Time-Split versus Process-Split Coupling of Parameterizations and Dynamical Core

David L. Williamson National Center for Atmospheric Research, Boulder, Colorado

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Abstract

Simulations are compared to determine the effect of the details of the coupling of the parameterization suite with the dynamical core on the simulated climate. Simulations based on time-split and process-split couplings are compared to a simulation with the original version of the NCAR Community Climate Model–3 (CCM3), which is a mixture of the two approaches. In the process-split coupling, the two components are based on the same state and their tendencies are added to produce the updated state. In the time-split coupling, the two components are calculated sequentially, each based on the state produced by the other. Overall the differences between simulations produced with the various coupling strategies are relatively small. Thus, with the time step used in the CCM3, the different time truncation errors introduced by the different coupling strategies have less effect on simulations than other arbitrary aspects of the model design. This does not imply that the time truncation errors are insignificant, just that they are similar in the cases examined here. There are, however, regions where the differences are statistically significant. The differences in the thermal balance are analyzed in these regions. The most notable differences occur between the time-split case and CCM3 over regions of Antarctica. In summer, although the temperature difference near the surface is modest, the balance of terms in the two cases is very different, with a difference in sign in the sensible heat flux between the two cases. In winter, the parameterization terms have a very strong grid-scale structure associated with parameterized clouds forming predominantly at a single grid level. The dynamics is unable to respond with a grid-scale structure. This draws into question whether the vertical resolution is adequate to properly model the physical processes.

Corresponding author address: Dr. David L. Williamson, NCAR/CGD, P.O. Box 3000, Boulder, CO 80307-3000. Email: wmson@ucar.edu

Abstract

Simulations are compared to determine the effect of the details of the coupling of the parameterization suite with the dynamical core on the simulated climate. Simulations based on time-split and process-split couplings are compared to a simulation with the original version of the NCAR Community Climate Model–3 (CCM3), which is a mixture of the two approaches. In the process-split coupling, the two components are based on the same state and their tendencies are added to produce the updated state. In the time-split coupling, the two components are calculated sequentially, each based on the state produced by the other. Overall the differences between simulations produced with the various coupling strategies are relatively small. Thus, with the time step used in the CCM3, the different time truncation errors introduced by the different coupling strategies have less effect on simulations than other arbitrary aspects of the model design. This does not imply that the time truncation errors are insignificant, just that they are similar in the cases examined here. There are, however, regions where the differences are statistically significant. The differences in the thermal balance are analyzed in these regions. The most notable differences occur between the time-split case and CCM3 over regions of Antarctica. In summer, although the temperature difference near the surface is modest, the balance of terms in the two cases is very different, with a difference in sign in the sensible heat flux between the two cases. In winter, the parameterization terms have a very strong grid-scale structure associated with parameterized clouds forming predominantly at a single grid level. The dynamics is unable to respond with a grid-scale structure. This draws into question whether the vertical resolution is adequate to properly model the physical processes.

Corresponding author address: Dr. David L. Williamson, NCAR/CGD, P.O. Box 3000, Boulder, CO 80307-3000. Email: wmson@ucar.edu

1. Introduction

The NCAR Community Climate Model (CCM) was recently restructured to change the coupling between the physical parameterizations and dynamical core. The change was introduced to allow interchange of dynamical cores without imposing a single coupling strategy on all cores, and to allow reordering of parameterization components within the parameterization package. The original CCM coupling strategy evolved over the life of the model with aspects of the coupling of the subgrid-scale parameterizations to the dynamical core traceable back 20 years or more. A result of the evolution was that the spectral transform dynamical core and physical parameterizations were entwined, making it difficult to modify the coupling or change the relative order of some of the parameterization components. The long evolution of the model lead to a hesitancy to change the coupling since ramifications of the coupling were not well understood, but it worked, and everyone was comfortable with it. The new structure clearly separates the parameterization suite from the dynamical core, and makes it easier to replace or modify each in isolation. The dynamical core can be coupled to the parameterization suite in a purely time-split manner or in a purely process-split one, as described later.

Split forms of approximations are common in atmospheric modeling. They provide an economical approach by allowing some degree of implicitness in approximations for faster processes in models without leading to extremely complex equations to solve. Beljaars (1991) and Lenderink and Holtslag (2000) provide arguments for applying time-split approximations when the timescales of the parameterizations are shorter than the time step of the model. Staniforth and Robert (1981) analyzed time splitting applied to explicit methods and concluded that split methods were more stable at the price of larger truncation error. However, they considered the larger error to be acceptable. Caya et al. (1998) showed that splitting the parameterizations and dynamical core may introduce serious errors when used in conjunction with longer time step three-time-level semi-Lagrangian methods. They illustrate the error in a complex prediction model, in a simplified model and by analysis of a simple equation. Murthy and Nanjundiah (2000) performed further analysis of the system introduced by Caya et al. (1998) and developed variants of their approximations that avoid certain splitting errors. In contrast to Caya et al. (1998), Chen and Bates (1996) found forecasts to be insensitive to time splitting the parameterizations and dynamical core with a two-time-level semi-Lagrangian formulation and 30-min time step.

The errors introduced by different splitting or coupling strategies are examined here in a practical sense by comparing the differences between long simulations with the dynamical core and parameterization suite of CCM3 combined via different coupling strategies. The differences identify some aspects of the uncertainty arising from time truncation error in the model. As will be seen, the differences tend to be small, which justifies allowing different coupling strategies in the model. Adopting different coupling strategies is only useful if the differences are relatively small so that the coupling strategies do not dominate the simulation. In the next section our definitions of time and process splitting are provided. Section 3 provides details of the model and simulations examined here, and section 4 discusses the differences between the simulations. Finally, section 5 presents some conclusions.

2. Process and time splitting

Consider the general prediction equation for a generic variable ψ,
i1520-0493-130-8-2024-e1
where ψ denotes a prognostic variable such as temperature or horizontal wind component. The dynamical core component is denoted D and the physical parameterization suite P. We consider two methods of coupling D and P. The first we denote process-split which for explicit schemes would approximate (1) by
i1520-0493-130-8-2024-e2
in which we employ a three-time-level notation to be consistent with the current CCM. The CCM actually adopts a spectral transform semi-implicit dynamical core that is centered in time and semi-implicit, and thus also depends on ψn+1. The numerical characteristics of the parameterizations are like those of diffusive processes rather than advective ones. It is therefore appropriate to approximate them as forward or backward approximations. In fact, several processes in the CCM parameterization package are approximated implicitly as backward approximations to avoid computational stability restrictions on the time step. Thus both D and P in (2) also depend on ψn+1, leading to a very complicated implicit equation to solve. To simplify the solution, the parameterization suite P is first solved alone implicitly, then the effective forcing from the parameterizations is applied to the dynamical core.
i1520-0493-130-8-2024-e3
where
ψψn−1tPψψnψn−1
Some CCM3 parameterizations have a mild dependency on ψn arising from a discrete time linearization to avoid nonlinear terms in ψn+1. The ψn can be replaced with ψn−1, which in fact is what is done in the time- and process-split versions of the CCM3 examined later.
The second method of coupling D and P we denote time-split and approximate (1) by
i1520-0493-130-8-2024-e5
The distinction is that in the process-split approximation the calculations of D and P are both based on the same past state, ψn−1, while in the time-split approximations D and P are calculated sequentially, each based on the state produced by the other. Although we employ a three-time-level notation in (2)–(6) to be consistent with the current CCM, Eqs. (2)–(6) would also apply to two-time-level approximations by converting the centered n term dependencies to n − 1, then replacing n − 1 by n and 2Δt by Δt.
The parameterization package can be applied to produce an updated field as indicated in (4) and (6). Thus we can write (6) with an operator notation
ψn+1Pψ
where we include only the past state in the operator dependency for notational convenience. The implicit predicted state dependency is understood. The process-split equation (2) can also be written in operator notation as
i1520-0493-130-8-2024-e8
where the first argument of D denotes the prognostic variable input to the dynamical core and the second denotes the forcing rate from the parameterization package, for example, the heating rate in the thermodynamic equation. Again we include only the past state in the operator dependency, with the implicit predicted state dependency left understood. With this notation the time-split system (5) and (6) can be written
ψn+1PDψn−1
The total parameterization package in CCM3 consists of a sequence of components, indicated by
PM,R,S,T
where M denotes (moist) precipitation processes, R denotes clouds and radiation, S denotes the surface model, and T denotes turbulent mixing. Each of these in turn is subdivided into various components: M includes an optional dry adiabatic adjustment normally applied only in the stratosphere, Zhang and McFarlane (1995) moist convection, Hack (1994) shallow convection, and large-scale stable condensation; R first calculates the cloud parameterization (Kiehl et al. 1998a) followed by the radiation parameterization; S provides the surface fluxes obtained from land (Bonan 1996), ocean, and sea ice models, or calculates them based on specified surface conditions such as sea surface temperatures and sea ice distribution. These surface fluxes provide lower flux boundary conditions for the turbulent mixing T, which is comprised of the planetary boundary layer parameterization (Holtslag and Boville 1993), vertical diffusion, and gravity wave drag. Details of those processes not specifically referenced above are provided in (Kiehl et al. 1996).
Defining operators following (7) for each of the parameterization components, the approximations as applied in CCM3 variants examined here can be summarized as
i1520-0493-130-8-2024-e11
The labels time-split and process-split refer to the coupling of the dynamical core with the complete parameterization suite. The components within the suite are coupled via time splitting in both forms. The CCM3 is seen to be a conglomeration of the two types of approximations. In (13) S uses the radiative fluxes from R, but it does not use the atmospheric temperature updated by the radiation. Notice that in all cases the relative ordering of the parameterization components is the same, and within the parameterization package mentioned earlier they are applied in a time-split manner in (11) and (12). In CCM3, the only non-time-split coupling within the parameterization package is S following R, however, that is only partial as the radiative fluxes are used for the surface flux calculation, but atmospheric temperatures updated by the radiative heating are not. In our experience that difference is relatively unimportant presumably because R is a relatively slow process. Reordering the parameterization components within the parameterization package is likely to produce larger differences in simulations than the differences to be discussed later between simulations based on approximations (11) through (13). Beljaars (1991) discusses the importance of the order of the processes. Notice that in our cases the moist precipitation process is always based on the state produced by the dynamical core. The main difference between the approximations (11) and (12) is the state seen by the dynamical core. In the time-split form the dynamical core forecasts from the state last modified by the turbulent mixing. In the process-split form it forecasts from the state produced by the dynamical core itself, which was forced by the parameterization package. The CCM3 form, detailed in Kiehl et al. (1996), is somewhere between those two.

The process-split form is convenient for spectral transform models. With time-split approximations extra spectral transforms are required to convert the updated momentum variables provided by the parameterizations to vorticity and divergence for the Eulerian spectral core, or to recalculate the temperature gradient for the semi-Lagrangian spectral core. The time-split form is convenient for dynamical cores such as the Lin–Rood finite-volume core (Lin and Rood 1997; Lin 1997), which adopts a Lagrangian vertical coordinate. Since the scheme is explicit and time step–restricted in its nonadvective component, it substeps the dynamics through a longer parameterization time step. With process-split approximations the forcing terms must be interpolated to an evolving Lagrangian vertical coordinate every substep of the dynamical core. Besides the expense involved, it is not completely obvious how to interpolate the parameterized forcing, which can have a vertical grid-scale component arising from vertical grid-scale clouds, to a different vertical grid.

3. Model description

We have carried out a set of long integrations to determine the effect of the details of the coupling of the parameterization suite with the dynamical core on the simulated climate. The time-split, process-split, and original versions of CCM3 were each run for 17 yr. The first 2 yr were discarded to avoid any spinup that might have occurred, and the analyses presented in the following are for 15 yr. In all cases, the parameterizations and dynamical core were identical (those of CCM3), and only the coupling between them was different. The individual parameterizations are described in the previous section. The dynamical core is that of the standard CCM3: Eulerian spectral transform dry dynamics with monotonic semi-Lagrangian moisture transport (Kiehl et al. 1996). The resolution for the simulations presented here conforms to the standard CCM3 configuration: T42 spectral truncation on a 128 × 64 quadraticly unaliassed Gaussian grid, 18 vertical levels and 20-min time step, with the parameterizations applied over the centered 2Δt = 40 min interval. The radiation is calculated every hour. That rate is applied as appropriate for that and the following two time steps.

An overview of CCM3 is provided in Kiehl et al. (1998a), the simulated energy budget of CCM3 is documented in Kiehl et al. (1998b), the hydrologic and thermodynamic structures simulated by CCM3 are documented in Hack et al. (1998), and Hurrell et al. (1998) document the dynamical aspects of the atmosphere simulated by CCM3.

The splitting approach implemented here is formally first-order accurate, integrating one operator then the other over the splitting time step. Strang (1968) proposed a symmetrized scheme, increasing the order of accuracy at the expense of calculating each operator twice over the splitting time step. However, classical error analysis may not be relevant for the CCM application. Splitting errors are often small in practical situations because of the stabilizing effect of stiffness in one of the operators. Sportisse (2000) shows that classical error analyses may fail when Δt is larger than the fastest timescale, such as when combining stiff and nonstiff operators. In that case, the particular sequential order of the splitting is crucial for accuracy and the stiff operator should finish the splitting scheme. In a sense, the fast process brings the solution closer to the true trajectory, while the slower processes move the solution away from the true trajectory. Beljaars (1991) also concludes from experiences with the European Centre for Medium-Range Weather Forecasts (ECMWF) model that the fastest processes should be calculated last.

Sportisse's (2000) analysis is for two processes in a deterministic evolution from specified initial conditions. In the climate simulation application, the statistics of the solution are examined after a sufficiently long time that the influence of initial conditions is lost. The climate statistics of the simulation are determined by external forcing rather than by the initial conditions. With just two processes the “order,” which is naturally defined relative to the initial conditions, is somewhat ill defined. For climate simulation, the equivalent of “last” in the deterministic forecast is the place in the split sequence when the data from the integration are sampled to calculate the statistics. For the statistics for all model configurations examined in the following, we sample the simulations immediately after the convective parameterization. These are relatively fast processes compared to the model time step. Atmospheric modelers intuitively applied the fast convective processes last from the earliest days of numerical weather forecasting. For example, convective adjustment was applied to eliminate supersaturation from the forecasts. This is almost a requirement for forecast/analysis cycles to ensure that the first guess is not inconsistent with observations.

One might also ask, Is formal accuracy with regard to, say, Strang-type splitting, relevant when subgrid-scale parameterizations are one of the components? In many split applications, the components are approximating resolved scales and an accurate solution of those equations is necessary. A typical problem is the advection/diffusion equation in pollution modeling. In climate modeling, the parameterizations are approximating the effect of unresolved scales on the resolved scales. Uncertainties or errors in the closure may outweigh the need for a formally accurate solution of the parameterized equations. Also, by definition, formal accuracy is associated with a limiting process going to small mesh sizes. The nature of the subgrid-scale parameterization should change when going to finer meshes, or even be eliminated approaching the zero limit as smaller and smaller scales are resolved.

In the following, we compare the time-split and process-split versions with the original CCM3. Since the CCM3 falls between the two purer forms in the way individual parameterization components are handled, this comparison provides some indication of which processes, when treated differently, contribute to any differences found in the simulations. The principal difference between time-split and CCM3 is that the radiation, surface fluxes, and turbulent mixing are not time split in the CCM3. The principal difference between process-split and CCM3 is that the moist processes are not process split in CCM3.

As will be seen below, the differences between the simulations occur in restricted regions and are relatively small compared to those often arising from model changes. Therefore, the simulations were extended to provide 15-yr averages to reduce the natural variability of the averages. Pointwise Student's t-tests are performed to identify regions over which the differences in the state might be significant and that warrant further analysis of the difference in the balance of physical and computational processes occurring there. The confidence level is chosen to be 99% in all tests. To examine the differences in the balance we consider regional averages, and also perform Student's t-tests on those averages in an attempt to avoid attaching physical causes to differences that are likely to be due to natural variability. We concentrate our analysis on temperature and the thermal balance creating the climate. We will comment on general characteristics on some other fields that might be thought to be more sensitive to the coupling changes following the thermal analysis.

4. Simulation differences

a. Global annual averages

Table 1 summarizes 15-yr climatological global annual averages of a few select variables from the simulations along with the standard deviations of the annual global averages. These are typical of variables that exhibit significant variation during model development and are often used to guide the final parameterization “tuning” before freezing a model. (However, we emphasize that no additional tuning beyond that for the original CCM3 was done for the time- and process-split versions.) The differences between the global average properties in Table 1 are physically insignificant as well as significantly smaller than those typically seen between different models. The difference in energy residual might at first be thought to be significant since the CCM3 and process-split are better than the time-split. However, the high degree of energy balance in the CCM3 and process-split cases is due to a fortuitous cancellation. The parameterization package itself contains known inconsistencies of the order of at least 0.2 W m−2.

b. December–January–February averages

Figure 1 shows the 15-yr December–January–February (DJF) average, zonal average temperature difference for the time-split and process-split simulations minus the CCM3. We will use the term differences in the time- (process-) split case to mean differences between the time- (process-) split case and the CCM3. The differences are calculated on model levels to avoid introducing vertical interpolation differences, and plotted as a function of model level expressed as a nominal pressure on the ordinate. The nominal pressure is the pressure of a model grid level in the absence of mountains, that is, calculated from the hybrid coordinate formula assuming a surface pressure of 1000 mb. In the following, nominal pressure is used as a tag in addition to model-level index to provide some indication of the relative physical location with respect to the surface of a particular level under consideration. The pointwise 99% confidence region is stippled in Fig. 1. The differences are less than 0.5 K except in the North Polar regions. Even there, the differences in the process-split case in the troposphere are at most 1.5–2 K, which although indicated as significant, are smaller than commonly seen between different models, for example, the Atmospheric Model Intercomparison Project (AMIP; Gates et al. 1995). The differences are slightly larger in the lower polar stratosphere in the time-split case, but again less than typical model differences. The regions where the zonal average differences might be considered statistically significant are very limited in extent. To examine the differences further we consider regional aspects.

Figure 2 shows the DJF temperature differences at the first model level above the surface, nominally 992 mb. All horizontal differences here are taken on model levels that are the same in all versions, to avoid the introduction of vertical interpolation differences. The levels are referred to by a vertical index and/or by their nominal pressure. Only 6% of the points satisfy the 99% pointwise significance test in the time-split case, and only 2% in the process-split case. The process-split case cannot be considered significantly different from CCM3 at this level; however, there are some coherent regions of differences in the time-split case to examine further. For example, the time-split difference at the edge of Antarctica in the Eastern Hemisphere, while small, O(1 K), is significant. It is reflected in a significant difference in the zonal average temperature (Fig. 1) at the first two model levels (indicated by the inner ticks in the figure) above the surface. The dynamical temperature tendency (not shown) also shows a significant difference there, exceeding 3.5 K day−1 locally.

To examine the nature of this difference, Fig. 3 shows vertical profiles averaged over the region where the difference in the dynamical temperature tendency at 992 mb in the time-split case is significant at the 99% level within the region bounded by −75 ≤ φ ≤ −60 and −30 ≤ λ ≤ 150. This masking region is snow-covered land bordering open ocean. The averaging mask is included in Fig. 3. The vertical profiles are plotted as a function of model level, labeled on the ordinate with the nominal pressure. Fig. 3a shows the vertical profile of the temperature difference. The curve is solid where the horizontal average difference is (vertical grid) pointwise significant at the 99% level, and dashed where it is not. It is only significant at the first two model levels (indicated by the inner ticks in the figure.) Although the temperature difference at 992 mb (first model level) seems modest at 0.7 K, the balance of terms in the two cases is very different. Figure 3b shows the difference in the time-averaged moist parameterizations, vertical diffusion, and the temperature tendency due to the dynamical core. Figure 3c shows the difference in the averaged shortwave and longwave radiative heating rates, and horizontal diffusion. The difference in the vertical diffusion at the first model level (which is off scale at −2.4 K day−1) balances the difference in the dynamical tendency (2.6 K day−1, also off scale). The difference in the other terms is much smaller.

In CCM3, the dynamical temperature tendency (Fig. 3d) at 992 mb of −2.2 K day−1 is balanced by the vertical diffusion (Fig. 3e) at 2.2 K day−1. Both terms are small in the time-split case (+0.44 and −0.09 K day−1, respectively) and are comparable in magnitude to the horizontal diffusion and moist processes there. The more dominant terms in the time-split case are the long- and short-wave heating rates (not shown) at −1.0 K day−1 and 0.45 K day−1, respectively. The morphology of the average temperature profile just above the surface including the surface temperature (Fig. 3f) is also very different in the two cases with an inversion forming at the first model level in the time-split case, but temperature decreasing monotonically in the CCM3. This different structure leads to sensible heat fluxes of opposite sign in the two cases that drive the vertical diffusion differences. Sensible heat flux is 1.9 W m−2 for CCM3 and −1.1 W m−2 for the time-split case. The latent heat flux difference is 0.1 W m−2, even though latent heat flux itself at 16 W m−2 dominates the sensible heat flux. Note, the different temperature profiles are not due to effectively looking at the state before vertical diffusion in one case (time-split) and after in the other (CCM3). Both cases are sampled after the convective parameterization. In addition, the temperature averages after the vertical diffusion look like those after the moist processes. The differences are from the surface exchange and vertical diffusion being time split in the time-split case and effectively process split in CCM3. The temperature profile from the process-split case parallels that of CCM3 without the inversion, with a rather uniform difference with CCM3 of 0.4 K. We do not have an explanation for why the CCM3 and process-split simulations have large and opposite vertical diffusion and dynamical tendencies while time-split forms an inversion and has small vertical diffusion and dynamical tendencies.

Returning to Fig. 2, as mentioned above, in the process-split case temperature differences exceed the 99% confidence limit at only 2% of the grid points making it dangerous to consider any of the differences as significant. Although the process-split case has larger differences than the time-split in the northern polar region, the natural variability is large there and the differences cannot be considered significant. At 500 mb, however (Fig. 4), there is a region over the edge of Antarctica in the process-split case where the temperature differences are around 1 K and are pointwise significant.

Figure 5 shows vertical profiles averaged over the region where the temperature difference at 500 mb in the process-split case is pointwise significant at the 99% level within the region bounded by −80 ≤ φ ≤ −50 and 0 ≤ λ ≤ 180. That averaging mask is included in Fig. 5. Very subtle changes in the terms in the thermodynamic balance lead to significant temperature differences. The temperature difference (Fig. 5a) maximizes at 1 K at 600 mb, but the individual terms in the temperature balance (Figs. 5b and 5c) differ by around 0.05 K day−1 throughout the column except longwave radiative cooling with a −0.1 K day−1 difference at 500 mb. The radiative difference follows from slightly different cloud fractions (less than +2% from 975 to 225 mb, but being closer to +1% from 800 to 400 mb). These differences are associated with relative humidity differences of +1% from 900 to 300 mb with slightly larger differences just above 300 mb (+2%) and below 900 mb (1.6%) in that region.

Figure 4 also shows a region over northern Canada with significant pointwise differences with the process-split case. Figure 6 shows vertical profiles averaged over the region where the temperature difference at 500 mb in the process-split case is pointwise significant at the 99% level within the region bounded by 50 ≤ φ ≤ 90 and 0 ≤ λ ≤ 360. The area-averaged temperature differences (Fig. 6a) are significant between 700 and 300 mb, and as large as 1.5 K through the lower half of that region. We note that this is one of the largest temperature difference seen, yet the differences in the terms creating the balance are rather small. The primary difference in the balance is between the longwave cooling and dynamical warming (Fig. 6b) with the differences being relatively small at 0.1 K day−1. The radiative changes are associated with small changes in cloud fraction. In both the arctic and antarctic regions the difference in longwave radiation is the dominant difference in the budget. The modeled longwave radiation in these regions is around −2 K day−1.

In Fig. 4 the time-split case shows significant differences in the tropical region, however, differences are more coherent at 200 mb as shown in Fig. 7. These differences are relatively small, around 0.5 K. Figure 8 shows vertical profiles averaged over the region where the temperature difference at 200 mb in the time-split case is pointwise significant at the 99% level within the region bounded by −30 ≤ φ ≤ 30 and −120 ≤ λ ≤ 80. The average temperature difference (Fig. 8a) is small, but significant, ranging from −0.2 K at 800 mb to −0.3 at 200 mb, making the time-split case less statically stable. The largest differences in the balance (Figs. 8b and 8c) are the moist processes that heat less in the time-split case, balanced by dynamics; however, the differences in both cases are less than 0.1 K day−1. The moist processes themselves are around 1.5 K day−1, but the dynamics varies from −0.25 to −0.75 K day−1. The temperature difference in Fig. 8a between 800 and 200 mb, which increases with height, is consistent with the difference between two moist adiabats anchored at the average temperatures of the two cases at 800 mb. The precipitation difference is −0.15 mm day−1. The difference in the specific humidity (not shown) decreases from −0.18 to −0.04 g kg−1 between 1000 mb and 850 mb, where the specific humidity itself decreases from 14 to 9 g kg−1.

Figure 2 and Fig. 7 also show significant differences in the time-split case over Australia at 992 mb and 200 mb, respectively. We consider this region separately from the preceeding South America–Africa region because the characteristics of the differences are different. Figure 9 shows vertical profiles averaged over the region where the temperature difference at 200 mb in the time-split case is pointwise significant at the 99% level within the region bounded by −45 ≤ φ ≤ −15 and 120 ≤ λ ≤ 160. The area average difference of ∼0.7 K is significant below 800 mb where vertical diffusion and dynamics balance in the difference. At 200 mb the area average difference is again significant but the change in the balance is much smaller, with moist processes balancing the dynamics.

Although the temperature difference at the surface over Australia is similar to that at the edge of Antarctica (Fig. 3), the vertical gradient from surface to atmosphere over Australia does not show the sign reversal seen over Antarctica, thus it does not have a large effect on the surface flux and vertical diffusion. Over Australia, the vertical gradient at the surface is slightly larger in the time-split case giving larger sensible heat flux in the time-split case (76 W m−2 vs 67 W m−2). This heat is transported above the first model level by the vertical diffusion, giving less net heating at the first model level by vertical diffusion in the time-split case but more at the levels above. The vertical diffusion difference is balanced primarily by dynamics, with moist processes being of secondary importance (Fig. 9b). The relative differences are again O(10%). At 200 mb, where the temperature difference is also significant, the changed balance is between moist processes balanced by dynamics and longwave radiation. However, these latter two differences are only about 0.06 K day−1.

c. June–July–August averages

Figure 10 shows the June–July–August (JJA) average, zonal average temperature difference for the time-split and process-split simulations minus the CCM3. The pointwise 99% confidence region is stippled. The largest differences occur in the time-split case at the South Pole at the third model level, nominally 930 mb. The process-split difference also maximizes there but at a smaller value and cannot be considered significant. Figure 11 shows the JJA temperature differences at the third model vertical level, nominally 930 mb. The differences in the process-split case are small with only 1% satisfying the pointwise significance test. Thus they cannot be considered significant and will not be pursued further. The time-split differences are large and significant near the South Pole.

Figure 12 shows vertical profiles averaged over the region over Antarctica where the temperature difference at 930 mb in the time-split case is significant at the 99% level within the region bounded by φ ≤ −60 and 0 ≤ λ ≤ 360. The temperature difference (Fig. 12a) is 3 K at third model level decreasing to ∼1 K at the first and second model levels, and to 0.4 K at the fourth model level. The difference clearly has a vertical grid-scale structure, although the temperature profiles themselves (Fig. 12d) show only a subtle change because the vertical temperature gradient itself is so strong, 20 K from the second to the fourth model level.

The grid-scale structure in the differences arises partly because the forcing in each simulation has a very dominant grid-scale structure. The overall balance in the two simulations is from radiation cooling (not shown) of −12 K day−1 at the second model level, while at the third model level it is around −4 K day−1, and at the first model level is just slightly positive. This strong grid signature comes from a maximum cloud layer of 75% (Fig. 12e) at the second model level, compared to 35%–40% at the level above and zero below. Vertical diffusion (Fig. 12f) transfers heat from third level down to the second level, but not enough to balance the radiation leaving a net cooling at the second and third levels from these two processes. The dynamics (not shown) provides the heating at these two levels to provide the balance.

Although the clouds are a maximum at the second model level, the difference in clouds is not at that level but above at the third and fourth levels (Fig. 12e); the time-split cloud fraction is 5% more at the third level and 2.5% less at the fourth. In the difference of the thermal balance (Figs. 12b,c), time-split dynamics heat the first model level relative to CCM3 by 3.5 K day−1 (off scale) and is balanced in the difference by relative cooling by the vertical diffusion (also off scale). Half of the vertical diffusion difference is associated with sensible heat flux which is −21.6 for time split and −19.5 for CCM3. The other half of the difference at the first level contributes to relative heating of the level above. The sensible heat difference would lead to −1.8 K day−1 more cooling in the time-split case if all the energy came from the first model level. Latent heat flux does not play a role, being 0.3 W m−2 in both cases. In the second model level, longwave radiation and vertical diffusion both heat the time-split case relative to CCM3 and are balanced by dynamics.

Although the temperature difference has a very strong grid-scale structure, the temperature itself does not indicate such a structure as seen in Fig. 12d. The strong grid-scale forcing in the clouds, radiation and vertical diffusion drive the dynamics (not shown) to attempt to produce a grid-scale heating, which it cannot calculate accurately given the nature of the approximations. Thus one must ask whether the vertical resolution is adequate to properly model these processes.

We have concentrated on thermal balance here, but other fields that might be thought to be more sensitive, for example, precipitation in the Tropics, do not show large differences. The precipitation differences, process split minus CCM3, have 1.7% of points indicating pointwise significance in DJF, and 1.1% in JJA. Once again, it would be dangerous to consider the differences significant. The time split minus CCM3 shows more points indicating significance, with 3.9% in DJF but not in large coherent regions. On the other hand, JJA has 5.1% in coherent tropical regions with values up to 2 mm day−1 in regions where precipitation values are 10–15 mm day−1. Nevertheless, it is difficult to see differences when comparing the fields themselves. Thus although there are significant differences, they are relatively small as was seen in the temperature field.

d. Process split within parameterization suite

The argument is sometimes made that process splitting all components of the model, including the individual components of the parameterization suite is desirable because all processes are then based on a correct resolved atmospheric state rather than an intermediate state that does not represent the atmosphere. We have also performed a simulation with such a configuration. The approximations can be written as
i1520-0493-130-8-2024-e14

The simulated climate from this configuration is very different from all simulation configurations discussed previously, in which the components within the parameterization suite are time split. A broad tropical tropopause with temperature less than 190 K forms spanning the 140 and 190 mb nominal pressure levels rather than at the 100 mb nominal pressure level. The temperature below this tropopause is 32 K colder than that in the dynamics/physics process-split simulation discussed above. This difference decreases rather uniformly in pressure to about 6 K at the first model level above the surface. In the arctic region below 900 mb, the new simulation is more than 20 K colder than the dynamics/physics process-split simulation, decreasing to 16 K colder at 500 mb. Needless to say, this new simulation in which the components within the parameterization suite are process split is not satisfactory.

Perhaps the poor quality of the simulation is not surprising since several of the parameterization components are capable of producing the same or very similar forcings if given the same state; the most obvious examples being the Zhang–McFarlane and the Hack convection parameterizations. Thus, when their full effects are accumulated, these two parameterizations force the dynamics at close to twice the rate obtained from sequential time-split application, in which the first convective parameterization applied is dominant and the second only provides a minor cleanup phase. It might be more appropriate, when there are competitive processes like these, to weight their contributions based on a fraction of the grid box over which they are assumed to work. However, the situation is not as simple as that. Simply omitting the Hack parameterization from the suite does not solve the problem. In this case the tropical tropopause with temperature below 190 K spans the 100-, 140-, and 190-mb nominal pressure levels and the tropical tropospheric temperature below the tropopause is also 32 K colder than that from the dynamics/physics process-split simulation. Some of the remaining parameterized processes may also be competitive. However, some parameterizations in the suite are likely to have been designed or tuned to work best from a state produced by the preceeding parameterizations, perhaps inadvertently during their development in the CCM framework. In fact there is growing sentiment in the community that the processes are all interrelated and not independent. If they were independent, process splitting might be more appropriate. For a process-split application, the CCM parameterizations would need to be redesigned and tuned to be based on states created by different processes than currently. Lenderink and Holtslag (2000) and Teixeira (2000) provide analyses of simple, but relevant, systems in which process splitting leads to incorrect equilibrium balances.

The original premise that all parameterizations should be based on a correct resolved atmospheric state may be flawed. Some parameterizations such as radiation are clearly designed to be based on the correct state. However, other parameterizations might need an unrealistic atmospheric state to be active. Large-scale stable condensation and moist adiabatic adjustment are examples, where in many formulations the relative humidity must exceed 100% for them to operate. The situation is less obvious with more modern convective parameterizations dealing with conditionally unstable states. In these cases the modeled state could be atmospheric-like both before and after application of the parameterization. However, some closures might require a nonphysical state in order to initiate convection. The dynamical component, or some other process must produce such a state for these parameterizations to be effective. There is no consensus in the community about whether to adopt process- or time-split approximations. Teixeira (2000) lists examples of both forms from major modeling centers.

5. Conclusions

We have examined the differences in simulations introduced by different splitting or coupling strategies between the dynamical core and parameterization suite of CCM3. We compare process-split and time-split strategies with the original CCM, which is a combination of the two. The principal algorithmic difference between time-split and CCM3 is that the radiation, surface fluxes, and turbulent mixing are not time split in the CCM3. The principal difference between process-split and CCM3 is that the moist processes are not process split in CCM3.

Overall the differences between the various coupling strategies are significantly smaller than differences introduced by changes in parameterizations such as from CCM2 to CCM3 (Kiehl et al. 1998a), differences introduced by changes within a parameterization such as Hack (1998), and differences introduced by the tuning of parameterizations during model development to achieve, say, reasonable top of atmosphere energy balances (which remain in the modeling lore but never appear in the literature). The differences are also smaller than seen between different models as in AMIP (Gates 1995; Gates et al. 1999) for example. In addition, the differences are neither consistently good nor bad in terms of simulation quality, although that aspect was not discussed here.

Thus, at least for the near term, with the time step used in the CCM3, the different time truncation errors introduced by the different coupling strategies allowed in the new code design have less effect on the simulations than other arbitrary aspects of the model design. This does not imply, however, that the time truncation errors are insignificant, just that they are similar in the cases examined here.

Although the differences are relatively small physically, there are regional differences that are statistically significant. In particular, we considered regional temperature and thermal balance differences. Most of the differences in the thermal balance are also small, but there is at least one example where processes creating the balance are behaving rather differently in two simulations yet produce a similar climate.

We summarize regions of small differences in the balance first. For example, the December–January–February average temperature has significant midtropospheric differences between the process-split case and CCM3 of around 1 K in regions in the arctic and antarctic. The largest difference in the thermal balance associated with the temperature differences is the longwave radiation, which has differences of around 0.1 K day−1. This difference is balanced to a large extent by differences in the dynamical heating rate, although other processes also contribute.

In the Tropics, over Africa and South America, the time-split simulation shows small but significant local temperature differences with CCM3 on the order of 0.25 K. These are accompanied by differences in the moist processes balanced by differences in the dynamical cooling. The relative differences are modest being ∼7% for the moist processes and ∼10% for the dynamics. Over Australia, the temperature difference is ∼0.7 K below 800 mb. The difference in balance is between moist processes and dynamics. The relative differences are ∼10%.

The most notable differences occur between the time-split case and CCM3 over regions of Antarctica. In December–January–February significant temperature differences occur just above the surface at the edge of Antarctica over snow-covered land bordering open ocean. Although the temperature difference is modest at 0.7 K, the balance of terms in the two cases is very different. An inversion forms at the first model level in the time-split case, but not in CCM3. This results in a difference in sign in the sensible heat flux between the two cases, positive for CCM3 and negative for the time-split case. The heating in the CCM3 is balanced by cooling by the dynamics.

In June–July–August the average temperature difference over the South Pole region of Antarctica between the time-split case and CCM3 is 3 K at the 930-mb nominal pressure level. The parameterization terms here have a very strong grid-scale structure associated with parameterized clouds forming predominantly at a single grid level. The dynamics is unable to respond with a grid-scale structure just because of the nature of the approximations. This draws into question whether the vertical resolution is adequate to properly model the physical processes.

Acknowledgments

I would like to thank John Truesdale for developing the code for these simulations and for running the experiments, James Hack for discussions on various aspects of the simulations, and the anonymous reviewers for suggestions for improvements of the paper. Preliminary results were presented at the Eleventh Annual BMRC Modelling Workshop, Parallel Computing in Meteorology and Oceanography, 9–11 November 1999, Melbourne, Australia. This work was partially supported by the Office of Biological and Environmental Research, U.S. Department of Energy, as part of its Climate Change Prediction Program.

REFERENCES

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Caya, A., R. Laprise, and P. Zwack, 1998: Consequences of using the splitting method for implementing physical forcings in a semi-implicit semi-Lagrangian model. Mon. Wea. Rev., 126 , 17071713.

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  • Chen, M., and J. R. Bates, 1996: Forecast experiments with a global finite-difference semi-Lagrangian model. Mon. Wea. Rev., 124 , 19922007.

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  • Gates, W. L., Ed.,. 1995: Proc. First Int. AMIP Scientific Conf.,. Monterey, CA. WCRP Rep. WCRP-92, WMO/TD-732, 532 pp.

  • Gates, W. L., and Coauthors. 1999: An overview of the results of the Atmospheric Model Intercomparison Project (AMIP). Bull. Amer. Meteor. Soc., 80 , 2955.

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    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Kiehl, J. T., J. J. Hack, G. B. Bonan, B. A. Boville, B. P. Briegleb, D. L. Williamson, and P. J. Rasch, 1996: Description of the NCAR Community Climate Model (CCM3). NCAR Tech. Note NCAR/TN-420+STR, 152 pp.

    • Search Google Scholar
    • Export Citation
  • Kiehl, J. T., D. L. Williamson, and P. J. Rasch, 1998a: The National Center for Atmospheric Research Community Climate Model: CCM3. J. Climate, 11 , 11311149.

    • Search Google Scholar
    • Export Citation
  • Kiehl, J. T., and J. W. Hurrell, 1998b: The energy budget of the NCAR Community Climate Model: CCM3. J. Climate, 11 , 11511178.

  • Lenderink, G., and A. A. M. Holtslag, 2000: Evaluation of the kinetic energy approach for modeling turbulent fluxes in stratocumulus. Mon. Wea. Rev., 128 , 244258.

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    • Export Citation
  • Lin, S-J., 1997: A finite-volume integration method for computing pressure gradient force in general vertical coordinates. Quart. J. Roy. Meteor. Soc., 123 , 17491762.

    • Search Google Scholar
    • Export Citation
  • Lin, S-J., and R. B. Rood, 1997: An explicit flux-form semi-Lagrangian shallow-water model on the sphere. Quart. J. Roy. Meteor. Soc., 123 , 24772498.

    • Search Google Scholar
    • Export Citation
  • Murthy, A. S. V., and R. S. Nanjundiah, 2000: Time-splitting errors in the numerical integration of semilinear systems of ordinary differential equations. Mon. Wea. Rev., 128 , 39213926.

    • Search Google Scholar
    • Export Citation
  • Sportisse, B., 2000: An analysis of operator splitting techniques in the stiff case. J. Comput. Phys., 161 , 140168.

  • Staniforth, A., and A. Robert, 1981: The computational stability of certain schemes for incorporating vertical fluxes in semi-implicit baroclinic primitive equation models. Division Rech. Prévision Numérique (RPN) Tech. Note, 19 pp.

    • Search Google Scholar
    • Export Citation
  • Strang, G., 1968: On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5 , 506517.

  • Teixeira, J. P. D. C. C., 2000: Boundary layer clouds in large scale atmospheric models: Cloud schemes and numerical aspects. Ph.D. thesis, Faculdade de Ciências, Universidade de Lisboa, ECMWF, 190 pp.

    • Search Google Scholar
    • Export Citation
  • Zhang, G. J., and N. A. McFarlane, 1995: Sensitivity of climate simulations to the parameterization of cumulus convection in the Canadian Climate Centre general circulation model. Atmos.–Ocean, 33 , 407446.

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

Dec–Jan–Feb (DJF) avg, zonal avg temperature diff for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K, significant diff stippled

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 2.
Fig. 2.

DJF avg temperature diff at first model level (992-mb nominal pressure) for time-split and process-split simulations minus the CCM3. Contour interval is 1.0 K centered around 0 K, significant diff stippled

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 3.
Fig. 3.

DJF avg vertical profiles of averages over a region on the edge of Antarctica. Differences of time-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance, and for each case (d) the dynamical temperature tendency, (e) vertical diffusion temperature tendency, and (f) temperature. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 4.
Fig. 4.

DJF avg temperature diff at 500-mb nominal pressure level for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K centered around 0 K, significant diff stippled

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 5.
Fig. 5.

DJF avg vertical profiles of averages over a region on the edge of Antarctica. Differences of process-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 6.
Fig. 6.

DJF avg vertical profiles of averages over a region in northern Canada. Differences of process-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 7.
Fig. 7.

DJF avg temperature diff at 200-mb nominal pressure level for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K centered around 0 K, significant diff stippled

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 8.
Fig. 8.

DJF avg vertical profiles of averages over regions in South America and Africa. Differences of time-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 9.
Fig. 9.

DJF average vertical profiles of averages over a region in Australia. Differences of time-split simulation minus the CCM3 of (a) temperature and (b), (c) terms in the thermodynamic balance, and for each case (d) the temperature. Temperature difference curve is solid at levels where difference is significant. Also included is the averaging mask

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 10.
Fig. 10.

Jun–Jul–Aug (JJA) avg, zonal avg temperature differ for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K, significant diff stippled

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 11.
Fig. 11.

JJA avg temperature diff at third model level (930-mb nominal pressure) for time-split and process-split simulations minus the CCM3. Contour interval is 1.0 K centered around 0 K, significant diff stippled

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Fig. 12.
Fig. 12.

JJA avg vertical profiles of averages over Antarctica. Differences of time-split simulation minus the CCM3 of (a) temperature and (b), (c) terms in the thermodynamic balance, and for each case the (d) temperature (e) cloud fraction, (f) vertical diffusion temperature tendency, and (g) moist processes. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

Citation: Monthly Weather Review 130, 8; 10.1175/1520-0493(2002)130<2024:TSVPSC>2.0.CO;2

Table 1. 

Fifteen-year avg, annual avg, global avg properties ± one std dev

Table 1. 

*

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Save
  • Beljaars, A. C. M., 1991: Numerical schemes for parameterizations. Proc. ECMWF Seminar on Numerical Methods in Atmospheric Models, Vol. 2. Reading, United Kingdom, ECMWF, 1–42.

    • Search Google Scholar
    • Export Citation
  • Bonan, G. B., 1996: A land surface model (LSM version 1.0) for ecological, hydrological, and atmospheric studies: Technical description and user's guide. NCAR Tech. Note NCAR/TN-417+STR, 150 pp.

    • Search Google Scholar
    • Export Citation
  • Caya, A., R. Laprise, and P. Zwack, 1998: Consequences of using the splitting method for implementing physical forcings in a semi-implicit semi-Lagrangian model. Mon. Wea. Rev., 126 , 17071713.

    • Search Google Scholar
    • Export Citation
  • Chen, M., and J. R. Bates, 1996: Forecast experiments with a global finite-difference semi-Lagrangian model. Mon. Wea. Rev., 124 , 19922007.

    • Search Google Scholar
    • Export Citation
  • Gates, W. L., Ed.,. 1995: Proc. First Int. AMIP Scientific Conf.,. Monterey, CA. WCRP Rep. WCRP-92, WMO/TD-732, 532 pp.

  • Gates, W. L., and Coauthors. 1999: An overview of the results of the Atmospheric Model Intercomparison Project (AMIP). Bull. Amer. Meteor. Soc., 80 , 2955.

    • Search Google Scholar
    • Export Citation
  • Hack, J. J., 1994: Parameterization of moist convection in the National Center for Atmospheric Research community climate model (CCM2). J. Geophys. Res., 99 , 55515568.

    • Search Google Scholar
    • Export Citation
  • Hack, J. J., . 1998: Sensitivity of the simulated climate to a diagnostic formulation for cloud liquid water. J. Climate, 11 , 14971515.

    • Search Google Scholar
    • Export Citation
  • Hack, J. J., J. T. Kiehl, and J. W. Hurrell, 1998: The hydrologic and thermodynamic structure of the NCAR CCM3. J. Climate, 11 , 11791206.

    • Search Google Scholar
    • Export Citation
  • Holtslag, A. A. M., and B. A. Boville, 1993: Local versus nonlocal boundary-layer diffusion in a global climate model. J. Climate, 6 , 18251842.

    • Search Google Scholar
    • Export Citation
  • Hurrell, J. W., J. J. Hack, B. A. Boville, D. L. Williamson, and J. T. Kiehl, 1998: The dynamical simulation of the NCAR CCM3. J. Climate, 11 , 12071236.

    • Search Google Scholar
    • Export Citation
  • Kiehl, J. T., J. J. Hack, G. B. Bonan, B. A. Boville, B. P. Briegleb, D. L. Williamson, and P. J. Rasch, 1996: Description of the NCAR Community Climate Model (CCM3). NCAR Tech. Note NCAR/TN-420+STR, 152 pp.

    • Search Google Scholar
    • Export Citation
  • Kiehl, J. T., D. L. Williamson, and P. J. Rasch, 1998a: The National Center for Atmospheric Research Community Climate Model: CCM3. J. Climate, 11 , 11311149.

    • Search Google Scholar
    • Export Citation
  • Kiehl, J. T., and J. W. Hurrell, 1998b: The energy budget of the NCAR Community Climate Model: CCM3. J. Climate, 11 , 11511178.

  • Lenderink, G., and A. A. M. Holtslag, 2000: Evaluation of the kinetic energy approach for modeling turbulent fluxes in stratocumulus. Mon. Wea. Rev., 128 , 244258.

    • Search Google Scholar
    • Export Citation
  • Lin, S-J., 1997: A finite-volume integration method for computing pressure gradient force in general vertical coordinates. Quart. J. Roy. Meteor. Soc., 123 , 17491762.

    • Search Google Scholar
    • Export Citation
  • Lin, S-J., and R. B. Rood, 1997: An explicit flux-form semi-Lagrangian shallow-water model on the sphere. Quart. J. Roy. Meteor. Soc., 123 , 24772498.

    • Search Google Scholar
    • Export Citation
  • Murthy, A. S. V., and R. S. Nanjundiah, 2000: Time-splitting errors in the numerical integration of semilinear systems of ordinary differential equations. Mon. Wea. Rev., 128 , 39213926.

    • Search Google Scholar
    • Export Citation
  • Sportisse, B., 2000: An analysis of operator splitting techniques in the stiff case. J. Comput. Phys., 161 , 140168.

  • Staniforth, A., and A. Robert, 1981: The computational stability of certain schemes for incorporating vertical fluxes in semi-implicit baroclinic primitive equation models. Division Rech. Prévision Numérique (RPN) Tech. Note, 19 pp.

    • Search Google Scholar
    • Export Citation
  • Strang, G., 1968: On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5 , 506517.

  • Teixeira, J. P. D. C. C., 2000: Boundary layer clouds in large scale atmospheric models: Cloud schemes and numerical aspects. Ph.D. thesis, Faculdade de Ciências, Universidade de Lisboa, ECMWF, 190 pp.

    • Search Google Scholar
    • Export Citation
  • Zhang, G. J., and N. A. McFarlane, 1995: Sensitivity of climate simulations to the parameterization of cumulus convection in the Canadian Climate Centre general circulation model. Atmos.–Ocean, 33 , 407446.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Dec–Jan–Feb (DJF) avg, zonal avg temperature diff for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K, significant diff stippled

  • Fig. 2.

    DJF avg temperature diff at first model level (992-mb nominal pressure) for time-split and process-split simulations minus the CCM3. Contour interval is 1.0 K centered around 0 K, significant diff stippled

  • Fig. 3.

    DJF avg vertical profiles of averages over a region on the edge of Antarctica. Differences of time-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance, and for each case (d) the dynamical temperature tendency, (e) vertical diffusion temperature tendency, and (f) temperature. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

  • Fig. 4.

    DJF avg temperature diff at 500-mb nominal pressure level for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K centered around 0 K, significant diff stippled

  • Fig. 5.

    DJF avg vertical profiles of averages over a region on the edge of Antarctica. Differences of process-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

  • Fig. 6.

    DJF avg vertical profiles of averages over a region in northern Canada. Differences of process-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

  • Fig. 7.

    DJF avg temperature diff at 200-mb nominal pressure level for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K centered around 0 K, significant diff stippled

  • Fig. 8.

    DJF avg vertical profiles of averages over regions in South America and Africa. Differences of time-split simulation minus the CCM3 of (a) temperature, and (b), (c) terms in the thermodynamic balance. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

  • Fig. 9.

    DJF average vertical profiles of averages over a region in Australia. Differences of time-split simulation minus the CCM3 of (a) temperature and (b), (c) terms in the thermodynamic balance, and for each case (d) the temperature. Temperature difference curve is solid at levels where difference is significant. Also included is the averaging mask

  • Fig. 10.

    Jun–Jul–Aug (JJA) avg, zonal avg temperature differ for time-split and process-split simulations minus the CCM3. Contour interval is 0.5 K, significant diff stippled

  • Fig. 11.

    JJA avg temperature diff at third model level (930-mb nominal pressure) for time-split and process-split simulations minus the CCM3. Contour interval is 1.0 K centered around 0 K, significant diff stippled

  • Fig. 12.

    JJA avg vertical profiles of averages over Antarctica. Differences of time-split simulation minus the CCM3 of (a) temperature and (b), (c) terms in the thermodynamic balance, and for each case the (d) temperature (e) cloud fraction, (f) vertical diffusion temperature tendency, and (g) moist processes. Temperature diff curve is solid at levels where diff is significant. Also included is the averaging mask

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