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Sylvie Gravel
and
Andrew Staniforth

Abstract

Within the context of a semi-Lagrangian shallow-water model, the dependence of forecast accuracy on the distribution of variable resolution and its robustness with respect to rapid variations in resolution is examined. This study also touches on the broader problem of designing a variable-resolution nested grid for regional modeling. It is demonstrated that the widely held belief that variable resolution induces severe noise problems at resolution interfaces—even for simple models—is not of universal applicability. In particular, no evidence of noise is found in the forecasts even when the resolution is changed abruptly by a factor of 3.5 across an internal boundary, thereby demonstrating the robustness of this particular variable-resolution technique. This result is achieved without any numerical smoothing technique other than that implicitly associated with the interpolation of a semi-Lagrangian scheme. The forecast produced on a uniform high-resolution mesh can be accurately reproduced for a limited time period on a subdomain at a fraction of the cost, by using a variable mesh where the resolution is gradually degraded away from this subdomain. The growth of the error variance when using such a mesh is an order of magnitude smaller than for one having the same number of degrees of freedom, except where the resolution changes abruptly at the boundary of the subdomain. It is concluded that variable resolution, using a smoothly varying mesh coupled with a semi-implicit, semi-Lagrangian integration scheme is an attractive approach to regional modeling.

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Sylvie Gravel
and
Andrew Staniforth

Abstract

By generalizing the algorithm of Priestley for passively advected fields, a mass-conserving scheme for the coupled shallow-water equations is obtained. It is argued that the interpolation step of semi-Lagrangian schemes is the principal reason for their lack of formal conservation. The corrections introduced by the proposed algorithm to achieve conservation appropriately reflect the localized nature of the interpolation errors that induce its violation.

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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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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

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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Jean Côté
,
Sylvie Gravel
,
André Méthot
,
Alain Patoine
,
Michel Roch
, and
Andrew Staniforth

Abstract

An integrated forecasting and data assimilation system has been and is continuing to be developed by the Meteorological Research Branch (MRB) in partnership with the Canadian Meteorological Centre (CMC) of Environment Canada. Part I of this two-part paper motivates the development of the new system, summarizes various considerations taken into its design, and describes its main characteristics.

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Jean Côté
,
Jean-Guy Desmarais
,
Sylvie Gravel
,
André Méthot
,
Alain Patoine
,
Michel Roch
, and
Andrew Staniforth

Abstract

An integrated forecasting and data assimilation system has been and is continuing to be developed by the Meteorological Research Branch (MRB) in partnership with the Canadian Meteorological Centre (CMC) of Environment Canada. Part II of this two-part paper presents the objective and subjective evaluations of the intercomparison process that led to the operational implementation of the new Global Environmental Multiscale model. The results of a “proof of concept” experiment and those of a meso-γ-scale simulation further demonstrate the validity and versatility of this model.

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Kao-San Yeh
,
Jean Côté
,
Sylvie Gravel
,
André Méthot
,
Alaine Patoine
,
Michel Roch
, and
Andrew Staniforth

Abstract

An integrated forecasting and data assimilation system has been and is continuing to be developed by the Meteorological Research Branch (MRB) in partnership with the Canadian Meteorological Centre (CMC) of Environment Canada. Part III of this series of papers presents the nonhydrostatic formulation and some sample results. The nonhydrostatic formulation uses Laprise's hydrostatic pressure as the basis for its vertical coordinate. This allows the departure from the hydrostatic formulation to be incorporated in an efficient switch-controlled perturbative manner. The time discretization of the model dynamics is (almost) fully implicit semi-Lagrangian, where all terms including the nonlinear terms are (quasi-) centered in time. The spatial discretization for the adjustment step employs a staggered Arakawa C grid that is spatially offset by half a mesh length in the meridional direction with respect to that employed in previous model formulations. It is accurate to second order, whereas the interpolations for the semi-Lagrangian advection are of fourth-order accuracy except for the trajectory estimation. The resulting set of nonlinear equations is solved iteratively using a motionless isothermal reference state that gives the usual semi-implicit problem as a preconditioner. The Helmholtz problem that needs to be solved at each iteration is vertically separable, the impact of nonhydrostatic terms being simply a renormalization of the separation constants. The convergence of this iterative scheme is not greatly modified by the nonhydrostatic perturbation. Three numerical experiments are presented to illustrate the model's performance. The first is a test to show that hydrostatic balance at low resolution is well maintained. The second one is a mild orographic windstorm case, where the flow should remain hydrostatic, to test that hydrostatic balance at high resolution is also well maintained. The third one is a convective case taken from the Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX). Although these results are encouraging, rigorous testing of the model's performance for strongly nonhydrostatic flows still remains to be done.

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Claude Girard
,
André Plante
,
Michel Desgagné
,
Ron McTaggart-Cowan
,
Jean Côté
,
Martin Charron
,
Sylvie Gravel
,
Vivian Lee
,
Alain Patoine
,
Abdessamad Qaddouri
,
Michel Roch
,
Lubos Spacek
,
Monique Tanguay
,
Paul A. Vaillancourt
, and
Ayrton Zadra

Abstract

The Global Environmental Multiscale (GEM) model is the Canadian atmospheric model used for meteorological forecasting at all scales. A limited-area version now also exists. It is a gridpoint model with an implicit semi-Lagrangian iterative space–time integration scheme. In the “horizontal,” the equations are written in spherical coordinates with the traditional shallow atmosphere approximations and are discretized on an Arakawa C grid. In the “vertical,” the equations were originally defined using a hydrostatic-pressure coordinate and discretized on a regular (unstaggered) grid, a configuration found to be particularly susceptible to noise. Among the possible alternatives, the Charney–Phillips grid, with its unique characteristics, and, as the vertical coordinate, log-hydrostatic pressure are adopted. In this paper, an attempt is made to justify these two choices on theoretical grounds. The resulting equations and their vertical discretization are described and the solution method of what is forming the new dynamical core of GEM is presented, focusing on these two aspects.

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