## 1. Introduction

The interest in accurately simulating trace constituent transport, both water vapor and chemical species, has increased greatly over the past several years, particularly with respect to GCMs and chemical transport models. In light of this current interest, several numerical methods have been developed to improve both the horizontal and vertical transport of trace constituents in atmospheric prediction models (Prather 1986; Carpenter et al. 1990; Thuburn 1993; Rasch 1994; Lin et al. 1994). One method receiving considerable attention is semi-Lagrangian transport (SLT) (Ritchie 1985; Williamson and Olson 1994; Huang 1994). Rasch and Williamson (1990a) provide a review of this method, including the conservative capabilities of both the conventional SLT and the shape-preserving SLT. The latter is the standard algorithm for simulating water vapor and other trace constituent transport in the National Center for Atmospheric Research (NCAR) Community Climate Model 2 (CCM2) (Hack et al. 1993).

As discussed by Rasch and Williamson (1990a), conservation of a property is not achieved with current SLT formulations. Conventional SLT algorithms generate spurious negative values and require borrowing or another negative value “fixer” similar to other transport methods, for example, second-order finite-difference algorithms, etc. Shape-preserving SLT methods preclude the development of negative values; however, a fixer is required to conserve the global integral of the property. Furthermore, the shape-preserving SLT is dispersive in the vicinity of extremes (Rasch and Williamson 1990b). Rasch and Williamson (1990a) also suggest that the lack of conservation in an SLT may be in part due to an inconsistency between the SLT vertical advection of water vapor and the finite-difference vertical advection of mass.

The utility of isentropic (*θ*) analysis for studies of long-range transport has been recognized for decades (Rossby 1937), although the use of *θ* coordinates in NWP and climate simulations remains relatively unexplored. The primary advantage of *θ* coordinates is that under isentropic conditions the three-dimensional transport of other coordinate systems (*z, p,* and *σ*) is reduced to two spatial dimensions and the vertical truncation error of a property is eliminated. Despite this important simplification, computational difficulties associated with coordinate intersections at the earth’s surface or another “interface,” as well as weak static stability in the PBL, have deterred many investigators from pursuing modeling in isentropic coordinates. Only a few models presently exist to simulate the atmosphere where at least part of the domain is represented by isentropic coordinates; some recent efforts are Black (1987), Benjamin (1989), Hsu and Arakawa (1990), Zhu et al. (1992), Bleck and Benjamin (1993), Johnson et al. (1993), and Zapotocny et al. (1993, 1994).

As discussed by Johnson et al. (1993), the advantages of *θ* coordinates for prediction are greatest on the global scale, where, for given resolution, the long-range transport of properties within baroclinic wave regimes is more accurately resolved in *θ* coordinates than in either *σ* or isobaric coordinates. This improved accuracy is important for diagnosing processes such as the cross-tropopause exchange of ozone or other properties, the transport of water vapor from evaporational source regions to condensational sink regions (Johnson 1989), and the exchange of chemical species and pollutants.

With the above considerations in mind, the two main objectives of this paper are to 1) compare the spatial evolution and filamentary structure of inert trace constituents in the University of Wisconsin (UW) hybrid isentropic–sigma (*θ*–*σ*) model against CCM2 and the UW *σ* model and 2) examine the ability of the models to conserve the initial trace constituent maxima during 10-day integrations. The work reported here follows Zapotocny et al. (1994), which documented development of the global UW *θ*–*σ* and *σ* models and compared their synoptic predictions against Goddard Earth Observing System (GEOS-1) assimilated data. Zapotocny et al. (1994) also examined the transport of an inert trace constituent in the UW *θ*–*σ* and *σ* models for 10-day isentropic simulations. In those comparisons, the primary difference between the two models was the vertical coordinate used; the numerics, time differencing, filters, and so on were all identical. The results displayed a decided advantage for the UW *θ*–*σ* model over the UW *σ* model simulations in coherency of stratospheric inert trace constituent transport and conservation of the initial maximum.

In this next step, the advantages of simulating transport processes in the UW *θ*–*σ* gridpoint model relative to “state of the art” *σ* coordinate spectral models are examined: CCM2 using SLT for the trace constituent, and a version of CCM2 using spectral transport for the trace constituent rather than SLT (hereafter CCM-S). The experiments here were formulated as a means of further testing the accuracy of trace constituent transport in *θ* versus *σ* coordinates. The CCM is used because it is well documented, easily accessible, incorporates sophisticated numerics, and is a community model used by a relatively large number of investigators.

The main point of these experiments is *not* to compare the accuracy of advection schemes in the classical one- and two-dimensional sense (e.g., Haltiner and Williams 1980; Williamson et al. 1992; Held and Suarez 1994; Pielke et al. 1995). Rather the main point is to highlight the relative advantages of using isentropic coordinates for transporting inert trace constituents within a baroclinic atmosphere where an analytic solution is not known. Unlike Zapotocny et al. (1994), which was limited to isentropic simulations and stratospheric comparisons, this paper examines trace constituent transport in the troposphere and stratosphere under both idealized isentropic conditions and with diabatic processes and physical parameterizations included. Although limited to a comparison of results from only three *σ* coordinate models, the results here, combined with those of Zapotocny et al. (1994) provide strong evidence that the dispersion characteristics found in the UW *σ* model and the CCM stem from three-dimensional transport in the presence of baroclinic advection and vertical wind shear. This dispersion should be found in all models based on sigma and/or isobaric coordinates, but less so in the UW *θ*–*σ* model due to the dominance of two-dimensional transport in *θ* coordinates.

Section 2 of this paper describes the initial data used in this study and outlines modifications to the models for the inert trace constituent experiments. Section 3 presents a comparison of the models’ mass, momentum, and zonal kinetic energy fields to demonstrate that the differences in trace constituent evolution do not develop from differences in the simulated circulation among the models. Section 4 outlines how the source-free inert trace constituents were specified in each model, and section 5 presents a comparison of each model’s ability to simulate coherent filamentary distributions and prognostically conserve the initial inert trace constituents maxima during 10-day isentropic and nonisentropic simulations. The results are discussed in section 6 and summarized in section 7.

## 2. The models and initial conditions

The UW *θ*–*σ* model horizontal resolution is identical to the GEOS-1 system (Schubert et al. 1993), except in section 5a(5) where it is reduced to more closely approximate the T42 CCM2 Gaussian grid. The vertical resolution in the *θ* domain of the UW *θ*–*σ* model is 8 K up to 360 K, and variable Δ*θ* above 360 K (see Fig. 1) with an upper bounding surface at 600 K. The initial interpolated pressure on 600 K, as obtained from the GEOS-1 data, is invariant throughout model integration. The PBL (*σ* domain) is a constant 150 mb with four predictive surfaces. This vertical structure results in an initial average of 17.8 vertical layers. The average number of layers in the UW *θ*–*σ* model is not an integer number since the number of *θ* layers is based on the vertical structure of temperature and pressure, which varies both spatially and temporally.

The north–south cross section from the UW *θ*–*σ* model (Fig. 1) schematically illustrates the meridional structure of the *θ* and *σ* model surfaces over a mountain range similar to the Himalayas in winter. The schematic portrays an extreme example of sloped interface orientation between *θ* and *σ* model domains: in oceanic basins and regions without orography the interface typically lies near 850 mb.

The governing equations for the UW *θ*–*σ* model are presented in Pierce et al. (1991). Vertical and horizontal transport in the UW *θ*–*σ* model is accomplished with a second-order flux-conservative box method (Kurihara and Holloway 1967) on the nonstaggered Arakawa A grid (Arakawa and Lamb 1977). The transport of properties across the interface of the *σ* and *θ* domains is determined by the vertical mass flux. The vertical mass flux is computed from the vertically integrated horizontal mass convergence in the *σ* domain; momentum, water vapor, and trace constituents transported by the mass flux are determined from interpolated values of properties to the interface from the highest *σ* and lowest *θ* information surfaces in each vertical grid column. Further details about the interface formulation, including a complete description on how isentropic surfaces submerge and emerge across the interface, may be found in Uccellini et al. (1979) and Pierce et al. (1991). A predictor-corrector time differencing scheme (Matsuno 1966) is used to suppress gravity waves. The UW *θ*–*σ* model uses a relaxed Arakawa–Schubert convective scheme (Moorthi and Suarez 1992) and the CCM2 radiation algorithm (Hack et al. 1993).

The UW *σ* model has 18 predictive surfaces and extends from the earth’s surface to a fixed upper boundary at 25 mb. The four layers in the lowest 150 mb replicate closely the PBL of the *θ*–*σ* model. The remaining 14 surfaces are equally spaced over the remaining depth of the atmosphere. Equations describing the UW *σ* model are presented in Johnson et al. (1993). For this study the horizontal grid spacing, time differencing, filters, physical parameterizations, borrowing, etc. are all identical to those of the UW *θ*–*σ* model.

The borrowing/filling algorithms for water vapor and trace constituent in both UW models have local and global components. The local component that is applied first borrows the needed amount as an equal percentage from the four nearest horizontal neighbors. The global component is applied only if the local component fails to eliminate all negative values. This component sums the remaining negative values and zeros them, by subtracting the integral of needed water or trace constituent proportionally from each grid box with a positive amount. The integral of water or trace constituent is conserved locally and globally by the respective components of the borrowing/filling algorithm.

The CCM uses a terrain following hybrid sigma-isobaric vertical coordinate with either three-dimensional semi-Lagrangian (CCM2) or spectral (CCM-S) transport for moisture and trace constituents. In this work, the spectral resolution for the CCM-S simulations is T42, and is T42, T63, and T106 for the CCM2 simulations. The 2° × 2.5° resolution used in the UW *θ*–*σ* and *σ* models is slightly greater than T42 resolution, less than T63 resolution, and less than half the resolution of T106. The 18 vertical layers used for all CCM simulations are comparable to the initial average of 17.8 layers in the UW *θ*–*σ* model and 18 layers in the UW *σ* model. All CCM simulations use the default value of 0.06 for the Asselin time filter and similar to the UW models use a mass fixer to ensure invariance of the global integral of the dry mass of the atmosphere. The CCM2 (version 2.1 was used) SLT shape-preserving scheme precludes the development of negative values of trace constituent. A fixer is applied only to conserve the global integral of moisture and trace constituent. The CCM-S simulation uses the standard spectral advection of trace constituent with a local scheme to borrow from the nearest longitude and height neighbors, but no global borrowing. Negative values not eliminated by the local borrowing scheme of CCM-S are allowed to persist. Rasch and Williamson (1990a) provide greater detail of the fixers used in the CCM.

For all simulations presented herein, with one exception, the initial conditions were 0000 UTC 1 February 1985, as obtained from archived GEOS-1 assimilated data (Schubert et al. 1993). The results from the exception use initial conditions obtained from the second year of a climate simulation by CCM2. The GEOS-1 system, based on *σ* coordinates with 20 layers extending from the earth’s surface to 10 mb, employs a latitude–longitude grid with a horizontal resolution of 2° north–south and 2.5° east–west. Initial data (*u, v, T,* and RH) from both the GEOS-1 data assimilation system and the CCM2 climate simulation were interpolated to *θ* surfaces of the UW *θ*–*σ* model linearly with respect to *p*^{κ} and to the *σ* surfaces of the UW *θ*–*σ* and *σ* models and all CCM2 models that were linear with respect to pressure.

For all results reported here the simulations use uninitialized initial conditions. However, to eliminate the possibility that spurious gravity waves from the uninitialized data could have substantive detrimental effects, corresponding tests were performed in all models with initialized data. No results are shown from the initialized simulations since the trace constituent evolution and conservation of initial maxima were nearly identical, with the largest difference in trace constituent maxima at day 10 being 2.5%.

Apart from the implicit dissipative effects of skin friction in all models and the horizontal diffusion of divergence, vorticity, and temperature in some CCM2 simulations, the models were run isentropically in all experiments reported here, except for the experiment in section 5c, which includes diabatic processes and a complete suite of physical parameterizations. Surface friction is used in all simulations to preclude excessive winds from developing in the PBL. The inclusion of explicit horizontal diffusion is to document the improvements achieved from introduction of these processes in CCM2. Additional conditions imposed on the simulations are that the trace constituents are source/sink-free and not filtered or diffused during the integration or plotting.

Isentropic conditions exclude precipitation, radiation, or any other physical parameterizations; the absence of convection and vertical diffusion precludes vertical transport of trace constituent by these processes. When isentropic conditions are prescribed, the primitive equations for the continuum require that the initial maxima of trace constituent on all *θ* surfaces (whether implied or explicit in any model) be conserved throughout the integration. Deviations from the initial maxima as functions of time, except for smoothing introduced by vertically interpolating the trace constituent to implied isentropic surfaces in the UW *σ* model and CCM, provide an estimate of error due to discretization and resolution in the model. Only one diabatic integration is presented (section 5c) since once diabatic processes are allowed the constraint of conservation of the initial maxima on isentropic surfaces is removed. Under diabatic conditions the global maxima should be constant, but the maxima on individual isentropic surfaces could be less than its initial value. The “inverse” interpolation of simulated distributions from isentropic to sigma coordinates in section 5a(1) clearly establishes that vertical interpolation is not a major source of error. The tests are limited to 10 days since that approximates the global residence time for water vapor (Peixoto and Oort 1992). Consequently, to “first-order considerations” these experiments test model accuracy in transporting water vapor from evaporational source to condensational sink regions.

## 3. Synoptic comparison of the models

This section compares mean sea level pressure, 200-mb zonal wind, and zonal wavenumber kinetic energy distributions to establish the kinematic and dynamical similarity of the simulations. The day-5 mean sea level pressure distributions produced by the UW models and the T42 and T63 CCM2 are shown in Fig. 2. An examination of the fields reveals that all four models have produced very similar forecasts. Intense cyclones (976 mb or less) are simulated in the central North Atlantic Ocean and northwest of the Caspian Sea. An intense anticyclone is located over the Aleutians with a central pressure in the UW models and T42 CCM2 near 1050 mb: it exceeds 1060 mb in the T63 CCM2 simulation. In the Southern Hemisphere, the surface pressure extremes are in general weaker than in the Northern Hemisphere, consistent with expectations for the summer hemisphere. All four of the models maintain an anticyclone greater than 1032 mb over the Antarctic Plateau and cyclones just west of the Antarctic Peninsula and southeast of the Cape of Good Hope.

Figure 3 shows the day-5 200-mb zonal wind distributions. All four models produce three pronounced Northern Hemispheric zonal wind maxima with values between 65 and 82 m s^{−l}. One maximum is over eastern North America, the second is south of the Caspian Sea, and the third is over southwestern Japan. The Southern Hemisphere has an elongated zonal wind maximum with the strongest values near 45°S in all models.

To further illustrate that the characteristics of trace constituent transport presented below do not stem from differences in the simulated flow fields, Fig. 4 shows the 300- and 700-mb specific kinetic energy spectra (Saltzman 1957) for zonal wavenumbers 1 through the Nyquist frequency of each model from the initial state and day-5 UW *θ*–*σ* model and T63 CCM2 simulations. The initial CCM2 distributions are not shown since they are nearly identical to the UW *θ*–*σ* distributions. Zonal and meridional wind components on the Gaussian grid are used for the CCM2 calculations, while the UW *θ*–*σ* calculations use winds from its latitude–longitude grid. Data in the *θ* domain of the *θ*–*σ* model were interpolated to isobaric coordinates assuming a linear variation with *p*^{κ}. In CCM2 and the *σ* domain of the UW *θ*–*σ* model data were interpolated linearly with respect to pressure. The kinetic energy spectra were then analyzed following Horn and Bryson (1963). As expected, the initial distributions show maximum kinetic energy at 300 mb in the extratropical jet stream regions of both hemispheres with the bulk of kinetic energy residing in low wavenumbers of the winter (Northern) hemisphere.

Although different in detail, the kinetic energy spectra from the UW *θ*–*σ* model and T63 CCM2 are in agreement in that both possess corresponding energy in medium and higher wavenumbers. At high wavenumbers, greater than 25, the UW *θ*–*σ* model retains substantially more kinetic energy at 300 mb in the baroclinic westerlies of each hemisphere and slightly more kinetic energy at 700 mb, particularly in the Southern Hemisphere.

These results indicate that similar scales are resolved in the UW *θ*–*σ* model and T63 CCM2, and that the differing transport characteristics of the two models discussed below cannot be attributed to excessive damping of smaller scales by one model relative to the other. These results also indicate that the UW *θ*–*σ* model does not suffer from an excessive cascade of energy to smaller scales, even though it uses the nonstaggered Arakawa A grid (Arakawa and Lamb 1977). Overall, the results indicate that the simulated kinematic and dynamical structure of all models evolve in a similar manner. Consequently the different trace constituent distributions that follow occur in essentially similar atmospheric circulations.

## 4. Specification of the inert trace constituents

Results from two experiments with different initial trace constituent distributions are examined. In the first experiment, the initial distribution is a pair of vertically invariant functions defined by bivariate normal distributions (hereafter referred to as vertical cylinders) specified analytically and centered in the middle latitudes of the Western Hemisphere (Fig. 5a). In the second experiment, the initial distribution consists of vertically and zonally invariant rings centered at 48° latitude, one in each hemisphere with meridional variation specified through univariate normal distributions (Fig. 5b). The vertical cylinder experiment examines transport characteristics of a localized distribution corresponding with material emitted by volcanic eruptions, cloud generation, chemical species created within clouds, chemical reactions, and especially water vapor transported vertically by mesoscale convective systems or narrow filaments moving poleward from the Gulf of Mexico. The zonal ring experiment examines a global distribution embedded within the circumpolar vortex.

*f*

_{max}is the maximum value (100.0 in these studies) and Λ and Φ are scaled spherical coordinates expressed by The coordinates of the maximum are

*λ*

_{0}and

*ϕ*

_{0}, while

*σ*

_{λ}and

*σ*

_{ϕ}(the standard deviations) scale the horizontal distributions. For the vertical cylinders (Fig. 5a),

*λ*

_{0}is 100.0°W,

*ϕ*

_{0}is 40.0°N and 40.0°S,

*σ*

_{λ}is 12.5°, and

*σ*

_{ϕ}is 10.0°. For the zonal rings (Fig. 5b), without functional dependence on longitude,

*ϕ*

_{0}is 48.0°N and 48.0°S,

*σ*

_{λ}is infinity, and

*σ*

_{ϕ}is 4.0°.

The initial trace constituent distributions are computed from (1), (2), and (3) directly on all model surfaces; neither horizontal nor vertical interpolation is required. In all simulations, except section 5a(5), the initial positions of the constituent maxima, exactly 100 units for both the vertical cylinders and zonal rings, are coincident with grid points of the UW models. In section 5a(5), which uses a modified horizontal grid resolution with grid points that do not exactly coincide with the position of the extremes just specified, the initial maxima are 99.7 units. The initial CCM maxima are also slightly less than 100 units since the Gaussian grid does not exactly resolve the position of the maxima. However, all CCM initial maxima exceed 99.3 units.

For experiments reported here, the fields of mass, momentum, and energy in the UW *θ*–*σ* and *σ* models are filtered as described by Zapotocny et al. (1994); the fields of trace constituent are not filtered nor diffused. The tables below present CCM2 trace constituent results for simulations with and without horizontal diffusion (labeled as D and ND, respectively, in the tables) of the divergence, vorticity, and temperature; the figures only show CCM2 results with diffusion. All CCM2 simulations also use the Asselin time filter with a constant of 0.06, the default. To provide a full set of comparisons, UW *σ* and CCM model results are either interpolated to isentropic surfaces or shown directly on the respective surfaces of each model; some UW *θ*–*σ* results are also interpolated to CCM2 model surfaces.

## 5. Inert trace constituent experiments

### a. Cylindrically distributed trace constituent

#### 1) UW *θ*–*σ* model and T63 CCM2

The comparisons presented in this section document the filamentary nature of inert trace constituent transport and conservation of initial maxima under isentropic conditions (except for the implicit effects of skin friction) in the UW *θ*–*σ* model and T63 CCM2. Results are also presented with and without diffusion of the divergence, vorticity, and temperature in some CCM2 simulations. Figures 6–9 show results on four isentropic levels: 555 K and 415 K in the stratosphere, 340 K at jet stream level, and 300 K in the troposphere. The primary reason for showing four levels is to illustrate the vertical structure of the trace constituent filaments and the extreme gradients that develop from the vertical wind shear of the westerlies over the 10 days of integration. Each figure portrays simulations at day 5 and day 10 from the UW *θ*–*σ* model (panels a and b, respectively) and T63 CCM2 (panels c and d, respectively). The trace constituent maximum on each isentropic surface is listed in the lower right-hand corner of each panel. (Note that as discussed in section 2, all models are constrained to ensure invariance of the global integral of trace constituent.) Interpolation of trace constituent from the CCM2 hybrid sigma–pressure surfaces to isentropic surfaces in the CCM processor is linear with respect to ln(*p*^{κ}).

The overall distributions of the constituent on the four layers displayed at day 5 and day 10 from both models are similar; however, the spatial patterns of all *θ*–*σ* distributions are superior to those from CCM2, particularly at day 10. The similarity of constituent patterns in both meridional and zonal extent is indicative of the similarity of the predicted mass and momentum fields discussed in section 3. However, the CCM2 distributions, particularly the day 10 results, portray a higher degree of numerical dispersion compared to the UW *θ*–*σ* model.

Several features stand out upon contrasting the patterns from these four layers. In Fig. 6, transport by relatively strong stratospheric easterlies is noticed in the summer (Southern) hemisphere at 555 K. At this level the day-10 trace constituent in the *θ*–*σ* model has been transported west in a narrow tongue from its initial position to the southern Indian Ocean. The CCM2 patterns also indicate easterlies, but the magnitude of the trace constituent maximum is much smaller. Another feature of interest occurs at jet stream level (340 K, Fig. 8), where the UW *θ*–*σ* transport algorithm maintains a narrow and nearly continuous tongue of trace constituent from northeast of Brazil, east to India, across the Northern Pacific Ocean and North America, to the central Atlantic Ocean.

At 300 K (Fig. 9), stronger meridional amplification but overall weaker horizontal transport of the trace constituent is clearly evident in the distributions from both models. The CCM2 distributions (Figs. 9c,d) demonstrate improved conservation of the initial maximum relative to higher layer maxima, implying less spurious numerical dispersion. The *θ*–*σ* distributions (Figs. 9a,b) are less regular than on higher isentropic surfaces since part of the 300-K surface is in the *σ* domain of the *θ*–*σ* model and trace constituent transport in this region is susceptible to the spurious vertical dispersion that occurs in the three-dimensional transport of sigma coordinate models.

Figure 10 presents east–west zonal distributions at 0, 2, 5, and 10 days. The UW *θ*–*σ* cross sections (Figs. 10a,b,d,f) are along 40°S; the T63 CCM2 cross sections (Figs. 10c,e,g) are slightly further north, along 39.375°S. These Southern Hemisphere cross sections are located near the most rapid eastward transport of trace constituent (see Figs. 7b,d). Results are displayed directly on the respective model surfaces; the initial cylindrical distribution for CCM2 is not shown since its structure is identical to that of the UW *θ*–*σ* model. Similar to the horizontal distributions presented above, the UW *θ*–*σ* model maintains more intense gradients of trace constituent and simulates larger relative maxima than in the corresponding CCM2 cross sections, especially at day 5 and day 10. The more rapid westward transport of trace constituent near the top of CCM2, most noticeable at day 5, results from stronger easterly flow in the top layers of the CCM, which extends much further into the stratosphere (the top CCM2 layer is near 2 mb, while the top of the UW *θ*–*σ* model is 555 K, which averages 37 mb).

To verify the Zapotocny et al. (1994) result that interpolation from one coordinate system to another was not a major source of error, Fig. 11 shows the day-5 inert trace constituent distributions interpolated from the UW *θ*–*σ* model to the 99.04H CCM2 surface (Fig. 11a) for direct comparison with the results from the T63 CCM2 simulation (Fig. 11b). Clearly the distribution and trace constituent maximum directly from CCM2 model surfaces is just as marginal as from the interpolated distribution in Fig. 7c. In contrast, the interpolated *θ*–*σ* constituent distribution and its maximum appear nearly as robust as when displayed directly on isentropic surfaces (Fig. 7a). Consequently, while the constituent maximum is decreased slightly through smoothing by the vertical interpolation from one coordinate system to another, this step is not the primary reason for differences between the distributions.

#### 2) T42 CCM2 and CCM-S

Five- and 10-day 415-K results from the T42 CCM2 (Figs. 12a,b) and T42 CCM-S (Figs. 12c,d) simulations are now compared to broaden the analysis to include different CCM resolutions (T42 versus T63) and transport methods (shape-preserving SLT versus spectral). As expected, the T42 CCM2 spatial distributions are very similar to the T63 CCM2 results of Fig. 7. The day-10 global trace constituent maximum is 5.7% less in the T42 simulation, indicating the impact of decreased horizontal resolution.

The 5- and 10-day T42 CCM-S distributions for 415 K (Figs. 12c,d) also evolve in a manner similar to the T42 and T63 CCM2 results. In fact, by day 10 the CCM-S simulation slightly improves conservation of the initial maximum on the 415-K surface relative to either the T42 or T63 CCM2 simulations. The CCM-S simulation at day 5 and day 10 has retained 59.4% and 38.0% of its initial maximum, respectively. However, the spectral transport results display substantial ringing (numerous small zero contours) from Gibbs phenomena (Navarra et al. 1994) throughout the model domain. Furthermore, as discussed in section 2 above, if the local longitude and height borrowing algorithm failed in CCM-S, the negative values were allowed to persist. Failure to eliminate the negatives in this simulation produced a trace constituent value slightly less than −9 units at day 5 near the intense gradient over southern Greenland.

All CCM2 results to this point have used the default shape-preserving SLT alogorithm. However, since Rasch and Williamson (1990b) indicate that such schemes are dispersive, the impact of the nonnegative shape-preserving component of the SLT on conservation of the trace constituent maxima is examined. When SLT without the shape-preserving component is utilized, a global fixer eliminates negatives. Figure 13 shows the day-5 inert trace constituent distributions on the 99.04H CCM2 surface from a simulation using SLT without shape preserving (NP, Fig. 13a), with shape preserving (P, Fig. 13b), and a difference field (Fig. 13c). As is evident, the horizontal distributions with and without shape preserving appear essentially identical, but the maximum value is 2.8% greater (63.1 versus 60.3) without shape preserving. Figure 13c indicates that the largest difference (6.9%) is southwest of Greenland, where a very intense gradient of trace constituent exists. Effects of the shape-preserving component of the SLT are further examined in section 5a(5).

#### 3) UW *σ* model

The clearest comparison of trace constiuent transport for the study of relative capabilities of sigma and isentropic coordinates is between the two UW models because both are gridpoint models with the same numerics, filters, etc. Since detailed results were presented in Zapotocny et al. (1994), only a brief comparison is presented here for completeness. The day-5 and day-10 415-K UW *σ* trace constituent maxima (Figs. 14a,b, respectively) are closer to the initial values than all three CCM simulations presented above; however, the spatial patterns of the UW *σ* trace constituent distributions are less coherent than those from CCM2 (cf. Figs. 14a,b, 7c,d, and 12a,b, respectively) and the UW *θ*–*σ* model (see Figs. 7a,b).

The respective 415-K maxima at day 10 are 65.8 for the UW *σ* model, 34.6 for the T63 CCM2, 28.9 for the T42 CCM2, and 38.0 for the T42 CCM-S. All are less than the day-10 415-K maximum of 102.5 for the *θ*–*σ* model. Reasons for the superior retention of inert trace constituent maximum and the development of filamentary structure with stronger gradients in the UW *θ*–*σ* model relative to the other four models are discussed in section 6.

#### 4) Summary of the cylindrical trace constituent experiments

Table 1 lists the maxima at day 5 and day 10 on 14 isentropic surfaces extending from the stratosphere (555 K) to near the earth’s surface (300 K) for seven models. The table includes the global maxima for CCM2 simulations with (D) and without (ND) horizontal diffusion of the divergence, vorticity, and temperature. The overall conclusion is that, relative to the other models, the *θ*–*σ* model is the best at conserving initial trace constituent maxima. However, several interesting features stand out at day 10 in this seven-way comparison. At jet stream level, 324 K and higher, both CCM2 results with diffusion are relatively uniform with most values ranging between 25% and 35% of the initial maximum of 100 units, while both CCM2 results without diffusion average near 15%. In lower layers and in the stratosphere, where the trace constituent filaments do not become as elongated, move as far, or develop as intense gradients, the maxima in both CCM2 simulations are much closer to the ideal with some values exceeding 75. The T42 CCM-S maxima exceed those of the T42 CCM2 for 12 of the 14 layers, with some decidedly better. Ten UW *σ* layer maxima exceed the CCM-S maxima and all are greater than those from the T42 and T63 CCM2. Some numerical difficulties from vertical gradients and advection in the UW *σ* gridpoint model results are indicated by the vertical oscillation with relative maxima below 308 K, at 348 K, and above 445 K, and relative minima at 332 K and 370 K. Furthermore, as discussed previously, the UW *σ* results are spatially less coherent than the CCM2 results.

The day-10 UW *θ*–*σ* results, clearly the best of the seven models, also display some interesting characteristics. As with the other six models, the UW *θ*–*σ* model displays poorest conservation of trace constituent maxima at jet stream level (332–390 K), where values decrease to 84.0 units. Additionally, some *θ*–*σ* levels reveal maxima greater than 100 units. Maxima greater than 100 units result from the numerics of extreme gradients by the second-order flux-conservative finite-difference algorithm.

#### 5) A comparison with climate initial conditions

A possible limitation of the above experiments is that the use of GEOS-1 assimilated data for initial conditions as opposed to “consistent” data from a climate simulation might negatively impact the CCM results relative to the UW *θ*–*σ* results. This section examines conservation of trace constituent maxima in the UW *θ*–*σ* model and T42 CCM2 using initial conditions from the second year of a T42 climate simulation. The initial dataset, as obtained from the NCAR archive was NCDATA = ‘/CSM/plx16/CSM/ccm2/379T42/SEP1P87’. Results from a CCM2 simulation using T106 horizontal resolution are also included to investigate whether the conservation of tracer maxima is improved when using substantially increased horizontal resolution. For this section only, horizontal resolution of the UW *θ*–*σ* model was decreased from 2° north–south and 2.5° east–west to 2.8125° north–south and east–west, a resolution closer to the Gaussian grid resolution of the T42 CCM2. In all other matters the models and location of the vertical cylinders are identical to those stated previously.

Figure 15 shows the day-5 and day-10 332-K distributions from the two models; 332 K is located in the middle and upper troposphere except near the Himalayas, where it extends into the lower troposphere. The day-5 and day-10 UW *θ*–*σ* model results (Figs. 15a,b, respectively) indicate very little meridional transport of trace constituent from the initial latitudes (40°N and 40°S). Rather, the trace constituent has been transported completely around the hemisphere in southern latitudes and halfway around the hemisphere in northern latitudes. The T42 CCM2 displays the same overall spatial evolution of trace constituent as the UW *θ*–*σ* model; however, the trace constituent gradients are much less intense.

Table 2 summarizes the trace constituent maxima for this experiment from the UW *θ*–*σ* model and four T42 CCM2 configurations: with (P) and without (NP) a shape-preserving SLT, and with (D) and without (ND) horizontal diffusion of the divergence, vorticity, and temperature. Clearly the day-5 and day-10 UW *θ*–*σ* maxima are higher than those from the four CCM2 simulations with day-10 *θ*–*σ* values ranging from 79.7 to 99.5. These UW *θ*–*σ* maxima tend to be slightly lower than the values presented in Table 1 using GEOS-1 data as initial conditions.

The results for conservation of trace constituent maximum for each isentropic layer from the four T42 CCM2 simulations and most isentropic layers from the T106 CCM2 simulation shown in Table 2 are all inferior relative to the UW *θ*–*σ* results. The best T42 CCM2 simulation using the climate initial state is the non-shape-preserving with diffusion case (labeled NP,D in the table); the two simulations without diffusion (ND) tie for the poorest conservation. The shape-preserving with diffusion CCM2 (P,D) maxima are only slightly less than the (NP,D) maxima. These results clearly indicate that relative to the other models, the UW *θ*–*σ* model improves conservation of the trace constituent maxima even when the initial atmosphere is from a CCM2 climate simulation and when the horizontal resolution of CCM2 is significantly greater than the UW *θ*–*σ* model. Additional investigation on the impact of diffusion and the shape-preserving component of the SLT algorithm is beyond the scope of this study.

### b. Zonally distributed trace constituent

One plausible presumption concerning the cylindrical distribution experiments is that the markedly favorable results of the *θ*–*σ* model might stem from an inherent weakness of spectral models to resolve a time-dependent simulation of a localized distribution. The experiment now presented addresses this issue in that, instead of localized distributions embedded within the circumpolar vortex, the initial distributions are global in scale. Note in Fig. 5b that the effective meridional scale of the initial distribution (hereafter called zonal rings) corresponds to wavenumber 8, while the zonal scale is infinite.

The day-5 415-K simulated distributions (Fig. 16) show substantial cross-polar transport in the Northern Hemisphere and wave–wave interactions in both hemispheres. Long coherent streamers of trace constituent are identifiable in the *θ*–*σ* simulation (Fig. 16a), such as the ones extending from eastern Canada to the Caspian Sea, from Venezuela northeast to the Mediterranean Sea, and from central Africa to India. The T63 CCM2 distribution (Fig. 16c), again while similar in pattern, does not maintain the integrity or intensity of the filamentary structure, and its gradients are weaker in most regions. The same conclusion is derived from the day-10 results in Figs. 16b,d.

The day-5 and day-10 UW *θ*–*σ* distributions in Figs. 16a,b reveal an increasingly small-scale pattern developing along a northeast–southwest band from northeast of South America to Spain and north of 60°N between Hudson Bay and Europe. While these patterns may in part represent horizontal transport by smaller scales, the structure suggests Gibbs aliasing (Navarra et al. 1994) caused by numerical inadequacies in resolving filaments being stretched through baroclinic amplification and compressed laterally through deformation. Potentially, the addition of weak horizontal diffusion would reduce the intense gradients and alleviate the aliasing. Regardless, the distributions produced by the *θ*–*σ* model are still superior to the CCM2 distributions and provide keen insight into the true nature of planetary-scale transport and the means by which filamentary structure and intense gradients of atmospheric trace constituents are generated.

The trace constituent maxima on isentropic surfaces for the zonal ring experiment (Table 3) once again show a decided advantage for the UW *θ*–*σ* model compared to CCM2. All *θ*–*σ* maxima are within 10% of their initial values, even at day 10, while most CCM2 maxima have dropped to near half their original value. For completeness, the CCM2 results are shown directly on the hybrid sigma–pressure surfaces of that model and also after interpolation to isentropic surfaces. Similar to the vertical cylinder, the CCM2 zonal ring results are best in the stratosphere and in the lowest layers while the *θ*–*σ* maxima tend to be more uniform, averaging near 100 units at 316 K and higher. Trace constituent values in the UW *θ*–*σ* model greater than 100 units likely result from reasons discussed in section 5a(4).

### c. Cylindrical distributions including diabatic process and parameterizations

The results in this section examine the impact of diabatic processes and physical parameterizations on vertical advection in the UW *θ*–*σ* model relative to the *σ* models. As stated in section 2, with the isentropic constraint relaxed, the invariance of trace constituent maxima within discrete model layers is no longer applicable, and thus conservation of the initial maxima is no longer available for objective comparison. In this diabatic experiment, individual isentropic layers should not have maxima greater than 100 units, but because of differential vertical transport the maxima on isentropic surfaces will likely be less than 100 units due to the introduction of a degree of freedom for vertical dispersion. As such, these diabatic comparisons, which include vertical transport of trace constituent by cumulus convection and other physical processes, concentrate on the spatial patterns of the fields rather than conservation of trace constituent maxima.

Figure 17 shows the day-5 and day-10 415-K stratospheric distributions produced by the UW *θ*–*σ* model and T63 CCM2, while Fig. 18 shows the 300-K tropospheric distributions. As expected, the day-5 415-K fields from each model (Figs. 17a,c) are similar to those when isentropic conditions are assumed (see Figs. 7a,c). In the Northern Hemisphere of each model, a band of trace constituent arches from Hudson Bay east-southeast to Saudi Arabia; while in the Southern Hemisphere portions of the initial field are advected both east and west. However, the trace constituent is not transported as far west over the tropical Pacific Ocean in either model.

At day 10 the large-scale features of the UW *θ*–*σ* diabatic simulation (Fig. 17b) are still quite similar to the corresponding isentropic simulation (Fig. 7b). The trace constituent in the diabatic simulation generally displays a smooth and coherent filamentary structure since only a few differences occur in the location of extremes from the isentropic simulation over the northern Pacific Ocean, northern Canada, and south of Africa. A comparison of the isentropic and diabatic CCM2 simulations (Figs. 7d and 17d, respectively) reveals similar differences in the higher latitudes of the Northern Hemisphere. Such differences are not unexpected since the circulations evolve differently with the inclusion of diabatic processes and physical parameterizations. The UW *θ*–*σ* results continue to maintain intense gradients, while the CCM2 results are relatively less coherent and the gradients are reduced.

In the troposphere, results from the *θ*–*σ* diabatic simulation (Figs. 18a,b) are for the most part similar to their isentropic counterpart (Figs. 9a,b), except that the spatial pattern is somewhat less regular. The maximum value on this isentropic surface in the lower troposphere is close to 100 units at day 5, but has decreased to 79.4 units by day 10. The CCM2 diabatic simulation (Figs. 18c,d) also produces a pattern similar to its isentropic counterpart. A complete list of tracer maxima on the coordinate surfaces of both models for the diabatic simulation is presented in Table 4; for completeness CCM2 results are also shown on isentropic surfaces. The comparison reveals that conclusions for the isentropic and diabatic simulations are similar.

Trace constituent degradation from vertical advection and spurious dispersion is not as severe in the *θ*–*σ* diabatic simulation as might be supposed from cursory arguments. Once the condensation process is initiated, trace constituents (including water vapor) develop a larger-scale vertical structure through the basic two-layered nature of tropospheric baroclinic systems. The corresponding development of a smooth profile in a saturated atmosphere over an extended region minimizes pronounced numerical dispersion since the vertical gradient of properties (∂*f*/∂*θ*) is minimized.

## 6. Discussion

The strategy utilized in this study of comparing the relative ability of models to simulate patterns and the maxima of inert trace constituent transport provides a substantive means to examine numerical dispersion. Several factors must be considered to understand the remarkably different trace constituent distributions simulated by isentropic- and sigma-coordinate models. First, if isentropic conditions are assumed, the initial extremes of a trace constituent oriented vertically within a continuum should ideally be conserved somewhere on each model surface even though the Lagrangian chain of fluid elements is tilted, stretched, and deformed. Within the *θ* domain of the UW *θ*–*σ* model, this ideal chain of fluid elements would be approximately represented by line segments connecting the maxima of each explicit isentropic layer. Exact conservation will not be achieved within the explicit isentropic layers of the *θ*–*σ* model since some dispersion occurs from the horizontal finite-difference algorithm. Neither will exact conservation be achieved in the UW *σ* model or any version of the CCM. While these models are subject to the isentropic constraint and attempt to approximate like distributions as the *θ*–*σ* model, dispersion occurs from both horizontal and vertical finite differencing.

The factor that compounds the issue for trace constituent transport in sigma-coordinate models is the complexity of, and inherent need for, resolving three-dimensional transport to represent the structure of isentropic processes within amplifying baroclinic waves of extratropical latitudes. In designing these experiments, the initial structure of trace constituent was purposely assumed to be vertically invariant in order that the results would not be prejudiced by the initial conditions. Under these conditions the vertical gradient of trace constituent is zero initially and minimal early in the integration. Thus, vertical advection is zero initially and minimal until vertical wind shear creates vertical gradients. The result is that the trace constituent distributions and local maxima obtained with the UW *σ* model and CCM remained in a relative sense more spatially coherent and larger at day 5 than at day 10. However, as integration proceeds the vertical gradient of trace constituent increases markedly in regions of strong baroclinic wave amplification, and spurious vertical dispersion occurs from inaccuracies in calculating the vertical advection *σ̇**f*/∂*σ*. After 10 days of integration the spatial pattern of the tracer field is dramatically reduced by spurious vertical dispersion.

Often one assumes that increased vertical resolution in sigma-coordinate models will increase accuracy, but this is not assured. With baroclinic amplification of waves and the backing and veering of wind with height, models with increased vertical resolution possess degrees of freedom that allow for intensification of both the vertical gradient of trace constituent and vertical motion over that simulated by lower-resolution models. Conceivably vertical dispersion may be further intensified in sigma-coordinate models with higher vertical resolution, particularly in regions of baroclinic amplification.

At the same time, while examining numerical simulations of the primitive equations, it is extremely important to recognize that in sigma-coordinate models the fields of momentum and energy retain higher order continuity relative to the distributions of a trace constituent (including water vapor). The prognostic equations for these properties possess degrees of freedom for exchange by pressure stresses and work apart from transport. As such, these fields do not develop the extreme vertical gradients evident in trace constituents, and spurious numerical dispersion associated with vertical advection of these properties is reduced substantially, regardless of the length of integration. Increased vertical resolution in conjunction with the proper selection of an aspect ratio aids in increasing accuracy of simulations.

The use of isentropic coordinates largely eliminates spurious vertical dispersion of trace constituents. In the absence of diabatic processes, the scale of water vapor and other trace constituents within inclined isentropic layers remains large relative to the scale in sigma coordinates, vertical advection across isentropic surfaces does not occur, and the proper selection of an aspect ratio is not a factor. The simplified two-dimensional quasi-horizontal form of transport in the isentropic domain of the *θ*–*σ* model, compared to the three-dimensional form of transport in sigma models and the sigma domain of the *θ*–*σ* model, results in the markedly improved simulation of the inert trace constituent.

With diabatic effects and physical parameterizations included, as in section 5c, vertical advection by *θ̇**f*/∂*θ* occurs in the isentropic domain of the *θ*–*σ* model. Under this condition, the constraint of conservation of trace constituent maxima within an isentropic layer is relaxed and the fields evolve differently than under isentropic conditions. However, the large-scale patterns at day 10 for the isentropic and diabatic comparison were quite similar, and departures of maxima from 100 units were not substantial. Once the condensation process is initiated, water vapor and other trace constituents develop a larger-scale vertical structure, weaker gradients of these properties, and less numerical dispersion. This sequence in regions of diabatic vertical advection alleviates the development of a severe degradation of trace constituent in the *θ*–*σ* simulations. Likewise, spurious vertical dispersion of tracer fields is alleviated in sigma-coordinate models once the condensation process induces a larger-scale vertical structure.

Since the two UW models are identical, except for vertical coordinates, the conclusion that the different transport characteristics of the two models results from spurious vertical dispersion in the UW *σ* model is firm. If the dispersion were associated with horizontal advection, the *θ* domain of the UW *θ*–*σ* model would be equally susceptible to the development of an incoherent tracer field and reduced conservation of maxima. Still, the different trace constituent transport characteristics between the UW *θ*–*σ* model and CCM cannot be solely attributed to vertical dispersion because of the substantial difference in model formulations (i.e., gridpoint versus spectral, flux-conservative versus SLT, etc.). For example, Table 2 shows up to a 10% improvement in conservation of trace constituent maximum when the shape-preserving component of the SLT is eliminated and up to a 21% decrease when diffusion is removed. Visual inspection of CCM2 temperature fields in the lowest level (not shown) reveals that diffusion eliminates the highly irregular small-scale pattern that is a reflection of Gibbs phenomena compounded by a cascade of energy to smaller scales. A more in-depth investigation of the conservation characteristics of the SLT algorithm in CCM2 with and without diffusion is beyond the scope of this study.

An additional consideration raised by these results is how the second-order centered in time and space advection algorithm of the UW models could more accurately simulate the filamentary structure and better conserve trace constituent maxima than the fourth-order SLT algorithm of CCM2. Rasch and Williamson (1990b) indicate that high formal accuracy, as evaluated from a Taylor series expansion of the alogorithm, does not guarantee an accurate solution to some important physical situations (shocks and fronts) where the phenomena to be modeled are quasi-discontinuous, and as known under these conditions error estimates evaluated from a Taylor series expansion are inappropriate. Furthermore, Zalesak (1981) points out that high-order finite-difference schemes may actually reduce the accuracy of a solution for distributions rich in high wavenumber. The day-10 trace constituent patterns described in this experiment, particularly in middle and high latitudes, were very rich in high wavenumber since the baroclinic processes of deformation, rotation, and vertical wind shear lead to filamentary structure of trace constituents.

A physical reason for the improved conservation of trace constituent maxima in the second-order accurate UW *θ*–*σ* model versus the fourth-order accurate CCM2 lies in the need to always resolve three-dimensional transport for baroclinic phenomena in CCM2, while only two dimensions need be resolved in isentropic integrations of the UW *θ*–*σ* model. Even though CCM2 is fourth-order accurate, semi-Lagrangian transport is dispersive (Rasch and Williamson 1990a) and there is still a small error each time step from evaluating transport. After 10 days of integration, the cumulative effects of this error take their toll. In contrast, under idealized isentropic conditions, the UW *θ*–*σ* model has no vertical transport. Furthermore, while the isentropic constraint imposed on most of the integrations presented here might seem to favor the UW *θ*–*σ* model, the constraint is not as severe as expected from cursory arguments. In a study utilizing satellite data over the western equatorial Pacific Ocean warm pool, Liu et al. (1995) determined that only 15% of clouds are precipitating at any one time in the Tropics. Apart from regions of active precipitation, atmospheric flow above the PBL is nearly isentropic for the scales resolved by these models, even in the presence of weak diabatic processes such as gentle subsidence associated with radiational cooling, cooling from evaporating cirrus, and weak, large-scale heating associated with stratiform clouds.

## 7. Summary

An issue fundamental to medium-range weather prediction and climate is increasing the accuracy of simulating the long-range transport of water vapor and inert trace constituents. Results from the long-range transport of an inert trace constituent were compared in this study for the period 1–11 February 1985 from numerical simulations by six different models. The six models for the comparison were the UW *θ*–*σ* gridpoint model, the nominally identical UW *σ* model, CCM2 with T42, T63, and T106 spectral resolution, and T42 CCM-S. The initial data for these simulations, with one exception, were supplied by the Goddard Laboratory for Atmospheres GEOS-1 data assimilation system. One set of experiments where the initial state was from the second year of a CCM2 climate simulation was also presented. The main objectives of this comparison were to 1) examine the spatial and temporal development of the filamentary structure of inert trace constituents prognostically simulated with the six models, and 2) examine the relative ability of each model to conserve initial trace constituent maxima during 10-day isentropic and nonisentropic simulations.

In simulating standard synoptic variables such as mean sea level pressure, 200-mb zonal wind, and zonal kinetic energy, the UW models and the CCM are subjectively equivalent. However, the UW *θ*–*σ* model is superior in simulating the filamentary structure of inert trace constituent transport throughout the 10-day integrations presented herein, particularly with respect to conserving initial maxima and maintaining spatial coherence of the fields. The improved patterns of filamentary transport and conservation of trace constituent initial maxima demonstrated by the *θ*–*σ* model relative to the *σ* coordinate models arises from the simplified nature of isentropic transport of trace constituents being confined to two dimensions along quasi-horizontal isentropic surfaces, versus three spatial dimensions in *σ* coordinates. Consequently, the UW *θ*–*σ* model intrinsically minimizes spurious numerical dispersion with respect to the vertical transport of trace constituents within a thermally stratified baroclinic atmosphere. Increasing vertical resolution in sigma-coordinate models may improve the accuracy of simulating atmospheric hydrologic processes and other trace constituent transport somewhat; however, the results presented here establish that, for equivalent vertical resolution, isentropic models predict remarkably more accurate trace constituent transport than sigma models, either gridpoint, spectral, or spectral with SLT.

One decided advantage of SLT models is the capability for implicit time differencing and an increased time step (Williamson and Olson 1994). For these simulations, the UW *θ*–*σ* model with explicit time differencing runs 213 times real speed on a Cray C90 using Unicos 8.0. On the same machine the T42, T63, and T106 CCM2 runs 1152, 506, and 40 times real speed, respectively. While the UW *θ*–*σ* model run time suffers relative to CCM2 for comparable resolution, the T106 CCM2 conservation of tracer maxima suffers relative to the UW *θ*–*σ* model when comparable computer time is utilized. As such, the sacrifice in accurately simulating water vapor and other trace constituent transport with current SLT schemes, as documented in this study, precludes unqualified endorsement. Generalized coordinate models with smooth transitions from sigma to isentropic coordinates have been developed (Bleck and Benjamin 1993; Hsu and Arakawa 1990) and eventually will accommodate implicit time differencing and SLT in isentropic coordinates.

The findings of this study, together with the results from Johnson et al. (1993) and Zapotocny et al. (1994), are important for understanding the problems of modeling hydrologic and trace constituent transport for climate and NWP. The findings also support the use of isentropic coordinates as the best choice for simulation of the long-range transport of water vapor, aerosols, and chemical constituents in studies aimed at improving our understanding of hydrologic and chemical processes.

## Acknowledgments

The authors gratefully acknowledge Tonya Ommodt for the preparation of graphics, Judy Mohr for technical typing assistance, and NASA Goddard’s Laboratory for Atmospheres and NCAR for ready access of the initial atmospheres. This research was sponsored by the Department of Energy under Grant DE-FG02-92ER61439, by NASA under Grants NAG5-1330 and NAG5-81, and by the National Science Foundation under Grant ATM-8922684. Computational support for the CCM simulations was provided by NCAR, which is sponsored by the National Science Foundation.

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The isentropically simulated day-5 and day-10 global maxima of inert trace constituent for the vertical cylinders as predicted by the UW *θ*–*σ* model, UW *σ* model, T42 CCM-S, and T42 and T63 CCM2 with shape-preserving SLT and with (D) and without (ND) horizontal diffusion of the divergence, vorticity, and temperature. Initial atmospheric conditions were specified using GEOS-1 assimilated data for 0000 UTC 1 February 1985.

The isentropically simulated day-5 and day-10 global maxima of inert trace constituent for the vertical cylinder as predicted by the UW *θ*–*σ* model and a combination of T42 and T106 CCM2 simulations with (P) and without (NP) the shape-preserving component of the SLT and with (D) and without (ND) horizontal diffusion of the divergence, vorticity, and temperature. CCM2 results were interpolated to isentropic surfaces, and the initial conditions were specified from the second year of a T42 CCM2 climate simulation.

The isentropically simulated day-5 and day-10 global maxima of the vertically invariant zonal rings as predicted by the UW *θ*–*σ* model and the T63 CCM2 with shape-preserving SLT and horizontal diffusion of the divergence, vorticity, and temperature. The CCM results are either displayed directly on model surfaces or interpolated to isentropic surfaces. Initial atmospheric conditions were specified using GEOS-1 assimilated data for 0000 UTC 1 February 1985.

The diabatically simulated day-5 and day-10 global maxima of inert trace constituent for the vertical cylinders as predicted by the UW *θ*–*σ* model and the T63 CCM2 with shape-preserving SLT and horizontal diffusion of the divergence, vorticity, and temperature. The CCM results are either displayed directly on model surfaces or interpolated to isentropic surfaces. Initial atmospheric conditions were specified using GEOS-1 assimilated data for 0000 UTC 1 February 1985.