## Abstract

Several numerical experiments have been conducted using the NCAR Community Atmosphere Model, version 3 (CAM3) to examine the impact of the time step on rainfall in the intertropical convergence zone (ITCZ) in an aquaplanet. When the model time step was increased from 5 to 60 min the rainfall in the ITCZ decreased substantially. The impact of the time step on the ITCZ rainfall was assessed for a fixed spatial resolution (T63 with L26) for the semi-Lagrangian dynamical core (SLD). The increase in ITCZ rainfall at higher temporal resolution was primarily a result of the increase in large-scale precipitation. This increase in rainfall was caused by the positive feedback between surface evaporation, latent heating, and surface wind speed. Similar results were obtained when the semi-Lagrangian dynamical core was replaced by the Eulerian dynamical core. When the surface evaporation was specified, changes in rainfall were largely insensitive to temporal resolution. The impact of temporal resolution on rainfall was more sensitive to the latitudinal gradient of SST than to the magnitude of SST.

## 1. Introduction

In the past decade several dynamical cores have been developed to increase the computational efficiency of the atmospheric GCM. Williamson and Olson (2003) examined the impact of the dynamical core on the simulation of climate in an atmospheric GCM. They used the Neale and Hoskins (2000) standard aquaplanet test suite with two variants of the National Center for Atmospheric Research (NCAR) Coupled Climate Model, version 3 (CCM3). The two variants differed in the formulation of dynamical cores, one was Eulerian (EUL) and the other was semi-Lagrangian (SLD). There were many differences between the two dynamical cores. Among these, time step size was one of the most important. Williamson and Olson (2003) carried out an extensive study and proved that the when the time step is changed, the impact on the simulated rainfall was high. They showed that the sensitivity was not due to the different time truncation errors, but primarily due to the impact of the time step on the cumulus parameterization.

Williamson and Olson (2003) have argued that the evaporation rate was the same for both the time steps so in the longer time step more moisture was put into the atmosphere by the surface exchange parameterizations. The primary focus of their work was on why the rainfall peak shifted when the model time step was changed.

In this paper, we have examined this issue further to understand why a decrease in the time step increases the rainfall in the ITCZ. We have carried out experiments with the NCAR Community Atmosphere Model, version 3 (CAM3), GCM with two different dynamical cores. As our main objective was to understand the sensitivity of the simulation to the time step used, we have chosen SLD for detailed analysis, because the maximum permissible time step size of SLD is larger than EUL. Hence, the time step size can be varied over a larger range in SLD.

The model components are briefly summarized in section 2, and the experiments are described in section 3. The results are discussed in sections 4 and 5. Section 6 verifies the response of the effect to different SST boundary conditions. Section 7 discusses the impact in Eulerian dynamical core. In section 8, the cause of the impact of time step has been traced out. Section 9 provides a summary of results.

## 2. Description of the model

Since a detailed description of CAM3 can be found in Collins et al. (2004), we will not discuss the model in detail in this paper. Nevertheless, parts of the model that are relevant to this work are explained in the following.

CAM3.0 is a three-dimensional global model. For this study, simulations are carried with two dynamical cores, namely EUL and SLD. The EUL dynamical core is a three-time-level, spectral transform applied at T42 truncation on a 128 × 64 quadratic grid with time step Δ*t* = 20 min. Moisture transport is monotonic semi-Lagrangian, which is time split in the horizontal and vertical directions. The trajectory calculation used for moisture transport uses a quasi-cubic interpolation. The SLD core is a two-time-level, spectral transform applied at T63 truncation on a 128 × 64 linear Gaussian grid with time step Δ*t* = 60 min. The trajectory calculation uses linear interpolation. The model uses the hybrid vertical coordinate, which is terrain following at the earth’s surface, but reduces to pressure coordinate at higher levels near the tropopause. The model has 26 levels in the vertical.

The same physical parameterization package is used for both EUL and SLD. The physical parameterization package consists of moist precipitation process (M), clouds and radiation processes (R), surface processes (S), and turbulent mixing processes (T). Each of these in turn is subdivided into various components. The M consists of deep convective, shallow convective, and stratiform processes. The physical parameterization schemes include those for deep convective (Zhang and McFarlane 1995), shallow convective (Hack 1994), and stratiform processes [Rasch and Kristjansson (1998) updated by Zhang et al. (2003)].

The dynamical core (EUL/SLD) is coupled to the physics package in a process-split approximation. Inside the physical parameterization package, individual parameterizations are time split. In the process-split technique, the calculations of dynamical and physical tendencies for prognostic variables are based upon the same past state. But, the tendencies for each subcomponents of the physics package are computed sequentially, each based upon the state produced by the other (Williamson 20002).

Essentially the computational sequences of the model is the following. Computation of M is followed by R, then S, and then finally by T. Computation of the M in turn involves three steps. In the first step it invokes the deep convective scheme; in the second step, the shallow convective scheme; and in the third step, the stratiform scheme. All the parameterizations except radiation parameterization are called during every time step. The frequency of calling of radiation parameterization is 1 h and is independent of the time step size used in the dynamical core.

## 3. Experimental strategy

To address the questions posed in the introduction, a series of experiments have been performed with a zonally symmetric SST profile as boundary condition. The distribution of SST used in the simulation is discussed in appendix A. The solar forcing with the diurnal cycle was set for equinoctial conditions (21 March) in the radiation code.

The initial condition for all simulations was from a previous aquaplanet simulation. All the simulations were performed for 18 months, and the last 12 months were considered for the analysis as suggested by Neale and Hoskins (2000). To verify whether 18 months of integration was sufficient, we carried out two integrations for 5 yr. The results were not very different from the 18-month integration. Hence, we integrated the model for 18 months in all the experiments discussed in this paper. The monthly and zonal mean value have been considered for analysis in this paper.

## 4. Results and discussion

### a. Response to temporal resolution

In Fig. 1, the meridional variation of rainfall (RF), evaporation (EVP), convergence (RF − EVP) and integrated water vapor (Pwat) is shown for integration with a time step of 5 and 60 min. We find that the largest difference occurs in the region 10°S–10°N. Hence, our analysis will focus primarily on this region. From Fig. 1a we see that the magnitude as well as the position of ITCZ is sensitive to temporal resolution. Table 1 shows the set of simulations carried out with SLD to investigate the response. Figure 2 shows that the mean rainfall in the ITCZ increases when the time step of integration is reduced. We find that the differences are statistically significant at the 99.8% level. The relative contribution of local evaporation and large-scale convergence over the ITCZ can be seen in Fig. 2. We find that (see the second row in Table 2) around 35% of the enhancement in rainfall was contributed by the enhancement in evaporation while 65% was contributed by the enhancement in the large-scale convergence [defined as rainfall − evaporation (RF − EVP) in Fig. 2]. Table 3 showsthat when the time step size is decreased from 60 to 20 min, the southern as well as the northern rainfall peaks have moved toward the equator by one grid point, which is consistent with those obtained by Williamson and Olson (2003).

### b. Impact of temporal resolution on components of precipitation

In the preceding section we discussed the response of total rainfall (TRF), which comprises deep convective, shallow convective, and large-scale rainfall. We now examine the sensitivity of the each component of rainfall to temporal resolution.

Figure 3 shows the change in each component of the total rainfall with temporal resolution. We have shown the changes with reference to the simulation with the 60-min time step. We find that the large-scale precipitation increases steadily when the time step decreases from 60 to 5 min. The deep convective precipitation does not show any such systematic increase and the shallow convective precipitation increases initially, but remains constant thereafter. Hence, we can conclude that the steady increase in total rainfall when the temporal resolution is increased is on account of the increase in large-scale precipitation. Moreover, it is noticed that, in the region of discussion, the fraction of the large-scale rainfall with SLD60 is less than 3% of the total rainfall, but the same contributes about 10% in SLD5. This impact of the time step on the simulation of the proportion of large-scale rainfall is important, since CAM3.0 underestimates the simulation of fraction of the large-scale rainfall (Rasch et al. 2006), which is crucial for the simulation of tropical transient activities.

### c. Vertical structure of large-scale precipitation

In CAM3 water exists in three states: water vapor (gaseous phase), liquid condensate (liquid phase), and ice condensate (solid phase), depending upon the pressure and temperature. Figure 4 shows the vertical structure of the large-scale rainfall, water vapor, liquid condensate, and ice in the ITCZ.

Figure 4a shows that large-scale rainfall in SLD5 is more than in SLD60 almost throughout the atmospheric column. Hence, it is expected that the simulation with SLD5 should have higher relative humidity (RH) in lower troposphere, more liquid condensate in the middle troposphere, and more ice condensate in the upper troposphere as seen in Figs. 4c and 4d, respectively.

Figure 4b shows that, there is an increase of RH by 10% in the lower troposphere (below 500 hPa). Figure 4c shows the vertical distribution of the liquid condensate is higher in SLD5 by 0.02 g kg^{−1} around 450 hPa (almost double of SLD60). Similarly, Fig. 4d shows the vertical distribution of ice condensate, which is higher by almost 0.0075 g kg^{−1} at 300 hPa in SLD5 in comparison with SLD60. This (Figs. 4b–d) is consistent with the vertical structure of the large-scale precipitation (Fig. 4a). Relative humidity is a function of both moisture content and temperature. Figures 5b,c show the corresponding specific humidity and temperature differences. It is seen that the profile of the difference of specific humidity resembles the profile of the difference of RH very closely especially in the lower troposphere, in the upper troposphere the reduction in specific humidity is further accentuated by an increase in temperature resulting in a significant reduction in RH.

### d. Lateral moisture transport into the ITCZ

Table 2 shows that around 35% of the enhancement of rainfall in the ITCZ is contributed by enhancement in the local surface evaporation and approximately 65% by the enhancement of the large-scale convergence. When the whole tropics (30°S–30°N) is considered, around 86% of the enhancement of rainfall is contributed by the enhancement in surface evaporation. In the region 5°S–5°N there is no difference in evaporation. Figure 6c shows that the difference in the Pwat over this region (5°S–5°N) is positive. Over this region the rainfall difference is positive although evaporation is largely the same. The excess moisture required in the ITCZ is contributed to by the meridional advection of moisture from the rest of the tropics.

Figure 7a shows the vertical profile of the net amount of moisture transported into the ITCZ in each simulation. (In this figure a positive value indicates moisture transport into the region, and a negative indicates moisture transport out of the region.) The figure shows that, in lower troposphere (below 800 hPa), SLD5 transports more amount of moisture into this region and also takes more amount of moisture out of this region, than SLD60. The strength of the circulation with a smaller time step is greater, and hence, the strength of the low-level convergence as well as the upper-level divergence are greater in SLD5 than in SLD60. As the specific humidity decreases rapidly with height, the amount of moisture flux entering the ITCZ through the low-level convergence is more than the amount of moisture flux leaving the ITCZ through the upper-level divergence. Hence, the moisture content in the ITCZ increases and this leads to higher large-scale rainfall.

## 5. Evaporation

### a. Impact of rainfall–evaporation feedback

In the previous section we have shown that with an increase in temporal resolution the surface evaporation increases. To verify whether evaporation plays an important role in the enhancement of rainfall we conducted an experiment in which the surface evaporation was specified. In this experiment the monthly mean evaporation obtained from the control simulation (SLD with a 60-min time step) was specified. Thus, evaporation was invariant in time but varied in space. This ensures that when latent heating of the atmosphere increases it does not automatically lead to an increase in surface evaporation (i.e., through an increase in wind speed). The result of the experiment is shown in Fig. 8. We find that the area-averaged total precipitation does not show a significant increase with the reduction in the time step. This is in contrast to the earlier simulation wherein the decrease in the time step caused a 10% increase in rainfall.

### b. Role of wind speed and specific humidity difference

We have argued that feedback between evaporation and large-scale precipitation plays an important role in the enhancement of rainfall. The model computes evaporation as explained in appendix B (Collins et al. 2004). Figure 9 shows that both the wind speed at the lowest level of the model and the surface humidity deficit (dq) increase lead to an increase in evaporation when the time step is reduced. However, the impact of the increase in wind speed is more dominant. Wind speed increases by about 10% at many latitudes, whereas dq increases by just 2%. Nonetheless, over some latitudes (±15°), though the change in wind speed is either negative or near zero, the evaporation over there is noticed to be increased, which is due to the increase in the dq (see Fig. 9).

## 6. Impact of SST

Does the impact of the time step on the simulation of ITCZ depend on the prescribed boundary condition? To investigate this issue, additional simulations were carried out for six different SST magnitudes and seven different SST gradients (see appendix A). Figure 10 shows the time mean zonal average precipitation of SLD5 and SLD60. We find that the differences occur primarily in the region 10°S–10°N. Figure 11 shows that rainfall in the ITCZ increases with increase in temporal resolution, irrespective of the strength of SST.

Figure 12 shows that the response of the rainfall to temporal resolution is sensitive to the latitudinal gradient of the prescribed SST. We find that when the SST gradient is greater at ±10°, the impact of the increase in temporal resolution on rainfall is greater (Fig. 13). Our previous discussion (i.e., section 4d) showed that 35% of the enhancement of rainfall over the ITCZ is contributed by the increase of local evaporation while 65% is contributed by the increase of large-scale convergence. In the SST7 simulation, the latitudinal gradient of SST over the deep tropics is very small (Fig. 14b), which leads to very weak surface winds (Fig. 14c). Hence, the local evaporation in the ITCZ is not affected much by the time step (Fig. 15b). Figure 14 shows the mean meridional circulation for SST1 and SST7, respectively. In the SST7 simulation, the rising limb of the Hadley cell shifts away from the equator and at the equator the circulation is weak. Weak circulation leads to reduced moisture convergence into the deep tropics and in turn causes precipitable water vapor to be low in this region (Fig. 15d). We notice that when the SST gradient is weak neither the local evaporation nor large-scale convergence (Fig. 15c) are sensitive to the temporal resolution. Hence, total rainfall is also insensitive to the size of the time step.

## 7. Simulations with the Eulerian dynamical core

We have shown that when the time step is decreased in the SLD there is an increase in ITCZ rainfall. Is this also true for the Eulerian dynamical core? To verify this we conducted two simulations with time steps of 5 and 20 min, respectively (see Table 4). Note that we cannot use the time step of 60 min with the Eulerian dynamical core. In Fig. 16 we show the magnitude of the area-averaged rainfall in the ITCZ. We find that the rainfall increases when the temporal resolution is increased from 20 to 5 min. From Fig. 16, it can also be noticed that the increase of the total rainfall is primarily due to the increase of the large-scale rainfall. Williamson and Olson (2003) showed that CCM3 simulates a single rainfall peak centered on the SST maximum with EUL, but with SLD it simulates a double rainfall peak straddling the SST maximum. They indicated that the successor of CCM3 forms a double peak with all the dynamical cores and they attributed this to the differences between the parameterizations suite of CCM3 and its successor. Here, we would like to highlight that in CAM3 (i.e., the latest publicly released model from NCAR and one of the successors to CCM3) the EUL dynamical core simulates a single peak when a time step is small (i.e., 5 min), but simulates two peaks at the larger time step of 20 min (Fig. 17).

## 8. Time step size, parameterization tendencies, and update of the state variables

To understand the root cause of the impact, we have analyzed the spinup period. Figure 18 shows that the impact occurs in the first time step of integration. The impact evolves during the initial 36 h of model integration, and persists thereafter. The model updates the state variables, as shown in Eq. (1), where *n* stands for the time point:

The computational sequences in the model is presented in Fig. 19. The state variables are updated after computation of every parameterization scheme and the updated variables are used for the computation of the tendencies in the next parameterization scheme (see Figs. 19a,b). We stored the state variables at the end of the computation of each module (see Fig. 19) and noticed that the impact originated inside the deep convection scheme. It is found that the theoretical upper bound used for the computation of the cloud-base mass flux depends on the time step size (see Fig. 19c). Because of the preceding cause, the tendency produced by the scheme in the two simulations are different. Moreover, the time step size used by both the simulations for updating the variables are different (i.e., SLD60 uses 3600 s and SLD5 uses 300 s). Thus, the second term of Eq. (1) is found to be different for both the simulations. As same initial conditions are used for both simulations the first term of Eq. (1) is the same; however, because of the difference in the second term, the updated states are different for the two simulations. This difference in the state variables produces different tendencies computed in the next parameterization scheme and so on to the next time step of integration. Furthermore, it is noticed that the theoretical upper bound is not the only factor used in the parameterization schemes, which uses the time step size in its formulation; there are many other factors. The evaporation limit for the precipitating rainfall is a function of the time step size. All these factors contribute to the impact during integration. Because of the nonlinear nature of the system (i.e., the presence of numerous feedback processes) later on (after 24 h of integration), the change in DRF becomes less important. It is seen that (not shown here) in the steady state (after the spinup period) over the deep tropics the moisture profile becomes steeper with SLD5 than with SLD60. This leads to a reduction of CAPE from 231 J kg^{−1} with SLD60 to 192 J kg^{−1} with SLD5. But, at the same time, higher evaporation with SLD5 leads to higher Pwat and cloud water in the atmosphere. However, the excess moisture pumped into the atmosphere by the higher evaporation has to be removed by some means (e.g., convective or large-scale rainfall). However, convective schemes are based on convective instability, which cannot remove the excess moisture since the convective instability is less in SLD5. So the atmosphere becomes more saturated (higher RH, see Fig. 5) and, hence, the LRF gets enhanced with SLD5.

We carried out two more numerical experiments (see Table 5). In one experiment the deep convective rainfall (DRF) scheme was switched off, but the shallow convective rainfall (SRF) and large-scale rainfall (LRF) schemes were operating. And in the second experiment, both the DRF and SRF schemes were switched off, but the LRF scheme was operating. The results are presented in Table 5. Table 5 shows that the difference in the rainfall between SLD5 and SLD60 is only reduced by 12%, while both the convective schemes were switched off. Hence, it implies that even at the absence of the convective schemes, the impacts do exist and are significant.

## 9. Summary and concluding remarks

We have shown that the time step adopted for a GCM has significant impact on the rainfall in the ITCZ. The total rainfall in the ITCZ increases steadily when the model time step is decreased. The large-scale rainfall was found to be the primary contributor to the steady increase in total rainfall. We have shown that at higher temporal resolution there is an increase in surface wind speed that leads to higher surface evaporation over the entire tropics. This higher surface evaporation leads to higher water vapor content and, hence, a higher meridional moisture transport into the ITCZ, which leads to higher large-scale rainfall. The higher large-scale rainfall leads to more latent heating and, hence, higher surface convergence. When this positive feedback process is disrupted deliberately by specifying the surface evaporation, the impact of the time step on the rainfall in the ITCZ is considerably reduced. The impact of higher temporal resolution on the ITCZ rainfall is more sensitive to the meridional gradient of SST than to the magnitude of SST. The increase in rainfall with increased time step size does not appear to be dependent on the dynamical core.

An important primary step, in this direction, has been made by showing that model simulation shows a monotonic sensitivity to the time step size. Since time step is an important factor associated with dynamical core and spatial resolutions, care should be taken when comparing the simulations with different dynamical cores or/and spatial resolutions. Also, our hypothesis is different from that proposed by Williamson and Olson, and could be related to the different parameterization suite used in the two models. It is necessary to verify this with other models and also in a real-planet framework.

## Acknowledgments

We thank Drs. Brian Mapes, Dave Williamson, Richard Neale, and Simona Bordoni for their valuable suggestions during the course of the work. Comments and suggestions from the two anonymous reviewers helped to clarify the exposition.

## REFERENCES

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*CCGC*M.

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### APPENDIX A

#### Mathematical Form of the SST Distributions

The equations below describe the functional form of the variation of SST in the experiments: Eq. (A1) SST1—the control experiment; Eq. (A2) SST2; Eq. (A3) SST3; Eq. (A4) SST4; Eq. (A5) SST5; Eq. (A6) SST6; Eq. (A7) SST7; Eq. (A8) SST1 + 1; Eq. (A9) SST1 + 2; Eq. (A10) SST1 + 3; Eq. (A11) SST1 + 4; and Eq. (A12) SST1 + 5, where *T _{S}* is the SST (°C),

*λ*is longitude, and Φ is latitude:

### APPENDIX B

#### Computation of the Surface Evaporation

In CAM3.0 the surface evaporation is estimated as follows:

where *E* is the surface evaporation, *ρ _{A}* is the atmospheric surface density, and Δ

**V**is the velocity of the lowest model level (

**V**

*). Since the model (CAM3.0) used for this work is an atmospheric model, it does not allow for motion of the ocean surface, the velocity difference between surface and atmosphere is Δ*

_{A}**V**:

where *u* is the zonal velocity at the lowest model level, *υ* is the meridional velocity at the lowest model level, and *C _{E}* is the transfer coefficient between the ocean surface and the atmosphere at a height

*Z*(i.e., the lowest model level).

_{A}Here *Q _{A}* is the specific humidity at the lowest model level and

*Q*(

_{S}*T*) is the saturation specific humidity at the sea surface temperature.

_{S}## Footnotes

*Corresponding author address:* Saroj K. Mishra, Centre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore 560 012, India. Email: saroj@caos.iisc.ernet.in