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Forward in time methods for global climate research. Final report

Description: Purpose is to demonstrate feasibility and utility of nonoscillatory forward-in-time (NFT) methods formodeling the global dynamics of the atmosphere and oceans. This includes development of new algorithms, construction of numerical models, and testing these models. One aspect of the research is to compare two variants of NFT methods, one based on Eulerian approximations and the other based on semi-Lagrangian approximations.
Date: May 1, 1996
Creator: Margolin, L.G. & Smolarkiewicz, P.K.
Partner: UNT Libraries Government Documents Department

Variational elliptic solver for atmospheric applications

Description: We discuss a conjugate gradient type method -- the conjugate residual -- suitable for solving linear elliptic equations that result from discretization of complex atmospheric dynamical problems. Rotation and irregular boundaries typically lead to nonself-adjoint elliptic operators whose matrix representation on the grid is definite but not symmetric. On the other hand, most established methods for solving large sparse matrix equations depend on the symmetry and definiteness of the matrix. Furthermore, the explicit construction of the matrix can be both difficult and computationally expensive. An attractive feature of conjugate gradient methods in general is that they do not require any knowledge of the matrix; and in particular, convergence of conjugate residual algorithms do not rely on symmetry for definite operators. We begin by reviewing some basic concepts of variational algorithms from the perspective of a physical analogy to the damped wave equation, which is a simple alternative to the traditional abstract framework of the Krylov subspace methods. We derive two conjugate residual schemes from variational principles, and prove that either definiteness or symmetry ensures their convergence. We discuss issues related to computational efficiency and illustrate our theoretical considerations with a test problem of the potential flow of a Boussinesq fluid flow past a steep, three-dimensional obstacle.
Date: March 1, 1994
Creator: Smolarkiewicz, P. K. & Margolin, L. G.
Partner: UNT Libraries Government Documents Department

On "spurious" eddies

Description: Recently several papers have appeared in the CFD literature, proposing an idealized instability problem as a benchmark for discriminating among numerical algorithms for two-dimensional Navier-Stokes flows. The problem is a double shear layer simulated at coarse resolution and with a prescribed interface perturbation. A variety of second-order accurate schemes have been tested, with all results falling into one of two solution patterns - one pattern with two eddies and the other with three eddies. In the literature, there is no fast-and-firm rule to predict the results of any particular algorithm. However it is asserted that the two-eddy solution is correct. Our own research has led to two conclusions. First, the appearance of the third eddy is tied up with small details of the truncation error; we illustrate this point by prescribing small changes that lead to reversal of the appearance/disappearance of the third eddy in several schemes. Second, we discuss the realizability of the two solutions and suggest that the three-eddy solution is the more physical. Overall, we conclude that this problem is a poor choice of benchmark to discriminate among numerical algorithms.
Date: January 1, 2001
Creator: Drikakis, D. (Dimitris); Margolin, L. G. & Smolarkiewicz, P. K. (Piotr K.)
Partner: UNT Libraries Government Documents Department

MPDATA: A positive definite solver for geophysical flows

Description: This article is a review of MPDATA, a class of methods for the numerical simulation of advection based on the sign-preserving properties of upstream differencing. MPDATA was designed originally as an inexpensive alternative to flux-limited schemes for evaluating the transport of nonnegative thermodynamic variables (such as liquid water or water vapour) in atmospheric models. During the last decade, MPDATA has evolved from a simple advection scheme to a general approach for integrating the conservation laws of geophysical fluids on micro-to-planetary scales. The purpose of this paper is to summarize the basic concepts leading to a family of MPDATA schemes, review the existing MPDATA options, as well as to demonstrate the efficacy of the approach using diverse examples of complex geophysical flows.
Date: March 1, 1997
Creator: Smolarkiewicz, P.K. & Margolin, L.G.
Partner: UNT Libraries Government Documents Department

Forward-in-Time Differencing for Fluids: Nonhydrostatic Modeling of Rotating Stratified Flow on a Mountainous Sphere

Description: Traditionally, numerical models for simulating planetary scale weather and climate employ the hydrostatic primitive equations-an abbreviated form of Navier-Stokes equations that neglect vertical accelerations and use simplified inertial forces. 1 Although there is no evidence so far that including nonhydrostatic effects in global models has any physical significance for large scale solutions, there is an apparent trend in the community toward restoring Navier-Stokes equations (or at least their less constrained forms) in global models of atmospheres and oceans. The primary motivation for this is that the state-of-the-art computers already admit resolutions where local nonhydrostatic effects become noticeable. Other advantages include: the convenience of local mesh refinement; better overall accuracy; insubstantial computational overhead relative to hydrostatic models; universality and therefore convenience of maintaining a single large code; as well as conceptual simplicity and mathematical elegancy--features important for education. The few existing nonhydrostatic global models differ in analytic formulation and numerical design, reflecting their different purposes and origins. Much of our present research improves the design of a high-performance numerical model for simulating the flows of moist (and precipitating), rotating, stratified fluids past a specified time-dependent irregular lower boundary. This model is representative of a class of nonhydrostatic atmospheric codes employing the an elastic equations of motion in a terrain-following curvilinear framework, and contains parallel implementations of semi-Lagrangian and Eulerian approximations selectable by the user. The model has been employed in a variety of applications; the quality of results suggest that modern nonoscillatory forward-in-time (NFT) methods are superior to the more traditional centered-in-time-and-space schemes, in terms of accuracy, computational efficiency, flexibility and robustness.
Date: March 31, 1999
Creator: Smolarkiewicz, P.K.; Grubisic, V. & Margolin, L.G.
Partner: UNT Libraries Government Documents Department

Semi-Lagrangian shallow water modeling on the CM-5

Description: We discuss the parallel implementation of a semi-Lagrangian shallow-water model on the massively parallel Connection Machine CM-5. The four important issues we address in this article are (i) two alternative formulations of the elliptic problem and their relative efficiencies, (ii) the performance of two successive orders of a generalized conjugate residual elliptic solver, (iii) the time spent in unstructured communication -- an unavoidable feature of semi-Lagrangian schemes, and (iv) the scalability of the algorithm.
Date: September 1, 1995
Creator: Nadiga, B.T.; Margolin, L.G. & Smolarkiewicz, P.K.
Partner: UNT Libraries Government Documents Department

A reduced grid model for shallow flows on the sphere

Description: The authors describe a numerical model for simulating shallow water flows on a rotating sphere. The model is augmented by a reduced grid capability that increases the allowable time step based on stability requirements, and leads to significant improvements in computational efficiency. The model is based on finite difference techniques, and in particular on the nonoscillatory forward-in-time advection scheme MPDATA. They have implemented the model on the massively parallel CM-5, and have used it to simulate shallow water flows representative of global atmospheric motions. Here they present simulations of two flows, the Rossby-Haurwitz wave of period four, a nearly steady pattern with a complex balance of large and small scale motions, and also a zonal flow perturbed by an obstacle. They compare the accuracy and efficiency of using the reduced grid option with that of the original model. The authors also present simulations at several levels of resolution to show how the efficiency of the model scales with problem size.
Date: September 1, 1995
Creator: Reisner, J.M.; Margolin, L.G. & Smolarkiewicz, P.K.
Partner: UNT Libraries Government Documents Department

Implicit turbulence modeling for high reynolds number flows.

Description: We continue our investigation of the implicit turbulence modeling property of the nonoscillatory finite volume scheme MPDATA. We start by comparing MPDATA simulations of decaying turbulence in a triply periodic cube with analogous pseudospectral studies. In the regime of direct numerical simulation, MPDATA is shown to agree closely with the pseudospectral model. As viscosity is reduced, the two model results diverge. We study the MPDATA results in the inviscid limit, using a combination of mathematical analysis and computational experiment. We validate the inviscid MPDATA results as representing the turbulent flow in the limit of very high Reynolds number.
Date: January 1, 2001
Creator: Margolin, L. G.; Smolarkiewicz, P. K. (Piotr K.) & Wyszogrodzki, A. A. (Andrzej A.)
Partner: UNT Libraries Government Documents Department

Forward-in-time differencing for fluids: Nonhydrostatic modeling of fluid motions on a sphere

Description: Traditionally, numerical models for simulating planetary scale weather and climate employ the hydrostatic primitive equations--an abbreviated form of Navier-Stokes` equations that neglect vertical accelerations and use simplified Coriolis forces. Although there is no evidence so far that including nonhydrostatic effects in global models has any physical significance for large scale solutions, there is an emerging trend in the community toward restoring Navier-Stokes` equations (or at least their less constrained forms) in global models of atmospheres and oceans. The primary motivation is that state-of-the-art computers already admit resolutions where local nonhydrostatic effects become noticeable. much of this present research aims to improve the design of a high-performance numerical model for simulating the flows of moist (and precipitating), rotating, stratified fluids past a specified time-dependent irregular lower boundary. This model is representative of a class of nonhydrostatic atmospheric codes that employs the anelastic equations of motion in a terrain-following curvilinear framework, and contains parallel implementations of semi-Lagrangian and Eulerian approximations selectable by the user. The model has been employed in a variety of application; the quality of results suggest that modern nonoscillatory forward-in-time (NFT) methods are superior to the more traditional centered-in-time-and-space schemes, in terms of accuracy, computational efficiency, flexibility and robustness. The authors have extended the Cartesian NFT model to a mountainous sphere and, consequently, have dispensed with the traditional geophysical simplifications of hydrostaticity, gentle terrain slopes, and weak rotation. In this paper, they discuss the algorithmic design, relative efficiency and accuracy of several different variants (hydrostatic, nonhydrostatic, implicit, explicit, etc.) of the NFT global model. They substantiate their theoretical discussions with the results of simulations of idealized global orographic flows and climates.
Date: December 31, 1998
Creator: Smolarkiewicz, P.K.; Grubisic, V.; Margolin, L.G. & Wyszogrodzki, A.A.
Partner: UNT Libraries Government Documents Department