing” error remains. Stability related to the time derivative term is investigated through analytic and numerical so...
344 citations