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Preliminary Results from aDry Global Variable-ResolutionPrimitive Equations ModelJean Côté a , Sylvie Gravel a , André Méthotb , Alain Patoine b , Michel Roch a & AndrewStaniforth aa Recherche en prévision numériqueb Centre météorologique canadien Service del'environnement atmosphérique , 2121 RouteTranscanadienne, porte 500, Dorval, Québec,Canada , H9P 1J3Published online: 26 Jul 2011.

To cite this article: Jean Côté , Sylvie Gravel , André Méthot , Alain Patoine ,Michel Roch & Andrew Staniforth (1997) Preliminary Results from a Dry GlobalVariable-Resolution Primitive Equations Model, Atmosphere-Ocean, 35:sup1,245-259, DOI: 10.1080/07055900.1997.9687350

To link to this article: http://dx.doi.org/10.1080/07055900.1997.9687350

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Preliminary Results from a Dry Global Variable-Resolution Primitive Equations Mode1

Jean Côté*, Sylvie Gravel*, André Méthott, Alain Patoinet, Michel Roch* and Andrew Staniforth*

‘Recherche en prévision numérique et t Centre météorologique canadien Service de l’environnement atmosphérique, 2121 Route Transcanadienne,

porte 500 - Dorval, Québec, Canada H9P 153

[Original manuscript received 4 February 1995; in revised form 17 July 19951

ABSTRA~ The viability of a proposed global variable-grid strategy was previously tested (Côté et al., 1993) using a shallow-water-equations protozype. It was demonstrated that by using a global variable-resolution mesh, a high-resolution short-tenn forecast cari be ob- tained for a region of interest at a fraction of the cost of using uniformly-high resolution everywhere. The prototype is generalized here to use the hydrostatic primitive equations. Preliminary results obtained with an adiabatic version of this baroclinic mode1 are pre- sented. They confrm the potential of the proposed strategy: dzjterences between the 48-h forecasts for the 500 hPa geopotential height and mean-sea-level’pressure jïekis obtained from a uniform 1.2” model, and those obtained from a variable-mesh mode1 with equivalent resolution on a 81.6” x 60” sub-domain, are small.

RÉSUMÉ Précédemment (Côté et al., 1993) un modèle modial des équations de St-Venant a permis de tester la viabilité d’une strategie à resolution variable et demontrer qu’une prévision de courte &Ch&ance à haute résolution peut être obtenue pour une région donnke à une fraction du cotît de la prkvision effectuke à résolution uniforme Equivalente. Ici, ce prototype est généralisé awr équations primitives hydrostatiques. Les resultats préliminaires de la version adiabatique de ce modèle barocline sont presentes. Ils conjrment le potentiel de la stratégie que nous proposons: les differences entre des previsions de 48 heures du géopotentiel à 500 hPa et de la pression au niveau moyen de la mer obtenues avec le modele à une résolution uniforme de 1,2”, et celles obtenues avec le modèle à resolution variable ayant une résolution équivalente sur une fenêtre de 816” x 60”, sont petites.

1 Introduction

A new mode1 is being developed to meet the operational weather forecasting needs of Canada for the coming years. These presently include short-range regional fore- casting, medium-range global forecasting and data assimilation. In the future they

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will probably include nowcasting at the mesoscale and dynamic extended-range forecasting on monthly to seasonal time-scales.

The Canadian Meteorological Centre currently runs two operational data assimi- lation and forecasting cycles, as do a number of other national weather forecasting centres. The first of these addresses medium-range needs by using a global data assimilation system (Mitchell et al., 1993) and a global spectral forecast model (Ritchie and Beaudoin, 1994). It also initializes the 12-h regional data assimilation spin-up system (Chouinard et al., 1994) of the second cycle, which provides the higher-resolution regional analyses used by the variable resolution finite-element model (Mailhot et al., 1994) to produce more detailed short-range (<2 days) fore- casts over N. America and some of its adjacent waters. This two-cycle strategy requires the maintenance, improvement and optimization of two sets of libraries and procedures. It has motivated the definition of a new strategy to consolidate these two complementary aspects of weather forecasting within a single model framework, and thereby rationalize resource utilization.

The essence of the approach (C&Z et al., 1993) is to develop a single highly efficient model that can be reconfigured at run time to either run globally at uniform resolution (with possibly degraded resolution in the southern hemisphere), or to run with variable resolution over a global domain such that high resolution is focused over an area of interest. To demonstrate the potential of this strategy, a shallow- water global variable-resolution prototype has been developed (C&C et al., 1993) that uses an arbitrarily rotated latitude-longitude mesh. In this way a high-resolution subdomain can be focused on any geographical area of interest, making it a more flexible strategy than the operational finite-element regional model, which is best suited to extra-tropical applications. This shallow water model has been successfully integrated on a variety of meshes right down to the meso-gamma scale (with 250 m resolution over a 100 km x 100 km subdomain). An important conclusion is that the overhead of using a model of global extent for short-range forecasting over an area of interest is no worse than that associated with the sponge regions of traditional limited-area models.

The goal of this paper is to briefly describe a generalization of the shallow water formulation to the dry hydrostatic primitive equations, and to give some preliminary results from this baroclinic prototype. This is an important step towards the ultimate goal, a full-physics non-hydrostatic global Euler equations model using a pressure- type hybrid vertical coordinate (Laprise, 1992).

2 Model formulation

a Governing Equations The governing equations (Kasahara, 1974; Simmons and Burridge, 1981) are the hydrostatic primitive equations for frictionless adiabatic flow on a rotating sphere:

dV dt+RTV lnp+V4+f(kxV)=O, (1)

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ap iln- +v.v+--0, I I

ti a-rl atl

(2)

where

and

$[ln(p-?)I = 0,

a+ a lnp -=-Ry-- a atl’

d a a Z = at -+v-v+q-,

ti

(3)

(4)

(3

q= p--pT PS --PT’

(6)

is the terrain-following vertical coordinate. In the above, (l)-(4) are respectively the horizontal momentum, continuity, thermodynamic and hydrostatic equations. Also: V is the horizontal velocity, T is temperature, p is pressure, ps and pT are its respective values at the surface and at the top of the model atmosphere, t$ = gz is the geopotential height,f is the Coriolis parameter, K = R/c,,, and R and cP are respectively the gas constant and specific heat of dry air.

The boundary conditions are periodicity in the horizontal; and no motion across the top and bottom of the atmosphere, where the top is at constant pressure pr. Thus

(7)

b Temporal Discretization As a preparatory step for the time discretization, the thermodynamic equation (3) is rewritten as

2 [In($) -icln(:)] -tcti-&(lnp*)=O, (8)

where

p* = p; + (P; - PFbl, (9)

and pi and p; = pr are the respective bottom and top constant pressures of a motionless isothermal (T* = constant) reference atmosphere. Equation (9) is a

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direct consequence of the definition (6) of the vertical coordinate, and (8) is written this way to ensure the computational stability of the gravitational oscillations.

The time discretization is fully-implicit/semi-Lagrangian. This was inspired by the introduction by Andre Robert of the semi-implicit semi-Lagrangian scheme (Robert, 198 1, 1982) and its further development (see Staniforth and CBtC, 1991; Staniforth, this volume). Indeed, half the papers in this Andre .I. Robert memo- rial volume were also inspired in part by his work on the subject (Benoit et al.; Cullen et al.; Daley; Geleyn and Bubnova, Kaas et al.; Laprise et al.; Leslie and Purser; Machenhauer and Olk; McGregor; Ritchie; Smolarkiewicz and Margolin; Temperton; Williamson).

Consider a prognostic equation of the form

dF z+G=O,

where F represents one of the prognostic quantities {V, ln]ap/&ll, ln(T/T*) - K ln(p/p*)}, and G represents the remaining terms, some of which are nonlinear. Such an equation is approximated by time differences and weighted averages along a trajectory determined by an approximate solution to

dx3 - = V3(x3, 0, dt

where x3 and V3 are the three-dimensional position and velocity vectors respec- tively. Thus

(F” - F”-1)

At =o, (12)

where !IP = ~(x3, t), v-“’ = @x3(t - mAt), t - mAt], w = { F,G}, t = nAt. The upstream position of the trajectory at time t - At is computed (Temperton and Staniforth, 1987; McDonald and Bates, 1987) using winds extrapolated at meshpoints from times t - At and t - 2At to time t -At/2. The upstream position of the trajectory at time t - 2Ar is then obtained (Rivest et al., 1994) by extending the trajectory back along a great circle.

Note that this scheme is decentred along the trajectory as in Rivest et al. (1994), in anticipation of the introduction of orography (not included in the results of the present paper). In this way the spurious resonant response arising from a cen- tred approximation in the presence of orography will be avoided. This decentred scheme nevertheless shares the O(At*) accuracy of the more usual centred schemes: an O(Ar) decentred scheme would also address the spurious resonance problem but would unduly limit the maximum permissible time step for reasons of accu- racy (Rivest et al., 1994). Cubic interpolation is used everywhere for upstream evaluations (c.f. 12) except for the trajectory computations (c.f. 1 l), where linear interpolation is used with no visible degradation in the results.

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Fig. 1 Schematic for horizontal placement of variables. Circled points: p. I$. Crossed points: V, T, +-I.

Grouping terms at the new time on the left-hand side and known quantities on the right-hand side, (12) may be rewritten as

(F+ ;*tG)” = F”-’ - $ &G”-2. (13)

This yields a set of coupled nonlinear equations for the unknown quantities at the meshpoints of a regular grid at the new time t, the efficient solution of which is discussed below. A fully-implicit time treatment, such as that adopted here, of the nonlinear terms has the useful property of being inherently computationally more stable than an explicit one (e.g. those of Bates et al., 1993 and McDonald and Haugen, 1992, whose computational stability is analysed in Gravel et al., 1993).

c Spatial Discretization A variable-resolution finite-element discretization, based on that described in Cot6 et al. (1993), is used in the horizontal with a placement of variables as shown schematically in Fig. 1. It has the advantage that only one set of trajectories is required since all of the right-hand sides of the equations are evaluated at cell centres before interpolation. Other placements are, however, possible and are currently

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TABLE I. Experimental configurations.

Mesh Coordinate System Experiment Uniform/Variable Rotated?

I Uniform No 2 Uniform Yes 3 Variable Yes

being examined. The vertical discretization is modelled after that of Tanguay et al. (1989).

d Solving the Coupled Nonlinear Set of Discretized Equations After spatial discretization the coupled set of nonlinear equations still has the form of (13). Terms on the right-hand side, which involve upstream interpolation, are evaluated once and for all. The coupled set is rewritten as a linear one (where the coefficients depend on the basic state) plus a perturbation which is placed on the right-hand side and which is relatively cheap to evaluate. This set is then solved iteratively using the linear terms as a kernel, and the nonlinear terms on the right-hand sides are re-evaluated at each iteration using the most-recent values. The linear set can be algebraically reduced to the solution of a three-dimensional elliptic-boundary-value problem. In practice the cost of solving the coupled set of nonlinear equations is only marginally more expensive than the iterative solution of the variable-coefficient linear set. The most significant contribution to the present cost of a time step is that of interpolation, which is the same regardless of whether the coupled set of equations is linear or nonlinear.

3 Preliminary results

a Methodology To make a preliminary assessment of the global variable-mesh strategy described above, we performed three 48-h integrations starting from the same initial data, the Canadian Meteorological Centre analysis valid at 12 UTC 12 February 1993. For all integrations, the (adiabatic) model was run with 23 vertical levels (pi = 10 hPa) using a 30-min time step and a Laplacian diffusion with a coefficient of 1.5 x 16 m2 s-‘. There was no topography and there were no heat or momentum fluxes.

The configurations of the three experiments are summarized in Table 1. The purpose of the uniform-resolution experiments is to validate a uniform-resolution ground truth (Expt. 2) against which the variable-resolution forecast of Expt. 3 can be compared and evaluated. Both uniform-resolution experiments were performed at 1.2” resolution (in both longitude and latitude), using either a mesh with the poles of the coordinate system coincident with respect to the geographical ones (Fig. 2a), or rotated (i.e., oriented as in Fig. 2b, but with resolution uniform everywhere). Note that for pictorial clarity, only every second latitude and longitude is plotted in Fig. 2. Ideally it would have been preferable to run these experiments at the

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Fig. 2 (a) The uniform 1.2” resolution 300 x 151 mesh used for Expt. 1 and (after rotatior

for clarity only every 2nd point in each direction is plotted. (b) A variable-resolution

me *sh having an 81.6” x 60’ window of uniform 1.2’ resolution, centred on (103”’

ant II used for Expt. 3: for clarity only every 2nd point in each direction is plotted.

I) Expt. 2:

119 x 87

W, 51’N),

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same 0.5” resolution as for the analogous ones presented in Cot6 et al. (1993) using the shallow-water prototype. However, at the present stage of development the use of computer memory by the model has not yet been sufficiently optimized to permit this for the uniform grid. This limits the agreement between the results of the experiments due to the more rapid growth of the spatial truncation errors outside the window of uniform resolution, and their advection into the region of interest.

The variable-resolution Expt. 3 was performed on the mesh depicted in Fig. 2b, where the poles of the coordinate system are rotated with respect to the ge- ographical ones. Its purpose is to demonstrate our thesis that we can reasonably well (given the above-mentioned current limit on resolution) reproduce the fore- cast over the 1.2” uniform resolution window at a fraction of the cost of using 1.2“ uniform resolution everywhere. The resolution of the variable mesh degrades smoothly away in each direction (each successive meshlength is approximately 11% larger than its predecessor) from an 81.6” x 60” uniform-resolution (1.2”) window centred on a point of the equator of a rotated coordinate system, located at (103”W, 51”N) in geographical coordinates. Uniform resolution again refers to uniform spacing in latitude and longitude: however, the meshpoints of the window are also almost uniformly spaced over the sphere with a meshlength that varies between approximately 114 and 132 km.

b Uniform Resolution Experiments The 500-hPa height and mean sea level pressure (mslp) fields of the initial analysis used for the experiments are shown in Fig. 3. The resulting 2-day forecasts of these fields for Expts 1 and 2 (i.e., using uniform 1.2” resolution on the unrotated and rotated meshes) are shown in Figs 4 and 5 respectively. The global r.m.s. forecast differences (computed after interpolating the forecast of Expt. 1 to the rotated mesh of Expt. 2) are small (4.9 m and 0.6 hPa respectively), showing that the effect of rotating the mesh while keeping its resolution uniform is small.

c Variable Resolution Experiment The integration of Expt. 2 (i.e., uniform resolution everywhere in the rotated coor- dinate system) is considered to be the ground truth for the purposes of validating the 48-h forecast of the variable resolution integration (Expt. 3): the meshes of both integrations are identical over the uniform resolution window of Fig. 2b. The 2-day variable resolution forecast is shown in Fig. 6 and may be compared to that of the control (Fig. 5). The two forecasts (Expts 2 vs. 3) are quite close over the uniform resolution area of interest (defined by the curvilinear rectangle of Fig. 6a). This confirms the thesis that the forecast over the 1.2” uniform resolution window can be well reproduced at a fraction of the cost of using 1.2” uniform resolution everywhere. However, they are significantly different over areas of low resolution, as indeed they should be. Quantifying this, the global r.m.s. differences between the forecasts of Expts 2 and 3 are 26.8 m and 3.4 hPa for the 500-hPa height

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Fig. 3 (a) Initial geopotential height at 500 hPa in dam on an orthographic projection; contour interval = 6 dam. (b) Initial mslp in hPa for a N. American window on a polar sterographic projection; contour interval = 4 hPa.

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a

Fig. 4 (a) Same as in Fig. 3a, but at 48-h for Expt. 1. (b) Same as in Fig. 3b. but at 48-h for Expt. 1.

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Fig. 5 (a) Same as in Fig. 3a, but at 48-h for Expt. 2. (b) Same as in Fig. 3b, but at 48-h for Expt. 2,

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7 1

Fig. 6 (a) Same as in Fig. 3a, but at 48-h for Expt. 3. (b) Same as in Fig. 3b, but at 48-h for Expt. 3,

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Fig. 7 Difference between 48-h forecasts of Expt. 2 and Expt. 3 on an orthographic projection for:

(a) 500 hPa geopotential height; contour interval = 6 m. (b) mslp; contour interval = 2 hPa.

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and mslp fields respectively, whereas they are only 7.0 m and 0.6 hPa over the curvilinear rectangle, where the meshpoints of the two grids are coincident.

Note that the spatial truncation errors associated with the variable resolution portion of the model’s grid propagate with the speed of the local wind. This has to be taken into account when defining a uniform resolution region of interest for the model. It has to be sufficiently large, so that the entire region is not unduly contaminated by the error advected from the variable portion of the grid during the time of integration. It is therefore a compromise between the width of this region and the length of the run. The differences between the forecasts of Expts 2 and 3 increase as a function of the proximity to the upstream boundaries of the uniform resolution window, due to the inflow from the coarser resolution outer domain (see Fig. 7). They would have been substantially smaller had it been possible to run the experiments at 0.5” resolution instead of 1.2”.

4 fiture work and conclusion All of the attributes (output program, postprocessors, time series, zonal diagnostic extractors, etc.) of a modem operational model are being added to the dynamical core of the described model, including topography. The physical parameterizations currently used by both Canadian operational models are being introduced into the new model using a plug-compatible approach, and each physical process will be individually tested. Once this has been successfully accomplished the model could then be considered as a replacement model at the Canadian Meteorological Centre for both operational models, the global spectral and the regional finite-element models. The next stage of development would then be to complete the coding of a non-hydrostatic version and to fully evaluate it for mesoscale applications where the hydrostatic assumption breaks down.

The preliminary results shown in the present work are encouraging. The strategy that was developed in a 2D prototype and recently extended to 3D shows promising results in progressing towards a unified data assimilation and forecast system, at the heart of which lies a single multipurpose and multiscale numerical model.

Acknowledgements

We gratefully acknowledge the helpful comments of the editorial committee and an anonymous reviewer.

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