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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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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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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
,
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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Bertrand Denis
,
Jean Côté
, and
René Laprise

Abstract

For most atmospheric fields, the larger part of the spatial variance is contained in the planetary scales. When examined over a limited area, these atmospheric fields exhibit an aperiodic structure, with large trends across the domain. Trying to use a standard (periodic) Fourier transform on regional domains results in the aliasing of large-scale variance into shorter scales, thus destroying all usefulness of spectra at large wavenumbers. With the objective of solving this particular problem, the authors have evaluated and adopted a spectral transform called the discrete cosine transform (DCT). The DCT is a widely used transform for compression of digital images such as MPEG and JPEG, but its use for atmospheric spectral analysis has not yet received widespread attention.

First, it is shown how the DCT can be employed for producing power spectra from two-dimensional atmospheric fields and how this technique compares favorably with the more conventional technique that consists of detrending the data before applying a periodic Fourier transform. Second, it is shown that the DCT can be used advantageously for extracting information at specific spatial scales by spectrally filtering the atmospheric fields. Examples of applications using data produced by a regional climate model are displayed. In particular, it is demonstrated how the 2D-DCT spectral decomposition is successfully used for calculating kinetic energy spectra and for separating mesoscale features from large scales.

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Andrew Staniforth
,
Nigel Wood
, and
Jean Côté

Abstract

Four schemes (referred to here as explicit, implicit, split-implicit, and symmetrized split-implicit) for coupling physics parameterizations to the dynamical core of numerical weather and climate prediction models have been studied in the context of a simplified, canonical model problem. This problem models the dynamics by a representation of the terms responsible for gravitational oscillations and models the physics by both a constant forcing term and a linear damping term, representative of horizontal or vertical diffusion. The schemes have been analyzed in terms of their numerical stability and accuracy. Two of the schemes (the explicit and split-implicit) have been studied previously in the context of a three-time-level discretization. Those results are confirmed here for a two-time-level discretization. The two other schemes (the implicit and the novel symmetrized split-implicit) are both found to be second-order accurate and unconditionally stable, and both represent improvements over the explicit and split-implicit schemes. The symmetrized split-implicit has the additional advantage over the implicit scheme, for simplicity and computational efficiency, of separating the physics and dynamics steps from each other. The canonical problem considered here is a considerable simplification of any real physics–dynamics coupling, which limits the generality of the conclusions drawn. However, such simplification allows detailed analysis of some important aspects and motivates further work on both broadening and deepening understanding of physics–dynamics coupling issues.

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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

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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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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