
The simulation of clouds has been a major source of uncertainty in projections of future climate using general circulation models (GCMs) (Stephens, 2005; Li et al., 2009; Bony et al., 2015). One limitation of cloud simulation is the coarse spatial resolution of GCMs (tens of kilometers to 100–200 km), which leaves clouds smaller than grid size unresolved (Barker et al., 2003; Randall et al., 2003). Consequently, clouds in GCMs usually cover only part of a grid layer and the overlap of fractional clouds in the vertical layers has to be addressed artificially in radiation calculations by imposing overlap assumptions (Tompkins and Di Giuseppe, 2015; Zhang and Jing, 2016). For a given vertical distribution of cloud fractions, the assumption of cloud overlap determines the total cloud cover or total cloud fraction (C_{tot}), which has a considerable influence on solar and terrestrial radiative transfer (Wang et al., 2016).
The cloud overlap assumption most widely used in recent decades is the maximum random overlap (MRO) assumption (Morcrette and Fouquart, 1986; Tian and Curry, 1989), in which clouds within layers that are vertically continuous are assumed to have a maximum overlap, whereas those that are separated by cloudfree layers are considered to overlap randomly. Such treatment is insufficient to represent the realistic features of cloud overlap as observed by groundbased radar (Hogan and Illingworth, 2000; Mace and BensonTroth, 2002) and depends largely on the vertical resolution of the host model (Bergman and Rasch, 2002).
In contrast with the simple, crude cloud overlap treatments such as the MRO assumption, Liang and Wang (1997) were among the first to explicitly depict the subgrid distribution of clouds of distinct physical types and to apply different treatments of vertical overlap for different types of clouds. This sophisticated, physically based treatment of cloud overlap (termed “mosaic”) has been demonstrated to improve cloud radiative forcing and radiative heating in both cloudresolving model (CRM) domains (Liang and Wu, 2005; Wu and Liang, 2005a, b) and climate simulations (Zhang F. et al., 2013).
Another ingenious approach is the analytical representation of cloud overlap proposed by Hogan and Illingworth (2000) and Mace and BensonTroth (2002) based on radar observations. This method is called general overlap (GenO). In GenO, for two layers of clouds at heights of Z_{k} and Z_{l} with cloud fractions of C_{k} and C_{l}, respectively, C_{tot} is defined as
$$ {C_{\rm {tot}}} = {\alpha _{k,l}}C_{k,l}^{\max } + \left( {1  {\alpha _{k,l}}} \right)C_{k,l}^{{\rm{ran}}}, $$ (1) where
$C_{k, l}^{\max } = \max \left( {{C_k},{C_l}} \right)$ and$C_{k,l}^{{\rm{ran}}} = {C_k} + {C_l}  {C_k}{C_l}$ are the total cloud fractions calculated using the maximum overlap and random overlap assumption, respectively, and${\alpha _{k,l}}$ is the overlap coefficient derived from$$ {\alpha _{k,l}} = \exp \left( {  \int_{{z_k}}^{{z_l}} {\frac{1}{{{L_{{\text{cf}}}}}}{\text{d}}Z} } \right). $$ (2) L_{cf} is the decorrelation length (in km) representing the distance at which
${\alpha _{k,l}}$ decreases to the value of e^{–1}. With Eqs. (1) and (2), the extent of overlap degrades exponentially from maximum overlap to random overlap as the vertical separation of clouds increases. This relationship of decreasing overlap with increasing distance has been reported from both radar observations and simulations by CRMs (Oreopoulos and Khairoutdinov, 2003). The merits of GenO are twofold: (1) it realistically depicts the distancerelated feature of cloud overlap and (2) it is independent of the vertical resolution of the host model and thus more widely applicable among models with various vertical configurations.In GenO, the extent of cloud overlap is determined by L_{cf}. For given fractional clouds in a vertical column, the use of larger values of L_{cf} results in smaller values of C_{tot} (prone to maximum overlap) and smaller values of L_{cf} result in larger values of C_{tot} (prone to random overlap). The parameter L_{cf} is highly variable both spatially and temporally because of variations in the shapes and formation processes of clouds. Therefore, when applying GenO, one challenge is to determine an optimum value of L_{cf} for each GCM grid point. Various attempts have been made to obtain detailed information about L_{cf} (e.g., Di Giuseppe, 2005; Kato et al., 2010; Shonk et al., 2010; Oreopoulos et al., 2012; Peng et al., 2013; Zhang H. et al., 2013). It has been demonstrated that L_{cf} is related to the cloud type and atmospheric dynamics (Naud et al., 2008; Di Giuseppe and Tompkins, 2015; Li et al., 2015) and that it has a global median value of approximately 2 km (Barker, 2008). Simplified expressions have also been extracted to represent L_{cf} in GCMs, either as a function of latitude and/or season without interannual variations (Shonk et al., 2010, Oreopoulos et al., 2012; Jing et al., 2016) or as a function of cloud type, which is affected by the limited cloud classification schemes of the host models (Zhang et al., 2014). These approaches either lack a direct link between L_{cf} and the instant largescale meteorological conditions that foster the clouds or address the dynamic (e.g., wind shear) impact on cloud overlap over the globe without considering the very different circulation conditions in different regions.
The formation and evolution of clouds are essentially associated with largescale circulation (Bony et al., 1997). Therefore one physically robust approach to describe L_{cf} is to establish a direct connection between L_{cf} and largescale circulation conditions. CRMs, because of their ability to simulate cloud micro and macrophysical structures as well as meteorological conditions in detail, have long been used as a tool to explore cloud physics and to obtain parameterizations applicable in GCMs (GEWEX Cloud System Science Team, 1993; Randall et al., 2003; Wu and Li, 2008). This study uses simulation results from a global CRM to explore the relationship between L_{cf} and atmospheric circulation. Unlike previous studies that attempted to explore such a relationship over the whole globe, we focus on the tropical region and vertical motion only, considering that there are large uncertainties in cloud radiative forcing due to vertical overlap treatment in the intertropical convective zone (ITCZ) (Barker and Räisänen, 2005; Zhang and Jing, 2010; Lauer and Hamilton, 2013) and that the formation and maintenance of clouds in this particular region are closely related to vertical convection. We will attempt to establish a statistical, mathematical description of the L_{cf}–convection connection, which is a novel application in GCMs, and then evaluate its effectiveness in improving the GCMscale cloud cover and radiation calculations.
The CRM data and the method used to derive L_{cf} are described in Section 2. Section 3 presents the analysis of L_{cf} versus atmospheric convection and evaluates the established representation of L_{cf}. The discussion and conclusions are presented in Section 4.

The vertical velocity in the midtroposphere (w_{500}) has been shown to be a representative indicator of tropical convection and cloud radiative forcing (Ichikawa et al., 2012). The relationship between L_{cf} and w_{500} is assessed in this subsection. As the overlap of clouds with very large or very small cloud fractions is of secondary importance for C_{tot} and radiation calculations, grids with maximum layer cloud fractions > 0.9 or < 0.1 are discarded.
Figure 4 shows the distributions of L_{cf} and w_{500} for each model day. Domains with a large and positive w_{500} (e.g., the western Pacific and South America) mostly have a large value of L_{cf} (typically 4–7 km and up to 10 km in extreme cases). By contrast, in domains with a small or negative w_{500}, L_{cf} is mostly about or below 2 km. Figure 5 shows the pattern correlation between the geographical distributions of L_{cf} and w_{500} for each snapshot. The pattern correlation stays at a moderate, but notable and constant level (from 0.61 to 0.66), implying a physically close association between L_{cf} and w_{500}.
Figure 4. Distributions of w_{500} (lefthand panels) and L_{cf} (righthand panels) in the tropics for each model day. Masked grids are those with maximum cloud fraction in the vertical direction of > 0.9 or < 0.1 and were thus eliminated from the analysis.
Figure 5. Pattern correlation between L_{cf} and w_{500} for each snapshot shown in Fig. 4.
Based on pattern similarity and the clear difference between areas of ascent (w_{500} > 0) and descent (w_{500} < 0), the relationship between L_{cf} and w_{500} will be explored separately for areas of ascent and descent.
Figure 6 shows the statistics for L_{cf} and w_{500} and their relationships in the ascending areas. A total of 4926 samples were used to derive these statistics. The gray lines in Fig. 6a show the median and the first and third quartiles of L_{cf} for each bin of L_{cf}. These lines clearly show a positive relationship between L_{cf} and w_{500} for areas of ascent: when w_{500} < 0.02 m s^{–1}, the most frequent occurrence of L_{cf} is around 2 km; as w_{500} increases to 0.08 m s^{–1}, the corresponding value of L_{cf} increases to as much as 6–8 km, although the occurrence probability of w_{500} > 0.08 m s^{–1} is very small (Fig. 6b). Linear regression is conducted (as shown by the black solid line and the regression equation in Fig. 6a) for L_{cf} as a function of w_{500}. The regression line captures very well the relationship illustrated by the yellow shaded area in Fig. 6a. The 95% confidence interval (blue dotted lines) and 95% prediction interval (red dashed lines) for this regression are also shown in Fig. 6a. The small 95% confidence interval (the mean of L_{cf} is 95% likely to fall into this interval for a given value of w_{500}) suggests that the linear regression is an excellent representation of the average relationship between L_{cf} and w_{500}; however, it should also be noted that the dispersion of the L_{cf} and w_{500} relationship is relatively large, as indicated by the 95% prediction interval (an individual L_{cf} for a given w_{500} falls into this interval with probability of 95%). The large dispersion of the L_{cf} and w_{500} relationship may stem primarily from the effect of other meteorological conditions on cloud overlap, which cannot be captured by a simple linear regression. Nevertheless, these results demonstrate that, in the statistical sense, L_{cf} and w_{500} are linearly related in the ascending regions of the tropics. The analysis in the following sections shows that addressing this statistical relationship will benefit the calculation of the cloud fraction and radiation fields.
Figure 6. Statistics of L_{cf} and w_{500} constructed for the whole 7day period. (a) The median (gray dashed line) and first and third quartiles (gray solid lines) of L_{cf} for each w_{500} bin (bin width 0.01 m s^{–1}) and the linear regression of L_{cf} as a function of w_{500} (black solid line) (the linear regression equation and correlation coefficient are also shown). The yellow shaded area is the interquartile range of L_{cf}; the blue dotted line and red dashed line are the 95% confidence interval and 95% prediction interval of the regression, respectively. (b) Probability distribution function of w_{500}.
The same analysis for areas of descent shows that L_{cf} changes very little with w_{500} (data not shown) and that a value of 2 km is a good representation for these areas regardless of the specific w_{500}. Consequently, L_{cf} can be approximated in the region 30°S–30°N depending on w_{500} as:
$$ {L_{{\rm{cf}}}} = \left\{ \begin{array}{l} 56.054 \times {w_{500}} + 1.951,\;\;\;{\rm{for}}\;\;{w_{500}} \geqslant 0.0,\\ 2.0,\qquad \qquad \qquad \qquad {\rm{for}}\;\;{w_{500}} < 0.0. \end{array} \right. $$ (3) It should be noted that although GenO is independent of the vertical resolution, Eq. (3) may be affected by the vertical resolution of the NICAM data. Thus we should be cautious when applying Eq. (3) in a model with a notably different vertical resolution from that of NICAM, especially in the troposphere.

The better representation of L_{cf} is primarily expected to be capable of generating a realistic C_{tot} for given cloud fraction profiles. We evaluate here the generated C_{tot} by using the L_{cf} parameterized from Eq. (3) (denoted as PARA). Two other cloud overlap treatments are also included to compare with PARA: the traditional MRO assumption, which has been widely used not only in GCMs (Räisänen, 1998; Collins, 2001), but also in model assessment tools such as the CFMIP Observation Simulator Package (BodasSalcedo et al., 2011); and the simplified GenO using a constant L_{cf} (2 km) (denoted as G2KM), which applies the framework of GenO, but with a constant L_{cf} suggested as a global mean value (Barker, 2008). These cloud overlap treatments are used to generate subgrid cloud fields from the cloud fraction profiles of the CRM and the values of C_{tot} of the generated fields are then compared with those of the original CRM fields.
Figure 7 shows the 7day mean biases of C_{tot} generated from the three cloud overlap representations relative to the reference value. It can be seen that the widely used MRO assumption remarkably underestimates C_{tot} in most regions. This is because there are few clear layers to separate the cloud layers in the vertical columns and therefore there is a maximum overlap of clouds in most cases. This demonstrates one drawback of the MRO technique: it depends greatly on the vertical resolution of the host model and is therefore usually nonequivalent among models.
Figure 7. Biases of generated C_{tot} using (a) the MRO assumption, (b) GenO with universal L_{cf} = 2 km (G2KM), and (c) GenO with dynamic representation of L_{cf} (PARA) relative to the true C_{tot} from the CRM (REF). Contour lines are shown for ±0.06.
The use of GenO significantly reduces the negative biases compared with the MRO assumption, even when a constant L_{cf} of 2 km is used (Fig. 7b). However, G2KM leads to notable C_{tot} biases in the ITCZ, especially in the western Pacific and Amazon regions, where clouds are generally more vertically organized because of systematic largescale upward motion. When the dynamic representation of L_{cf} for regions of ascent is used, the cloud fraction errors in the ITCZ are remarkably reduced (Fig. 7c).

Radiation calculations are performed for the generated cloud fields and the original CRM cloud fields in the evaluation using the BCC–RAD correlatedk distribution radiation model (Zhang et al., 2003, 2006a, b ), which has been implemented in the GCM of the Beijing Climate Center (BCC_AGCM2.0.1) (Zhang et al., 2014). For both the generated and original cloud fields, the cloud water/ice content in each GCM grid is obtained as a grid mean value to eliminate the effect of the horizontal distributions of cloud water/ice. Because only the cloud and dynamic output from NICAM are stored (due to the large amount of data), we apply in the radiation calculation the atmospheric pressures, temperatures, and gas concentrations from the tropical atmosphere of the US Air Force Geophysics Laboratory atmospheric models (Anderson et al., 1986) and the broadband surface albedos for the ocean (0.08, Jin et al., 2004) and solid surfaces (0.28, Liang, 2001), respectively.
Figure 8 shows the biases in the generated net longwave (LWTOA) and shortwave (SWTOA) radiative fluxes at the top of the atmosphere relative to the CRM results. The largest radiation biases occur in the ITCZ, where the largest C_{tot} biases are also seen. The MRO assumption shows significant negative biases for LWTOA and positive biases for SWTOA (mostly > 10 and > 25 W m^{–2}, respectively) (Figs. 8a, b) due to the underestimation of C_{tot}. Figures 8c and 8d show that the use of GenO with a constant L_{cf} of 2 km reduces the biases in subtropical regions, but also introduces notable biases in the ITCZ with opposite signs to the biases of the MRO assumption. The dynamic representation of L_{cf} largely reduces the negative biases in the ITCZ (Figs. 8e, f), with considerably fewer regions having absolute biases > 10 and > 25 W m^{–2} for LWTOA and SWTOA, respectively.
Figure 8. Biases of generated LWTOA (lefthand panels) and SWTOA (righthand panels) using (a, b) the MRO assumption, (c, d) GenO with L_{cf} = 2 km universally, and (e, f) GenO with dynamic representation of L_{cf}, relative to those calculated directly from the CRM fields. The downward direction is defined as positive. The contour lines for ±10 and ±25 W m^{–2} are shown for LWTOA and SWTOA, respectively.
Figure 9 shows the biases in the net longwave (LWSFC) and shortwave (SWSFC) radiative fluxes at the surface. By comparing Fig. 9 with Fig. 8, it is seen that the main features of the biases at the surface are similar to those at the TOA, except that G2KM and PARA have much smaller (larger) biases in LWSFC over (outside) the ITCZ. PARA has the smallest errors among the three treatments of cloud overlap over the ITCZ for both LWSFC and SWSFC. It is therefore evident that the dynamic representation of L_{cf} yields the best spatial patterns of the radiation fields.
Figure 9. As in Fig. 8, but for (a, c, e) LWSFC and (b, d, f) SWSFC.
Figures 10 and 11 are scatter diagrams comparing the generated and reference radiation fields at the TOA and surface, respectively. Systematically negative biases in LWTOA and positive biases in SWTOA are shown for the MRO assumption (Figs. 10a, d) and the opposite for G2KM (Figs. 10b, e). PARA shows little systematic bias in LWTOA and SWTOA—that is, the points are distributed more symmetrically around the reference lines (Figs. 10c, f). Similar features are also shown at the surface (Fig. 11), except that G2KM and PARA resemble each other for the LWSFC (Figs. 11b, c), consistent with their similarity in Fig. 9.
Figure 10. Comparisons between LWTOA and SWTOA calculated from generated cloud fields with different cloud overlap treatments (MRO, G2KM, and PARA) and the values calculated directly from the CRM fields (REF) for the tropical areas.
Figure 11. As in Fig. 10, but for (a, b, c) LWSFC and (d, e, f) SWSFC.
Figure 12 shows the tropicalaveraged radiation biases for all areas (solid fill) and areas of ascent only (hatched fill). As expected, the MRO assumption shows the largest errors relative to the reference, especially for the shortwave fluxes (about 16 W m^{–2} at both the TOA and the surface). G2KM performs much better than the MRO assumption, with absolute errors of approximately 2 W m^{–2} for LWTOA and SWTOA. PARA reduces the errors more significantly, especially for shortwave fluxes. The error in LWSFC of PARA is more negative than that of G2KM (Fig. 12c); this is because PARA reduces the positive errors in the ITCZ (as shown in Fig. 9) that compensate for negative errors in other regions. It should be stressed that, for most of these variables, PARA reduces the allarea mean errors as effectively as it does in the areas of ascent only, implying that unrealistic cloud overlap treatment in areas of ascent is a major source of radiation error in tropical areas. Considering that these areas typically have large radiation biases in GCM simulations (Lauer and Hamilton, 2013), the introduction of PARAlike overlap treatment could possibly reduce the uncertainty in tropical radiation calculations.
Figure 12. Tropical mean biases in (a) LWTOA, (b) SWTOA, (c) LWSFC, and (d) SWSFC for generated clouds fields with different cloud overlap treatments (MRO, G2KM, and PARA) compared with those for the CRM fields (REF) over all areas (solid fill) and areas of ascent only (hatched fill).
The treatment of vertical cloud overlap also influences the radiative heating rate in the atmosphere, a property important for atmospheric stability and circulation. Figure 13 compares the effects of different overlap treatments on the ascending area mean longwave (LWHTR) and shortwave (SWHTR) heating rates. The LWHTRs of different overlap treatments are similar to each other in the middle to higher troposphere, but near the surface the MRO assumption shows remarkable longwave cooling (Fig. 13b) and G2KM and PARA show overestimated longwave heating. This is probably due to the underestimated (overestimated) cloud fraction of the MRO assumption (G2KM and PARA) over these regions (as shown in Fig. 7). For the SWHTR, the MRO assumption underestimates heating around altitudes of 2 and 10 km, whereas G2KM overestimates heating around 10 km height and underestimates heating in the lower troposphere. PARA has a similar SWHTR to G2KM, but the bias in the upper troposphere is remarkably reduced and the bias in the lower troposphere is also reduced to some extent. These changes in the radiative heating rates caused by applying PARA, when coupled with a circulation model, will exert an influence on atmospheric stability in the vertical direction and consequently change the dynamic circulation. These aspects will be explored in future studies.
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Jing, X. W., H. Zhang, M. Satoh, et al., 2018: Improving representation of tropical cloud overlap in GCMs based on cloudresolving model data. J. Meteor. Res., 32(2), 233–245, doi: 10.1007/s1335101870959. 
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