Figure 1 illustrates the PDF-based scheme with a uniform PDF and a triangular PDF of total water substance qt. By assuming that the clear region is free of condensates and that the cloudy region is fully saturated, the cloudy region (b) becomes the area where qt is larger than the saturation value qs (shaded area). The PDF-based scheme automatically retains consistency between CF and condensates because it is derived from the same PDF. Here, we used the uniform PDF to demonstrate the relationship between RHc and the width of the PDF. Using a derivation extended from Tompkins (2005), 2b=12δ(qt‾+δ-qs). It is evident that, with the uniform PDF, 3δc=qs1-RHc. Therefore, RHc=1-δcqs. Thus, if the width δ of the uniform PDF is determined, then RHc can be determined accordingly. This relation reveals that the RHc assumption of the RH-based scheme actually assumes the width of the uniform PDF to be δc from the PDF-based scheme. As noticed by Tompkins (2005), the RHc used by Sundqvist et al. (1989) for cloud generation can be linked to the statistical cloud scheme with a uniform distribution. Building upon this finding, we eliminated the assumption of RHc by determining the P(qt) with information about qv‾ and ql‾ provided by the base model. Please note that uniform temperature is assumed over the grid for the GTS scheme.

With uniform PDF as denoted in Fig. 1a, the liquid cloud fraction (bl) and grid-mean cloud liquid mixing ratio (ql‾) can be integrated as follows: 4bl=∫qs∞Pqtdqt=12δql‾+qv‾+δ-qs, and 5ql‾=∫qs∞(qt-qs)Pqtdqt=14δq‾t+δ-qs. Given ql‾, qv‾, and qs, the width of uniform PDF can be determined as follows: 6δ=ql‾+qs-qv‾2. Therefore, we can calculate the liquid cloud fraction from Eq. (4).

In addition to the application of a PDF-based approach for liquid CF parameterization, the GTS scheme also uses the same concept for parameterizing the ice CF (bi) as follows: 7bi=12δ(q‾i+q‾v+δ-sup⋅qsi), where q‾i, q‾v, and qsi denote the grid-mean cloud ice mixing ratio, water vapor mixing ratio, and saturation mixing ratio over ice, respectively. In Eq. (7), qsi is multiplied by a supersaturation factor (“sup”) to account for the situation in which rapid saturation adjustment is not reached for cloud ice. In the present version of the GTS scheme, sup is temporarily assumed to be 1.0. Sensitivity tests regarding sup will be discussed in Sect. 5.6. Values of qi‾ and qv‾ used to calculate Eq. (7) are the updated state variables before calling the cloud macrophysics process.

A more complex PDF can be used for P(qt) instead of the uniform distribution in our derivation. For example, the Community Atmosphere Model version 5.3 (CAM5.3) macrophysics model adopts a triangular PDF instead of a uniform PDF to represent the subgrid distribution of the total water substance (Park et al., 2014). Mathematically, the triangular distribution is a more accurate approximation of the Gaussian distribution than the uniform distribution and it may also be more realistic. Therefore, we followed the same procedure to diagnose the CF by forming a triangular PDF with ql‾, qv‾, and qs‾ provided. Moreover, by using a triangular PDF, we can obtain results that are more comparable to the CAM5.3 macrophysics scheme because the same PDF was used. By considering the PDF width, the CF (b) and liquid water content (ql‾) can be written as follows: 8b=121-ss2ifss>01-121+ss2ifss<0, and 9ql‾δ=16-ss26+ss36-ssbifss>0-16-163ss2-2ss3-ssbifss<0, respectively, where ss=qs-qt‾δ. From these two equations, we can derive the width of the triangular PDF and calculate the CF (b) based on qs, qt‾, and qv‾ instead of RHc. Detailed derivations of Eqs. (8) and (9) can be seen in Appendix A. Notably, the PDF width for the total water substance can only be constrained when the cloud exists. Therefore, the RHc is still required when clouds start to form from a clear region. To simplify the cloud macrophysics parameterization, value of RHc in the GTS scheme is assumed to be 0.8 instead of RHc varying with height in the default Park scheme. The GTS scheme still uses the default prognostic scheme for calculating cloud condensates (Park et al., 2014), and it has effects only on the stratiform CFs. Although the GTS scheme is presumed to have good consistency between CF and condensates, the consistency check subroutines of the Park scheme are still kept in the GTS scheme to avoid “empty” and “dense” clouds due to the usage of the Park scheme for calculating cloud condensates, and the GTS schemes still need RHc when clouds start to form.

In this study, GTS schemes utilizing two different PDF shape assumptions are evaluated: uniform (hereafter, U_pdf) and triangular (hereafter, T_pdf). These two PDF types are specifically formulated to evaluate the effects of the choice of PDF shape. A triangular PDF is the default shape used for cloud macrophysics by CAM5.3 (hereafter, the Park scheme). The T_pdf of the GTS scheme is numerically similar to that of the Park scheme except for using a variable width for the triangular PDF once clouds are formed.

The GTS schemes described in this study were implemented into CAM5.3 in the Community Earth System Model version 1.2.2 (CESM 1.2.2), which is developed and maintained by Department of Energy (DOE) University Corporation for Atmospheric Research/National Center for Atmospheric Research (UCAR/NCAR). Physical parameterizations of CAM5.3 include deep convection, shallow convection, macrophysics, aerosol activation, stratiform microphysics, wet deposition of aerosols, radiation, a chemistry and aerosol module, moist turbulence, dry deposition of aerosols, and dynamics. References for the individual physical parameterizations can be found in the NCAR technical notes (Neale et al., 2010). The master equations are solved on a vertical hybrid pressure–sigma coordinate system (30 vertical levels) using the finite-volume dynamical core option of CAM5.3.

We conducted both the single-column tests and stand-alone global-domain simulations with CAM5.3 physics. The single-column setup provides the benefit of understanding the responses of physical schemes under environmental forcing of different regimes of interest. Here, we adopt the case of Tropical Warm Pool – International Cloud Experiment (TWP-ICE), which was supported by the ARM program of the Department of Energy and the Bureau of Meteorology of Australia from January to February 2006 over Darwin in northern Australia. Based on the meteorological conditions, the TWP-ICE period can be divided into four shorter periods: the active monsoon period (19–25 January), the suppressed monsoon period (26 January to 2 February), the monsoon clear-sky period (3–5 February), and the monsoon break period (6–13 February; May et al., 2008; Xie et al., 2010). To take advantage of previous studies of cloud-resolving models and single-column models, we followed the setup of Franklin et al. (2012) to initiate the single-column runs starting on 19 January 2006 and running for 25 d.

Stand-alone CAM5.3 simulations of the CESM model, forced by climatological sea surface temperature for the year 2000 (i.e., CESM compset: F_2000_CAM5), are conducted to demonstrate global results. The horizontal resolution of the CESM global runs is set at 2∘. Individual global simulations are integrated for 12 years, and the output for the last 10 years is used to calculate climatological means and annual cycles in global means. Because we made changes largely with respect to CF, we also conducted corresponding simulations using the satellite-simulator approach to provide CF for a fair comparison with satellite CF products and typical CESM model output. This was done using the Cloud Feedback Model Intercomparison Project (CFMIP) Observation Simulator Package (COSP) built into CESM 1.2.2 (Kay et al., 2012). In addition to the default monthly outputs, daily outputs of several selected variables are also written out for more in-depth analysis.

Cloud field comparisons are critical for modifications to our system with respect to cloud macrophysical schemes. Therefore, we use the products from CloudSat/Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) to provide CF data for evaluating the modeling capabilities of the default and modified GTS cloud macrophysical schemes. This dataset (provided by the AMWG diagnostics package of NCAR) is used to compare with CF simulated by the COSP satellite simulator of CESM 1.2.2. Notably, this dataset is different from the one below which also includes cloud water content (CWC).

In addition to cloud observations, observational radiation fluxes from the Clouds and the Earth's Radiant Energy System – Energy Balanced and Filled (EBAF) product (CERES-EBAF) are also used to investigate whether simulations using our system will improve radiation calculations for both shortwave and longwave radiation flux, as well as their corresponding cloud radiative forcings. Precipitation data are compared with Global Precipitation Climatology Project data and several other climatic parameters, e.g., air temperature, RH, precipitable water, and zonal wind, are evaluated against the reanalysis data (ERA-Interim). All these observational data are also obtained from the AMWG diagnostics package provided by NCAR and their corresponding datasets can be found in the NCAR Climate Data Guide (https://climatedataguide.ucar.edu/collections/diagnostic-data-sets/ncar-doe-cesm/atmosdiagnostics, last access: 8 January 2021). The time periods used to calculate the climatological means are simply following the default setup of the AMWG diagnostics package.

We further evaluate the performance of the three macrophysics schemes by using the approach of Su et al. (2013), which compares CF and CWC sorted by large-scale dynamical and thermodynamic parameters. The CF products are based on the 2B-GEOPROF R04 dataset (Marchand et al., 2008), while the CWC data are based on the 2B-CWC-RO R04 dataset (Austin et al., 2009). The methodology from Li et al. (2012) is used to generate gridded data. Two independent approaches (i.e., FLAG and PSD methods) are used in Li et al. (2012) to distinguish ice mass associated with clouds from ice mass associated with precipitation and convection. The PSD method is used in this study (Chen et al., 2011). In total, 4 years of CloudSat/CALIPSO data, from 2007 to 2010, are used to carry out the statistical analyses. These data are used to obtain overall climatological means to compare to those obtained from model simulations instead of undergoing rigorous year-to-year comparisons between observations and simulations. Monthly data from ERA-Interim for the same 4 years are used to obtain the dynamical and thermodynamic parameters used in Su et al.'s approach. These parameters include large-scale vertical velocity at 500 mbar and RH at several vertical levels.

To evaluate the impact of assumptions of CF distributions for the RH- and PDF-based schemes, we conducted offline calculations of the CF by using the reanalyzed temperature, humidity, and condensate data from ERA-Interim. As the differences in CF characteristics do not change from month to month, the results for July are shown in Fig. 2 as an example. The ERA-Interim reanalysis performed by Dee et al. (2011) using a 0.75∘ resolution from 1979 to 2012 is used in the calculation. With this offline approach, we can observe the impacts of these macrophysics assumptions with a balanced atmospheric state provided by the reanalysis.

Using the U_pdf of GTS scheme as an example to elaborate on the details of calculation procedures, we simply obtain the cloud liquid mixing ratio (ql‾), water vapor mixing ratio (qv‾), and air temperature (to calculate qsl‾) from the ERA-Interim as input variables to calculate the liquid CF via using Eqs. (6) and (4) when ql‾ is greater than 10-10 (kgkg-1). When ql‾ is smaller than 10-10 (kg kg-1) and if RH>RHc, CFs are calculated based on Eq. (3) and the liquid CF parameterization of Sundqvist et al. (1989), and if RH<RHc, CFs are equal to zero. Ice CFs are calculated similarly to those of liquid CFs but using Eq. (7), q‾i, q‾si, and sup of 1.0. Procedures for calculating CFs diagnosed by the T_pdf of the GTS scheme are similar to those of U_pdf but using the equation set of the triangular PDF. Values of RHc used in the U_pdf and T_pdf of GTS schemes are assumed to be 0.8 and height independent. The maximum overlapping assumption is used to calculate the horizontal overlap between the liquid CF and ice CF.

Overall, the geographical distributions from the two GTS schemes are similar to that of the ERA-Interim reanalysis shown in Fig. 2. In July, high clouds corresponding to deep convection are shown over south and east Asia where monsoons prevail. The diagnosed clouds of the GTS scheme have a maximum level of 125 hPa, which is consistent with those of the ERA-Interim reanalysis but also have a more extensive cloud coverage of up to 90 %. Below the freezing level at approximately 500 hPa, the CF diagnosed by the GTS scheme is comparable to that diagnosed by ERA-Interim reanalysis. The most substantial differences in CF between the GTS scheme and ERA-Interim are observed in the mixed-phase clouds, such as the low clouds over the Southern and Arctic oceans. Such differences suggest that more complexity in microphysics assumptions may be needed to describe the large-scale balance of mixed-phase clouds. It is interesting to note that the U_pdf simulates CFs at the lower levels in closer agreement with those of ERA-Interim and the U_pdf obtains similar magnitude of CFs to those of the T_pdf at the upper levels. The potential reason for such differences could be related to the nature of the two PDFs. The U_pdf is likely to calculate more CFs compared to T_pdf given similar RH and cloud liquid mixing ratio in the lower atmospheric levels. The diagnosed CF for the Park macrophysics scheme is also shown in the right column of Fig. 2. We found that the cloud field diagnosed by the Park macrophysics scheme was considerably different from that diagnosed by ERA-Interim reanalysis and the GTS schemes. The Park scheme diagnosed overcast high clouds of 100–125 hPa with coverage of up to 100 % over the warm pool and Intertropical Convergence Zone, but very little cloud coverage below 200 hPa, suggesting that the assumptions of the Park scheme are probably not suitable for large-scale states of the ERA-Interim reanalysis.

However, such a calculation does not account for the feedback of the clouds to the atmospheric states through condensation or evaporation and cloud radiative heating. Therefore, we further extended our single-column CAM5.3 experiments to examine the impact of the cloud PDF assumption.

GTS schemes tend to produce smaller RMSE values for most of the global mean values of the radiation flux, cloud radiative forcing, and CF parameters shown in Table 4, suggesting that the GTS scheme is capable of simulating the variability of these variables. Furthermore, the assumed U_pdf shape appears to perform better for outgoing longwave radiation flux, longwave cloud forcing (LWCF), and CF at various levels, whereas the T_pdf assumption is better for simulating net and shortwave radiation flux at the top of the atmosphere as well as shortwave cloud forcing (SWCF) (Table 4). On the other hand, the Park scheme is better for simulating clear-sky net shortwave radiation flux and precipitation. Smaller RMSE values can also be seen for parameters such as total precipitable water, total-column cloud liquid water, zonal wind at 200 mbar (hereafter, U_200), and air temperature at 200 mbar (hereafter, T_200) when U_pdf of GTS is used. For global annual means, U_pdf simulates net radiation flux at the top of the atmosphere, all- and clear-sky outgoing longwave radiation flux, and precipitable water as well as U_200 and T_200 in closer agreement with observations. In contrast, the Park scheme is better for simulating global mean variables such as net shortwave radiation flux at the top of the atmosphere, longwave cloud forcing, and precipitation. T_pdf simulates SWCF closest to the observational mean.

Overall, the averaged RMSE values of the 10 parameters are 0.97 and 0.96 for U_pdf and T_pdf, respectively, in the GTS schemes (Fig. 10), suggesting that using the GTS schemes would result in global simulation performance more or less similar to that of the Park scheme. It is also worth noting that the biases in RH are smallest when U_pdf in the GTS scheme is used (Table S3 in the Supplement). In contrast, T_pdf results in the smallest biases for SWCF, sea level pressure, and ocean rainfall within 30∘ N–30∘ S. On the other hand, the Park scheme produces the smallest biases regarding mean fields such as LWCF, land rainfall within 30∘ N–30∘ S, Pacific surface stress within 5∘ N–5∘ S, zonal wind at 300 mbar, and temperature.

Comparisons of latitude–height cross-sections of RH and ERA-Interim show that the GTS schemes tend to simulate RH values smaller than the default scheme does, especially for high-latitude regions (>60∘ N and 60∘ S), as shown in Fig. 11. In general, in terms of RH, using T_pdf in the GTS scheme results in better agreement with ERA-Interim (Table S4). Figure 12 shows that the Park and T_pdf schemes are wetter than ERA-Interim almost everywhere and that the uniform scheme is sometimes drier. Table S5a further suggests that specific humidity simulated by the GTS schemes is slightly more consistent with ERA-Interim than the Park scheme. Comparisons of air temperature show that the three schemes tend to have cold biases almost everywhere. However, it is interesting to note that the cold biases are reduced to some extent when using the GTS schemes compared to the default scheme, as is evident in the smaller values of RMSE shown in Table S5b. These effects on moisture and temperature are likely to result in changes in the annual cycle and seasonality of climatic parameters. Such observable changes in RH, clouds (both CF and CWC), and cloud forcing suggest that the GTS scheme will simulate cloud macrophysics processes in GCMs quite differently from the Park scheme, due to the use of a variable-width PDF that is determined based on grid-mean information.

Figure 13 shows the annual cycle of precipitable water simulated by the three schemes. The magnitude of precipitable water simulated by the GTS schemes is closer to the ERA-Interim data than the Park simulation is (Table S6). Interestingly, U_pdf results in slightly better agreement with ERA-Interim than T_pdf for the region 60∘ N–60∘ S. This implies that the GTS scheme would alter the moisture field for both RH and precipitable water in GCMs. These results are relatively more realistic with respect to both the moisture field and CF and CWC (Figs. 8 and 9), and are likely to yield a more reasonable cloud–radiation interaction in the GCMs. It is therefore also worth examining any differences in dynamic fields, for example, in the annual U_200 cycle, between the three schemes and the ERA-Interim data (Fig. 14). Like the annual cycle of precipitable water, U_200 simulated by the GTS schemes is closer to that of ERA-Interim than that simulated by the Park scheme (Table S6). Furthermore, the U_pdf assumption results in a better annual U_200 cycle than the T_pdf assumption, especially for 60∘ N–60∘ S. This further supports the argument that this GTS scheme can effectively modulate global simulations, with respect to both thermodynamic and dynamical climatic variables.

Figure 15 displays the global mean annual cycles of several parameters simulated by the three schemes and the corresponding parameters from observational data. The GTS scheme simulations of total precipitable water (TMQ) are close to that of ERA-Interim; indeed, U_pdf almost exactly reproduces the ERA-Interim TMQ. However, we must admit that such good agreement of the global mean is partly due to offsetting wet and dry differences from ERA-Interim. The GTS schemes also produce a more reasonable global mean annual cycle for outgoing longwave radiation (FLUT). It is probably due to the reduced CF simulated by the GTS scheme compared to the Park scheme even though the cloud top heights simulated by GTS are lower than observations in the tropics. Interestingly, for SWCF, T_pdf yields a simulation closer to the observations than the other two schemes, which is consistent with the features of the global annual mean of SWCF shown in Fig. 10 and Table S3. However, for LWCF, the annual cycle simulated by Park is closest to the observations. The U_pdf of the GTS scheme also results in improvements in U_200 and T_200 (Fig. 15). The RMSEs for all of these comparisons confirm these results (Table S7).

As mentioned in Sect. 5.1, usage of the GTS cloud macrophysics schemes would affect the cloud fields, i.e., CF and CWC. This, in turn, is likely to affect global simulations with respect to both mean climatology and the annual cycles of many climatic parameters (as discussed in Sect. 5.2 and 5.3) through cloud–radiation interaction. Figure 16 compares CF, radiation heating rate (i.e., longwave heating rate plus shortwave heating rate, hereafter QRL + QRS), and temperature tendencies due to moist processes (hereafter, DTCOND) for each pair-wise combination of the three schemes. Qualitatively consistent changes in CF are apparent for the GTS schemes, e.g., an increase in the highest clouds over the tropics and a decrease below them, a decrease in 150–400 mbar clouds over the midlatitudes, a decrease in 300–700 mbar clouds over the high latitudes, an increase in 300–700 mbar clouds over the tropics to midlatitudes, and an increase in low clouds over the high-latitude regions. The GTS schemes also yield a significant increase in CF at atmospheric levels higher than 300 mbar over the high-latitude regions (Fig. 16). These changes affect the radiation calculations to some extent. In addition, CWC is also affected by the GTS schemes (Figs. 8 and 9). The combined effects of the changes in CF and CWC are likely to result in changes in cloud–radiation interaction. In addition, although there are significant changes in CF at high atmospheric levels in the high-latitude regions, the combined effect of CF and CWC on QRL + QRS is quite small, due to the low CWC values over this region. The changes in moisture processes, i.e., DTCOND (Fig. 16), also suggest that the combined effects of the changes in the thermodynamic and dynamical fields occur as a result of changes in cloud–radiation interaction within the GCMs from GTS schemes.

The bottom row in Fig. 16 shows the differences in CF, QRL + QRS, and DTCOND between the two GTS schemes. Relative to T_pdf, U_pdf simulates a greater CF for 300–1000 mbar clouds within 60∘ N–60∘ S but a smaller CF for all three cloud levels for the high-latitude regions. Furthermore, the CWC vertical cross-section also differs for the two GTS schemes (data not shown for limitations of space). Combining the changes in CF and CWC, the corresponding changes in QRL + QRS and DTCOND, particularly the increase of low clouds over the midlatitude region, are clear with an obvious decrease of high clouds over the tropical to midlatitude region. It is also evident that DTCOND simulated by the U_pdf is stronger than that simulated by the T_pdf below 700 hPa. Such enhanced condensation heating is probably contributed by the enhanced shallow convection as a result of changes in cloud–radiation interaction. However, more process-oriented diagnostics are needed to understand the complicated interaction of the moist processes.

Observable changes in large-scale circulations are likely, given the various changes in QRL + QRS and DTCOND resulting from applying different cloud macrophysics. Accordingly, both the mean and variability of the climate simulated by the GCMs differ among the three schemes, as shown in the previous subsections. These results emphasize the importance of improving cloud-related parameterization to provide better simulations of the cloud–radiation interaction within GCMs. Furthermore, as previously shown, the cloud–radiation interaction is highly sensitive to the assumptions of the CF parameterization used in the macrophysical scheme in the GCMs, even if there is only a small change in the CF parameterization. The uniqueness of the GTS scheme is in its application of a variable PDF width to calculate CF in the default PDF-based CF scheme of the CESM model. Further systematic experiments are necessary to improve our understanding of the sensitivity of the GTS scheme, and some are presented in Sect. 5.6.

Observable changes in clouds and radiation fluxes after adopting the GTS scheme were clearly shown in the previous subsections. It is thus worth examining features in cloud radiative forcings caused by the GTS scheme that produce such changes, as compared to those of the default Park scheme. Figure 18 shows the difference in total cloud fraction, SWCF, LWCF, CF, and averaged cloud water content, as well as the averaged RH at the three levels, i.e., 100–400, 400–700, and 700–1000 mbar, derived from the T_pdf of GTS with the Park results subtracted. One can readily observe that changes in SWCF (Fig. 17b) are quite consistent with those for total CF, showing a decrease in the total CF over the area within 30∘ N and 30∘ S with an increase everywhere else (Fig. 17a). Such prominent changes in latitudinal distribution of SWCF can be further related to the changes in the low (Fig. 17e) and middle (Fig. 17f) CFs particularly associated with low clouds.

On the other hand, changes in the high CF (Fig. 17d) are also quite consistent with those in LWCF (Fig. 17c), showing an overall decrease of high clouds especially over the tropical convection areas. As expected, changes in cloud water condensates (Fig. 17g–i) are closely related to changes in the CF at the three levels except for the middle clouds. Therefore, according to the evidence shown in Fig. 17a–i, it is clear that use of the GTS scheme would cause significant changes in the spatial distribution of low, middle, and high clouds (both in CF and cloud water condensates) that would result in corresponding changes in cloud radiative forcings (both for SWCF and LWCF).

Surprisingly, changes in RH at the three levels (Fig. 17j–l) are relatively less consistent with changes in the CF and condensates, especially for middle and low clouds over the mid- and high-latitude areas. Such results also indicate that there are complicated factors accounting for changes in RH in the GCMs. We suggest that, in addition to the active roles of the GTS scheme in redistributing/modulating moisture between clouds (i.e., cloud liquid or ice) and environment (water vapor) in GCM grids, thermodynamic and dynamical feedback resulting from cloud–radiation interaction also contributes to RH changes. At the present stage, we cannot quantify these individual contributions. More in-depth analysis is needed to unveil the detailed mechanisms of why GTS schemes tend to produce less low clouds over the tropics while more low clouds over the mid- and high latitudes compared to the default Park scheme, as well as observable changes regarding middle and high clouds.