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Gary S. Dietachmayer

Abstract

The continuous dynamic adaptive grid (CDGA) technique has been shown to yield significant improvements in solution accuracy over equivalent fixed-grid methods. In this paper we consider the related question of efficiency; for a given degree of solution accuracy is the CDGA method faster than the fixed-grid approach, and if so, by how much? A new grid-generation algorithm based on discrete equidistribution principles is proposed, which, while it has the disadvantage of producing nonorthogonal grids, is extremely simple to implement and fast to execute. The algorithm is applied to a barotropic model designed to examine the motion of interacting multiple vortices. Numerical tests confirm the efficacy of the proposed algorithm and show that the CDGA method can produce solutions as accurate as those obtained from a fixed-grid model in only one-third the time.

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Lance M. Leslie and Gary S. Dietachmayer

Abstract

Over the years there have been a number of studies comparing the relative merits of semi-Lagrangian and Eulerian schemes. These studies, which continue to appear in the literature up to the present, almost invariably conclude that semi-Lagrangian schemes are superior in accuracy, and produce less noise, than Eulerian schemes. It is argued in this note that such conclusions are not justified because they have compared semi-Lagrangian and Eulerian schemes of different orders of accuracy. Typically, the semi-Lagrangian schemes tested have employed cubic spatial interpolation (and therefore are third order) in space, whereas the Eulerian schemes have usually been second order (and sometimes fourth order) in space. It is shown here that when semi-Lagrangian and Eulerian schemes of the same order are applied to the test case, namely, that of “warm bubble” convection, there are almost indiscernible differences between the simulations. The contention presented here, therefore, is that it is the order of the scheme that is of primary importance, not whether it is semi-Lagrangian or Eulerian.

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Gary S. Dietachmayer and Kelvin K. Droegemeier

Abstract

The continuous dynamic grid adaption (CDGA) technique developed in astrophysics and aeronautics is applied, to our knowledge, for the first time to meteorological modeling. The aim of CDGA is to improve the accuracy of numerical solutions of partial differential equations (typically those governing fluid flow) by the use of nonuniform grids that have higher local resolution in regions where the numerical error is presumed to be large. Conceptually, CDGA has some relationship to the well-known technique of grid stretching, but its power lies in its ability to determine an appropriate spatial distribution of grid points automatically and to update this distribution in response to changes in the evolving numerical solution. Application of the technique is facilitated by transforming the governing equations from physical space in which the grid is nonuniform, nonorthogonal and for which the individual grid points are in continuous motion to computational space, which by definition has both a regular and stationary distribution of grid points. The distribution of grid points is found by the solution of “grid-generator” equations, which in turn can be derived as a weighted combination of several variational problems, each of which attempts to enforce a particular desirable property of the grid. These properties include the smoothness and orthogonality of the gridpoint distribution and its response to the user-defined “weight function,” which is a quantitative measure of where the local resolution is to be increased.

The method is applied to several problems of meteorological relevance. The first, Burgers’ equa`tion in one dimension, is used primarily to illustrate the method in a simple context, but also illuminates several features of CDGA, one of which is its ability to improve the accuracy of a numerical solution purely by inducing motion of the grid points. A kinematic frontogenesis problem is used to extend the method to two dimensions, and with the aid of a readily available exact solution, shows the very considerable gains in accuracy that may be achieved over fixed-grid methods. A surprising observation is that the formal order of accuracy of the adaptive results is, for certain parameters, actually greater than for the fixed-grid results. The ability of the technique to allocate multiple zones of high resolution is demonstrated by experiments in which several (four) “cones” are advected by a field of solid-body rotation. The final application is to the evolution of a slab-symmetric thermal in a neutral environment. Again, considerable improvements in accuracy over fixed-grid calculations are achieved, and it is shown that the problem of spurious numerical oscillations associated with rapid variation in an advected field, a problem that has received a great deal of attention in recent times, is greatly alleviated by the CDGA formulation.

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Greg J. Holland, Lance M. Leslie, Elizabeth A. Ritchie, Gary S. Dietachmayer, Peter E. Powers, and Mark Klink

Abstract

The design concept and operational trial of a fully interactive analysis and numerical forecast system for tropical-cyclone motion are described. The design concept emphasizes an interactive system in which forecasters can test various scenarios objectively, rather than having to subjectively decide between conflicting forecasts from standardized techniques. The system is designed for use on a personal computer, or workstation, located on the forecast bench. A choice of a Barnes or statistical interpolation scheme is provided to analyze raw or bogus observations at any atmospheric level or layer mean selected by the forecaster. The track forecast is then made by integration of a nondivergent barotropic model.

An operational trial during the 1990 tropical-cyclone field experiments in the western north Pacific Ocean indicated that the system can be used very effectively in real time. A series of case-study examples is presented.

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Noel E. Davidson, Yi Xiao, Yimin Ma, Harry C. Weber, Xudong Sun, Lawrie J. Rikus, Jeff D. Kepert, Peter X. Steinle, Gary S. Dietachmayer, Charlie C. F. Lok, James Fraser, Joan Fernon, and Hakeem Shaik

Abstract

The Australian Community Climate and Earth System Simulator (ACCESS) has been adapted for operational and research applications on tropical cyclones. The base system runs at a resolution of 0.11° and 50 levels. The domain is relocatable and nested in coarser-resolution ACCESS forecasts. Initialization consists of five cycles of four-dimensional variational data assimilation (4DVAR) over 24 h. Forecasts to 72 h are made. Without vortex specification, initial conditions usually contain a weak and misplaced circulation pattern. Significant effort has been devoted to building physically based, synthetic inner-core structures, validated using historical dropsonde data and surface analyses from the Atlantic. Based on estimates of central pressure and storm size, vortex specification is used to filter the analyzed circulation from the original analysis, construct an inner core of the storm, locate it to the observed position, and merge it with the large-scale analysis at outer radii.

Using all available conventional observations and only synthetic surface pressure observations from the idealized vortex to correct the initial location and structure of the storm, the 4DVAR builds a balanced, intense 3D vortex with maximum wind at the radius of maximum wind and with a well-developed secondary circulation. Mean track and intensity errors for Australian region and northwest Pacific storms have been encouraging, as are recent real-time results from the Australian National Meteorological and Oceanographic Centre. The system became fully operational in November 2011. From preliminary diagnostics, some interesting structure change features are illustrated. Current limitations, future enhancements, and research applications are also discussed.

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