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Jean Côté and Andrew Staniforth

Abstract

In Côté and Staniforth the efficiency of a semi-implicit spectral model of the shallow-water primitive equations was significantly improved by replacing the usual three-time-level Eulerian treatment of advection by a two-time-level semi-Lagrangian one. The Côté and Staniforth model nevertheless suffers from the important disadvantage of all global spectral models; viz. prohibitive expense at high resolution associated with the cost of the Legendre transforms.

A scheme is proposed that considerably improves the computational performance of the Côté and Staniforth scheme at high resolution. This is achieved by replacing the spectral discretization by a finite-element one, which is referred to as pseudo-staggering. This spatial discretization scheme uses an unstaggered grid yet doesn't propagate small-scale energy in the wrong direction, and no ad hoc measures are taken to avoid pole problems.

The proposed model was tested by comparing its forecasts with those of both the Côté and Staniforth model and an independent high-resolution Eulerian spectral control model. It was found that one can stably and accurately integrate the new model with time steps as long as three hours (which is approximately 18 times longer than the limiting time step of an Eulerian spectral model at equivalent resolution), without recourse to any divergence damping and with no evidence of any pole problem.

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Abdessamad Qaddouri and Jean Côté

Abstract

A direct elliptic boundary value problem solver used for meteorological applications has been optimized. The problem to be solved is symmetric under a parity operation, and this is preserved by discretization. Therefore if the mesh possesses this symmetry, then the discretized problem will share this symmetry as well. The direct method can make use of this symmetry on a variable mesh to reduce the cost associated with the slow transform, a matrix product, by half. It is also shown that this can be combined with the Strassen–Winograd algorithm for even better results.

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Andrew Staniforth and Jean Côté

Abstract

The semi-Lagrangian methodology is described for a hierarchy of applications (passive advection, forced advection, and coupled sets of equations) of increasing complexity, in one, two, and three dimensions. Attention is focused on its accuracy, stability, and efficiency properties. Recent developments in applying semi-Lagrangian methods to 2D and 3D atmospheric flows in both Cartesian and spherical geometries are then reviewed. Finally, the current status of development is summarized, followed by a short discussion of future perspectives.

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Jean Côté and Andrew Staniforth

Abstract

Recently, it has been demonstrated that the semi-implicit semi-Lagrangian technique can be successfully coupled with a three-time-level spectral discretization of the barotropic shallow-water equations. This permits the use of time steps that are much larger than those permitted by the Courant-Friedrichs-Lewy (CFL) stability criterion for the corresponding Eulerian model, without loss of accuracy.

In this paper we show that it is possible to further quadruple the efficiency of semi-implicit semi-Lagrangian spectral models beyond that already demonstrated. A doubling of efficiency accrues from the use of the stable and accurate two-time-level scheme described herein. For semi-implicit semi-Lagrangian spectral models a further doubling of efficiency can be achieved by using a smaller computational Gaussian grid than the usual one, without incurring the significant loss of stability and accuracy that is observed for the corresponding Eulerian spectral model in analogous circumstances.

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Jean Côté, Michel Béland, and Andrew Staniforth

Abstract

The linear stability of vertical discretization schemes for semi-implicit primitive-equation models is thoroughly investigated. The equations are linearized about a stationary rotating basic state atmosphere that has a vertically shearing zonal wind. The amplification matrix of the finite-element model is constructed and its eigenvalues examined for possible instability. Investigating, the small time step limit of that matrix, we identify two operators whose eigenvectors are the “physical” and “computational” modes of the semi-implicit method, respectively, and whose eigenvalues are their frequencies. It is further shown that if the frequencies are real then the respective modes are stable. Switching off the rotation and horizontal advection in the above operators, we are able to state conditions on the implicit and explicit temperature profiles such that the unconditional instability is avoided (e.g., the so-called semi-implicit instability). These stability criteria may be easily extended to any type of vertical discretization.

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Ahmed Mahidjiba, Abdessamad Qaddouri, and Jean Côté

Abstract

Local conservation with the Semi-Lagrangian Inherently Conserving and Efficient (SLICE) transport method with a new trajectory algorithm is studied. Validation results of 1D and 2D passive advection with this new algorithm, which converges twice as fast as the old one, on the Arakawa C grid of a model in Cartesian coordinates are obtained. The effects of numerically computed divergence and trajectories on the results were also investigated. Random small-scale errors due to the divergence, especially with realistic winds, can be observed. The total mass is conserved, however, and is not affected since the results show clearly that SLICE ensures a perfect local conservation. This work represents the first step toward implementation of SLICE in the operational Canadian Global Environmental Multiscale (GEM) model.

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Jean Côté, Sylvie Gravel, and Andrew Staniforth

Abstract

It is known that straightforward finite-difference and finite-element discretizations of the shallow-water equations, in their primitive (uv) form, can lead to energy propagation in the wrong direction for the small scales. Two solutions to this problem have been proposed in the past. The first of these is to define the dependent variables on grids which are staggered with respect to one another, and the second is to use the governing equations in their differentiated (vorticity-divergence) form.

We propose a new scheme that works with the primitive form of the equations, uses an unstaggered grid but doesn't propagate small-scale energy in the wrong direction, works well with variable resolution, and is as computationally efficient as staggered formulations using the Primitive form of the equations. We refer to this approach as pseudostaggering since it achieve the benefits of a staggered formulation without a staggered placement of variables.

The proposed method has been tested using the two-time-level variable resolution finite-element semi-Lagrangian model of the shallow-water equations proposed by Temperton & Staniforth (1987). Our new pseudostaggered scheme yields high accuracy with time steps as long as three hours; the rms height and wind differences are smaller than or comparable to those of the Temperton and Staniforth scheme as well as to those of its semi-implicit Eulerian analogue with a much smaller time step. It leads to a 20% reduction in computational cost of the very efficient two-time-level semi-Lagrangian Temperton & Staniforth algorithm, and is an order-of-magnitude faster than its semi-implicit Eulerian analogue.

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Michel Béland, Jean Côté, and Andrew Staniforth

Abstract

The accuracy of a slightly modified version of the finite-element vertical discretization scheme first described in Staniforth and Daley is studied with respect to a set of Rossby and gravity analytical normal modes obtained as solutions of a linearized primitive equation model. The scheme is also compared to a second-order, staggered, finite-difference vertical discretization scheme. The results of these comparisons are in favor of the finite-element method as far as accuracy is concerned. In terms of computation time, both methods are identical.

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Sylvie Gravel, Andrew Staniforth, and Jean Côté

Abstract

The computational stability of a family of recently introduced semi-Lagrangian schemes for baroclinic models is analyzed to better explain their observed behavior and to provide additional theoretical justification. The linear stability analysis is a generalization of that presented in Bates et al. that includes the important impact of evaluating certain (nonlinear) terms using extrapolated quantities.

There are three sets of physical modes, namely, the usual gravity and slow (“Rossby”) modes, corresponding to the three solutions of a third-order (in time) normal-mode differential equation. For one-, two-, and three- term extrapolation of quantities, there are also zero, one, and two computational modes, respectively, since the normal-mode difference equation is then of higher order than third. The following conclusions hold equally well for both the Bates et al. and McDonald and Haugen model formulations, which although different in detail behave very similarly.

The slow mode is stable and slightly damped (by interpolation) for all schemes, both with and without extrapolated terms, and the gravity modes are unconditionally stable in the absence of extrapolated terms. When the extrapolated terms are included, however, the gravity modes become unstable in the absence of damping mechanisms. Introducing both divergence damping and a time decentering of the scheme (with a judicious choice of coefficients) stabilizes these modes. The time decentering is the more efficient of these two damping mechanisms, and the values of the coefficients required for computational stability as determined from our analysis agree well with those determined empirically in the Bates et al. and McDonald and Haugen studies. Two-term extrapolation is to be preferred to both one- and three-term extrapolation, since the former is insufficiently accurate, whereas the latter requires unacceptably large damping coefficients.

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Jean Côté, Sylvie Gravel, and Andrew Staniforth

Abstract

The one-parameter three-time-level family of Ot 2)-accurate schemes, introduced in Rivest et al. to address the problem of the spurious resonant response of semi-implicit semi-Lagrangian schemes at large Courant number, has been generalized to a two-parameter family by introducing the possibility of evaluating total derivatives using an additional time level. The merits of different members of this family based on both theory and results are assessed. The additional degree of freedom might be expected a priori to permit a reduction of the time truncation errors while still maintaining stability and avoiding spurious resonance. Resonance, stability, and truncation error analyses for the proposed generalized family of schemes are given. The subfamily that is formally Ot 3)-accurate is unfortunately unstable for gravity modes. Sample integrations for various members of the generalized family are shown. Results are consistent with theory, and stable nonresonant forecasts at large Courant number are possible for a range of values of the two free parameters. Of the two methods proposed in Rivest et al. for computing trajectories, the one using a piecewise-defined trajectory is to be preferred to that using a single great-circle arc since it is more accurate at a large time step for some members of the generalized family.

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