PUBLICATIONS. Journal of Advances in Modeling Earth Systems

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1 PUBLICATIONS Journal of Advances in Modeling Earth Systems RESEARCH ARTICLE /2017MS Key Points: An economical analytical equation was derived for the integrated vertical overlap of cumulus and stratus The new analytical equation is much more efficient than the previously proposed heuristic version and induces no truncation error The radiation scheme incorporating the subgrid variation of water vapor substantially increases the longwave cloud radiative forcing Correspondence to: S. Park, Citation: Park, S. (2018). An economical analytical equation for the integrated vertical overlap of cumulus and Stratus. Journal of Advances in Modeling Earth Systems, 10, Received 5 OCT 2017 Accepted 26 FEB 2018 Accepted article online 2 MAR 2018 Published online 23 MAR 2018 VC The Authors. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. An Economical Analytical Equation for the Integrated Vertical Overlap of Cumulus and Stratus Sungsu Park 1 1 School of Earth and Environmental Sciences, Seoul National University, Seoul, South Korea Abstract By extending the previously proposed heuristic parameterization, the author derived an analytical equation computing the overlap areas between the precipitation (or radiation) areas and the cloud areas in a cloud system consisting of cumulus and stratus. The new analytical equation is accurate and much more efficient than the previous heuristic equation, which suffers from the truncation error in association with the digitalization of the overlap areas. Global test simulations with the new analytical formula in an offline mode showed that the maximum cumulus overlap simulates more surface precipitation flux than the random cumulus overlap. On the other hand, the maximum stratus overlap simulates less surface precipitation flux than random stratus overlap, which is due to the increase in the evaporation rate of convective precipitation from the random to maximum stratus overlap. The independent precipitation approximation (IPA) marginally decreases the surface precipitation flux, implying that IPA works well with other parameterizations. In contrast to the net production rate of precipitation and surface precipitation flux that increase when the cumulus and stratus are maximally and randomly overlapped, respectively, the global mean net radiative cooling and longwave cloud radiative forcing (LWCF) increase when the cumulus and stratus are randomly overlapped. On the global average, the vertical cloud overlap exerts larger impacts on the precipitation flux than on the radiation flux. The radiation scheme taking the subgrid variability of water vapor between the cloud and clear portions into account substantially increases the global mean LWCF in tropical deep convection and midlatitude storm track regions. 1. Introduction Some physics parameterizations used in the General Circulation Models (GCMs; e.g., radiation, cloud microphysics, and aerosol wet deposition schemes) require information on the vertical cloud overlap in each grid column. The treatment of vertical cloud overlap in many GCMs, however, is rudimentary and inconsistent vertical cloud overlap assumptions are used in different physics parameterizations. For example, in the case of the Community Atmosphere Model version 5 (CAM5; Park et al., 2014), the radiation scheme combines cumulus and stratus into a single cloud and assumes maximum vertical cloud overlap within each regime below 700 hpa (low-level), hpa (midlevel), and above 400 hpa (high-level) but random vertical overlap between the three regimes (Collins, 2001). When computing the evaporation rate of convective precipitation, the convective precipitation area is set to one for the entire cumulus layers, without taking vertical cumulus overlap into account. The stratus microphysics scheme assumes maximum vertical stratus overlap; however, the corresponding stratiform precipitation area is simply set to the maximum stratus fraction in the layers above the current layer, without taking the evaporation of stratiform precipitation and the associated decrease in the stratiform precipitation area into account. To compute the wet deposition rate of aerosols, the convective (stratiform) precipitation area is set to the vertical integral of the cumulus (stratus) fraction in the layers above, weighted by the net production rate of convective (stratiform) precipitation. For internal consistency, it is necessary that all physics parameterization schemes operate in a unified vertical cloud overlap structure. Numerous studies have been devoted to understanding the observed vertical cloud overlap. In an attempt to describe the degree of vertical cloud overlap between maximum and random overlaps, Geleyn and Hollingsworth (1979) and Hogan and Illingworth (2000) (HI2000 hereinafter) proposed an inverse exponential expression as a function of the vertical separation distance. Although other functional forms have also been suggested (e.g., Brooks et al., 2005; Neggers et al., 2011), the inverse exponential expression suggested by PARK 826

2 HI2000 has been widely used for studying vertical cloud overlap and also widely applied in operational models. Subsequent studies tested the validity of HI2000 s hypothesis with various observational and modeling analyses (e.g., Barker, 2008a, 2008b; Jakob & Klein, 2000; Mace & Benson-Troth, 2002; Naud et al., 2008; Oreopoulos & Khairoutdinov, 2003; Oreopoulos & Norris, 2011; Tian & Curry, 1989) and showed that the e-folding decorrelation length scale Dz 0, the key parameter in the formulation of HI2000, is a function of geographical location and seasons. They concluded that Dz 0 needs to be defined for individual cloud types with geographical and seasonal variations, instead of being defined as a constant for a single cloud type. Based on these observational studies, Park (2017) (hereinafter P2017) developed a vertical cloud overlap parameterization to handle the contrasting vertical overlap structures of cumulus and stratus in an integrated way. His parameterization assumes that cumulus is maximum-randomly overlapped with adjacent cumulus with a decorrelation length scale of cumulus, Dz c ; stratus is maximum-randomly overlapped with adjacent stratus with a decorrelation length scale of stratus, Dz s ; and radiation and precipitation areas are grouped into convective, stratiform, mixed, and clear areas. The classic definition of maximum-random overlap assumes that overlap is maximum between adjacent cloudy layers, and random between cloudy layers separated by clear air. Here a different definition is used, as introduced in P2017, following the decorrelation length method. The parameterization in P2017 provides a new framework to handle the vertical overlap of multiple cloud types and potentially address the characteristics of the observed vertical cloud overlap mentioned above if Dz c and Dz s are appropriately parameterized as a function of other relevant factors, such as vertical shear of grid mean horizontal wind and grid mean vertical velocity. The key ingredient of P2017 is to compute the overlap area (a K l ) between the precipitation or radiation area at the model interface (a K ) and the cloud fraction at the layer midpoint (a l ). P2017 computed a K l in a heuristic way by dividing the individual grid box into N subpixels and counting all possible ways to allocate the discretized precipitation (Na K ) and cloud pixels (Na l )tonpixels such that the assumed overlap assumption for cumulus and stratus is satisfied. Although conceptually correct, P2017 s method requires too much computation time and leads to a truncation error in association with the discretization of Na K, Na l, and Na K l into the nearest integer. In this paper, we suggest a set of new analytical equations computing a K l that is computationally efficient and does not have a truncation error. The structure of this paper is as follows: in section 2, a set of economical analytical equations computing a K l is derived. Section 3 describes the global test simulation setting in an offline mode. Section 4 provides various simulation results: (1) comparison of the simulations with the new analytical equation and the previous heuristic equation, (2) the sensitivity of global mean precipitation and radiation processes to Dz c and Dz s, and (3) the sensitivity of LWCF at the top of the atmosphere to subgrid fluctuation of water vapor within the grid layer. A summary and conclusions are provided in section Analytical Equation for the Vertical Overlap Between the Radiation or Precipitation Areas and Cloud Areas In this section, a brief summary is provided on the P2017 s vertical cloud overlap scheme. In P2017, it is assumed that each grid layer contains two types of clouds, cumulus and stratus, which are horizontally nonoverlapped within the grid layer but can be vertically overlapped between the grid layers. The fractional areas occupied by cumulus, stratus, and clear portion at the layer midpoint are a c, a s, and a r, respectively, and P 3 l51 a l51 for l5c; s; r. The precipitation or radiation areas at the model interface are a K with the superscripts K 5 C, S, M, and R reflecting convective, stratiform and mixed precipitation, and clear portion, respectively, and P 4 K51 ak 51 (see Figure 1). It is assumed that cumulus is maximum-randomly overlapped with the cumulus of the adjacent vertical layer and, similarly, stratus is maximum-randomly overlapped with the stratus of the adjacent vertical layer. The overlap areas a K l between the precipitation or radiation area a K at the top interface of an individual grid layer and the cloud area a l at the layer midpoint (see Figure 1) are a K l j max;c max;s for maximum cumulus and maximum stratus overlaps, ak l j max;c ran;s for maximum cumulus and random stratus overlaps, a K l j ran;c max;s for random cumulus and maximum stratus overlaps, and ak l j ran;c ran;s for random cumulus and random stratus overlaps, respectively. The final a K l is obtained by: h i h i a K l 5k c k s a K l j max;c max;s 1ð12k sþa K l j max;c ran;s 1ð12k c Þ k s a K l j ran;c max;s 1ð12k sþa K l j ran;c ran;s ; (1) and the weighting factors for cumulus (k c ) and stratus (k s ) overlap are parameterized as: PARK 827

3 Figure 1. Diagram illustrating the vertical overlaps of cloud and radiation or precipitation areas. The variable a l at the layer midpoint is the cloud fraction with l 5 c (cumulus), s (stratus), and r (clear portion); a K at the top interface is the radiation or precipitation area with K 5 C (convective), S (stratiform), M (mixed), and R (clear portion); and a K l is the overlap area between the radiation or precipitation area at the top interface and the cloud fraction at the layer midpoint; and a K l" is the overlap area between the radiation or precipitation area at the top interface and the cloud fraction in the layer above; and Dz is the vertical distance between two adjacent grid layers. Based on the consistency requirement, P K ak 51; P l a l51; P K ak l 5a l, and P l ak l 5a K. The upward arrow denotes the value of the adjacent upper layer, while the downward arrow marks the value at the base interface. See the text for more details. k c 5exp ð2dz=dz c Þ; k s 5exp ð2dz=dz s Þ; (2) where Dz is the vertical separation distance between two adjacent grid layers and Dz c and Dz s are the decorrelation length scales for cumulus and stratus, respectively. Using a K l, P2017 computed the precipitation areas at the base interface a K#, where the downward arrow denotes the base interface of the individual grid layer; grid mean production rates of precipitation within cumulus (P c ) and stratus (P s ), grid mean evaporation rate of precipitation in the clear portion (E r where the overbar denotes the mean value averaged over the grid box); and grid mean precipitation fluxes at the base interface f K#, using the methods described in Appendix C of P2017. Similar computations for radiation are explained in Appendix B of P2017. P2017 computed a K l j max;c max;s ; ak l j max;c ran;s ; ak l j ran;c max;s, and ak l j ran;c ran;s in a heuristic way by dividing the grid box into N subgrid pixels and counting possible ways to allocate the discretized Na K and Na l to N pixels such that the assumed vertical overlaps of cumulus and stratus are satisfied. Here any N can be chosen regardless of the vertical cloud overlaps being assumed. Although conceptually correct, P2017 s approach has a weakness in praxis. To use P2017 s method, Na l ; Na K, and Na K l should be discretized into the nearest integer, which inevitably leads to a truncation error. For example, if a c is smaller than that might frequently happen (see Figure 3a of P2017), the discretization procedure considers Na c 50, a c 5 0, and a K c 50. Although the cumulus fraction is much smaller than the stratus fraction, cumulus dominantly contributes to precipitation production in the tropics, as noted by P2017. A truncation error generated in a certain grid layer also influences the overlap computations in the layers below. Thus, the truncation error associated with the discretization may induce a nonnegligible error in the simulated precipitation. It may be possible to reduce the truncation error by increasing N, which substantially increases the computation time: the calculation of the individual a K l requires an order of N 2 operations (see Appendix A of P2017). To use the P2017 cloud overlap scheme as a practical parameterization for GCM, a K l should be computed in an efficient way without the truncation error, an issue that will be addressed hereafter. First, we defined the following variables to describe the maximum vertical overlap of cumulus and stratus, respectively, m c minða c" ; a c Þ; m s minða s" ; a s Þ; a c a c 2m c ; a s a s 2m s ; a c" a c" 2m c ; a s" a s" 2m s ; I c 12m c ; I s 12m s ; I cs 12m c 2m s ; (3) where m c (m s ) denotes the overlap area between cumulus (stratus) in the adjacent layers when cumulus (stratus) is maximally overlapped in vertical direction; a denotes the cloud fraction excluding the overlap PARK 828

4 area; the upward arrow denotes the adjacent upper layer; and I is the net fraction excluding the overlap area. Let us assume that O(a, b) is the overlap area between the areas a and b. To compute a K l 5Oða K ; a l Þ, a K is decomposed into the sum of individual overlap fractions, a K 5a K c" 1aK s" 1aK r". Then, a K l 5Oða K c" ; a lþ1oða K s" ; a lþ1oða K r" ; a lþ; (4) and based on the theory of conditional probability, it becomes Oða K c" ; a lþ5oða c" ; a l Þða K c" =a c"þ [and similarly for Oða K s" ; a lþ and Oða K r" ; a lþ], which is true because a K and a l are conditionally independent given a c", that is, Oða c" ; a l Þ are determined independently from a K c", working down layer by layer from the top. As a result, a K l 5Oða c" ; a l Þða K c" =a c"þ1oða s" ; a l Þða K s" =a s"þ1oða r" ; a l Þða K r" =a r"þ; (5) which can be written for each l5c; s, and r, as shown below: a K c 5Oða c"; a c Þ ak c" a c" 1Oða s" ; a c Þ ak s" a s" 1Oða r" ; a c Þ ak r" a r" ; a K s 5Oða c"; a s Þ ak c" a c" 1Oða s" ; a s Þ ak s" a s" 1Oða r" ; a s Þ ak r" a r" ; (6) a K r 5Oða c"; a r Þ ak c" a c" 1Oða s" ; a r Þ ak s" a s" 1Oða r" ; a r Þ ak r" a r" : The remaining work is to compute Oða l" ; a m Þ for each combination of l5c; s; r and m5c; s; r using the vertical overlap assumptions of cumulus and stratus. In the used procedure, the maximum overlap fraction (m c, m s ) is determined for each cumulus and stratus fraction, the random overlap area between the cloud fractions without the maximum overlap fraction is computed, and the random overlap area is normalized. In the case of maximum cumulus and stratus overlaps, the resulting overlap areas between the cloud fractions in the adjacent vertical layers become: Oða c" ; a c Þ5m c ; Oða s" ; a c Þ5 a s" a c ; Oða r" ; a c Þ5 a r" a c ; I cs I cs Oða c" ; a s Þ5 a c" a s ; Oða s" ; a s Þ5m s ; Oða r" ; a s Þ5 a r" a s ; I cs I cs Oða c" ; a r Þ5 a c" a r ; Oða s" ; a r Þ5 a s" a r ; Oða r" ; a r Þ5 a r" a r : I cs I cs I cs (7) In the case of maximum cumulus and random stratus overlaps, they become: Oða c" ; a c Þ5m c ; Oða s" ; a c Þ5 a s" a c ; Oða r" ; a c Þ5 a r" a c ; I c I c Oða c" ; a s Þ5 a c" a s ; Oða s" ; a s Þ5 a s" a s ; Oða r" ; a s Þ5 a r" a s ; I c I c I c Oða c" ; a r Þ5 a c" a r ; Oða s" ; a r Þ5 a s" a r ; Oða r" ; a r Þ5 a r" a r : I c I c I c (8) In the case of random cumulus and maximum stratus overlaps, they become: Oða c" ; a c Þ5 a c" a c I s ; Oða s" ; a c Þ5 a s" a c I s ; Oða r" ; a c Þ5 a r" a c I s ; Oða c" ; a s Þ5 a c" a s I s ; Oða s" ; a s Þ5m s ; Oða r" ; a s Þ5 a r" a s I s ; (9) Oða c" ; a r Þ5 a c" a r I s ; Oða s" ; a r Þ5 a s" a r I s ; Oða r" ; a r Þ5 a r" a r I s : In the case of random cumulus and stratus overlaps, they become: PARK 829

5 Journal of Advances in Modeling Earth Systems /2017MS Figure 2. Annual mean precipitation flux at the (left) SFC and (right) LWCF at the TOA from (a and e) the simulation using the analytical equation (ANALYTICAL), (b and f) the simulation based on the heuristic equation (HEURISTIC), (c and g) the online SAM0 simulation (SAM0), and (d and h) observations. The observations are the 20 year climatology of the surface precipitation rate from CMAP (January 1979 to December 1998; Xie & Arkin, 1996) and the 10 year climatology of LWCF from CERES-EBAF (March 2000 to February 2010; Loeb et al., 2009). The area-weighted global mean value is denoted as mean in the top left of the individual plots with corresponding unit and color scales in the top right and bottom of each plot, respectively. Similar plotting rules were used for the following plots. PARK 830

6 Oða c" ; a c Þ5a c" a c ; Oða s" ; a c Þ5a s" a c ; Oða r" ; a c Þ5a r" a c ; Oða c" ; a s Þ5a c" a s ; Oða s" ; a s Þ5a s" a s ; Oða r" ; a s Þ5a r" a s ; Oða c" ; a r Þ5a c" a r ; Oða s" ; a r Þ5a s" a r ; Oða r" ; a r Þ5a r" a r : (10) Figure 3. Annual zonal mean, grid mean (a) convective precipitation flux f C based on the analytical equation, (b) f C from the heuristic equation, (c) evaporation rate of precipitation within clear sky E r from the analytical equation, and (d) E r from the heuristic equation. PARK 831

7 Journal of Advances in Modeling Earth Systems /2017MS Figure 4. Difference maps of the precipitation flux at the SFC (Df ) between the control simulation (CTRL: Dzs 52; 000 m and Dzc 51 with the integrated treatment of convective and stratiform precipitation without IPA) and various sensitivity simulations for (a and e) maximum stratus overlap (Dzs 51), (b and f) random stratus overlap (Dzs 50), (c and g) maximum-random cumulus overlap (Dzc 5200 m), and (d and h) process-splitting sum with IPA from the simulations using the (left) analytical equation and (right) the heuristic equation. PARK 832

8 These analytical equations satisfy the mandatory consistency condition, that is, a c 5 P 4 K51 ak c ; a s5 P 4 K51 ak s, and a r 5 P 4 K51 ak r. This completes the computation of the economical analytical ak l without truncation error, which replaces the expensive heuristic a K l with truncation error (equation (4) and Appendix A in P2017). Not shown but we checked that when driven by the same input conditions, the analytical and heuristic formula produced very similar a K l. 3. Simulation Setting We repeated the simulations of P2017 with the analytical a K l instead of the heuristic a K l. The vertical cloud overlap parameterization is implemented into the Seoul National University Atmosphere Model version 0 (SAM0; P2017, Park et al., 2017) in an offline mode such that it does not interfere with internal model integration. At each time step in each grid column, the vertical profiles of the fractional area and in-cloud Figure 5. Global annual mean (a) convective precipitation area, a C, (b) stratiform precipitation area, a S, (c) mixed precipitation area, a M, and (d) clear area, a R,asa function of Dz s and Dz c. PARK 833

9 Figure 6. Global annual mean overlap areas (a K l ) between the precipitation area (a K ) and cloud area (a l ) for each combination of K5C; S; M; R and l5c; s; r,asa function of Dz s and Dz c. PARK 834

10 condensate mass of cumulus and stratus generated by the online SAM0 are fed into the vertical cloud overlap equations to compute a K, a K l ; P c ; P s ; E r ; H l (grid mean radiative heating rate), and f K (grid mean radiation and precipitation flux). A global standalone simulation was conducted for 1 year, forced by the observed climatological sea surface temperature and sea-ice fraction, with an annual cycle at a horizontal resolution of 0.98 latitude longitude, 30 vertical layers, and a model integration time step of Dt530 min, as described in Park et al. (2014). The simulation results obtained by using the analytical a K l were compared with that obtained with the heuristic a K l, online SAM0 simulation, and observation. Based on the economical analytical equation, we also examined the sensitivities of the global mean precipitation and radiation processes to the assumed Dz s and Dz c values. Finally, to quantify the error associated with the use of horizontally homogeneous water vapor in the radiation scheme in typical GCMs, we ran the radiation scheme with different q v in the cloudy and clear portions. The simulated radiation with inhomogeneous q v was compared with that with homogeneous q v. 4. Results 4.1. Comparison of the Simulations Using the Analytical and Heuristic a K l Precipitation Flux at the SFC and LWCF at the TOA Figure 2 shows the annual mean precipitation flux at the surface (SFC) and the LWCF at the top of the atmosphere (TOA) based on the simulations using the analytical and heuristic a K l and the online SAM0 compared with that obtained from observations. The LWCF simulated with the analytical equation is very similar to that simulated with the heuristic equation with nearly identical global mean LWCFs. However, although the patterns are similar, the global mean precipitation flux at the SFC simulated with the analytical a K l is about 5% smaller than that obtained with the heuristic a K l with relatively large differences in the tropical deep convection region. To understand the cause of this discrepancy in precipitation, we compared various precipitation processes of the two simulations. Figure 3 shows the vertical cross sections of the precipitation flux and the grid mean evaporation rates of precipitation. The grid mean evaporation rate simulated by the analytical equation is about 50% larger than that using the heuristic equation and the tropical deep convection regions show large differences. As a result, the precipitation flux simulated by the analytical equation is smaller than that using the heuristic equation. The differences of the precipitation production between the two simulations (i.e., P c and P s ) are much smaller than those of the evaporation of precipitation (not shown). It is speculated that the truncation of the overlap areas in the heuristic equation (i.e., the discretization of Na K l into the nearest integer) caused the difference of the evaporation rate of precipitation. Because the cumulus fraction is much smaller than the stratus fraction, the discretization has a larger impact on the cumulus component. Because the cumulus dominantly contributes to precipitation processes in the tropical region, the discretization causes large errors in the precipitation flux in the tropical region. As noted by P2017, stratus dominantly contributes to radiation, so that the discrepancy of the radiation flux between the two simulations is Figure 7. Global annual mean, grid mean (a) production rate of precipitation within stratus (P s ), (b) evaporation rate of precipitation within clear sky (E r ), and (c) total production rate of precipitation (P5P c 1P s 2E r ) as a function of Dz s and Dz c. Not shown but the grid mean production rate of precipitation within cumulus (P c ) is a constant, 0:159 ðgkg 21 d 21 Þ. PARK 835

11 much smaller than that of the precipitation flux. Although some quantitative differences exist, the overall patterns of various overlap and precipitation areas between the analytical and heuristic simulations were quite similar to each other (not shown) Sensitivity of Surface Precipitation Flux to Vertical Cloud Overlap and IPA Figure 4 shows the differences of the precipitation flux at the SFC between the control and sensitivity simulations obtained using the analytical a K l (left plots) and heuristic a K l (right plots). The figures in the right plots are identical to Figure 10 in P2017. In general, the overall anomaly patterns based on the analytical equation are similar to those from the heuristic equation, but some differences exist. In contrast to the heuristic equation, which shows an increase of the surface precipitation flux over the land when the vertical overlap of cumulus changes from maximum to maximum-random (Figure 4g), the analytical equation simulates uniform decrease of the surface precipitation flux over both the ocean and land (Figure 4c). The unified Figure 8. Global annual mean, grid mean (a) convective precipitation flux (f C ), (b) stratiform precipitation flux (f S ), (c) mixed precipitation flux (f M ), and (d) total precipitation flux (f 5f C 1f S 1f M ) at the surface as a function of Dz s and Dz c. PARK 836

12 convection scheme used in our simulations (UNCON; Park, 2014a) is configured to simulate a smaller cumulus fraction over the land than over the ocean (Park, 2014b). As a result, the cumulus fraction over land is more susceptible to the digitalization of Na K l than that over the ocean, which seems to result in the erroneous positive anomalies of surface precipitation flux over land in the heuristic simulation. Another notable difference is the weakening or reversal of positive anomalies in the midlatitude storm track from the heuristic to analytical simulations (compare Figure 4a with Figure 4e). Since UNICON is designed to simulate extratropical convection as well as tropical convection, the cumulus fraction and associated precipitation processes in the midlatitude storm track are also susceptible to the digitalization of Na K l. Compared with the heuristic simulation, the overall magnitude of the anomalies of the random stratus and maximum-random cumulus overlaps substantially increased in the analytical simulation (Figures 4b and 4c). However, the anomalies of the simulation with an independent precipitation approximation (IPA) substantially reduced (Figure 4d). Not shown but we repeated the heuristic simulations by changing the way how N Figure 9. Global annual mean, grid mean LW radiative heating rate of (a) cumulus (Q c 52H c =C p ), (a) stratus (Q s ), (c) clear sky (Q r Þ, and (d) all sky (Q5Q c 1Q s 1Q r ) as a function of Dz s and Dz c. PARK 837

13 a K l was digitalized and found that the anomalies shown on the right plots of Figure 4 are sensitive to the digitalization method. In contrast to the heuristic simulation, the analytical simulation is free from any truncation error Sensitivity of Global Mean Precipitation and Radiation Processes to Dz s and Dz c Based on the computationally efficient analytical equation, we explored the sensitivity of global mean precipitation and radiation processes to the vertical overlap structures of cumulus and stratus. Figure 5 shows the global mean precipitation areas as a function of the decorrelation length scales of stratus (Dz s ) and cumulus (Dz c ). The random and maximum vertical overlaps correspond to Dz50 anddz51, respectively. If stratus is more randomly overlapped, stratiform (a S ) and mixed precipitation areas (a M )increase but convective (a C ) and clear areas (a R ) decrease. Similarly, if cumulus is more randomly overlapped, both a C and a M increase but a S decreases. Figure 6 shows a plot for the overlap areas (a K l ) between the precipitation and cloud areas. Roughly speaking, the sensitivities of a K l to Dz s and Dz c follow those of a K.Exceptions are the reductions of a S s and am c in the small Dz c regime and increase of a R s when stratus is more randomly overlapped and the reductions of a C c and am c and increase of as c when cumulus is more randomly overlapped. These changes of the overlap areas directly influence the precipitation and radiation processes. Figure 7 shows the sensitivity of the global mean production and evaporation rates of precipitation to Dz s and Dz c. The global mean production rate of precipitation within cumulus is a constant (P c 50:159 ðgkg 21 d 21 Þ) that is independent of Dz s and Dz c because accretion within cumulus is neglected in our simulation. The global mean production rate of precipitation within stratus (P s ) increases when stratus is more maximally overlapped and cumulus is more randomly overlapped. The former is associated with the increase of a S s in which the accretion of stratus droplets by stratiform precipitation occurs (Figure 6e) and the latter is due to the enhanced accretion of stratus droplets by convective precipitation. The global mean evaporation rate of precipitation (E r ) increases if the cumulus is more randomly overlapped but, surprisingly, decreases if stratus is more randomly overlapped, which is likely due to the reductions of a C r and the associated evaporation of convective precipitation. Note the similarity between Figures 7c and 6c. Quantitatively, E r is more sensitive to vertical cloud overlaps than P c and P s, such that the global mean net production rate of precipitation (P) increases if cumulus is more maximally overlapped and stratus is more randomly overlapped. Figure 8 shows shows the grid mean precipitation fluxes at the surface. Both convective (f C ) and stratiform (f S ) precipitation fluxes increase when cumulus and stratus are maximally overlapped, while the mixed precipitation flux (f M ) shows the opposite pattern. The resulting net precipitation flux at the surface (f in Figure 8d) increases Figure 10. Global annual mean, grid mean (a) LWCF at the TOA and (b) LWCF at the SFC as a function of Dz s and Dz c. PARK 838

14 Journal of Advances in Modeling Earth Systems /2017MS if the cumulus is more maximally overlapped and stratus is more randomly overlapped. The difference between the maximum and minimum f is 0.44 (mm/d), that is about 15% of the mean f. Figure 9 shows the sensitivity of the global mean radiative heating rate to the vertical overlap of cumulus and stratus. The global mean radiative cooling by cumulus (stratus) increases as the cumulus (stratus) is more randomly overlapped but the radiative cooling of the clear sky shows the opposite pattern. This results in the increase of net radiative cooling as cumulus and stratus are randomly overlapped. The sensitivities of LWCF shown in Figure 10 reveal that both LWCFs at the SFC and at the TOA increase if cumulus and stratus are randomly overlapped. The difference between the maximum LWCF and minimum LWCF is 0.18 (W/m2) at the TOA and 0.3 (W/m2) at the SFC, corresponding to about 0.8 and 3% of the mean values, respectively, which are much smaller than that of the net precipitation flux at the surface. This indicates that the vertical cloud overlap has a larger impact on the precipitation flux than on the radiation flux Sensitivity of LWCF at the TOA to the Subgrid Fluctuation of Qv Similar to other GCMs, the radiation scheme in SAM0 assumes that the water vapor specific humidity (qv) within each grid layer is horizontally homogeneous. In reality, however, qv is larger in the cloudy portion than in the clear portion. To assess the error associated with the use of a homogeneous qv in the radiation scheme, we ran the radiation scheme by assuming that both cumulus and stratus are 100% saturated and water vapor in the clear sky is qv;clr 5ðq v 2as qs Þ=ð12as Þ, where qs is the saturation specific humidity, as is Figure 11. Annual mean (a) LWCF at the TOA from the simulation using an inhomogeneous qv within the grid layer and difference maps of LWCF at the TOA (D LWCF) between the (b) simulations with inhomogeneous qv and homogeneous qv, (c) the simulation with inhomogeneous qv and the observation, and (d) the simulation with homogeneous qv and the observation. In Figures 11c and 11d, r and rmse at the top of each plot denote the global pattern correlation and rootmean-square error between the simulation and observation, respectively. PARK 839

15 the stratus fraction, and q v is the grid mean water vapor excluding the cumulus portion. Because the ice stratus fraction in SAM0 is empirically computed without direct association to the subgrid variability of relative humidity, the above-mentioned q v;clr can be negative (Park et al., 2014). To prevent a too small or negative q v;clr, we used an additional constraint, that is, q v;clr is larger than 0:5 q v. Figure 11a shows the resulting LWCF at the TOA with inhomogeneous q v. The LWCF is defined as the difference between the clear-sky and all-sky LW fluxes. In computing the clear-sky LW flux as well as all-sky component, water vapor in the clear portion is set to q v;clr not q v, such that the simulated LWCF with inhomogeneous q v is defined in the same way as the satellite-derived LWCF. On the other hand, the simulated LWCF with homogeneous q v is computed using q v both for the clear-sky and all-sky LW fluxes. Compared with the simulation with homogeneous q v shown in Figure 2e, the global mean LWCF substantially increased from 22.6 to 26.9 (W m 22 ), which is roughly similar to the observed value (26.1 (W m 22 )). As shown in Figure 11b, the increases are concentrated over tropical deep convection and midlatitude storm track regions, where the vertical integral of stratus fraction is large. Although the bias of the global mean LWCF is reduced, substantial regional biases still exist. The comparison of Figures 11c and 11d show that the strongly negative biases in subsiding branches of the Hadley circulation along 308S(N) are reduced using an inhomogeneous q v, but positive biases in tropical deep convection and midlatitude storm track regions are amplified. Further efforts are necessary to reduce these biases. 5. Summary and Conclusion P2017 developed a cloud overlap parameterization to handle the contrasting vertical overlap structures of cumulus and stratus in an integrated way. In his study, the overlap area (a K l ) between the precipitation area at the top interface of an individual grid layer (a K ) and the cloud area at the layer midpoint (a l ) was computed in a heuristic way by dividing the individual grid box into N subpixels and counting all possible ways to allocate the discretized precipitation (Na K ) and cloud pixels (Na l )ton pixels such that the assumed overlap assumption for cumulus and stratus is satisfied. Although conceptually correct, P2017 s method suffered from the truncation error in association with the discretization of Na K, Na l, and Na K l into the nearest integer. In addition, too much computation time is required for the use as a practical parameterization for GCMs. To address this issue, the author derived a set of new analytical equations computing a K l that is computationally efficient without suffering from the truncation error. Similar to P2017, the new analytical equations were implemented into SAM0 in an offline mode and the global simulation results were compared with the ones of the previous heuristic equation of P2017. Based on the economical analytical equation, we also examined the sensitivities of global mean precipitation and radiation processes to the assumed decorrelation length scales of cumulus (Dz c ) and stratus (Dz s ). Finally, we explored the sensitivity of simulated radiation fields to subgrid variations of water vapor (q v ) by using different q v values for the cloudy and clear portions in the radiation scheme. The LWCF simulated using the analytical equation is very similar to that based on the heuristic equation, indicating that the analytical a K l is formulated correctly. However, the surface precipitation flux simulated by the heuristic a K l is larger than that of the analytical a K l, mainly due to the underestimation of the grid mean evaporation rate of precipitation in association with the discretization of the overlap areas in the heuristic equation. We examined if the sensitivities of surface precipitation flux to various model configurations (i.e., maximum stratus overlap, random stratus overlap, maximum-random cumulus overlap, and IPA) explored in P2017 can be similarly reproduced based on the analytical equation. In general, the analytical equation reproduced the sensitivities of surface precipitation flux simulated by the heuristic equation. However, in contrast to the heuristic equation, the analytical equation shows a decrease of the surface precipitation flux over the entire globe when the vertical cumulus overlap changes from maximum to maximum-random. The overall magnitude of the precipitation anomalies in association with the use of random stratus and maximum-random cumulus overlaps are substantially enhanced from the heuristic to the analytical simulations. However, the anomalies associated with IPA are substantially reduced, implying that IPA works well with other parameterizations. Based on the computationally efficient analytical equation, we also explored the sensitivity of global mean precipitation and radiation processes to the decorrelation length scales of cumulus and stratus. As stratus becomes more randomly overlapped, stratiform and mixed precipitation areas increase but convective and clear areas decrease. As cumulus becomes more randomly overlapped, both convective and mixed PARK 840

16 precipitation areas increase but the stratiform precipitation area decreases. The global mean production rate of precipitation within stratus increases when stratus becomes more maximally overlapped and cumulus becomes more randomly overlapped. The global mean evaporation rate of precipitation increases when cumulus is more randomly overlapped but decreases when stratus is more randomly overlapped. The global mean net production rate of precipitation and the associated surface precipitation flux increase as the cumulus becomes more maximally overlapped and the stratus becomes more randomly overlapped. On the other hand, the global mean net radiative cooling and associated LWCF increase when cumulus and stratus are randomly overlapped but decreases when cumulus and stratus are maximally overlapped. Vertical cloud overlap has a larger impact on the precipitation flux than on the radiation flux. Compared with the simulation with homogeneous q v within the grid layer, the simulation with different q v values in the cloudy and clear portions substantially increased the global mean LWCF to a level similar to the observation. However, positive biases over tropical deep convection and midlatitude storm track regions further amplified. So far, the integrated cloud overlap parameterization has been tested in an offline mode. However, to examine the full impact with interactive feedback, it is necessary to consistently implement the cloud overlap parameterization in an online mode in all relevant physics parameterizations, such as radiation, cloud microphysics, and aerosol wet scavenging schemes. In principle, our overlap framework can be extended to handle more cloud types with other overlap functionalities different from the classic maximum-random overlaps (e.g., vertical tilting of cumulus by the vertical shear of horizontal wind) if the corresponding overlap fraction a K l is computed in an appropriate way. The author is currently working on these issues. Acknowledgments This work was supported by the Research Resettlement Fund for the new faculty of Seoul National University, the Creative-Pioneering Researchers Program of the Seoul National University (SNU) (grant ), and the research project of the Korea Polar Research Institute titled Development and Application of the Korea Polar Prediction System (KPOPS) for Climate Change and Disastrous Weather Events (PE18130). The author expresses his gratitude to Siyun Kim for her help with plot preparation. The data used in this paper are available at snu.ac.kr/datapublic/james2017c/. References Barker, H. W. (2008a). Overlap of fractional cloud for radiation calculations in GCMs: A global analysis using CloudSat and CALIPSO data. Journal of Geophysical Research, 113, D00A01. Barker, H. W. (2008b). Representing cloud overlap with an effective decorrelation length: An assessment using CloudSat and CALIPSO data. Journal of Geophysical Research, 113, D Brooks, M. E., Hogan, R. J., & Illingworth, A. J. (2005). Parameterizing the difference in cloud fraction defined by area and by volume as observed with radar and lidar. Journal of the Atmospheric Sciences, 62(7), Collins, W. D. (2001). Parameterization of generalized cloud overlap for radiative calculations in general circulation models. Journal of the Atmospheric Sciences, 58(21), Geleyn, J., & Hollingsworth, A. (1979). An economical analytical method for the computation of the interaction between scattering and line absorption of radiation. Beitraege zur Physik der Atmosphaere, 52, Hogan, R. J., & Illingworth, A. J. (2000). Deriving cloud overlap statistics from radar. Quarterly Journal of the Royal Meteorological Society, 126(569), Jakob, C., & Klein, S. (2000). A parametrization of the effects of cloud and precipitation overlap for use in general-circulation models. Quarterly Journal of the Royal Meteorological Society, 126(568), Loeb, N., Wielicki, B., Doelling, D., Smith, G., Keyes, D., Kato, S., et al. (2009). Toward optimal closure of the earth s top-of-atmosphere radiation budget. Journal of Climate, 22(3), Mace, G. G., & Benson-Troth, S. (2002). Cloud-layer overlap characteristics derived from long-term cloud radar data. Journal of Climate, 15(17), Naud, C. M., Del Genio, A., Mace, G. G., Benson, S., Clothiaux, E. E., & Kollias, P. (2008). Impact of dynamics and atmospheric state on cloud vertical overlap. Journal of Climate, 21(8), Neggers, R. A., Heus, T., & Siebesma, A. P. (2011). Overlap statistics of cumuliform boundary-layer cloud fields in large-eddy simulations. Journal of Geophysical Research, 116, D Oreopoulos, L., & Khairoutdinov, M. (2003). Overlap properties of clouds generated by a cloud-resolving model. Journal of Geophysical Research, 108(D15), Oreopoulos, L., & Norris, P. (2011). An analysis of cloud overlap at a midlatitude atmospheric observation facility. Atmospheric Chemistry and Physics, 11(12), Park, S. (2014a). A unified convection scheme (UNICON). Part I: Formulation. Journal of the Atmospheric Sciences, 71(11), Park, S. (2014b). A unified convection scheme (UNICON). Part II: Simulation. Journal of the Atmospheric Sciences, 71(11), Park, S. (2017). A heuristic parameterization for the integrated vertical overlap of cumulus and stratus. Journal of Advances in Modeling Earth System, 9, Park, S., Baek, E.-H., Kim, B.-M., & Kim, S.-J. (2017). Impact of detrained cumulus on climate simulated by the community atmosphere model version 5 with a unified convection scheme. Journal of Advances in Modeling Earth System, 9, Park, S., Bretherton, C. S., & Rasch, P. J. (2014). Integrating cloud processes in the community atmosphere model, version 5. Journal of Climate, 27(18), Tian, L., & Curry, J. A. (1989). Cloud overlap statistics. Journal of Geophysical Research, 94(D7), Willen, U., Crewell, S., Baltink, H. K., & Sievers, O. (2005). Assessing model predicted vertical cloud structure and cloud overlap with radar and lidar ceilometer observations for the Baltex Bridge Campaign of CLIWA-NET. Atmospheric Research, 75(3), Xie, P., & Arkin, P. (1996). Analyses of global monthly precipitation using gauge observations, satellite estimates, and numerical model predictions. Journal of Climate, 9(4), PARK 841