Andrew Staniforth

The Operational CMC–MRB Global Environmental Multiscale (GEM) Model. Part I: Design Considerations and Formulation

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.

Semi-Lagrangian Integration Schemes for Atmospheric Models—A Review

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.

The Operational CMC–MRB Global Environmental Multiscale (GEM) Model. Part II: Results

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.

The Conversion of Semi-Lagrangian Advection Schemes to Quasi-Monotone Schemes

Abstract A method to convert conventional semi-Lagrangian schemes into quasi-monotone schemes is given. Numerical examples with linear and nonlinear transport equations demonstrate the ability of our modified semi-Lagrangian schemes to better maintain the shape of the solution in the presence of shocks and discontinuities.

The CMC–MRB Global Environmental Multiscale (GEM) Model. Part III: Nonhydrostatic Formulation

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 Laprises 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 interpolat...

A Variable-Resolution Finite-Element Technique for Regional Forecasting with the Primitive Equations

Abstract A barotropic primitive-equation model using the finite-element method of space discretization is generalized to allow variable resolution. The overhead incurred in going from a uniform mesh to a variable mesh having the same number of degrees of freedom is found to be approximately 20% overall. The variable-mesh model is used with several grid configurations, each having uniform high resolution over a specified area of interest and lower resolution elsewhere to produce short-term forecasts over this area without the necessity of high resolution everywhere. It is found that the forecast produced on a uniform high-resolution mesh can be essentially reproduced for a limited time over the limited area by a variable-mesh model having only a fraction of the number of degrees of freedom and requiring significantly less computer time. As expected, the period of validity of forecasts on variable meshes can be lengthened by refining the mesh in the outer region. It is concluded that from the point of view ...

Spurious resonant response of semi-Lagrangian discretizations to orographic forcing: Diagnosis and solution

Abstract Semi-Lagrangian semi-implicit techniques are now well established and used by an increasing number of meteorological centers. However, it is demonstrated by both analysis and numerical integration that there is a serious problem incorporating orographic forcing into semi-Lagrangian models, since spurious resonance can develop in mountainous regions for Courant numbers larger than unity. A solution, consisting of two classes of schemes, is proposed, analyzed, and then evaluated using a global shallow-water model. Simply off-centering the semi-implicit scheme eliminates the spurious resonances. Although this can be achieved with a first-order scheme, it is at the expense of decreased accuracy, and therefore a second-order scheme is recommended.

A Two-Time-Level Semi-Lagrangian Semi-implicit Scheme for Spectral Models

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.

Finite elements for shallow-water equation ocean models

The finite-element spatial discretization of the linear shallow-water equations on unstructured triangular meshes is examined in the context of a semi-implicit temporal discretization. Triangular finite elements are attractive for ocean modeling because of their flexibility for representing irregular boundaries and for local mesh refinement. The semi-implicit scheme is beneficial because it slows the propagation of the high-frequency small-amplitude surface gravity waves, thereby circumventing a severe time step restriction. High-order computationally expensive finite elements are, however, of little benefit for the discretization of the terms responsible for rapidly propagating gravity waves in a semi-implicit formulation. Low-order velocity/surface-elevation finite-element combinations are therefore examined here. Ideally, the finite-element basis-function pair should adequately represent approximate geostrophic balance, avoid generating spurious computational modes, and give a consistent discretization of the governing equations. Existing finite-element combinations fail to simultaneously satisfy all of these requirements and consequently suffer to a greater or lesser extent from noise problems. An unconventional and largely unknown finite-element pair, based on a modified combination of linear and constant basis functions, is shown to be a good compromise and to give good results for gravity-wave propagation.

The Canadian Regional Data Assimilation System: Operational and Research Applications

Abstract The Canadian regional data assimilation system is described. It is a spinup cycle designed to provide the regional finite-element forecast model with more detailed analyses in a dynamically consistent manner. Its operational performance is evaluated using performance statistics, and a case study is presented to highlight some of the benefits. These include analyses that better fit the data and more detailed and accurate forecasts, particularly for precipitation. The system also benefits research applications. To illustrate this the authors describe the preparation of the first set of analysts for the international COMPARE (Comparison of Mesoscale Prediction and Research Experiments) Project. The scientific interest of this explosive marine cyclogenetic case is discussed, together with a useful methodology for determining the minimum domain size required by a regional model to avoid forecast contamination from lateral boundaries.