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Claude Girard and Yves Delage

Abstract

The intensity of vertical mixing in atmospheric models generally depends on wind shear and static stability, making the diffusion process nonlinear. Traditional implicit numerical schemes, which treat the variables to be diffused implicitly but the diffusion coefficients explicitly, are shown to be only conditionally stable. Instability arises in statically stable conditions with an increase of the vertical resolution or of the time step. Stable schemes are derived whose principal characteristic is to take into account the variation of the diffusion coefficient with respect to the basic variables. One scheme looks like a traditional scheme in which the parameter that determines how implicit the calculations are done is made to vary locally instead of being a constant. This insures stability and at the same time provides optimum accuracy. This scheme did remove spurious oscillations found in the Canadian spectral weather forecasting model.

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René Laprise and Claude Girard

Abstract

A numerical scheme for the vertical discretization of primitive equations in a generalized pressure-type coordinate is developed through application of the Galerkin formalism with piecewise-constant finite elements: this methodology affords an elegant—and direct—mean of formulating conservative discretization schemes without the arbitrariness that usually characterizes the development of finite differences. The form of the resulting semidiscrete equations is equivalent to some second-order accurate finite-difference approximation to the continuous equations. Flexibility of this scheme in the choice of different layers for projecting the thermodynamic and momentum variables effectively allows for staggering of these variables in the vertical.

Numerical integrations performed with this scheme at various vertical resolutions have revealed the sensitivity of the simulated circulation to resolution in the lower stratosphere. We found that application of the “lid” upper boundary condition at a finite height alleviates a documented bias in the estimation by this scheme of the thermal wind relationship at upper level with coarse resolution, and this is accomplished here without sacrificing the conservation properties of the scheme.

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Syed Zahid Husain and Claude Girard

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Inconsistencies may arise in numerical weather prediction models—that are based on semi-Lagrangian advection—when the governing dynamical and the kinematic trajectory equations are discretized in a dissimilar manner. This study presents consistent trajectory calculation approaches, both in the presence and absence of off-centering in the discretized dynamical equations. Both uniform and differential off-centering in the discretized dynamical equations have been considered. The proposed consistent trajectory calculations are evaluated using numerical experiments involving a nonhydrostatic two-dimensional theoretical mountain case and hydrostatic global forecasts. The experiments are carried out using the Global Environmental Multiscale model. Both the choice of the averaging method for approximating the velocity integral in the discretized trajectory equations and the interpolation scheme for calculating the departure positions are found to be important for consistent trajectory calculations. Results from the numerical experiments confirm that the proposed consistent trajectory calculation approaches not only improve numerical consistency, but also improve forecast accuracy.

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Claude Girard, Robert Benoit, and Michel Desgagné

Abstract

The Canadian Mesoscale Compressible Community (MC2) model provided daily forecasts across the Alps at 3-km resolution during the Mesoscale Alpine Programme (MAP) field phase of 1999. Among the results of this endeavor, some have had an immediate impact on MC2 itself as it increasingly became evident that the model was spuriously too sensitive to finescale orographic forcing. The model solves the Euler equations of motion using a semi-implicit semi-Lagrangian scheme in an oblique terrain-following coordinate. To improve model behavior, typical approaches were tried at first. These included a generalization of the coordinate transformation to make the terrain influence decay much more quickly with height as well as the introduction of nonisothermal basic states to diminish the amplitude of numerical truncation errors. The concept of piecewise-constant finite elements was invoked to reduce coding arbitrariness. But it was later pointed out that the problem was very specific and due to a numerical inconsistency. The true height of model grid points is fixed and known in height-based coordinates. Nevertheless, it was discovered that for this semi-Lagrangian scheme to be consistent, the departure height is an unknown that must be obtained in the same manner as the other unknowns.

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Vincent Vionnet, Stéphane Bélair, Claude Girard, and André Plante

Abstract

Numerical weather prediction (NWP) systems operational at many national centers are nowadays used at the kilometer scale. The next generation of NWP models will provide forecasts at the subkilometer scale. Large impacts are expected in mountainous terrain characterized by highly variable orography. This study investigates the ability of the Canadian NWP system to provide an accurate forecast of near-surface variables at the subkilometer scale in the Canadian Rocky Mountains in wintertime when the region is fully covered by snow. Observations collected at valley and high-altitude stations are used to evaluate forecast accuracy at three different grid spacing (2.5, 1, and 0.25 km) over a period of 15 days. Decreasing grid spacing was found to improve temperature forecasts at high-altitude stations because of better orography representation. In contrast, no improvement is obtained at valley stations due to an inability of the model to fully capture at all resolutions the intensity of valley cold pools forming during nighttime. Errors in relative humidity reveal that the model tends to overestimate relative humidity at all resolutions, without improvement with decreasing grid spacing. Wind speed forecasts show large improvements with decreasing grid spacing for high-altitude stations exposed to or sheltered from wind. However, no systematic improvement with decreasing grid spacing is found for all stations, which is similar to previous studies. In addition, the model’s sensitivity at subkilometer grid spacing is investigated by evaluating the effects of (i) accounting for additional drag generated by subgrid orographic features, (ii) considering slope angle and aspect on surface radiation, and (iii) using high-resolution initialization for the surface fields.

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Ron McTaggart-Cowan, Claude Girard, André Plante, and Michel Desgagné

Abstract

The importance of stratospheric influences for medium-range numerical weather prediction (NWP) of the troposphere has led to increases in the heights of global model domains at operational centers around the world. Grids now routinely extend to 0.1 hPa (approximately 65 km) in these systems, thereby covering the full depth of the stratosphere and the lower portion of the mesosphere. Increasing the vertical extent of higher-resolution limited-area models (LAMs) nested within the global forecasts is problematic because of the computational cost of additional levels and the possibility of inaccuracy or instability in the high-speed stratospheric jets. An upper-boundary nesting (UBN) technique is developed that allows information from high-topped driving grids to influence the evolution of a lower-topped (~10 hPa) LAM integration in a manner analogous to the treatment of lateral boundary conditions.

A stratospheric vortex displacement event in the winter of 2007 is used to study the effectiveness of the UBN technique. Tropospheric blocking over Europe leads to the development of an amplifying planetary-scale wave in the lower stratosphere that culminates in an anticyclonic wave break over Asia and a marked increase of wave-1 asymmetry. The rapid evolution of stratospheric potential vorticity (PV) is poorly represented in low-topped models, resulting in PV-induced forecast height errors throughout the depth of the troposphere on time scales as short as 2–5 days. Application of the UBN technique is shown to be an effective way for low-topped configurations to benefit from stratospheric predictability without the problems associated with the inclusion of the stratospheric flow in the higher-resolution model domain. The robustness and relative ease of implementation of the UBN technique may make this computationally inexpensive strategy attractive for a wide range of NWP applications.

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Stéphane Bélair, Jocelyn Mailhot, Claude Girard, and Paul Vaillancourt

Abstract

The role and impact that boundary layer and shallow cumulus clouds have on the medium-range forecast of a large-scale weather system is discussed in this study. A mesoscale version of the Global Environmental Multiscale (GEM) atmospheric model is used to produce a 5-day numerical forecast of a midlatitude large-scale weather system that occurred over the Pacific Ocean during February 2003. In this version of GEM, four different schemes are used to represent (i) boundary layer clouds (including stratus, stratocumulus, and small cumulus clouds), (ii) shallow cumulus clouds (overshooting cumulus), (iii) deep convection, and (iv) nonconvective clouds. Two of these schemes, that is, the so-called MoisTKE and Kuo Transient schemes for boundary layer and overshooting cumulus clouds, respectively, have been recently introduced in GEM and are described in more detail.

The results show that GEM, with this new cloud package, is able to represent the wide variety of clouds observed in association with the large-scale weather system. In particular, it is found that the Kuo Transient scheme is mostly responsible for the shallow/intermediate cumulus clouds in the rear portion of the large-scale system, whereas MoisTKE produces the low-level stratocumulus clouds ahead of the system. Several diagnostics for the rear portion of the system reveal that the role of MoisTKE is mainly to increase the vertical transport (diffusion) associated with the boundary layer clouds, while Kuo Transient is acting in a manner more consistent with convective stabilization. As a consequence, MoisTKE is not able to remove the low-level shallow cloud layer that is incorrectly produced by the GEM nonconvective condensation scheme. Kuo Transient, in contrast, led to a significant reduction of these nonconvective clouds, in better agreement with satellite observations. This improved representation of stratocumulus and cumulus clouds does not have a large impact on the overall sea level pressure patterns of the large-scale weather system. Precipitation in the rear portion of the system, however, is found to be smoother when MoisTKE is used, and significantly less when the Kuo Transient scheme is switched on.

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Christoph Schär, Daniel Leuenberger, Oliver Fuhrer, Daniel Lüthi, and Claude Girard

Abstract

Most numerical weather prediction models rely on a terrain-following coordinate framework. The computational mesh is thus characterized by inhomogeneities with scales determined by the underlying topography. Such inhomogeneities may affect the truncation error of numerical schemes. In this study, a new class of terrain-following coordinate systems for use in atmospheric prediction models is proposed. Unlike conventional systems, the new smooth level vertical (SLEVE) coordinate yields smooth coordinates at mid- and upper levels. The basic concept of the new coordinate is to employ a scale-dependent vertical decay of underlying terrain features. The decay rate is selected such that small-scale topographic variations decay much faster with height than their large-scale counterparts. This generalization implies a nonlocal coordinate transformation. The new coordinate is tested and compared against standard sigma and hybrid coordinate systems using an idealized advection test. It is demonstrated that the presence of coordinate transformations induces substantial truncation errors. These are critical for grid inhomogeneities with wavelengths smaller than approximately eight grid increments, and may overpower the regular-grid truncation error of the underlying finite-difference approximation. These results are confirmed by a theoretical analysis of the truncation error. In addition, the new coordinate is tested in idealized and real-case numerical experiments using a nonhydrostatic model. The simulations using the new coordinate yield a substantial reduction of small-scale noise in dynamical and thermodynamical model fields.

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Syed Zahid Husain, Claude Girard, Abdessamad Qaddouri, and André Plante

Abstract

A new dynamical core of Environment and Climate Change Canada’s Global Environmental Multiscale (GEM) atmospheric model is presented. Unlike the existing log-hydrostatic-pressure-type terrain-following vertical coordinate, the proposed core adopts a height-based approach. The move to a height-based vertical coordinate is motivated by its potential for improving model stability over steep terrain, which is expected to become more prevalent with the increasing demand for very high-resolution forecasting systems. A dynamical core with height-based vertical coordinate generally requires an iterative solution approach. In addition to a three-dimensional iterative solver, a simplified approach has been devised allowing the use of a direct solver for the new dynamical core that separates a three-dimensional elliptic boundary value problem into a set of two-dimensional independent Helmholtz problems. The issue of dynamics–physics coupling has also been studied, and incorporating the physics tendencies within the discretized dynamical equations is found to be the most acceptable approach for the height-based vertical coordinate. The new dynamical core is evaluated using numerical experiments that include two-dimensional nonhydrostatic theoretical cases as well as 25-km resolution global forecasts. For a wide range of horizontal grid resolutions—from a few meters to up to 25 km—the results from the direct solution approach are found to be equivalent to the iterative approach for the new dynamical core. Furthermore, results from the different numerical experiments confirm that the new height-based dynamical core is equivalent to the existing pressure-based core in terms of solution accuracy.

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Abdelaziz Beljadid, Abdolmajid Mohammadian, Martin Charron, and Claude Girard

Abstract

In this paper, theoretical and numerical analyses of the properties of some complex semi-Lagrangian methods are performed to deal with the issues of the instability associated with the treatment of the nonlinear part of the forcing term. A class of semi-Lagrangian semi-implicit schemes is proposed using a modified TR-BDF2 method, which is the combination of the trapezoidal rule (TR) and the second-order backward differentiation formula (BDF2). The process used for the nonlinear term includes two stages as predictor and corrector in the trapezoidal method and one stage for the BDF2 method. For the treatment of the linear term, the implicit trapezoidal method is employed in the first step, the explicit trapezoidal method in the second step, and the implicit BDF2 method in the third step. The combination of these techniques leads to a family of schemes that has a large region of absolute stability, performs well for the purely oscillatory cases, and has good qualities in terms of accuracy and convergence. The use of the explicit method for the linear term in the second step makes the proposed class of schemes competitive in terms of efficiency compared to some well-known schemes that use two steps. Numerical experiments presented herein confirm that the proposed class of schemes performs well in terms of stability, accuracy, convergence, and efficiency in comparison with other, previously known, semi-Lagrangian semi-implicit schemes and semi-implicit predictor–corrector methods. The potential practical application of the proposed class of schemes to a weather prediction model or any other atmospheric model is not discussed and could be the subject of other forthcoming studies.

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