THE CSIRO CONFORMAL-CUBIC ATMOSPHERIC GCM 8u ...

THE CSIRO CONFORMAL-CUBIC ATMOSPHERIC GCM

JOHN L. MCGREGOR AND MARTIN R. DIX CSIRO Atmospheric Research PBl Aspendale 3195 Austmlia

1. Introduction

A grid was devised cubic-conformal Which some thought was rather abnormal But by technique~ semi-Lagrangian With conversions Cartesian The results turn out better than normal

Global atmospheric models are usually formulated upon latitude-longitude grids. Near the poles, these grids have disproportionately high resolution, which may severely constrain the time step of integration or require special filtering. Advection problems may also occur near the poles of such grids. Quite recently, a global conformal-cubic grid was devised by Rancic et al. (1996). This grid avoids the disadvantages of latitude-longitude grids, but does require careful selection of numerical techniques to account for the eight vertices of the grid (McGregor, 1996).

At CSIRO, a semi-Lagrangian atmospheric general circulation model (GCM) has been suc­cessfully developed on the conformal-cubic grid, complete with physical parameterizations. This paper describes several aspects of the model formulation, and the model performance will be demonstrated for the early part of a simulation being undertaken for the AMIP II atmospheric GCM intercomparison experiment.

2. Formulation

The primitive equations on the grid resemble those on a polar stereographic projection, except that the map factor, m, takes values appropriate to the conformal-cubic grid:

8u 8u 8u 8u 8¢ 8lnps - + mu- + mv- + &- + m- + mRT-- = Iv + Nu (1) 8t 8x 8y 8a 8x 8x

Horizontal Vertical advection advection

Pressure gradient

Coriolis Physics force

8v 8v 8v. 8v 8rt> 8lnps - +mu-+mv- +a- +m-+mRT-- = -Iu + Nv (2) 8t 8x 8y 8a 8y 8y

P.F. Hodnett (ed.),

Horizontal Vertical advection advection

Pressure gradient

8T 8T 8T .8T - +mu-+mv- +a-8t 8x 8y 8a

RTw

Coriolis Physics force

= NT

Horizontal advection

Vertical Adiabatic Physics advection term

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IUTAM Symposium on Advances in Mathematical Modeling of Atmosphere and Ocean Dynamics, 197-202. © 2001 Kluwer Academic Publishers.

(3)

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a¢ aa

RT a

--+mu--+mv-- + -+m ---+--- =0 alnp. alnp. alnp. au 2 [a(u/m) a(v/m)]

at ax ay aa ax ay Horizontal advection

Vertical advection

Divergence

(4)

(5)

where u and v are horizontal velocity components on the grid, T is temperature, ¢ is geopotential height, a is the normalized-pressure vertical coordinate, P. is surface pressure, and R is the gas constant. A semi-Lagrangian semi-implicit procedure is adopted, where the horizontal material time derivatives are evaluated along trajectories, whilst the other terms are averaged along the trajectories. The semi-Lagrangian technique (as reviewed by Staniforth and Cote, 1991) allows large advective time steps. The calculation of the departure points follows the procedure advocated by McGregor (1993). To avoid turning problems, which might lead to errors near the vertices, the vector equations for the horizontal wind components (u, v) are solved in terms of the three equivalent equations for the corresponding three-dimensional Cartesian wind components; the updated values are projected back on to the surface of the sphere to give the updated values of (u, v). This Cartesian representation is also used during the calculation of horizontal diffusion. Although weak horizontal diffusion is employed to represent the cascade of energy to smaller scales, the model performs without spurious noise even for zero horizontal diffusion.

As is common for semi-implicit models, the discretized primitive equations are linearized and combined using vertical eigenvectors . This leads to a set of vertically-decoupled Helmholtz equations in terms of a height variable. The grid lends itself to solving each of these Helmholtz equations by efficient three-colour successive over-relaxation; the rationale for the vectorization is similar to that of two-dimensional red-black schemes (e.g., Young, 1971). The three-colouring of the grid is illustrated in Fig. 1 for a very coarse C5 grid (5 x 5 grid points on each panel). During each iteration all the points of the same colour are updated before proceeding to the other colours.

Figure 1. An example of a conformal-cubic grid. The shading illustrates the three-colour arrangement used for solution of the Helmholtz equations.

Another unusual and attractive feature of the model is the use of reversible staggering for the wind components. All model variables are located and stored at the centre of the grid cells,

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already illustrated in Fig. 1. However, during the semi-implicit solution procedure it is desirable to transform the wind components so that they are located normal to the mid-points of the cell boundaries, in an Arakawa-C configuration; this is the preferred arrangement for accurate calculation of divergence. A compact procedure has been devised to allow this transformation to be performed reversibly between the mid-points and centres of the cells; the cyclic properties of the conformal-cubic grid are beneficial for this procedure. It can be shown that superior dispersion properties ensue from the reversible-staggering treatment of the winds. A fairly comprehensive set of physical parameterizations has been incorporated into the model, as listed in Table 1.

TABLE 1. Physical parameterizations

GFDL ~arameterization for long and short wave radiation (Schwarzkopf and Fels, 1991) Interactive diagnosed cloud distributions Arakawa/Gordon mass-flux cumulus convection scheme (McGregor et al., 1993) Tiedtke shallow convection Evaporation of rainfall Stability-dependent boundary layer and vertical diffusion, following Louis (1979) Deformation-based horizontal diffusion (Smagorinsky et al., 1966) Vegetation/canopy scheme with six layers for soil temperatures and soil moistures.

3. Results

In order to demonstrate the model 's capabilitites, we have commenced simulations for the AMIP II intercomparison (for more information see http://www-pcmdi,l1nl.gov/amipl), where sea surface temperatures and sea-ice cover are prescribed from 1979 to 1996. A C48 grid is being used, as shown in Fig. 2, having an average resolution of about 200 km; the model has 18 vertical levels.

Figure 2. The C48 conformal-cubic grid .

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Figure 3a. Mean sea level pressures for November 1979 from the NCEP reanalysis.

Figure 3b. Mean sea level pressures for November 1979 from the C48 model simulation.

Several months of simulation have been performed so far, starting November 1 1979. Figure 3a shows the average mean sea level pressures for that month from the NCEP reanalysis, whilst Fig. 3b shows the simulated values. There is good agreement for important features. For example, there are strong pressure gradients near Antarctica, there is a well-defined Aleutian low, and an acceptable Icelandic low. Figure 4a show the average observed precipitation for the same month (Xie and Arkin, 1996), whilst Fig. 4b shows the modelled field. Again there is good agreement of the tropical rainfall patterns and the precipitation over each of the continents, although the rainfall over western Australia is somewhat excessive.

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2 4 8 16

Figure 4a. Rainfall for November 1979 from the Xie and Arkin analysis (mm/day).

2

Figure 4b. Rainfall for November 1979 from the C48 model simulation (mm/day).

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4. Conclusions

A primitive equations model has been formulated on the conformal-cubic grid, with a fairly complete set of physical parameterizations. Simulations are being undertaken for the AMIP 2 intercomparison. Results so far are very encouraging.

The following advantages are anticipated compared to spectral GCMs (these advantages are probably also available to other grid-point models):

there is freedom from non-zero ocean orography and other effects caused by the Gibb's phenomenon specialized treatment is possible for various horizontal gradient terms near orography deformation-based horizontal diffusion schemes can be used.

The following advantages are also anticipated compared to typical grid-point GCMs:

the grid has relatively uniform resolution, avoiding the requirement for filtering near the poles the grid is isotropic reversible staggering of the winds is available, providing good dispersion properties the grid permits the use of a simple efficient Helmholtz solver the panel structure of grid has advantages for code parallelization.

References

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