NOVEMBER 1997 FOX-RABINOVITZ ET AL. 2943

A Finite-Difference GCM Dynamical Core with a Variable-Resolution Stretched Grid

MICHAEL S. FOX-RABINOVITZ AND GEORGIY L. STENCHIKOV University of Maryland at College Park, College Park, Maryland

MAX J. SUAREZ Goddard Space Flight Center, Greenbelt, Maryland

LAWRENCE L. TAKACS General Sciences Corporation, Laurel, Maryland

(Manuscript received 30 December 1996, in ®nal form 2 April 1997)

A ®nite-difference atmospheric model dynamics,ABSTRACT or dynamical core using variable resolution, or stretched grids, is developed and used for regional±global medium-term and long-term integrations. The goal of the study is to verify whether using a variable-resolution dynamical core allows us to represent adequately the regional scales over the area of interest (and its vicinity). In other words, it is shown that a signi®cant downscaling is taking place over the area of interest, due to better-resolved regional ®elds and boundary forcings. It is true not only for short-term integrations, but also for medium-term and, most im- portantly, long-term integrations. Numerical experiments are performed with a stretched grid version of the dynamical core of the Goddard Earth Observing System (GEOS) general circulation model (GCM). The dynamical core includes the discrete (®nite-difference) model dynamics and a Newtonian-type rhs zonal forcing, which is symmetric for both hemispheres about the equator. A ¯exible, portable global stretched grid design allows one to allocate the area of interest with uniform ®ne-horizontal (latitude by longitude) resolution over any part of the globe, such as the U.S. territory used in these experiments. Outside the region, grid intervals increase, or stretch, with latitude and longitude. The grids with moderate to large total (global) stretching factors or ratios of maximum to minimum grid intervals on the sphere are considered. Dynamical core versions with the total stretching factors ranging from 4 to 32 are used. The model numerical scheme, with all its desirable conservation and other properties, is kept unchanged when using stretched grids. Two model basic horizontal ®ltering techniques, the polar or high-latitude Fourier ®lter and the Shapiro ®lter, are applied to stretched grid ®elds. Two ®ltering approaches based on the projection of a stretched grid onto a uniform one are tested. One of them does not provide the necessary computational noise control globally. Another approach provides a workable monotonic global solution. The latter is used within the developed stretched grid version of the GEOS GCM dynamical core that can be run in both the middle-range and long-term modes. This ®ltering approach allows one to use even large stretching factors. The successful experiments were performed with the dynamical core for several stretched grid versions with moderate to large total stretching factors ranging from 4 to 32. For these versions, the ®ne resolutions (in degrees) used over the area of interest are 2 ϫ 2.5, 1 ϫ 1.25, 0.5 ϫ 0.625, and 0.25 ϫ 0.3125. Outside the area of interest, grid intervals are stretching to 4 ϫ 5or8ϫ10. The medium-range 10-day integrations with summer climate initial conditions show a pronounced similarity of synoptic patterns over the area of interest and its vicinity when using a stretched grid or a control global uniform ®ne-resolution grid. For a long-term benchmark integration performed with the ®rst aforementioned grid, the annual mean circulation characteristics obtained with the stretched grid dynamical core appeared to be profoundly similar to those of the control run with the global uniform ®ne-resolution grid over the area of interest, or the United States. The similarity is also evident over the best resolved within the used stretched grid northwestern quadrant, whereas it does not take place over the least-resolved southeastern quadrant. In the better-resolved Northern Hemisphere, the jet and Hadley cell are close to those of the control run, which does not take place for the Southern Hemisphere with coarser variable resolution. The stretched grid dynamical core integrations have shown no negative computational effects accumulating in time. The major result of the study is that a stretched grid approach allows one to take advantage of enhanced resolution over the region of interest. It provides a better representation of regional ®elds for both medium- term and long-term integrations.

Corresponding author address: Dr. Michael Fox-Rabinovitz, University of Maryland, Suite 200, 7501 Forbes Blvd., Seabrook, MD 20706. E-mail: [email protected]

᭧1997 American Meteorological Society

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The developed stretched grid model dynamics is supposed to be the ®rst stage of the development of the full diabatic stretched grid GEOS GCM. It will be implemented, within a portable stretched grid approach, for various regional studies with a consistent representation of interactions between global and regional scales and phenomena.

1. Introduction only a marginal to moderate impact on the coarse-reso- lution ®elds within a buffer zone around the area of in- The advantage of using nonuniform grids with enhanced terest. Formally, boundary conditions could be of a two- resolution over the area of interest has been discussed in way-interaction type for which the aforementioned prob- the context of regional modeling for years (e.g., Charney lems still occurs. Along with using ®ne resolution, the 1966; Anthes 1970). The downscaling with the nested- success of nested grid models is based mostly on intro- grid approach was applied ®rst to regional forecast models ducing a ®ne-resolution boundary condition forcing. Also, (e.g., Phillips 1979; the review by Koch and McQueen it is possible, within this approach, to use nonhydrostatic 1987) and later to regional climate simulations (e.g., Pielke approximations for regional models. et al. 1992). The different versions of the Penn State± Notice that periodic updates of initial conditions makes NCAR mesoscale model (MM4 and MM5) are most wide- the regional integration procedure quite similar, in a sense, ly used. to that of an intermittent data assimilation with longer Exceptions are the French (Courtier and Geleyn 1988; periods between data insertion. Namely, regional simu- Yessad and Benard 1996) and Canadian (Staniforth and lations are not continuous but intermittent, with regional Mitchell 1978; Staniforth et al. 1991; Cote et al. 1993) climate diagnostics produced in between outer boundary operational models with variable-resolution stretched grids and initial data insertions. The results depend on the type and the models developed by Paegle (1989) and the Krish- namurti group (Hardiker 1997). of boundary conditions used and the type and quality of For limited-area and nested-grid models, the problem inserted data. of formulating well-posed or mathematically correct lateral Recently, the adaptive grid approach, originally devel- boundary conditions, is far from resolved in spite of the oped for astrophysics, aeronautical, and other computa- four-decade-long effort since the pioneering paper of Char- tional ¯uid dynamics problems has been discussed and ney et al. (1950). This dif®cult problem has been elo- applied to simple atmospheric models (e.g., Dietachmayer quently discussed by Robert and Yakimiw (1986), Oliger and Droegemeier 1992; Berger and Oliger 1984; Ska- and Sundstrom (1978), Yakimiw and Robert (1990), marock 1989; Skamarock et al. 1989). The ®rst attempt Browning et al. (1973), Williamson and Browning (1974), to introduce a speci®c spherical adaptive grid to an at- and Staniforth (1995). As practical compromise solutions, mospheric model was undertaken by Kurihara (1965). Ap- some versions of the Davies (1976) technique are most parently, the adaptive grid schemes can be bene®cial for widely used. short-term forecasting, paying special attention to mete- Successful nested-grid model integrations are known to orological phenomena (like fast-moving cyclones) requir- be performed for short-term forecasts. After that, the fore- ing enhanced resolution. However, for regional climate cast ®elds, especially near the ®ne-resolution area bound- simulations, along with using ®ner-resolution ®elds, the ary, become contaminated by noise due to the abrupt most important impact is obtained by using ®ne-resolution change of resolution at the boundary. Applying strong orographic and other physical forcings over the entire area ®lters will inevitably result in deteriorating the solution of interest. Such an impact can be negatively affected by quality. In other words, the longer-term integrations suffer the gridpoint redistributions applied for adaptive grids to from an inaccurate interaction at the boundaries of the area keep the computational cost under control. of interest. Therefore, a nested-grid approach does not Another kind of variable-resolution model to be dis- allow one to perform continuous long-term integrations cussed in this study is based on a stretched grid approach. without updating initial and boundary conditions. With such an approach, grid intervals outside a uniform However, nested-grid models are the ®rst ones to be ®ne-resolution area of interest are stretched uniformly over used successfully in recent pioneering studies on regional the rest of the globe. As a result, a single global variable- climate (e.g., Pielke et al. 1992). The only way of using resolution grid is obtained. As an option, the spherical grid these models to produce regional climate diagnostics is to can be rotated so that the area of interest is located, on periodically update initial and/or lateral boundary condi- our choice, over the new, rotated equator area. Such an tions after a short period of integration, that is, 1±2 days. approach was introduced by Staniforth for ®nite-element Either observational analyses or global (or larger area) models (e.g., Staniforth and Mitchell 1978; Cote et al. model forecast ®elds can be used for this purpose. These 1993; Gravel and Staniforth 1992; and Staniforth 1995). periodic updates of initial and lateral boundary conditions For a gridpoint model with variable resolution, the are a dominating forcing that results in a predominantly stretched grid approach with the same model used over one-way interaction between the coarse- and ®ne-resolu- the entire globe seems to be attractive because it is free tion areas. Actually, the ®ne-resolution ®elds may provide of the previously mentioned ill-posed problem typical for

Unauthenticated | Downloaded 09/28/21 03:43 PM UTC NOVEMBER 1997 FOX-RABINOVITZ ET AL. 2945 nested grids. The associated additional computational cost erties, is kept unchanged when using stretched grids. The due to stretching versus nesting is not overwhelming. For approach developed by Held and Suarez (1994) for testing example, with a practically acceptable local stretching rate dynamical cores is used in the study. Experimenting with of about 5%±10%, which is a grid interval change between a model dynamical core provides an ultimate test of com- the adjacent grid points, the signi®cant part or even ma- putational properties of a model numerical scheme and its jority of global grid points are usually located inside the dependence on resolution, or more speci®cally, on stretch- area of interest for a variety of resolution versions. ing factors. Long-term dynamical core integrations are per- Schmidt (1977) developed another stretched grid ver- formed with the Newtonian-type rhs forcing (Held and sion for spectral models (e.g., Courtier and Geleyn 1988). Suarez 1994) as a substitution for model physics. Using Within this approach, a conformal coordinate transfor- model dynamical core integrations allows one to investi- mation is introduced. For the French operational model, gate the speci®c impacts of discretized model dynamics one transformed hemisphere with the pole located at cen- and resolution on full model errors, independently of full tral France contains western Europe, while another trans- model physics. formed hemisphere comprises the rest of the globe. For In the context of verifying the key hypothesis outlined this model, using large total stretching factors of 7±10 above, when testing the stretched grid dynamical core, we works only for short-term forecasting (J.-F. Geleyn 1995, are pursuing the following goals. We are comparing, in personal communication). Small to moderate stretching both a medium-range forecast mode and a long-term cli- factors of 2 to 4 are used for promising, successful climate mate mode, the results of stretched grid runs against the simulation with the spectral model (DeÂque and Piedelievre corresponding basic control runs with ®ne uniform global 1995). grids, especially over the area of interest and its vicinity. Having established the attractive advantages of the glob- We should verify whether there is any degradation of the al stretched grid approach, the legitimate question to ask synoptic-scale patterns, produced by the control runs, by is what its limitations are. They should be considered in the stretched grid runs. We are also making sure that in a the context of various applications to regional forecasting, long-term integration mode there is no development of data assimilation, and climate simulation, as compared to cumulative undesirable effects of a numerical nature re- the corresponding control runs with time-consuming glob- lated to stretched grid approximation of model dynamics. al models with uniform ®ne resolution. Outside the ®ne- A particular emphasis is placed on ultimately using the resolution area of interest, all scales are approximated with stretched grid approach for future regional climate mod- larger truncation errors; however, the small-scale motions eling applications. dissipate quickly and their impact on the regional scales Section 2 is devoted to discussing the basic numerical is small. Similar to the situation with limited-area models, problems for variable-resolution, or nonuniform, stretched one can assume that it takes a limited time for the incoming grids. A brief description of the numerical scheme and ¯ow approximated on a variable-resolution stretched grid ®ltering procedures is given in section 3. Application of to reach, affect, and in¯uence the ¯ow over the area of the ®ltering techniques and the results of long-term nu- interest. The regional ¯ow will depend on interactions with merical experiments with the stretched grid dynamical core such an incoming large-scale background circulation. A are presented in section 4. Middle-range integrations with downscaling due to better resolved regional scales and large stretching factor grids are discussed in section 5. regional forcing as well as their interactions will contin- Concluding remarks are given in section 6. uously play an important role over the area of interest. Notice that for the uniformly stretched grid, the incoming ¯ow is better resolved and more effectively represented 2. A brief discussion of basic numerical by gradually changing resolution than by a nested grid problems for ®nite-difference approximations with an abrupt resolution change. Therefore, for stretched with variable-resolution stretched grids grids, the ¯ow coming into the area of interest is better resolved, especially its large-scale and medium-scale com- The rationale for using irregular (nonuniform or ponents. nonequidistant) grids for approximations of atmo- The key hypothesis to verify in this study is whether a spheric model dynamics has been discussed in the signi®cant downscaling or an adequate representation of introduction. However, there are basic computational regional scales is taking place over the area of interest problems arising from grid irregularity. It affects the (and its vicinity) for both short-, medium-, and, most im- accuracy of approximation, computational stability, portantly, long-term integrations with a variable-resolution computational dispersion, and other properties of ®- (stretched grid) dynamical core. nite-difference schemes. These problems have been In this study, the stretched grid approach is developed discussed since the 1970s in computational ¯uid dy- and tested for a ®nite-difference model. Namely, it was namics publications (e.g., Roache 1976; Vichnevet- implemented to the Goddard Earth Observing System sky 1987; Oliger and Sundstrom 1978; Fletcher 1988) (GEOS) general circulation model (GCM) dynamical core for advection equations. For atmospheric models, the (Suarez and Takacs 1995). The GEOS GCM numerical additional numerical problems arising for adjustment scheme, with all its desirable conservation and other prop- or wave equations using stretched grids are equally

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of approximation is lost completely. For a moderately

smoothly stretched grid with rj close to 1,

FIG. 1. The 1D nonuniform (nonequidistant) grid. rj ϭ 1 ϩ O(⌬x). (3) Although the second order of approximation is for- mally lost, the accuracy of approximation is close to important (e.g., Anthes 1970, 1983; Staniforth 1995; that of the second order. A similar loss in the accuracy Fox-Rabinovitz 1988). of approximations takes place for higher derivatives In this section, we will make comments on the basic of a discrete function and for higher orders of the computational problems for ®nite-difference approx- accuracy of approximation. imations related to grid irregularity for the 1D case. b. Coordinate transformation from a physical to a. Accuracy of approximation for spatial computational space derivatives The coordinate transformation for the example of

To de®ne a nonuniform, nonequidistant, stretched a uniformly stretched grid with rj ϭ r ϭ constant can grid we need to introduce the local stretching rate or be derived from geometric progression expressions for stretched grid intervals (see Fig. 1), the ratio rj of the adjacent grid intervals ⌬xj and ⌬xjϪ1 (see Fig. 1): j ⌬xj ϭ r ⌬xo, (4)

⌬xj and for the coordinates of stretched grid points, rj ϭ , (1) ⌬xjϪ1 rj Ϫ 1 jϪ1 xjoϭ⌬x(1 ϩ r ϩ ´´´ϩr )ϭ⌬x o , where ⌬xj ϭ xjϩ1 Ϫ xj, ⌬xjϪ1 ϭ xj Ϫ xjϪ1, and x is an rϪ1 independent variable. Note that r ϵ 1 for a uniform j jϭ1,2,..., (5) grid, and rj is constant for all j's for a uniformly stretched grid. For the latter, grid intervals change where ⌬xo is the ®rst grid interval starting from which exponentially according to a geometric progression. the grid is stretching rightward, in the original phys-

Note also that rj Ͼ 1 corresponds to a grid with ical space or coordinate system. In the transformed stretching, or increasing grid intervals in the direction coordinate system, the original nonuniform grid be- of increasing x (rightward), and rj Ͻ 1 to stretching comes a uniform one, with the constant grid intervals in the opposite direction. In the study, we use uni- for all j's. formly stretched grids with constant rj's. (Actually, a piecewise constant r 's will be used within a global j c. Comments on computational stability, dispersion, stretched grid). and wave propagation and re¯ection The local stretching rate rj is the basic, de®ning local characteristic for a stretched grid. Another basic, For the same centered-difference schemes applied fundamental parameter for a stretched grid is the total to both uniform and nonuniform grids, the CFL (Cour- global stretching factor, or the ratio of the maximal ant±Friedrichs±Lewy) computational stability crite- to minimal grid intervals for the entire grid: rions have similar forms. It is also true for compu- tational dispersion characteristics. However, the sub- ⌬x R ϭ max . (2) stantial difference is that space intervals ⌬x are func- ⌬xmin tions of x for nonuniform grids. For the 1D linear advection equation approximated This parameter represents the total amount of stretch- with a centered-difference scheme, the CFL compu- ing for a grid. tational stability criterion and the computational dis- Using irregular (nonuniform, nonequidistant) grids persion relation have the corresponding forms results in a loss of the accuracy of approximation (see, e.g., Roache 1976; Thompson 1984; Thompson et al. ⌬x ⌬t Յ min 1985; Fletcher 1988). For example, the well-known c centered ®nite-difference approximation of the ®rst derivative of a discrete function provides the second- and order accuracy only for a uniform grid for which ⌬xj c ϭ⌬x and r ϭ 1. ␯ ϭ sin[␬(x)⌬x(x)], (6) jϪ1 j ⌬x(x) For a nonuniform grid with ⌬xj ± ⌬xjϪ1 and rj ± 1, the accuracy of approximation becomes only of the where c ϭ const is the advection speed, ␯ the fre- ®rst order. For a nested grid, which is characterized quency, and ␬ is a wavenumber. by an abrupt change in resolution and, correspond- Therefore, a time step is controlled by the ®ne- ingly, in the local stretching rate rj, the second order resolution grid interval over the area of interest for

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which ⌬xmin ϭ⌬xo (Fig. 1). Note that using semi- allow for implementing the ``plug-compatible'' rules implicit schemes is desirable for variable-resolution introduced in Kalnay et al. (1989) for structuring models to improve their computational ef®ciency. model physics parameterizations. The designed dy- Propagation and re¯ection of a wave group enve- namical core is nearly ``plug compatible.'' It is used lope, or wave packet, in a uniformly stretched grid as a model dynamics component in the GEOS GCM (Fig. 1) for the linear 1D advection equation approx- (Takacs and Suarez 1996). imated with either a centered-difference or the Crank± For our experiments, we adopted the approach de- Nicolson scheme, has been thoroughly investigated in veloped by Held and Suarez (1994) for calculations a seminal paper by Vichnevetsky (1987). with dynamical cores. Along with model dynamics, It is worth noting that an interesting computational all model ®lters and a time stepping, or a state variable re¯ection effect for a wave packet takes place for the update, are implemented. Moreover, as a substitution grid with the ®nest grid interval in the vicinity of x for model physics, it also includes a simple linear ϭ 0 and with grid increments increasing monoton- damping of the velocities in lower model layers and ically and symmetrically in both directions from that temperature relaxation to a prescribed ``radiative point. For the initially rightgoing wave group, the equilibrium'' (Held and Suarez 1994). This Newto- re¯ection occurs at the point xR with the group ve- nian-type temperature forcing depends on latitude and locity G ϭ 0. The resulting motion becomes leftgoing pressure and is symmetric for both hemispheres about and the second re¯ection occurs at a symmetric point the equator.

ϪxR where also G ϭ 0. Therefore, the energy is More speci®cally, the previously mentioned damp- trapped in the domain (ϪxR, xR) in a kind of a ``well.'' ing and relaxation terms are introduced in the mo- This effect of the double re¯ection or `` wave trap- mentum and energy equations, respectively, in the ping'' was ®rst shown by Giles and Thompkin (1985). form (Held and Suarez 1994) vץ Finally, for nonuniform grids, a spurious scattering effect takes place. It is enhanced for a nonuniformly ϭ ´´´ Ϫ␬ (␴)v t vץ stretched grid with rj ± const, as discussed by Vi- Tץ .(chnevetsky and Turner (1991 A general solution for these computational prob- ϭ ´´´ Ϫ␬v(␸,␴)[T Ϫ Teq(␸, p)], (7) tץ lems arising from grid irregularity is to introduce dif- fusion-type ®lters and uniformly stretched grids. where ␸ is the latitude, p the pressure, v the wind vector, These are necessary tools for controlling the different T the temperature, and kinds of computational noise discussed in this section (e.g., Vichnevetsky 1987; Fox-Rabinovitz 1988). T ϭ max 200 K, 315 K Ϫ (⌬T) sin 2␸ Speci®cally, small-scale components of the ¯ow and eq Ά [ y their transformations should be carefully controlled. ␬ pp Ϫ(⌬␪) log cos 2␸ , z pp ΂΃00] ΂΃· 3. The GEOS GCM dynamical core with a stretched grid ␴Ϫ␴ 4 b ␬Taϭ␬ϩ(␬ asϪ␬) cos ␸ max 0, , The study is done using the GEOS GCM dynamics ΂΃1Ϫ␴b or dynamical core with a stretched grid. The model numerical scheme, with all its desirable conservation ␬vfϭ␬max[0, (␴ Ϫ ␴ b)/(1 Ϫ ␴ b)], and other properties, is kept unchanged when using 1 Ϫ1 Ϫ1 stretched grids. Using the dynamical core allows us ␴bfϭ0.7, ␬ ϭ 1 day , ␬ aϭ day , 40 to address the numerical problems due to grid irreg- ularity on the sphere in the medium-range and long- and term integration modes. The ®nite-difference dynam- 1 Ϫ1 ical core originally developed by Suarez and Takacs ␬syzϭ day , (⌬T) ϭ 60 K, (⌬␪) ϭ 10 K, (1995) for a uniform grid, with a possibility to run it 4 Ϫ1Ϫ1 on nonuniform grids, is used in the study. The design p0 ϭ1000 mb, ␬ ϭ 2/7, cp ϭ 1004 J kg K . of the ®lters to run on stretched grids is one of the main objectives of the study. (8)

The temperature Teq is symmetric about the equator. The friction and relaxation coef®cients ␬ and ␬ decrease a. Brief description of the dynamical core v f with height and in the boundary layer reach the maxi- The model dynamical core has been developed by mum values of 1 dayϪ1 and ¼ dayϪ1, respectively. Suarez and Takacs (1995) as a module updating the For a ®nite-difference approximation, the horizontal time tendencies to include the adiabatic effects of the staggered Arakawa C grid (Mesinger and Arakawa dynamics. The strict modular approach is intended to 1976) and the vertical staggered Lorenz (1960) grids

Unauthenticated | Downloaded 09/28/21 03:43 PM UTC 2948 MONTHLY WEATHER REVIEW VOLUME 125 are used. Basically, the numerical scheme is close to that of the UCLA GCM. The equations of motion are written in the form used by Sadourny (1975), Burridge and Haseler (1977), and Arakawa and Lamb (1981). Using some simpli®ed assumptions, the scheme pro- vides conservation of energy and potential enstrophy. The continuity thermodynamic and moisture equation approximations provide conservation of mass, potential temperature, and moisture (Suarez and Takacs 1995). For a uniform grid, the scheme is of a second-order, except for the horizontal advection of vorticity and po- tential temperature by the rotational component of the ¯ow, which is of a fourth-order accuracy (Suarez and Takacs 1995; Takacs and Suarez 1996). In the vertical, there are 20 sigma levels (Phillips 1957) spaced as in the GEOS GCM (Takacs et al. 1994). The vertical differencing is done according to the Ar- akawa and Suarez (1983) conservation scheme. The time-integration scheme employs the leapfrog scheme and a Robert±Asselin time ®lter (Robert 1966; Asselin 1972). It is combined with the economical explicit scheme (Brown and Campana 1978; Schuman 1971; Fox-Rabinovitz 1974). Following Brown and Campana (1978), a three-time-level averaging operator is applied to the pressure gradient force to provide stricter control of gravity wave instability. It results in using larger time steps and does not require any signi®cant changes in the numerical scheme, except for introducing the certain order in which the model equations are integrated. The dynamical core structure allows for the pole rotation (Takacs and Suarez 1996) so that the area of interest FIG. 2. The latitudinal (a) and longitudinal (b) stretching factors as can be located away from the poles and near the equator, piecewise constant functions. for example. A stretched grid design and the corre- sponding polar and Shapiro ®lter modi®cations for a stretched grid are described below. local stretching rates r. More speci®cally, for the grid de- sign, the grid intervals increase from the right boundary to the farthermost, coarse grid interval with a constant b. Stretched grid generation local stretching rate r␭. If one continues to move from the Relying upon considerations discussed in section 2, the coarse grid interval to the left boundary, the grid intervals smooth, uniformly stretched grid has been designed for decrease with the constant local rate r ϭ 1/r␭. As a result, dynamical core experiments. In other words, the local the local stretching rate for a latitudinal circle covering stretching rate rj [Eq. (1)] is kept constant and within not the area of interest is actually a piecewise-constant function more than a 5%±10% deviation from unity. The total glob- (Fig. 2a). Namely, r ϵ 1 over the area of interest, r ϭ al stretching factor R [Eq. (2)], or the ratio of the coarsest const Ͼ 1 from the right boundary to the farthermost, to ®nest grid increments used in the experiments, ranges coarse grid interval, and r ϭ 1/const Ͻ 1 from the far- from moderate to large values of 4 to 32. thermost, coarse grid interval to the left boundary. The area of interest contains a uniform ®ne-resolution For the longitudinal circles covering the area of interest, latitude±longitude grid. It allows us to resolve maximally the stretching procedure is quite similar. Starting from the the regional prognostic ®elds as well as orographic and area boundaries, the grid stretching is done with constant other physical boundary forcings. On the opposite side of local stretching rates in both directions toward the poles. the latitudinal circles covering the area of interest, we have Therefore, for longitudinal circles covering the area of only one coarse-resolution grid interval. In between the interest, the local stretching rate is also a piecewise-con- area boundaries and this maximally remote grid interval, stant function (Fig. 2b). As a result, a stretched grid is the grid intervals are uniformly stretched (Fig. 2). Namely, de®ned globally, with arbitrary local stretching rates in the grid intervals h(x) depend on x as h(x) ϭ hmin ϩ ␣x, both latitudinal and longitudinal directions. When the area where ␣ ϭ r Ϫ 1 and hmin is the minimal grid interval of interest is shifted, or is nonsymmetric in respect to the over the area of interest. This stretching on both left and equator, and the local stretching rates are the same in both right sides from the area of interest is done with constant directions, we obtain larger meridional grid intervals over

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FIG. 3. The global stretched grid with the rectangular area of interest with ®ne resolution located over the United States region, or from 25Њ to 50ЊN and 125Њ to 75ЊW. one of the poles than over the other. Therefore, when the on the size of the area of interest (it could be only a part area of interest is located near the pole, the coordinate of the United States, for example). Also, the remote coarse- rotation is desirable. resolution area may contain not just one grid box, as con- Practically, when generating a stretched grid, the total sidered above, but a signi®cant part of the globe (e.g., global stretching factor is introduced as the ®rst input pa- Staniforth and Mitchell 1978; Cote et al. 1993). For dif- rameter, de®ned by setting the ®ne resolution over the area ferent practically feasible stretched grids, the gains in com- of interest and the coarse resolution over the farthermost putation time are ranging from several times to 1±2 orders grid interval. Then, the input parameters for a rectangular of magnitude. In other words, using the stretched grid (in latitude by longitude coordinates) area of interest are approach allows one to achieve such a variable ®ne re- de®ned. The local stretching rates are calculated auto- gional±global resolution that is not possible otherwise (i.e., matically by positioning the uniformly stretched grid for uniform ®ne-resolution grids), even with the most pow- points between the area of interest boundary and the re- erful computers available now and in the foreseeable fu- mote coarse grid interval. The developed stretched grid ture. generator automatically de®nes the grid with prescribed stretching parameters. Such a portable automated stretched c. The polar and Shapiro ®lters grid generation procedure, with the previously mentioned Two horizontal ®lters are used at every time step (Tak- input parameters, allows one not only to design stretched acs et al. 1994). The ®rst is a high-latitude, or polar, Fourier grids with different areas of interest and stretching param- ®lter applied to all dependent variable tendencies. The eters, but also to introduce time-dependent or moveable polar ®lter allows us to avoid linear computational insta- stretched grids an option similar to adaptive grids. The bility due to the convergence of the meridians near the stretched grid used in this study is presented in Fig. 3. poles. In other words, it prevents decreasing the time step The area of interest is the rectangle covering the United due to a violation of a CFL criterion near the poles. The States. For this stretched grid, the northwestern quadrant ®lter is applied poleward of 45Њ latitude. Its strength varies has the best variable resolution, whereas the southeastern gradually when approaching the poles, by increasing the quadrant is the least-resolved one. Two other quadrants number and damping rate of affected zonal wavenumbers. have a medium variable resolution that is enhanced either The Fourier transformation coef®cients for a latitudinal in latitudinal or longitudinal direction. circle are scaled by a ®ltering function For the grids considered in section 5, the computational 2 gains are approximately 6, 10, and 16 times for the total cos␸ 1 stretching factors of 8, 16, and 32, correspondingly. The F ϭ Ά·cos(␲/4) sin[(i/lM)(␲/2)] reduction of computation time for stretched grids versus the corresponding uniform ®ne-resolution grids depends i ϭ 1,2,...,lM, Ϫ␲/2 Յ ␸ Յ ␲/2. (9)

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FIG. 4. The 6-day integration with the stretched grid dynamical core for the sea level pressure (a) and surface temperature (b), with ®ltering on the global uniform ®ne-resolution grid. The contour interval for the sea level pressure is 3 hPa and for surface temperature 3 K. Here, SG (2 ϫ 2.5 to 8 ϫ 10) stands for the stretched grid with uniform 2Њϫ2.5Њ resolution over the area of interest stretching to 8Њϫ10Њ. Note that in the absence of orography the sea level pressure and surface pressure are the same.

The ®lter strength can be increased by modifying the It provides not only stronger but also spectrally wider ®ltering function (12) in the following way: ®ltering for the small-scale spectral range. It was used

22only for the experiments described in section 5 with the cos␸ 1 FЈϭ . (10) 0.5Њϫ0.625Њ and 0.25Њϫ0.3125Њ resolution over the cos(␲/4) sin[(i/l )(␲/2)] Ά·[]M area of interest but not for coarser-resolution runs.

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FIG. 5. The 6-day integration with the stretched grid dynamical core for the sea level pressure (a) and surface temperature (b), with direct ®ltering on the global stretched grid (2Њϫ2.5Њ stretching to 8Њϫ10Њ), using a coordinate transformation approach.

qqF Ϫqץ Another ®ltering technique used is the Shapiro (1970) ®lter. It is applied to all dependent variables, ϭ , (11) tTץ namely, to the winds, potential temperature, and sur- ΂΃SF face pressure to damp globally small-scale dispersive where q and qF are un®ltered and ®ltered quantities, waves. It also prevents the computational nonlinear correspondingly, and the parameter T is an adjustable instability (Phillips 1959) to occur. A Shapiro ®lter timescale. Thus, only a fraction ⌬t/T of the full Sha-

,t)SF for a quantity q is piro ®lter is incorporated at each time step. Actuallyץ/qץ) tendency

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FIG. 6. The zonal annual mean (for the third year of the dynamical core integration, or year 3) vertical distribution of the zonal (a) and meridional (c) wind components for the stretched grid dynamical core run, with ®ltering on the stretched grid, using a coordinate transformation approach. (b) Same as in (a), and (d) same as (c) but for the control dynamical core run with the global uniform 2Њϫ2.5Њ grid. The SG and UG stand for stretched grid and uniform ®ne- resolution grid, respectively. The negative values are shown by dashed lines. The contour interval for (a) and (b) is 5 m sϪ1 and for (c) and (d) 0.5 m sϪ1.

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FIG.6.(Continued)

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T ϭ 1.5 h is adjusted in such a way to remove the on the global uniform ®ne-resolution ®ltering. Name- smallest two-grid-interval wave in approximately 6 ly, the tendency ®elds obtained at every time step of h. The ®lter is applied separately in the longitudinal the stretched grid model integration are interpolated and latitudinal directions. ®rst onto the global uniform ®ne-resolution grid. Its discrete form is Next, standard polar and Shapiro ®lters are applied to the uniform grid. Then the resulting ®ltered ®elds qF ϭ [1 Ϫ (F 2 )n ][1 Ϫ (F 2 )]n q , (12) i,j ␸␭i,j are interpolated back onto the stretched grid. where Another version of the ®ltering procedure was in- terpreted above in terms of the coordinate transfor- 1 2 F ␭(qi,j) ϭϪ (q iϩ1, ji,jiϪ2q ϩqϪ1, j), mation approach. When applied for ®lters only, it pro- 4 vides uniform ®ltering directly on a stretched grid. 1 Namely, the polar and Shapiro ®lters are applied di- 2 F (qi,j) ϭϪ (q i,j 1 Ϫ2qi,jϩq i,j 1), rectly to the discrete or grid functions de®ned as glob- ␸ 4 ϩ Ϫ al stretched grid ®elds, without interpolations to and and n is a ®lter parameter. Note that 2n is a ®lter from the global uniform ®ne grid. Actually, the spe- order. For 4Њϫ5Њand 2Њϫ2.5Њ resolutions, the ci®c coordinate transformation following from (5) for sixteenth- and eighth-order ®lters are applied, re- grid points is introduced by a stretched grid generator. spectively (Takacs et al. 1994). Lower-order ®ltering In the transformed coordinate system, the stretched corresponds to stronger damping. Usually, stronger grid becomes uniform. All ®ltering is to be done for noise is generated by a discretized model nonlinear the grid. dynamics when using ®ner resolution. Actually, it is not even necessary to make a formal Since the Shapiro ®lter is not applied to surface transformation because the ®lters (9)±(12) do not de- pressure ®elds, the conservation of mass is not af- pend on spatial coordinates (the same is true for the fected. The polar ®lter is applied only to the tenden- inverse transformation back to the physical space). cies, but not forecast variables, and is conservative The ®lters are applied to the grid functions or the in the zonal direction (for which it is applied). The stretched grid ®elds assuming that the grid is uniform polar and Shapiro ®lters have only a small effect on in the transformed computational space. As a result, total energy and potential enstrophy conservation. All we control noise in a small-scale spectral range, which that is true for the dynamical core with both uniform depends on variable resolution or stretched grid in- and variable resolution. These conservation proper- tervals ⌬x(x). ties were carefully checked for long-term integrations. For the stretched grid versions of the dynamical 4. Basic experiments with the stretched grid core considered below, the strength of ®ltering, and, dynamical core correspondingly, the ®lter order, is adjusted depend- ing on horizontal resolution over the area of interest. The goals of the experiments presented in this and the next section are as follows. We investigate the initial condition, or Cauchy problem, by running sev- d. Filtering on a nonuniform grid eral-day integrations with realistic mean initial con- There are two practical ways to account for com- ditions and with two ®ltering techniques described putational problems arising due to grid irregularity above. We are running long-term benchmark integra- effects. In one approach, the coordinate transforma- tions to analyze how they stabilize in time, compared tion is applied to model governing equations and the to uniform grid runs such as those performed by Held corresponding numerical scheme. A grid transfor- and Suarez (1994). The long-term experiments are mation from nonuniform physical coordinates to or- essential for the future regional climate applications. thogonal, uniform computational coordinates (see, For medium-range integrations, the possibility of us- e.g., Anthes 1970; Wilhelmson and Chen 1982) can ing large total global stretching factors without gen- be used. In another approach, the model governing erating noise is tested. The integrations allow us to equations and corresponding numerical scheme are analyze the quality of the resulting ®elds versus the not modi®ed. This approach is used, for example, for corresponding control experiments. the adaptive re®nement method (e.g., Berger and Oli- As a result of the aforementioned testing, the ger 1984; Skamarock 1989; Skamarock et al. 1989). stretched grid dynamical core is prepared to intro- In this study, the second approach with a smooth, duce, at the future stages of the study, the orographic uniformly stretched grid and no model numerical and other physical forcings within a stretched grid scheme modi®cation is used. To account for actual GCM. grid irregularity and to control related computational All experiments with stretched grid model versions noise, two simple versions of the existing model ®l- are performed with the grid shown in Fig. 3, with the tering techniques designed for uniform grids are test- rectangular area of interest located approximately ed. One of them, the interpolation approach, is based over the United States, or from 25Њ to 50ЊN and 75Њ

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FIG. 7. The annual mean (year 3) differences between the stretched grid and control runs for the zonal U (a) and meridional V (b) wind components. The contour interval for (a) is 2 m sϪ1 and for (b) 0.1 m sϪ1. The negative values are shown by dashed lines. The GL stands for the global domain.

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FIG. 8. The annual mean (year 3) differences between the stretched grid and control runs for the zonal wind component at the surface (a) and the ␴ ϭ 0.5 or midtroposphere level (b). The contour interval is 1 msϪ1 for (a) and 2 m sϪ1 for (b). to 125ЊW. A bilinear interpolation is used to create dent variables. Initial conditions of this kind allow us initial conditions on a stretched grid. to reduce the model spinup period as compared to a For testing numerical procedures suitable for state of the rest (with zero initial winds) initial con- stretched grid model dynamics, we have chosen the ditions used by Held and Suarez (1994). The clima- July mean, or climate ®elds obtained from the 10-yr tological initial conditions used here produce strong Atmospheric Model Intercomparison Project (AMIP) disturbances in the beginning of the model integra- (Gates 1992) run with the GEOS GCM, with 4Њϫ5Њ tion, which are due to the differences between the horizontal resolution, as initial conditions for depen- GCM and the dynamical core such as the absence of

Unauthenticated | Downloaded 09/28/21 03:43 PM UTC NOVEMBER 1997 FOX-RABINOVITZ ET AL. 2957 orography and simpli®ed diabatic forcing. It allows periment, with globally uniform ®ne-resolution ®l- us to investigate the robustness of the numerical tech- tering, demonstrate the development of noise due to nique used. grid irregularity that cannot be reliably controlled by The results of any stretched grid model run are sup- this version of the ®ltering procedure for a stretched posed to be compared with those of the corresponding grid model. control run performed with a global uniform ®ne-res- olution grid. The purpose of the validation is to assess b. Uniform ®ltering directly on a stretched grid how well the stretched grid model is capable of de- scribing the instantaneous regional and global scales The workable solution for the aforementioned com- and phenomena during medium-range integrations putational problem related to grid irregularity is ob- and representing mean circulation characteristics dur- tained by applying both the polar and Shapiro ®lters ing long-term integrations. Also, for both medium- directly to the global stretched grid ®elds, without range and long-term integrations, it has to be veri®ed interpolating back and forth from the global uniform that no noise and/or other unrealistic features are gen- ®ne grid. As described above, the coordinate trans- erated. formation following from (5) is introduced for grid points when using a stretched grid generator. All ®l- tering is applied directly to the grid functions de®ned a. Filtering on a uniform ®ne-resolution global on the stretched grid, or for the grid transformed into grid a uniform one in the transformed coordinate system. As described above, in this case the stretched grid The eighth-order Shapiro ®lter is used as in the pre- model is run with ®ltering on a ®ne-resolution global vious experiment. grid with the back and forth interpolation to the The results of the run with this ®ltering procedure stretched grid at every time step. The 2Њϫ2.5Њ res- are presented in Fig. 5. There is no indication of any olution stretching to 8Њϫ10Њ is used for the exper- computational noise generated during several days of iment. The eighth-order Shapiro ®lter is applied. The the stretched grid model integration. Actually, this experiment is important for exposing the numerical noise-free integration is continued in a climate mode, problems related to grid irregularity. This seemingly the results of which are discussed in the next section. simple ®ltering technique appeared to be incapable of The obtained results show that computational noise providing the reliable noise control. generated by grid irregularity can be effectively con- After several days of model integration, the prog- trolled by basically the same ®ltering procedures as nostic ®elds are contaminated with strong small-scale in the original dynamical core applied here to computational noise (Fig. 4). The area of interest over stretched grid prognostic ®elds. the United States is not contaminated by noise. How- ever, the noise pattern is most pronounced over the c. Long-term benchmark integrations North Pole within the coarse-resolution grid boxes (Fig. 4a). Also, strong noise occurs over South Asia The experiment discussed in section 4b and the cor- and the surrounding areas with strong initial gradients responding control one with global uniform 2Њϫ2.5Њ and, most importantly within the coarse latitudinal resolution have been run in a climate mode for 3 yr. resolution grid boxes (Fig. 4b). The goal is to compare the benchmark experiment Actually, such a result should not be surprising. performed with a stretched grid and the control run Small-scale features have been introduced as a result with the uniform ®ne-resolution grid, in terms of tem- of the ®nal interpolation back onto a stretched grid poral mean ®elds. from the global uniform ®ne-resolution grid for which Vertical distributions of the zonal-mean zonal wind all ®ltering has been actually done. In other words, component are presented in Figs. 6a,b. Because of the the ®ltering is done only for ®ne-resolution scales but symmetric hemispheric rhs forcing used in the dy- not for those of the stretched grid. The small-scale namical core, both hemispheric jets are practically noise features appearing within the stretched grid identical for the control experiment (Fig. 6b). The jet ®elds are not resolved for coarser-resolution gridbox- cores are positioned at approximately the ␴ ϭ 0.25 es outside the ®ne-resolution area of interest. Namely, level in the vertical and near the 40Њ±45Њ latitude re- the ®ltering applied to the ®ne grid is not enough for gion in both hemispheres. the grid boxes with coarser resolution in one or both For the stretched grid run (Fig. 6a), with ®ner res- directions. Such noise is exacerbated over the areas olution in the Northern Hemisphere according to Fig. with strong gradients resolved inadequately for the 3, the hemispheric jets are no longer symmetric. This elongated grid boxes. The solution is not destroyed is due to the fact that both computational errors and momentarily, as would be the case for the linear com- the actual rhs forcing in the dynamical core depend putational CFL instability. Instead, noise grows dur- on resolution. For the Southern Hemisphere with ing several days of model integration before the so- coarser resolution, the jet gradients, pattern and con- lution is completely destroyed. The results of this ex- ®guration, especially in the Tropics, are somewhat

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FIG. 9. The vertical ␴-level distributions of the annual (year 3) mean wind component for (a) the zonal wind over the area of interest for the stretched grid run, (b) same as (a) but for the control uniform ®ne-resolution run, (c) the meridional wind over the area of interest for the stretched grid run, and (d) same as (c) but for the control uniform ®ne-resolution run. The contour interval for(a)and(b)is5msϪ1, and for (c) and (d) 0.2 m sϪ1. The R (in SG-R and UG-R) stands for the region, or area of interest, located between 25Њ and 50ЊN, and 75Њ and 125ЊW.

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FIG.9.(Continued) different from those of the control run (Fig. 6b). Most are obtained for the vertical distributions of the zonal importantly, the Southern Hemisphere jet core is over- mean meridional wind component (Figs. 6c,d). In the estimated by 5 m sϪ1 and is latitudinally shifted equa- better-resolved Northern Hemisphere, the meridional torward by approximately 10Њ. For the better resolved wind distribution for the Tropics and midlatitudes in within the stretched grid Northern Hemisphere, the the vicinity of the ␴ ϭ 0.25 level for the stretched jet position and characteristics are very close to those grid run (Fig. 6c) is close to that of the control run of the control experiment (Figs. 6a,b). Similar results (Fig. 6d). It means that the Hadley cell is well rep-

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FIG. 10. Same as in Fig. 9 but for the area located within the least-resolved southeastern quadrant, between 10Њ and 80ЊS, and 0Њ and 150ЊE. The contour interval for (a) and (b) is 5 m sϪ1, for (c) 0.3 m sϪ1, and for (d) 0.5 m sϪ1. The RC (in SG-RC and UG-RC) stands for the region with coarse resolution. resented in the stretched grid run (Fig. 6c). For the cell for the Southern Hemisphere with coarser reso- Southern Hemisphere, the corresponding meridional lution. Therefore, the results for each hemisphere wind distribution for the stretched grid run is under- strongly depend on hemispheric resolution. estimated (Fig. 6c). It results in weakening the Hadley The differences between vertical distributions of

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FIG.10.(Continued) the wind components shown in Fig. 6 are presented from the control run hereafter are called errors. We in Figs. 7a,b. These differences are calculated for the realize that these errors include a redistribution of the simulated stretched grid ®elds against the correspond- uncertainty due to, for instance, the differences be- ing uniform ®ne resolution or control run ®elds. We tween integrations started from slightly different ini- are supposed to verify how close the results are of tial conditions. the stretched grid and control runs. The deviations For the error calculation, the stretched grid ®elds

Unauthenticated | Downloaded 09/28/21 03:43 PM UTC 2962 MONTHLY WEATHER REVIEW VOLUME 125 here and hereafter are interpolated onto the global importantly in the context of this study, they show uniform ®ne-resolution grid used for the control run. the ability of a stretched grid model to reproduce cli- In the better resolved Northern Hemisphere, the errors mate patterns, especially over the area of interest and are fairly small, mostly within Ϯ2msϪ1 for the zonal its vicinity, which are close to those of the control wind U and Ϯ0.1 m sϪ1 for the meridional wind V. run with ®ne uniform global resolution. It appears that For the Southern Hemisphere with coarser resolution, the regional circulation patterns and distributions are the errors are signi®cant, almost an order of magni- only moderately affected but not really overwhelmed tude larger due to the equatorward shift of the jet and by the incoming relatively poorly resolved ¯ow. underestimation of the Hadley cell. The annual mean errors for the stretched grid zonal 5. Dynamical core integrations with enhanced winds for the ␴ ϭ 1 and ␴ ϭ 0.5 levels calculated as variable resolution and large total global the differences between the stretched grid and control stretching factors runs are presented in Figs. 8a,b. The errors are smaller over the best-resolved northwestern quadrant, for The results of stretched grid integrations presented which they are mostly within Ϯ1±2 m sϪ1 at the sur- above have been obtained with a total global stretch- face (Fig. 8a) and Ϯ2±4 m sϪ1 for the middle tro- ing factor of 4 and rather coarse variable resolution posphere (Fig. 8b). Within the northwestern quadrant, for the grid stretching from 2Њϫ2.5Њ to 8Њϫ10Њ.In the errors are minimal for the area of interest over view of the basic problems related to ®nite-difference the United States, for which they are at least twice approximations with nonuniform grids outlined in smaller than for the entire quadrant (Figs. 8a,b). Over section 2, we should carefully verify the robustness other quadrants, especially over the Southern Hemi- of the numerical technique for stretched grids with sphere, the errors are signi®cantly larger. larger total stretching factors. In this section, we pres- This conclusion can also be corroborated by con- ent results of medium-range integrations of the sidering the time-averaged zonal mean distributions stretched grid dynamical core with ®ner variable res- of wind components over the ®ne-resolution area of olution and larger total global stretching factors rang- interest (Fig. 9) located within the northwestern quad- ing from 4 to 32. The same area of interest as in Fig. rant, and over the large area located in the least re- 3 is used for the experiments. The following variable solved southeastern quadrant (Fig. 10). For the area horizontal resolution grids are used: 1Њϫ1.25Њ of interest, the patterns of the zonal and meridional stretching to 4Њϫ5Њ;1Њϫ1.25Њ stretching to 8Њϫ wind vertical distribution are similar for both the 10Њ; 0.5Њϫ0.625Њ stretching to 4Њϫ5Њ; 0.5Њϫ0.625Њ stretched grid and control experiments. The vertical stretching to 8Њϫ10Њ; 0.25Њϫ0.3125Њ stretching to and latitudinal position and gradients of the jet core 4Њϫ5Њ; and 0.25Њϫ0.3125Њ stretching to 8Њϫ10Њ. practically coincide (Figs. 9a,b), whereas the jet core For validation purposes, two control experiment re- strength is overestimated by 5 m sϪ1 in the stretched sults, with ®ne uniform global 1Њϫ1.25Њ and 0.5Њϫ grid run. The corresponding patterns of the meridional 0.625Њ horizontal resolutions, are used. All these ver- wind vertical distribution are also close (Figs. 9c,d), sions of ®ne-resolution stretched and uniform grids although the negative feature is slightly underesti- are run with the stronger fourth-order Shapiro ®lter mated by approximately 0.2 m sϪ1, and its center is and stronger polar ®lter [Eq. (10)]. Initial conditions slightly shifted by approximately 1Њ±2Њ (i.e., by less are the same as in the benchmark integrations (the than one grid interval) equatorward for the stretched July mean ®elds from the AMIP run). grid run. The results of the 10-day integrations with the total For the coarse-resolution area in the southeastern stretching factors of 4 and 8 for the global domain quadrant, the same circulation characteristics are pro- are presented in Fig. 11. Over the area of interest and foundly different for the stretched grid and control the entire northwestern quadrant, both the control uni- experiments (Fig. 10). The jet core is shifted in the form grid and stretched grid runs are close to each vertical upward by approximately 25±50 hPa and lat- other in terms of both the amplitude and phase of itudinally by approximately 8Њ±9Њ equatorward for the synoptic features even for the strong gradient area stretched grid run (Figs. 10a,b). The jet patterns and over the United States (Fig. 11a,b). The differences gradients are also signi®cantly different. The same is between the stretched grid runs and the control run even more true for the meridional wind distributions over the area of interest are small, less than 1 K (Figs. (Figs. 10c,d). These results are consistent with those 11c,d). The differences over the entire northwestern obtained for the GEOS GCM by Takacs and Suarez quadrant are also small, mostly within 2 K. At the (1996) on the impact of coarser resolution or lower same time, for other quadrants, especially over the order of the accuracy of approximation. least-resolved southeastern quadrant, the differences The obtained results show the model sensitivity to are much larger. variable horizontal resolution (over the Northern and For a more detailed comparison, we consider the Southern Hemispheres, for the global quadrants and results of the 10-day integrations, with the total over the area of interest) within a stretched grid. Most stretching factors of 4, 8, and 16, for the major part

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FIG. 11. The global surface potential temperature ®elds (a) and (b) and their differences (c) and (d) after 10-day integrations: (a) with the global uniform 1Њϫ1.25Њ resolution grid (the control run); (b) with the grid stretching from 1Њϫ1.25Њ resolution to 4Њϫ5Њ;(c)the difference ®eld between (b) and (a); (d) the difference ®eld between the run with the grid stretching from 1Њϫ1.25Њ to 8Њϫ10Њ and (a). The contour intervals for (a) and (b) are 3 K, and for (c) and (d) 1 K. The SG and UG stand for a stretched grid and uniform grid, correspondingly. The negative values are presented by dashed lines. of the best-resolved northwestern quadrant that in- area of interest (Figs. 12a,c,e). Notice that the com- cludes the area of interest and its large vicinity (Fig. putational costs are reduced by the factors of 6 and 12). Two control experiment results, with uniform 10 for the total stretching factors of 8 and 16, rela- global resolution of 1Њϫ1.25Њ (Fig. 12a) and 0.5Њϫ tively (as compared to the corresponding uniform grid 0.625Њ (Fig. 12b), appeared to be well reproduced by control runs). the corresponding stretched grid runs [Figs. 12c,e (the For the stretched grid runs, the deviations from the left column) and Figs. 12d,f (the right column)] in corresponding control runs or rms errors are calcu- terms of the position and depth for all centers, and in lated for all model sigma levels (Fig. 13). For both terms of the gradient strength. The ®elds obtained grids with total stretching factors of 4 and 8, the rms with the larger total stretching factors of 8 and 16 errors over the area or region of interest are small, using the grids stretching from 1Њϫ1.25Њ to 8Њϫ although they are larger for the grid with the total 10Њ (Fig. 12e) and 0.5Њϫ0.625Њ to 8Њϫ10Њ (Fig. stretching factor of 8. Over the entire best-resolved 12f) are only slightly smoother than those of the cor- northwestern quadrant, the rms errors (Fig. 13c) are responding control runs (Figs. 12a,b). The results of larger than those of the area of interest but are still the integrations with the grids stretching from 1Њϫ quite moderate. Over the least-resolved southeastern 1.25Њ to 4Њϫ5Њ(Fig. 12c) and 0.5Њϫ0.625Њ to 4Њ quadrant, the rms errors are approximately an order ϫ 5Њ (Fig. 12d) represent even minor details of the of magnitude larger than those of the area of interest. corresponding control runs (Figs. 12a,b). The ®elds For the stretched grid run with ®ner 0.5Њϫ0.625Њ obtained with ®ner 0.5Њϫ0.625Њ resolution over the resolution over the area of interest stretching to 4Њϫ area of interest (Figs. 12b,d,f) have smaller-scale fea- 5Њ, or with the total stretching factor of 8, the rms tures and stronger centers and gradients than those errors (calculated against their corresponding control, obtained with coarser 1Њϫ1.25Њ resolution over the or uniform 0.5Њϫ0.625Њ resolution run) are similar

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FIG. 12. The surface zonal wind ®elds after 10-day integrations for the large part of the northwestern quadrant (10Њ to 70ЊNand20Њto 170ЊW) using the grids with (a) global uniform 1Њϫ1.25Њ resolution (the ®rst control run), (b) global uniform 0.5Њϫ0.625Њ resolution (the second control run), (c) 1Њϫ1.25Њ resolution stretching to 4Њϫ5Њ(the TSF of 4), (d) 0.5Њϫ0.625Њ resolution stretching to 4Њϫ5Њ(the TSF of 8), (e) 1Њϫ1.25Њ resolution stretching to 8Њϫ10Њ (the TSF of 8), and (f) 0.5Њϫ0.625Њ stretching to 8Њϫ10Њ (the TSF of 16). The TSF stands for the total stretching factor. The contour interval is 2 m sϪ1. but slightly smaller than those of the grid with the tions, the large- and medium-scale spectral range (for same total stretching factor of 8 but with the coarser the total wavenumbers less than or equal to 8) are 1 ϫ 1.25Њ resolution over the area of interest. very close to that of the control run. The result is Let us compare the spherical harmonic spectra for important. These global spectral ranges should not be stretched grid integrations and the control one pre- signi®cantly affected by model variable resolution for sented in Fig. 14. It is important to ®nd out which medium-range integrations. The smaller scales show spectral ranges are affected by introduction of global the impact of model variable resolution and ®ltering. variable resolution. For both stretched grid integra- The impact is stronger for the run with the larger total

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FIG. 13. The rms errors for 10-day inte- grations for (a) zonal wind ®elds over the area of interest obtained from two stretched grid runs, (b) same as (a) but for potential temperature, and (c) same as (b) but for the entire northwestern quadrant. The rms er- rors are shown for the grid stretching from 1Њϫ1.25Њ to 4Њϫ5Њby dark circle lines, and for the grid stretching from 1Њϫ1.25Њ to 8Њϫ10Њ by open circle lines. Wind errors are in meters per second and potential tem- perature errors in kelvins.

stretching factor of 8. Notice that prior to spectra cal- lution over the area of interest. They contain smaller-scale culations the stretched grid ®elds are interpolated onto features and stronger centers and gradients. However, it the corresponding global uniform ®ne-resolution grid appeared to be too costly in terms of both the computer that affects the smaller-scale spectral range. time and memory to perform the corresponding control Two more successful integrations have been performed run with the uniform global 0.25Њϫ0.3125Њ resolution using the grids with the very ®ne 0.25Њϫ0.3125Њ reso- needed for calculating errors similar to those of Fig. 13. lution over the area of interest stretching to 4Њϫ5Њ(with Note that the requirements on consistent horizontal the total stretching factor of 16) and to 8Њϫ10Њ (with and vertical resolution pointed out by Lindzen and the total stretching factor of 32). (The last grid has ap- Fox-Rabinovitz (1989) should be applied within the proximately the same number of grid points as the global stretched grid approach. Obviously, the resolution uniform 1Њϫ1.25Њ resolution grid.) The resulting ®elds consistency within a stretched grid should be de®ned are basically similar to those obtained with coarser reso- for the area of interest. The simplest way to implement

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thoroughly tested. It allows one to reliably control computational noise arising from grid irregularity. The medium-range and long-term benchmark in- tegrations of the stretched grid dynamical core have shown no negative computational effects ac- cumulating in time. The positive impact from in- creased horizontal resolution, especially over the area of interest and its vicinity, is obtained. The regional prognostic ®elds and circulation patterns and distributions are not overwhelmed by the in- coming relatively poorly resolved ¯ow. For long-term benchmark integrations, the par- ticular emphasis is placed on ultimately using the stretched grid approach for regional climate mod- eling applications. FIG. 14. The spherical harmonic spectra for the temperature ®elds 3) With adjusted basic ®lter parameters, stretched at the ␴ ϭ 0.5 level after 10-day integrations using the grids with grids with medium to large total global stretching the global uniform 1Њϫ1.25Њ resolution (the control run) (solid line), factors of 4 to 32 have been used. Successful me- the 1Њϫ1.25Њ resolution stretching to 4Њϫ5Њwith the total stretching factor of 4 (dotted line), and the 1Њϫ1.25Њ resolution stretching to dium-range integrations with summer climate ini- 8Њϫ10Њ with the total stretching factor of 8 (dashed line). tial conditions are performed for the grids with ®ne resolution over the area of interest (the United States) of 2 ϫ 2.5Њ,1Њϫ1.25Њ, 0.5Њϫ0.625Њ, and it for the enhanced horizontal resolution region within 0.25Њϫ0.3125Њ stretching outside the area of in- a stretched grid is to consistently increase model ver- terest to 4Њϫ5Њor 8Њϫ10Њ. The profound sim- tical resolution globally. The associated additional ilarity to the corresponding control runs with uni- computational cost is much less than that of increased form ®ne global resolution is obtained. horizontal resolution (with the corresponding de- 4) Stretched grid integrations are performed at sig- crease of a time step). Experiments on consistent res- ni®cantly reduced computational costs as com- olution are planned to be performed later. pared to those of the corresponding global uniform The close similarity between the stretched grid in- ®ne-resolution ones. This allows one to use such tegrations and those of the corresponding control runs ®ne resolution over the area of interest that is not in a medium-range forecast mode, even for large total possible otherwise for a foreseeable future. stretching factors presented in this section, shows the 5) The stretched grid dynamical core is planned to robustness and high quality of the numerical solutions be introduced into the future stretched grid GEOS obtained with variable resolution stretched grids. GCM. The results presented in this and previous sections with various stretched grid model versions are en- couraging. We can use the basic ®ltering procedure, Acknowledgments. The authors would like to thank readjusting only its parameters depending on reso- Mr. R. Govindaraju for his programming support and lution. Using the stretched grid approach allows us to Ms. M. Hall for typing the manuscript. The study has take advantage of enhanced resolution and the cor- been supported by NASA Grant 578-41-11-20 and responding downscaling over the area of interest for DOE Grant LWT6212307506. We would like to thank medium-range and long-range integrations. Dr. Kenneth Bergman, the NASA Program Manager, for his support of the study. 6. Conclusions The following conclusions are obtained as the result REFERENCES of the study. Anthes, R. A., 1970: Numerical experiments with a two-dimen- 1) The key result of the study with the variable-res- sional horizontal variable grid. Mon. Wea. Rev., 98, 810±822. olution dynamical core is that an adequate repre- , 1983: Regional models of the atmosphere in middle lati- tudes. Mon. Wea. Rev., 111, 1306±1335. sentation of regional scales, that is, a signi®cant Arakawa, A., and V. R. Lamb, 1981: A potential enstrophy and downscaling, is taking place over the area of in- energy conserving scheme for the shallow water equations. terest (and its vicinity) not only for short-term in- Mon. Wea. Rev., 109, 18±36. tegrations but also for medium-range and, most , and M. J. Suarez, 1983: Vertical differencing of the primitive importantly, long-term integrations. equations in sigma coordinates. Mon. Wea. Rev., 111, 34± 45. 2) The stretched grid version of a ®nite-difference Asselin, R., 1972: Frequency ®lter for time integrations. Mon. dynamical core with effective ®ltering techniques Wea. Rev., 100, 487±490. for nonuniform, stretched grids, is developed and Berger, M., and J. Oliger, 1984: Adaptive mesh re®nement for

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