Linking Deep and Shallow Convective Mass Fluxes via an Assumed Entrainment Distribution in CAM5‐CLUBB: Parameterization and Simulated Precipitation Variability

We modify the Zhang‐McFarlane (ZM) deep convection scheme in the Community Atmosphere Model version 5 to couple it with a unified parameterization for boundary‐layer turbulence and shallow convection, that is, Cloud Layers Unified by Binormals (CLUBB). By assuming a lognormal distribution of entrainment rate across the entire moist convective regimes, we link mass fluxes between shallow and deep convection, which are partitioned by the entrainment rate of the shallowest deep convective plume. Hence, a new deep convective closure is established which is coupled to the sub‐grid vertical motion variability in CLUBB. The convection feedback (or memory) effects are also considered to decrease the entrainment spectrum width and enhance the vertical velocity variability that further affect deep convection. Results show that the revised scheme improves the precipitation simulations in terms of the mean state and variability at various timescales, such as the alleviated double‐intertropical convergence zone and more realistic simulations of the seasonal variation of monsoon precipitation over East Asia, Madden‐Julian Oscillation, and precipitation diurnal phase propagations downstream of large terrains. The improvements are still seen in many aspects such as the mean‐state precipitation when turning off the convection feedback impacts in the revised scheme, emphasizing the benefits of using the modified mass‐flux closure. However, the convection feedbacks have considerable effects on the precipitation diurnal cycle simulations over regions with late‐afternoon precipitation peaks. Overall, the revised scheme provides a unified treatment for sub‐grid vertical motions across regimes of boundary‐layer turbulence, shallow convection, and deep convection, leading to better‐simulated precipitation at various timescales.


Introduction
Cumulus convection mainly induced by surface fluxes, large-scale convergence, and atmospheric radiative cooling can occur at a wide range of temporal and spatial scales in the globe (Arakawa, 2004;Rio et al., 2019). It is of vital importance to cloud and precipitation formation and can interact with the latent heating associated with precipitation process. Moreover, cumulus convection is key for redistributing atmospheric heat and moisture, and modulating the energy and hydrological cycles on the Earth (Arakawa & Schubert, 1974;Raymond, 1994). However, it is hard to represent convection in both global and regional climate models (GCMs and RCMs) that are generally unable to explicitly resolve the fine-scale convective processes. As a result, convective parameterizations have to be used in these models to formulate the statistical effects on the grid-mean atmospheric properties of convection (Arakawa, 2004).
However, undoubtedly, parameterization itself is a summary of our knowledge about the real atmosphere and its performance can reflect how well we understand the physical processes it represents (Rio et al., 2019). Recently, K. Emanuel (2020) pointed out that we need to make efforts to find out the reason why climate models with convective parameterizations fail to produce the large-scale precipitation phenomenon if such phenomenon is "parameterizable" (i.e., not sensitive to the detailed arrangement of individual convective cells), suggesting that we might compute too much and think too little. In addition, due to the needs for long integration length and multiple ensemble members, climate simulations adopted in climate change projection still use relatively coarse grid spacing (Eyring et al., 2016). Therefore, it is still necessary to continue the development of convection parameterizations.
Many efforts have been devoted to improving the behaviors of convective parameterizations in the past decades. For example, Zhang and Wang (2006) applied a convective closure based on the large-scale generation of convective available potential energy (CAPE) in the free troposphere (Zhang, 2002) in the Zhang-Mc-Farlane (ZM) convective parameterization in the National Center for Atmospheric Research (NCAR) Community Climate System Model version 3. Compared to the original version that used the total CAPE consumption as a closure, their new closure reduced the sensitivity of convection to surface forcing and thus alleviated the double-ITCZ problem. Meanwhile, several studies have shown that the simulated precipitation diurnal cycle can be improved via using different functions for convection triggering (Tawfik et al., 2017;Xie et al., 2019;S. Xie & Zhang, 2000), employing new diagnostic or prognostic quantities for convective closure (Fuchs & Raymond, 2007), or modifying the formulation of convective entrainment rates (Piriou et al., 2007;Stirling & Stratton, 2012). It was also reported that structural modifications or parameter optimizations in convective parameterizations can improve the simulations of MJO and ENSO (Boyle et al., 2015;Liu et al., 2019;Lu & Ren, 2016;Neale et al., 2008) in GCMs.
In convective parameterizations, one of the well-known deficiencies is the unrealistic transition from shallow to deep convection, which is thought to be responsible for many of the cloud and precipitation biases mentioned above. Various approaches have been proposed to overcome this problem. For example, YANG ET AL. 10.1029/2020MS002357 2 of 23 Bechtold et al. (2014) modified the CAPE-based closure in which parts of the CAPE production are consumed by shallow convection under particular conditions. Alternately, Rio et al. (2009) linked the triggering and intensity of deep convection to boundary-layer thermals that can reach the lifting condensation level (LCL). Both the above two approaches help improve the precipitation diurnal cycle simulations over land. Park (2014b) developed a new parameterization that represents shallow and deep convection in a unified way, which can better simulate the continental precipitation diurnal variation and the MJO. Several fully unified parameterizations were also presented in some studies that included boundary-layer turbulence and all types of convection in a single framework (Storer et al., 2015;Tan et al., 2018;Suselj et al., 2019). For example, Suselj et al. (2019) extended the eddy-diffusivity mass-flux (EDMF) approach to contain multiple plumes representing dry, shallow, and deep convective updrafts and precipitation-driven downdrafts. The new parameterization can well simulate the transitions across various convective regimes in a single-column model (SCM).
In our recent work (Yang et al., 2020), we modified the ZM deep convective closure in the Community Atmosphere Model version 5 (CAM5; Neale et al., 2010) coupled with a third-order turbulence closure parameterization, that is, Cloud Layers Unified by Binormals (CLUBB; Golaz et al., 2002;Larson & Golaz, 2005) (hereafter CAM5-CLUBB). In the modified closure, a part of CAPE diagnosed from the heating and moistening profiles in CLUBB is reserved for shallow convection, which helps suppress deep convection until shallow convection has sufficiently moistened the lower troposphere, leading to improved precipitation diurnal cycle simulation. However, compared to the original closure, the modified one still shows similar bias features in many aspects, such as the spurious double-ITCZ, too weak MJO, and unrealistic monsoon migration. Besides, inconsistency still exists between different components within the ZM scheme. Specifically, the closure is based on CAPE which is calculated with a fixed entrainment rate for an air parcel lifted from the convective launch layer, while the updraft cloud properties are based on an ensemble of plumes with a range of different entrainment rates. To improve the consistency between shallow and deep convection, as well as between the deep convective closure and updraft plumes, here we modify the closure of the ZM convective parameterization and couple it with CLUBB to provide a unified treatment for sub-grid vertical motions across regimes of boundary-layer turbulence, shallow convection, and deep convection. A lognormal distribution is assumed to represent the spectrum of entrainment rates across the entire moist convective regimes, with the spectrum width varying with the development of convection. The entrainment rate of the shallowest plume for deep convection is used to partition mass fluxes between deep and shallow convection, with the latter diagnosed from CLUBB. We will show that the revised ZM scheme markedly improves the simulated precipitation variability at various timescales in CAM5-CLUBB.
Section 2 depicts the modifications in the ZM convective parameterization scheme, including those related to the assumed distribution of entrainment rate, deep convective closure, and impacts of convection feedback. The model experiments and observational datasets are introduced in Section 3. In Section 4 we evaluate the impacts of the modified scheme on the precipitation simulation at various timescales. We first assess the results from the SCM experiments briefly and then focus on the GCM simulated precipitation in terms of the mean state, seasonal evolution, MJO, and diurnal variation. A summary of the main results is given in Section 5.

Convective Closure Based on Assumed Entrainment Distribution
In the original ZM scheme, the updraft cloud ensemble consists of multiple updraft plumes featured by different fractional entrainment rates () to represent clouds reaching different heights. The shallowest deep convective plume has the maximum entrainment rate  0 and detrains at height 0 z corresponding to the minimum saturated moist static energy in the vertical. Therefore, the updraft plumes with entrainment rate exceeding  0 and detraining below 0 z can be categorized as shallow convection, which is not parameterized in ZM but implicitly represented by CLUBB that provides a unified treatment for boundary-layer turbulence and shallow convection. Previous studies have used the entrainment rates diagnosed from large-eddy simulations to evaluate the distributions of entrainment rate in convective parameterizations (e.g., Plains (SGP) and deep convective clouds over the western Pacific, respectively. Their results revealed that in both regimes the entrainment rate (; in units of km −1 ) follows a lognormal distribution: but the values of  and  (i.e., the mean value and standard deviation of    ln , respectively) are different between the shallow and deep convective regimes. Here, we add a scaling factor  in Equation 1 that is set to 1.0 by default, aiming to adjust the mean value of    ln from  to      ln . Therefore, we can use one lognormal distribution to cover the entire moist convective regimes by allowing the width of the spectrum (i.e., ) to decrease with the development of convection (see Section 2.2). In Equation 1,  and  are set to 0.5 and 0.65, respectively. As  decreases, the distribution gradually evolves from the shallow convective regime as in Lu et al. (2012) toward the deep convective regime as in Guo et al. (2015).
The value of  0 corresponding to the shallowest deep convective plume is used to partition mass fluxes between deep and shallow convection (i.e., deep M and shallow M ) (Figure 1), and those with entrainment rates above and below  0 are categorized as shallow and deep convection, respectively. The partitioning is performed at the height of LCL (i.e., LCL z ) where the mass flux diagnosed in CLUBB has a close relationship with the strength of moist shallow convection. According to the entrainment distribution    f in Equation 1, the relationship between deep M and shallow M at LCL is given by: where  min is the minimum entrainment rate for deep convection. Because  is inversely proportional to cloud radius as indicated by previous studies (e.g., de Rooy et al., 2013;Kain, 2004), the value of  min corresponds to the largest cloud that needs to be parameterized. When the model grid spacing approaches the convection-permitting scale, some large deep convective clouds can be explicitly resolved, and  min should increase accordingly for scale-aware purpose. Given that the probability for entrainment rate between 0 and  0.1 is below 0.1%, which is almost negligible, here we set     1 min 0.1 km for the experiments performed in this study that are all configured with a resolution of 2° (see Section 3). Then, the mass flux of deep convection can be obtained from Equation 2 once we know the mass flux associated with shallow convection derived from the distribution of vertical velocity (w) primarily provided by CLUBB: YANG ET AL.

Impacts of Convection Feedback
Convection has its own memory. Idealized cloud-resolving model simulations by Colin et al. (2019) revealed that the spatial variability in water vapor and temperature, especially in the lower troposphere induced by convection can affect convection during the following 2-3 h (24 h) for unorganized (organized) cases. By contrast, the impacts of hydrometeors or winds are much weaker. Precipitation itself has negligible memory and removing in-cloud hydrometeor information in the model has very weak effect on the upcoming convection. However, surface precipitation rate can be used as an effective indicator of convection intensity, convective cloud size, as well as interactions among multiple processes that are potentially related to convection memory, such as the evaporation-driven cold pool and spatial variability in water vapor and moist static energy. For example, Harrop et al. (2018) parameterized the effects of convective gustiness as a function of surface convective precipitation rate. Mapes and Neale (2011) used precipitation evaporation as a source for convection organization. In this study, we use the surface deep convective precipitation rate ( d P ) from the previous time step to parameterize the deep convection memory (A): The first and second terms on the right-hand-side of Equation 6 represent the production and dissipation rates of A, respectively. Here,  is set to 7200 s, close to the recovery time of 2-3 h in Colin (2019) for unorganized convection. The value of A is carried over time but reset to 0 when its value is below 0.05. We set  = 2/3 mm −1 . Given that the ratio of global mean precipitation to precipitation evaporation is ∼3:1, the generation rate of A in Equation 6 is comparable to that parameterized by Mapes and Neale (2011), in which the increase of the organization degree per unit of evaporation is 2 mm −1 . However, it is more suitable to YANG ET AL.
10.1029/2020MS002357 5 of 23 consider A in Equation 6 as an index for the maturity of individual convective cells rather than organization at the mesoscale, which generally represents the spatial aggregation of convective cells and is accompanied by remarkable contrasts between moist and dry regions in a model grid point.
In this study, A can affect deep convection in two ways. First, cloud size (or radius), which can increase with the development of convection, is a key factor governing the width of the entrainment distribution  (in Equation 1). Here we let cloud radius increase by a factor of a when A (in Equation 6) increases from 0 to 1, which further leads to decreased entrainment rate (de Rooy et al., 2013;Kain, 2004). We set  0.5 a , producing a relative change in cloud radius half of that applied in the Kain-Fritsch convective parameterization, which uses a range of 1-2 km for cloud radius (Kain, 2004). However, one can image that when shallow cumuli dominate, they rarely produce surface precipitation (that is used to parameterize A), but can still grow horizontally as shallow convection develops, further increasing the probability for convective plumes extending above z 0 (corresponding to the shallowest plume in ZM). To consider this process,  is allowed to vary with the ratio of water vapor flux of shallow convection at z 0 to that at LCL, that is, lcl 0 QFX / QFX z provided by CLUBB. This vertical-scale ratio can approximately reflect the changes in the shallow plume width because larger clouds sampled at LCL will more likely reach the level of z 0 , with a higher lcl 0 QFX / QFX z due to weaker dilution from entrainment. The above assumption is similar to that applied in the MYNN (i.e., Mellor-Yamada-Nakanishi-Niino)-EDMF scheme (Olson et al., 2019), which also links the shallow plume widths to a vertical scale, that is, the boundary-layer height. As deep convective clouds start to rain, the entrainment distribution should become more dependent on the characteristics of deep convection (i.e., A in this study). Finally, the evolution of  is given by: where b is a tunable parameter and is set to 2 in this study. It indicates that  is determined by the smaller one of the two terms on the right-hand side of Equation 7 to reflect the different roles of shallow and deep convection at different stages.  0 represents the value of  when convection just starts, the impacts of which will be discussed in Section 4.1.
Second, as deep convection develops, the evaporation of precipitation can drive cold pool that further enhances the spatial variability of the upward velocity w. This is mainly because of the increased boundary-layer inhomogeneities in temperature (Mapes & Neale, 2011) and secondary updrafts due to strong outflows associated with the gust front (Feng et al., 2015), both of which can be assumed to rise in proportion to convection memory or surface precipitation rate (Harrop et al., 2018;Mapes & Neale, 2011). The impact of cold pool on the standard deviation of w at LCL (i.e.,  wlcl) is formulated as: where  clubb is the standard deviation of w provided by CLUBB and  cold _ pool is that contributed by cold pool, which is parameterized as . We set  0 to 0.5 m s −1 , with its magnitude comparable to the dailymaximum value of  clubb .

Unrestricted Launch Level for Deep Convection
In the original ZM parameterization, the launch level of deep convection is restricted within the boundary layer. However, earlier studies suggested that nocturnal convection is often elevated, with the most unstable air located above the boundary layer and the launch level decoupled from the surface over the US Great Plains (Geerts et al., 2017;Marsham et al., 2011;Xie et al., 2014). Wang et al. (2015) and Xie et al. (2019) found that using an unrestricted launch level (ULL) trigger that allows the convective updrafts to be launched above the boundary layer can greatly improve the simulations of nocturnal convection.
In this study, we follow the ULL concept in the revised ZM scheme that searches the convection launch level within the lowest 400 hPa above the surface rather than in the boundary layer.
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Experimental Design
We use the CESM version 1.2.1 with the CAM5 model physics (Neale et al., 2010). The PBL turbulence, shallow convection, and macrophysics are represented by CLUBB (Bogenschutz et al., 2013). First, two SCM cases (ARM95 and TWP06) are selected to test the performance of the revised ZM scheme. The large-scale forcing data for the SCM experiments are derived from the CESM input repository (https://svn-ccsm-inputdata. cgd.ucar.edu/trunk/inputdata/atm/cam/scam/iop/), containing meteorological fields collected during two Intensive Observation Periods (IOPs) at the SGP site (262.5°E, 36.6°N) from July 18, 1995 to August 3, 1995 and at the Tropical western Pacific (TWP) site (130.891°E, 12.425°S) from January 17, 2006to February 12, 2006. For each case, we conduct three experiments (i.e., CTL_ARM95/REV1_ARM95/REV2_ARM95 for the ARM95 case and CTL_TWP06/REV1_TWP06/REV2_TWP06 for the TWP06 case; see Table 1). Taking the ARM95 case as an example, CTL_ ARM95 uses the original ZM scheme, while both REV1_ ARM95 and REV2_ ARM95 use the revised ZM scheme but with the parameter  0 (Equation 7) set to 1 and 0.5, respectively.
Then, two GCM experiments (i.e., CTL and REV; Table 2) are conducted, which use the original and revised ZM scheme, respectively. Different from the SCM experiments, we set   0 0.5 over land and   0 1 over oceans in the GCM experiment with the revised scheme (i.e., REV). The reason is explained in Section 4.1. We also conduct two additional GCM experiments, that is, REV_ULL0 and REV_CF0. In REV_ULL0, the revised ZM scheme is used except that the convection launch level is still searched within the boundary layer. In REV_CF0, the value of A (Equation 6) is always set to zero, so as to isolate the effects of the revised mass-flux closure from that of the deep convection feedback. To make the averaged entrainment rate and thus the convection intensity comparable to that in REV, in REV_CF0 we set   0 3 / 8 over land and   0 3 / 4 over oceans, both of which are in the varying range of  in REV. All the GCM experiments are run for 11 years forced by the observed climatological-mean monthly sea surface temperature (SST), with the last 10 years used for the analyses. The model is configured with a horizontal resolution of 2.5° longitude × 1.9° latitude and 30 vertical hybrid levels.

Observational Datasets
The Tropical Rainfall Measuring Mission (TRMM; Huffman et al., 2007) 3B42 version 7 (available at ftp:// disc2.nascom.nasa.gov/data/TRMM/Gridded) precipitation data from 2001 to 2010 are used for model evaluation on various scales. The TRMM data has a grid spacing of 0.25° with at a 3-h interval. The daily precipitation data from the Global Precipitation Climatology Project (Adler et al., 2003) is used to evaluate the simulated MJO precipitation. In addition, the hourly rain-gauge data from the National Meteorological Information Center of China Meteorological Administration is used for evaluating the precipitation diurnal cycle simulation over China.
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Analysis Method
All of the observational data are interpolated onto the model grids using bilinear interpolation before comparisons. At each grid point, the intraseasonal component of precipitation associated with MJO is derived via a band-pass filter of 20-90 days.
When evaluating the simulated precipitation diurnal variation, precipitation at each hour of a day (i.e., P h ( )) is normalized by the daily-mean precipitation (i.e., mean P ): where N(h) is the normalized hourly precipitation.

Precipitation Simulations in SCM
In this subsection, we briefly evaluate the performance of the revised scheme during the two IOPs of ARM95 and TWP06. The observed and simulated time series of precipitation are given in Figure 2. For both the ARM95 and TWP06 cases (top and bottom rows in Figure 2, respectively), the simulations with the original ZM scheme (red lines) can capture the main precipitation events as in the observations (black lines) although some apparent model biases exist. The simulated precipitation using the revised scheme (with   0 1) (blue lines) is slightly different from that using the default scheme, but the overall model skills with the two schemes are comparable with each other. Meanwhile, both of the two schemes can realistically capture the vertical distributions of the total diabatic heating associated with precipitation ( Figures S1 and S2), with strong heating rate at levels from 700 to 200 hPa during periods when strong precipitation events are present.
In the revised scheme,  0 governs the overall strength of the convective entrainment. We find that in the two selected SCM cases, adjusting the value of  0 from 1 to 0.5 has a weak effect on the precipitation simulations based on the time series ( Figure S3). Figure 3 shows the observed and simulated diurnal variations of precipitation averaged during the IOP of ARM95. We can see that the observed precipitation (black line) peaks at 15 Local Solar Time (LST). By contrast, the experiment with the original scheme produces two peaks at 13 LST and 20 LST (red line), respectively, with the first peak mainly contributed by convective precipitation (figure not shown) (because the CAPE-type closure often simulates a too-early onset of convection; Zhang, 2002). When using the revised scheme, the simulated precipitation (blue lines) peaks at around 18 LST, later than that in observation. Decreasing the value of  0 from 1.0 (blue dashed line) to 0.5 (blue solid line) tends to make convection initiate earlier.
In Figure 4 we examine the impacts of    (Table 2). In the following subsections, the analyses are mainly based on the GCM experiments.

Precipitation Climatology
In Figure 5 we evaluate the annual mean precipitation from simulations (averaged over the last 10 years) with different convective schemes against the TRMM observation (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010). Compared to the TRMM result (Figure 5a), clear biases are seen in the CTL experiment with the original scheme (Figure 5b), such as the underestimated precipitation over the TWP and the Bay of Bengal (BOB), as well as the overestimated precipitation over the southeastern Pacific (i.e., the double-ITCZ problem). Over East Asia, the CTL experiment underestimates the midlatitude storm track precipitation over oceans to the south and east of Japan. Meanwhile, it underestimates (overestimates) the precipitation over the southern coastal (inland) regions of China, failing to capture the south to north and coastal to inland pattern of decreasing precipitation there. In the REV experiment with the revised scheme (Figure 5c), the above precipitation biases are all reduced to some degree, making the precipitation simulations agree more with the observation in terms of both magnitude and spatial pattern in many regions. For example, the double-ITCZ problem is less obvious, which can also be clearly seen in the meridional distribution of precipitation averaged over the eastern Pacific (160ºW-90ºW) ( Figure 6). However, the northern ITCZ is overestimated by using the revised closure (blue line in Figure 6). When turning off the ULL option in the revised scheme (i.e., the REV_ULL0 experiment), the results (Figure 5d) are generally consistent with those in REV, suggesting that the improved precipitation climatology in REV compared to that in CTL is mainly due to the modifications related to the mass-flux closure and the convection feedback effects, with the two individual impacts discussed in Section 4.6.
We further compare the simulated precipitation averaged in June-July-August (JJA) and December-January-February (DJF) against the observation (Figure 7). In JJA, large precipitation amounts are observed over the BOB, eastern equatorial Indian Ocean (EIO), northern-ITCZ, and TWP (Figure 7a). There exists a clear rain-belt from China to Japan, which is associated with the East Asian summer monsoon (EASM). YANG ET AL.  The CTL experiment (Figure 7c) fails to capture the strong precipitation over the TWP, with the rainfall center shifted southward to the Maritime Continent compared to the observation. Meanwhile, the EASM precipitation is too weak. In addition, the CTL experiment underestimates (overestimates) the precipitation over the eastern (western) EIO, exhibiting an Indian Ocean dipole-like bias that is also seen in many other GCMs (Li et al., 2015). The REV experiment with the revised scheme better simulates the precipitation distribution over the broad western Pacific region, including the EASM region, Maritime Continent, and TWP ( Figure 7e). However, the underestimated precipitation over the eastern EIO becomes more apparent which deserves further investigations.
In DJF, large precipitation amounts are observed over the Maritime Continent, the northern-ITCZ, and the South Pacific Convergence Zone (SPCZ) (Figure 7b). A strong rainfall center is also seen in the Amazon region. The CTL experiment underestimates the SPCZ precipitation, especially in the areas to the east of YANG ET AL.  Australia (Figure 7d). This bias is alleviated to some extent in the REV experiment ( Figure 7f).
Overall, the above results indicate that the revised ZM scheme can realistically capture the main features in the mean state of precipitation and perform better than the original scheme in many aspects.

Seasonal Variation of Precipitation
Rainfall centers experience strong south-north oscillation following the seasonal variation in SST that is primarily driven by the solar radiation. Previous studies revealed that the strength of the double ITCZ varies with seasons (Zhang et al., 2019). Figure 8 shows the month-latitude distributions of precipitation over the eastern Pacific (160°W-90°W) in both the TRMM observation and simulations. In the observation (Figure 8a   its northernmost and southernmost location in September and March, respectively. Compared to the northern component, the precipitation in the southern oceans is much weaker, with its maximum magnitude around 4 mm day −1 from March to April. The CTL experiment (Figure 8b) can reasonably simulate the seasonal variation of precipitation in the northern ITCZ, but largely overestimates the southern ITCZ precipitation in March and April, consistent with the results shown in Figures 5 and 6. By contrast, the intensity of the southern ITCZ is markedly reduced in the REV experiment (Figure 8c). The result from REV_ULL0 is not presented here because it is largely consistent with that from REV.
The seasonal migration of precipitation is a prominent phenomenon over the monsoon regions. Around the globe, East Asia is a distinct region where the precipitation climate is dominated by the evolution of monsoon. Figure 9 presents the month-latitude distributions of precipitation over East Asia and its adjacent ocean areas (105°E-130°E). In the observation (Figure 9a), the precipitation maximum is located south of the equator from December to April and gradually migrated to 20°N from May to August, corresponding to the onset of the Western North Pacific summer monsoon. Then, it retreats back to the south. A separate rain-belt forms in May between 20°N and 30°N and extends to 40°N in July and August, corresponding to the northward advance of the EASM precipitation. The CTL experiment (Figure 9b) produces large biases in the seasonal variation of precipitation over East Asia, especially to the south of 20°N. For example, the precipitation maximum is always located around the equator throughout the year, consistent with the too weak precipitation over TWP in JJA (Figure 7c). Meanwhile, the EASM precipitation is largely underestimated, particularly in July and August. In contrast, the REV experiment (Figure 9c) can well capture the seasonal variation of precipitation in the tropical oceans and better simulate the EASM precipitation in terms of both mean magnitude and seasonal migration. Figure 10 shows the time lag-longitude distributions of correlation coefficients of the intraseasonal component of precipitation averaged from 10°S to 10°N with that averaged in two reference regions, i.e., the Indian Ocean region (70°E−90°E, 5°S-5°N; left column of Figure 10) and the western Pacific Ocean (130°E−150°E, 5°S-5°N; right column of Figure 10), which are often applied to characterize the MJO activity (e.g., Boyle et al., 2015). The plots are derived based on the precipitation data from November to April when the MJO activity is the most prominent. In the observation (Figures 10a and 10b), we can see a notable eastward propagation of the MJO precipitation anomaly from Indian Ocean to the western Pacific Ocean. However, the CTL experiment (Figures 10c and 10d) simulates a westward propagation of the precipitation anomalies, opposite to that in the observation. The MJO simulation in REV (Figures 10e and 10f) is improved to some degree compared to that in CTL, although the eastward propagation is still less obvious than in the observation. We consider that the low skill of the MJO simulation in CTL is largely because convection often occurs too easily in the original ZM scheme. Previous studies have shown that tuning several key parameters in the convective scheme can considerably improve the MJO simulation (Boyle et al., 2015;Liu et al., 2019). Our preliminary test experiments also indicate that the MJO simulation in REV can be further improved when the overall intensity of convection is suppressed via using a different value for  0 or a in

Precipitation Diurnal Variation
Many previous studies have demonstrated the important roles of the transition from shallow to deep convection in regulating the precipitation diurnal variations (Rio et al., 2009;Yang et al., 2020). Figure 11 shows the spatial patterns of the diurnal peak time (i.e., the hour corresponding to the daily-maximum precipitation) for the annual mean precipitation in the TRMM observation and different simulations. In the TRMM (Figure 11a), precipitation often peaks at around 17 LST over land, except for some subtropical and midlatitude areas, such as those adjacent to the Tibetan Plateau (TP) over East Asia and east of Rocky Mountains over North America, where nighttime precipitation maximum can be observed as revealed in previous studies (e.g., Dai et al., 1999). Over oceans, precipitation generally peaks during nighttime or early morning.
All the simulations with different convective schemes can well capture the nighttime precipitation peaks over oceans as in the observation (Figures 11b-11d). However, large biases and inter-model differences are found over land. In the CTL experiment (Figure 11b), precipitation over tropical land often shows noontime diurnal peaks, which are much earlier than that in the observation. Meanwhile, the CTL experiment fails to produce the nighttime precipitation peak over regions east of TP or Rocky Mountains.
In the REV experiment (Figure 11c), adopting the revised scheme improves the simulations of the precipitation diurnal features over many tropical land regions, with the peak time delayed from 12 to around 15 LST which is still about 2-3 h earlier than that in the observation. Moreover, the revised scheme can reproduce the nighttime peaks of precipitation over East Asia and North America. However, the areas with nighttime peaks are much wider than in the observation. In the REV_ ULL0 experiment with the ULL option turned off (Figure 11d), the nighttime precipitation peaks are still clearly seen over East Asia and North America but the associated areas are largely reduced compared to the REV simulations. However, the simulated diurnal peaks over the tropical land are too early, similar to that in CTL. These results indicate that both the ULL configuration and the modifications related to the mass-flux closure and convection feedback impacts can affect the simulations of the precipitation diurnal variations.
We further evaluate the simulated diurnal variations of the summertime mean (i.e., JJA) precipitation over the US (Figure 12), which is usually hard to represent by the traditional convective parameterizations (Xie et al., 2019). In the TRMM (Figure 12a), precipitation is featured by late-afternoon diurnal peaks over the western and southeastern parts of the US and nighttime peaks over the Great Plains, exhibiting a propagating feature of the diurnal peak time east of Rocky Mountains. The CTL experiment with the original ZM scheme (Figure 12b) well-captures the late-afternoon precipitation peaks over the western and southeastern regions but fails to simulate the nighttime peaks over the Great Plains. By contrast, in the REV experiment with the revised scheme (Figure 12c), the spatial distribution of the diurnal peak time is improved to a large degree. In REV_ULL0 (Figure 12d), the nighttime peak is also seen but only over a limited region east of Rocky Mountains. We also conduct a 11-year experiment using the original ZM scheme but with the ULL option turned on, in which the nighttime diurnal peaks are better simulated but still weaker than in the TRMM (figure not shown). These results confirm that different modifications to the ZM scheme can individually affect the simulation of the precipitation diurnal variation over this region. YANG ET AL.

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13 of 23   (Figure 13a), precipitation shows a strong diurnal peak at around 18 LST over Rocky Mountains (i.e., west of 105°W). To the east (i.e., 105°W-90°W), the diurnal peak time gradually shifts from late afternoon (i.e., 18 LST) to early morning (i.e., 05 LST). The CTL experiment (Figure 13b) fails to simulate the propagation. By contrast, the REV experiment (Figure 13c) with the revised scheme well-captures such feature, consistent with the results shown in Figure 12c. In REV_ULL0, the shift in the precipitation diurnal peak time is also partly reproduced but the propagation signal disappears at around 102°W.
YANG ET AL. In Figure 14 we compare the simulated diurnal variations of precipitation over the EASM region against the TRMM and rain-gauge records. Previous studies have revealed that the EASM precipitation is characterized by complex diurnal features (Yu et al., 2007). The distributions of the diurnal peak time are different in the TRMM and in the rain-gauge records (Figure 14a vs. Figure 14e). For example, the rain-gauge precipitation over China is featured by late-afternoon peaks over the southeastern coastal areas and nighttime peaks over most inland areas to the south of 40°N. While in the TRMM, the nighttime precipitation peak is rarely seen to the east of 110°E. The convective scheme has strong impacts on the precipitation diurnal variation simulations over East Asia (Figures 14b-14d). The CTL experiment (Figure 14b) simulates a too early peak over southeastern China and fails to capture the nighttime peaks to the north. By contrast, the revised scheme (Figure 14c) slightly delays the late afternoon peaks over southeastern China. It can also reproduce the nighttime peaks but over a much wider region than in the observations (Figures 14a and 14e).  The simulated precipitation diurnal features in REV_ULL0 are generally consistent with that in CTL, except that REV_ULL0 performs better in capturing the nighttime precipitation peaks over the Sichuan Basin (around 105°E and 30°N).
The simulated propagation of the precipitation diurnal feature east of TP is also compared against observations in Figure 15. In both the TRMM (Figure 15a) and rain-gauge records (Figure 15e)  diurnal peak gradually shifts from around 20 to 06 LST downstream of TP (i.e., 100°E−110°E). As revealed previously (e.g., Yang et al., 2020), the original ZM scheme has difficulty in simulating the propagating feature and nighttime peaks along the regions east of TP (Figure 15b). Notably, the REV experiment performs much better in this regard (Figure 15c). The REV_ULL0 experiment can also well-capture the propagating feature of the precipitation diurnal peak there (Figure 15d).

Impacts of Convection Feedback
In the above subsections we have demonstrated the benefits of using the revised ZM scheme in simulating the mean state and variability of precipitation. However, it is unclear how the two modifications related to the mass-flux closure (i.e., Section 2.1) and the convection feedback impacts (i.e., Section 2.2) contribute to the improved model behaviors individually. Here, an additional experiment REV_CF0 with the convection feedback effects turned off is used to further understand the individual impacts of the two components. Figure 16 shows the annual mean precipitation distribution simulated in REV_CF0, which is generally consistent with that in REV (Figure 5c) in terms of both magnitude and spatial pattern, except that the YANG ET AL.  precipitation amount over Amazon is smaller in REV_CF0 than in REV. The precipitation distributions averaged in JJA and DJF are also largely consistent between the two simulations ( Figure S5 vs. Figures 7e and 7f), indicating that the improved mean state of precipitation in REV is mainly contributed by the modification related to the mass-flux closure. Figure 17 shows the lag-longitude distributions of correlation coefficients of band-pass filtered (20-90 days) precipitation (averaged for 10°S-10°N) with precipitation averaged over the Indian Ocean region and the western Pacific region in REV_CF0. We can see that in REV_CF0 the simulated MJO precipitation is improved to a large degree compared to that in CTL (Figures 10c and 10d) and even better than that in REV (Figures 10e and 10f). We find that the overall convection intensity is weaker in REV_CF0 than in REV although a smaller value of  0 is applied. The relatively weaker convection in REV_CF0 is likely responsible for the better-simulated MJO precipitation.
It has been suggested that the convection feedback impacts are of vital importance to the simulations of precipitation diurnal cycle across various climate regimes. In Figure 18 we show the spatial distribution of the diurnal peak times for the JJA mean precipitation over the US simulated by REV_CF0. Notably, with the convection feedback impacts turned off, the model can still capture the nighttime peaks of precipitation downstream of Rocky Mountains and thus the propagating feature there. However, the simulated diurnal peaks are too early over the southeastern region of the US. Similarly, the REV_CF0 experiment also fails to capture the late-afternoon peaks of precipitation over southeastern China ( Figure S6   and the convection feedback impacts need to be explicitly or implicitly parameterized in climate models.

Summary and Discussions
In this study, we modified the closure of the ZM convective parameterization in CAM5 and coupled it with CLUBB to provide a unified treatment for sub-grid vertical motions across regimes of boundary-layer turbulence, shallow convection, and deep convection. Instead of using CAPE as a closure for the ZM scheme, by assuming a lognormal distribution to represent the spectrum of entrainment rates across the entire moist convective regimes, we link mass flux for deep convection to that of shallow convection, which is determined by the sub-grid information of vertical velocity in CLUBB. The modified mass-flux closure improves the consistency between shallow and deep convection, as well as between the deep convective closure and updraft plumes in the ZM scheme. In addition, the convection feedback effects were considered to decrease the entrainment spectrum width and enhance the vertical velocity variability that further affect deep convection. We also included a ULL concept that allows the ZM scheme to search the convection launch level above the boundary-layer height so as to improve the precipitation diurnal variation.
First, the performance of the revised ZM scheme was evaluated with two SCM experiments, in which the simulated precipitation and associated atmospheric heating profiles are generally consistent with that using the original ZM scheme and with the observation. Then, we focused on the global simulations of precipitation in terms of the mean state and variability at various timescales. The revised scheme greatly improves the precipitation climatology, such as the alleviated double-ITCZ problem, and better-simulated south-north distributions of precipitation over the TWP and East Asia. Meanwhile, the seasonal variations of precipitation associated with the Western North Pacific summer monsoon and EASM are improved to a large degree. The MJO precipitation is also improved by using the revised scheme.
The revised scheme can considerably improve the precipitation diurnal variations over land. For example, the diurnal peak time of precipitation is delayed from 12 to around 15 LST, in better agreement with the observation (i.e., 17 LST). Meanwhile, the revised scheme can realistically capture the propagating features of the precipitation diurnal variation downstream of TP and Rocky Mountains. When turning off the ULL option in the revised scheme, the model fails to capture the late-afternoon precipitation peaks over the tropical land, but still shows some skills in simulating the propagating features downstream of TP and Rocky Mountains, indicating that both the ULL configuration and the modifications related to the massflux closure and the convection feedback impacts can affect the precipitation diurnal variation simulations.
An additional experiment turning off the convection feedback effects was conducted to investigate how the modified mass-flux closure and the convection feedbacks contribute to the improved model behaviors individually. It showed that without the convection feedback impacts the improvements can still be seen in many aspects, such as the better-simulated precipitation climatology, MJO, and nighttime precipitation diurnal peaks downstream of Rocky Mountains, emphasizing the benefits of using the revised mass-flux closure. However, the simulated diurnal peaks are often too early over regions with late-afternoon precipitation peaks, likely because the convection feedback impacts are important for the gradual transition from shallow to deep convection over these regions.
Overall, in this study we proposed to modify the ZM scheme that couples deep convection to shallow convection based on the sub-grid information in CLUBB, which helps improve the consistency between shallow and deep convection. The convection feedback impacts were also considered but large uncertainty is expected in the formulations (Colin et al., 2019). Further studies are needed to better characterize such effects using high-resolution model simulations. In addition, the organization of convection at the mesoscale was not included, which could potentially be parameterized via using the sub-grid atmospheric variability (i.e., humidity contrast between moist and dry regions) provided by CLUBB or produced by convection itself (e.g., Wing et al., 2017). YANG ET AL.