This article was corrected on 01 AUG 2014. See the end of the full text for details.
Corresponding author: K. Suzuki, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA. (Kentaro.Suzuki@jpl.nasa.gov)
 This study demonstrates how aerosols influence the liquid precipitation formation process. This demonstration is provided by the combined use of satellite observations and global high-resolution model simulations. Methodologies developed to examine the warm cloud microphysical processes are applied to both multi-sensor satellite observations and aerosol-coupled global cloud-resolving model (GCRM) results to illustrate how the warm rain formation process is modulated under different aerosol conditions. The observational analysis exhibits process-scale signatures of rain suppression due to increased aerosols, providing observational evidence of the aerosol influence on precipitation. By contrast, the corresponding statistics obtained from the model show a much faster rain formation even for polluted aerosol conditions and much weaker reduction of precipitation in response to aerosol increase. It is then shown that this reduced sensitivity points to a fundamental model bias in the warm rain formation process that in turn biases the influence of aerosol on precipitation. A method of improving the model bias is introduced in the context of a simplified single-column model (SCM) that represents the cloud-to-rain water conversion process in a manner similar to the original GCRM. Sensitivity experiments performed by modifying the model assumptions in the SCM and their comparisons to satellite statistics both suggest that the auto-conversion scheme has a critical role in determining the precipitation response to aerosol perturbations and also provide a novel way of constraining key parameters in the auto-conversion schemes of global models.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Atmospheric aerosols are known to have a profound influence on cloud and precipitation through their role as cloud condensation nuclei. The so-called aerosol indirect effect has been classified by previous studies primarily into two main types. Aerosol-induced changes of cloud optical properties under the assumption of fixed cloud water content are referred to as the first indirect effect or the Twomey effect [Twomey, 1974]. The second indirect effect or the cloud lifetime effect of aerosols [Albrecht, 1989] results from aerosol perturbations that modify precipitation formation. These indirect effects of aerosols have received much attention in the recent years and have been extensively studied with observational [e.g., Nakajima et al., 2001; Breon et al., 2002; Sekiguchi et al., 2003; Matsui et al., 2004; Lebsock et al., 2008], theoretical [e.g., Stevens and Feingold, 2009; Koren and Feingold, 2011] and modeling [e.g., Suzuki et al., 2004; Quaas et al., 2009; Wang et al., 2011] approaches. Despite these efforts, attempts to estimate the climate forcing due to these aerosol indirect effects, or the indirect radiative forcing, remain highly uncertain [Forster et al., 2007], and a large diversity in estimates of the indirect radiative forcing currently exists [e.g. Quaas et al., 2009]. Overcoming these uncertainties requires a more solid understanding of the influence of aerosol on cloud microphysical processes. This understanding requires an observational underpinning because of the diversity and complexity of the processes involved.
 Recent advances in the use of multi-sensor satellite observations such as the CloudSat [e.g. Stephens et al., 2008] and A-Train constellation [e.g. Stephens et al., 2002] provides unprecedented capability of simultaneously measuring different aspects of the precipitation formation process. Combined analysis of the active and passive sensors included in A-Train has offered new insights into the precipitation formation process and the interaction of clouds and precipitation with aerosols [e.g., Stephens and Haynes, 2007; Suzuki and Stephens, 2008; Lebsock et al., 2008; Kubar et al., 2009; L'Ecuyer et al., 2009; Suzuki et al., 2010; Kawamoto and Suzuki, 2012]. These studies devised new methodologies of combining active and passive multi-sensor satellite observations to examine the microphysical processes of warm liquid clouds, which are thought to be particularly susceptible to aerosol perturbations.
 Significant progress has also occurred in numerical modeling of the aerosol-cloud interactions, particularly given the emergence of high-resolution global models such as the approach of multi-scale modeling framework (MMF) [e.g. Tao et al., 2009; Wang et al., 2011] and that of global cloud-resolving model (GCRM) [Satoh et al., 2008; Suzuki et al., 2008], as well as the progress in state-of-the-art global climate modeling of the aerosol indirect effect [e.g. Quaas et al., 2009; Golaz et al., 2011]. Given the new capabilities of the CloudSat and A-Train satellite observations mentioned above, microphysical processes in these models can be better constrained than has previously been possible with passive satellite observations alone.
 One of the recent examples for such a study is Wang et al.  who investigated the dependency of the probability of precipitation (POP) on aerosol amount obtained from both A-Train observations and global climate models and found the model biases in representing the dependency. To understand the possible cause for such biases, model representations of the precipitation formation process need to be examined in more detail. Suzuki et al.  applied some of the new methodologies developed to examine the process to both satellite observations and cloud-resolving models to identify fundamental biases in the representation of the cloud-to-rain water conversion process in the models. Since these methodologies are capable of examining the fundamental process-level characteristics of warm cloud microphysics, their applications to the analysis of the aerosol influence on liquid precipitation are expected to provide insight into how differing aerosol conditions modulate the precipitation process in both the real and modeled atmospheres.
 In this paper, we extend the model diagnostic approach of Suzuki et al.  to evaluation of the aerosol indirect effect due, in particular, to the second indirect effect. We apply the new methodologies that exploit both the CloudSat/A-Train satellite observations and GCRM simulations to construct the statistics that depict how aerosols affect the warm rain process and to identify model biases in representing the second indirect effect. The model biases thus found are further explored to understand how the model discrepancy in the form of such statistics relates to fundamental model assumptions in microphysics parameterizations with the aid of a simplified single-column model (SCM) framework. Such model-satellite comparison analyses are also extended into a general theoretical framework that provides a context in which the aerosol-cloud-precipitation interaction is simply but quantitatively described and its model representation is evaluated with satellite observations. The latter aspect of the framework demonstrates how the microphysics parameterization formula could be constrained by the new satellite observations.
2 Satellite data
 The observational data used in this study is provided by CloudSat and the Moderate Resolution Imaging Spectroradiometer (MODIS) both flying in the A-Train. We use pixel-level data from the CloudSat radar whose footprint size is about 1.4 km (across the track) and 1.7 km (along the track), on which the MODIS cloud product with horizontal resolution of about 1 km is also matched. The RO4 CloudSat products of radar reflectivity profile (2B-GEOPROF, Mace et al. ; Marchand et al. ) and precipitation rate (2C-PRECIP-COLUMN, Haynes et al. ) are analyzed. The path-integrated attenuation (PIA) estimated from CloudSat by the method of Lebsock et al.  is also employed to derive the total liquid water path (W) containing contributions from both cloud and drizzle water. The analysis is restricted to global ocean since the PIA estimate is currently limited to ocean.
 We also employ the MODIS collection 5.1 level 2 MYD06 cloud product of effective particle radius (re), optical thickness (τc) and cloud top temperature (Tc) [Platnick et al., 2003]. The re and τc are used to estimate the cloud liquid water path (Wc) that contains only contributions from cloud water. The cloud liquid water path (Wc) is estimated as
where ρw denotes the liquid water density. The cloud top temperature (Tc) is used to determine the warm-topped liquid clouds as those having Tc warmer than 273.15K. We also restricted the analysis to single-layered clouds using the cloud mask in the CloudSat 2B-GEOPROF product.
 For aerosols, we employ the level 3 MYD08-D3 aerosol product of optical thickness (τa) and Ångström Exponent (α) [Remer et al., 2005] derived from MODIS and matched to the radar footprint of CloudSat. The aerosol retrievals of daily and 1.0 degree averages are used and matched to the CloudSat footprint [Lebsock et al., 2008]. The aerosol parameters (τa and α) are used to estimate the aerosol index (AI = τaα), which is a proxy for columnar aerosol particle number concentration [Nakajima et al., 2001]. The satellite-observed data for the period of June-July-August for 2007 and 2008 are used for analysis to match the season to the model simulation period of July.
3 The model data
 The simulated data from the aerosol-coupled global cloud-resolving model Nonhydrostatic ICosahedral Atmospheric Model (NICAM)-Spectral Radiation-Transport Model for Aerosol Species (SPRINTARS) [Suzuki et al., 2008] is used in this study. Suzuki et al.  performed a global simulation with horizontal resolution of 7 km for the period of July 1-8, 2006, and found that the global characteristics of aerosols and clouds are realistically simulated in comparison with satellite observations. The model data of clouds and aerosols are employed in this study to construct statistics and to compare with satellite observations.
Suzuki et al.  showed that the NICAM-SPRINTARS model reproduced the global-scale signatures of aerosol effect on liquid clouds with some success. This includes a realistic simulation of geographical distributions of aerosol loading and effective particle radius, and a successful simulation of vertical growth pattern of cloud particles and its modulation with aerosols. These results demonstrate that the model appropriately represents some aspects of the aerosol-cloud interaction although the study did not analyze the aerosol effect on the precipitation formation process; the present study investigates this aspect of the model representation of aerosol indirect effect.
 For studying the precipitation formation process with the methodologies mentioned above and described in detail below, the radar reflectivity factor is simulated from the model output to make the model statistics directly comparable to CloudSat observations. For this purpose, the CloudSat radar simulator QuickBeam [Haynes et al., 2007] is employed and applied to the model output to simulate the radar reflectivity profiles that are used to construct the statistics. The aerosol parameters (i.e. τa and α) simulated in the model at the clear sky grid box are also used to estimate the aerosol index (AI = τaα) for consistent comparisons with satellite observations. The clear sky condition is determined as the grid box where the total cloud optical thickness is less than 1. We also upscaled the aerosol index originally obtained 3 hourly at 7 km resolution in the model to the daily and 1.0 degree averages to match to the satellite observations. This makes the correlation statistics of cloud and precipitation with regard to aerosols more directly comparable between the model and observations.
4 Aerosol effect on the warm rain process
4.1 Probability of precipitation
 The CloudSat-observed precipitation rate, when combined with information of cloud water retrieved from MODIS shortwave measurement, provides key information to examine the cloud-to-rain water conversion process and to evaluate its model representation. Given different sensitivities of CloudSat and MODIS to particles sizes, i.e. the former and the latter being primarily sensitive to drizzle and cloud particles, respectively, the combined CloudSat and MODIS analysis provides useful information of the interrelationship between cloud and rain.
 The rain formation process can be conveniently described by the probability of precipitation (POP) as first introduced by Lebsock et al.  and L'Ecuyer et al. , who analyzed POP as a function of cloud liquid water path (LWP) and found that POP tends to monotonically increase with increasing LWP and is modified by differing aerosol conditions. The POP is defined as the fractional occurrence of precipitation rate greater than a specified threshold value, which is assumed to be 0.1 mm hr-1 in this study. As has been demonstrated by Lebsock et al.  and L'Ecuyer et al. , the POP-LWP statistic dictates how the cloud water tends to be converted to rain water and how the conversion process tends to be influenced by aerosols. Figure 1 shows such statistics that are obtained from the satellite observations (Fig. 1a) and the NICAM-SPRINTARS model (Fig. 1b) when classified according to the aerosol index (AI) into pristine (AI = 0.01-0.1), moderate (AI = 0.1-0.3) and polluted (AI = 0.3-1.0) aerosol conditions. The satellite statistics are based on the CloudSat-derived POP and the MODIS-derived cloud liquid water path (Wc), and the model statistics are also constructed from the cloud-only liquid water path and the POP defined using the surface precipitation rate simulated in the model.
 For appropriate comparisons between the observations and the model whose native horizontal resolutions are different, we upscaled the CloudSat observations with native resolution of 1.75 km to NICAM resolution of 7 km, following a simple averaging method of Stephens et al. . Each orbit of the satellite observations is divided into segments of length that corresponds to the model resolution, and the number of individual POP and LWP observations within each segment is determined. For each segment of data, the precipitation is defined to exist when any one CloudSat radar profile within the segment has a precipitation rate above the specified threshold of 0.1 mm hr-1. The LWP is also averaged over the given segment. The precipitation occurrence thus determined is plotted as a function of the averaged LWP in Fig. 1a and compared with corresponding model statistics in Fig. 1b.
 The POP is found to generally increase with increasing LWP, demonstrating the water conversion characteristics similar to those found by Lebsock et al.  and L'Ecuyer et al. . Although POP generally increases with increasing LWP, one can find weaker increasing and even slight decreasing tendencies over the LWP range larger than 300 gm-2, possibly due to less stable statistics associated with relatively smaller number of data as also shown in Fig. 1 as dashed curves. Figure 1a also shows that the observed POP at a given LWP value tends to decrease with increasing AI, a tendency consistent with the reduced precipitation in more polluted conditions. This suggests that the cloud-to-rain water conversion is suppressed in more polluted conditions, which is also consistent with findings by Lebsock et al.  and L'Ecuyer et al. . It should be noted that some differences can be found between this study and the studies by Lebsock et al.  and L'Ecuyer et al. , possibly associated with differences in detailed procedure of analysis including the definition of aerosol conditions and the data product used. Regarding the definition of aerosol conditions, different AI values are used to distinguish the pristine and polluted cases among the studies. Lebsock et al.  used AI = 0.1 as the separation value, and L'Ecuyer et al.  defined AI = 0-0.5 and 0.15-1.0 as the pristine and polluted cases, respectively. The magnitude of the POP change between the pristine and polluted cases found by those two studies is comparable to the difference found in Fig. 1a between the cases of AI = 0.01-0.1 and AI = 0.1-0.3. This is found to be relatively smaller than the difference between the cases of AI = 0.1-0.3 and AI = 0.3-1.0, implying that the POP change with regard to changing aerosols may vary over different AI ranges. This would deserve further investigation in future studies. Another major difference of this study from Lebsock et al.  and L'Ecuyer et al.  is the different data product of LWP: Lebsock et al.  and L'Ecuyer et al.  used the AMSR-E microwave-retrieved LWP whereas this study uses the MODIS cloud product. The difference in horizontal resolution between these two satellite sensors may also influence the statistics. To avoid this possible uncertainty arising from the different horizontal resolutions, we upscaled the CloudSat-MODIS analysis to the NICAM native resolution scale for our model comparisons here. This issue should nevertheless be kept in mind for future investigations of similar statistics.
 The model statistics (Fig. 1b) also show the increasing tendency of POP with increasing LWP, which is qualitatively similar to those observed (Fig. 1a). The simulated increasing rate of POP, however, is found to be substantially larger than observed when compared for a same AI range. This implies that the cloud-to-rain water conversion takes place faster in the model than the observations as pointed out by Suzuki et al. . The model statistics also indicate a quantitative discrepancy from observations in terms of the POP difference among different aerosol conditions. Although the aerosol influences on the conversion process are qualitatively represented in a manner similar to the observations (i.e. smaller values of POP for more polluted conditions at a given LWP value), the simulated POP change among different aerosol conditions is substantially smaller than that observed particularly between the moderate (AI = 0.1-0.3) and polluted (AI = 0.3-1.0) cases.
4.2 Process transition
 The cloud-to-rain water conversion process that governs the POP relationships of Fig. 1 is further examined in the context of particle growth processes when the multi-sensor satellite observations are combined in another way. Suzuki and Stephens  proposed a combined analysis of the radar reflectivity (Ze) and the effective particle radius (Re) as a way of identifying the signatures of microphysical processes using satellite-observed cloud parameters. The interrelationships between Ze and Re are interpreted in the context of microphysical particle growth processes as proposed by Suzuki and Stephens  since these two quantities are theoretically related in a different way under circumstances dominated by different microphysical processes, i.e. the condensation growth and the collision-coalescence process.
 When the radar reflectivity Ze is found to be proportional to sixth power of particle radius Re as , the number concentration N is suggested to be constant, and the sixth-power relationship is interpreted as a proxy for the condensation process that conserves N. The radar reflectivity Ze is also theoretically related to Re according to the cubic relationship of when the mass concentration q is constant as is the case when the coalescence process takes place. The cubic relationship is then interpreted as a proxy for the coalescence process. These two different Ze − Re relationships are written more explicitly under an assumption of droplet size spectrum as given in Suzuki and Stephens , and are used to interpret the observed relationships between Ze and Re. We apply this methodology to investigation of aerosol effect on microphysical processes.
 To analyze the observed and simulated relationships in terms of these theoretical relationships, we estimate the layer-mean radar reflectivity and the columnar effective particle radius Re,column. The latter is derived from the total liquid water path (W) and the cloud optical thickness (τc) as [Masunaga et al., 2002]
where the total liquid water path (W) contains combined contributions from cloud and drizzle water and is derived from the radar-derived PIA with the method of Lebsock et al. . This provides an estimate of the column-mean particle radius including combined contributions from cloud and drizzle water as proposed by Masunaga et al. , who contrasted Re,column against the shortwave-retrieved cloud-top effective radius to examine vertical stratification of the cloud particle size. This columnar effective radius can be interpreted as the vertical average of effective radius profile with weight of layered cloud optical depth. For the corresponding model analysis, we estimated the Re,column in a manner consistent with the satellite analysis according to the definition above, based on the model-simulated total liquid water path and the cloud optical thickness.
 To compare with the Re,column consistently, we estimate the layer-mean radar reflectivity also as the vertical average of reflectivity profile with weight of layered cloud optical depth, which is scaled as h2/3 with regard to the geometric height h under the assumption of adiabatic growth [Suzuki et al., 2010]. We combine Re,column with the layer-mean radar reflectivity thus estimated to investigate microphysical process signatures that can be identified by the method described above [Suzuki and Stephens, 2008]. It is worth noting that the Re,column and are derived from independent source of information given that the PIA is estimated from the surface radar reflectivity and is therefore independent of the radar reflectivity profile. The analysis is restricted to the liquid clouds of τc > 15 to screen out thin clouds, for which the solar radiation at the MODIS 2.1 micron channel is likely to penetrate beyond the cloud bottom [Nakajima et al., 2010a] and therefore the microphysical signatures may be more ambiguous.
 Shown in Figure 2 is the joint probability density functions of and Re,column obtained from the A-Train and NICAM-SPRINTARS for pristine (AI = 0.01-0.1) and polluted (AI = 0.3-1.0) aerosol conditions with the theoretical relationships for specified values of number and mass concentrations superimposed. The upper panels (Figs. 2a and 2b) illustrate that the observed correlations among these quantities tend to systematically differ between pristine and polluted cases. In the pristine case (Fig. 2a), the majority of data are located around Re,column = 15 ~ 30µm and and are found to follow the cubic relationship, implying that the coalescence process is dominant. By contrast, the majority of data for the polluted case (Fig. 2b) fill in the region of to -10dB and Re,column = 10 to 15µm, and the correlation between Ze and Re more closely follows the sixth-power relationship. This implies that the particle growth is dominated by the condensation process. We interpret the difference found between pristine and polluted cases as an observed signature of transitional change in microphysical processes that occurs associated with the rain suppression due to aerosols.
 The corresponding model statistics shown in the lower panels (Figs. 2c and 2d) illustrate a qualitative similarity to but a quantitative difference from those observed. It is found that the simulated correlations between and Re,column tend to show transitional change from cubic relationship in the pristine condition to sixth power relationship in the polluted condition, which is qualitatively similar to the features found in the observations. There is, however, a quantitative difference from the observations in terms of magnitude of the change in reflectivity between pristine and polluted cases. The model statistics indicate that the majority of the radar reflectivity value does not change so widely as compared to observations, whereas the particle size Re,column tends to significantly shift toward smaller values for polluted conditions. This is a clear contrast to the observations where both Re,column and tend to be affected by differing aerosol amount. The radar reflectivity change in the model among different aerosol conditions is substantially smaller than that observed. The simulated change in peak values of reflectivity between pristine and polluted conditions (Figs. 2c and 2d) is limited within the approximate range of confined to the light drizzle occurrence, while the observed change shown in Figs. 2a and 2b takes place more widely between precipitating and non-precipitating regimes for differing aerosols. This model difference from the observations is consistent with the discrepancy found in the POP-LWP analysis described above and in Fig. 1, confirming that the precipitation response to perturbed aerosols is substantially smaller in the model than inferred from the observations.
4.3 Vertical microphysical structure
 A reason for the much weaker change of radar reflectivity in the model shown above is further explored by analyzing the vertical profile of reflectivity. For this purpose, we employ the methodology developed by Suzuki et al.  and Nakajima et al. [2010b] that rescales the radar reflectivity profile as a function of in-cloud optical depth (ICOD) in the form of the contoured frequency by optical depth diagram (CFODD). Such a depiction of the reflectivity profile is found to illustrate clear transitions of vertical microphysical structure from non-precipitating through drizzling to raining clouds [Suzuki et al., 2010; Nakajima et al., 2010b].
 The CFODD diagram is constructed from the probability density function (PDF) of radar reflectivity at each ICOD bin, and shows the PDFs as contoured frequency of radar reflectivity as a function of ICOD. The ICOD is determined as a vertical slicing of the total cloud optical thickness τc into each radar bin according to the adiabatic growth assumption that scales the in-cloud optical thickness τd as τd ∝ h5/3 with respect to the geometric height h from the cloud base [Suzuki et al., 2010]. The ICOD value thus determined at each radar height bin is used to define the ICOD bin to which the radar reflectivity belongs. The radar reflectivity values belonging to each ICOD bin are then collected to compute the PDF normalized at each ICOD bin. The PDFs are shown in the form of the contoured frequency of radar reflectivity as a function of ICOD. In our analysis, the ICOD range from 0 to 60 is divided homogeneously into 30 bins and the reflectivity range from -30 to 20 dBZ is divided into 25bins. We utilized the MODIS cloud product of τc > 15 following Suzuki et al.  and Nakajima et al. [2010b] to avoid ambiguity resulting from thin clouds of τc < 15 for which the solar radiation at 2.1 micron tends to penetrate beyond the cloud bottom [Nakajima et al., 2010a]. The analysis is restricted to the warm-topped single-layered clouds over global ocean as determined from the MODIS cloud top temperature and the CloudSat cloud mask product.
 Shown in Fig. 3 are the statistics thus obtained from the A-Train and NICAM-SPRINTARS classified according to the AI into pristine (AI = 0.01-0.1), moderate (AI = 0.1-0.3) and polluted (AI = 0.3-1.0) conditions. The number of data used to construct the PDF statistics at each ICOD bin is also shown as a function of ICOD in Fig. 4. The A-Train observations (Fig. 3 upper panels) illustrate that the vertical microphysical structure tends to systematically change with increasing aerosol burden. The pristine case (Fig. 3a) shows a downward increase in radar reflectivity with increasing the optical depth, reaching around Ze = 5 − 10dB at optical depth of 60. This suggests that the particle growth takes place downward via accretion to form rain drops in the lower layer of clouds. The rain formation is found to be suppressed in the polluted case (Fig. 3c) where the reflectivity increase with increasing ICOD tends to be smaller than the pristine (Fig. 3a) and moderate (Fig. 3b) cases. The reflectivity profile in Fig. 3c also illustrates that the cloud-to-drizzle mode in the upper cloud layer (ICOD ~ 0-20) is more pronounced and the drizzle mode in the lower layer (ICOD ~ 40-60) is less pronounced than in Figs. 3a and 3b. These observed variations in vertical cloud structure are consistent signature of rain suppression due to aerosols.
 The corresponding statistics obtained from the NICAM-SPRINTARS model (Fig. 3 lower panels), in contrast, show a much smaller variation of the reflectivity profile with respect to differing aerosol amount. Although one could find only a slight change in the reflectivity profile among different aerosol cases that is qualitatively similar to observations, the extent of change in the model is much smaller than observed. This is clearly depicted by a major difference found in the polluted cases (Figs. 3c and 3f), where satellite observation shows a substantial inhibition of rain formation (Fig. 3c) whereas the model shows a significant rain formation (Fig. 3f). These model characteristics also mean that the overall rain formation in the model is much faster than reality.
 These model discrepancies from observations in Fig. 3 are also consistent with those shown above in Figs. 1 and 2. Since the vertical average of the reflectivity profile in Fig. 3 provides the layer-mean reflectivity value in Fig. 2, the much smaller change of the simulated reflectivity profile in Fig. 3 is reflected to the analysis of Fig. 2 that also shows the substantially smaller reflectivity change in response to aerosol perturbations. These model characteristics reflect the fundamental model deficiency in representing the warm rain formation process as noted in Suzuki et al.  that identified the model bias of much faster water conversion from cloud to rain compared to reality. The comparison in Fig. 3 demonstrates that this model bias directly influences the representation of rain suppression due to aerosols. It is worth noting that the model horizontal and vertical resolutions are relatively coarse to represent the boundary-layer clouds. The coarse resolution may bias the precipitation formation from such clouds due to an inherent deficiency of representing the condensation process that is closely coupled to smaller scale turbulence in such clouds. The condensation in this context, however, is more responsible for the conversion from water vapor to cloud water, while the microphysical statistics examined here is more dictated by the conversion from cloud water to rain water which is primarily determined by microphysical process representations in the model. It is nevertheless important to be aware of the relatively coarse resolution used in the model, and future studies will be necessary to examine the resolution dependency of the statistics shown here.
5 Single-column model analysis
 To understand the discrepancies between the model and the observations in terms of cloud microphysical processes and to seek a possible path to improvement of the representation of these processes in models, we further investigate the behavior of cloud microphysics parameterizations employed in the NICAM-SPRINTARS model. To this end, we construct a simplified model framework in the form of one-dimensional single-column model (SCM) that mimics the aerosol-cloud microphysical processes represented in the NICAM-SPRINTARS model. The simple model framework distills the complicated interactions of cloud microphysics with dynamics and the other physical processes in the original NICAM-SPRINTARS model to the fundamental microphysical processes that govern the cloud-to-rain transition. This framework also facilitates various sensitivity experiments changing the model assumptions, allowing us to examine the behavior of the parameterization schemes in relation to aerosol effects and obtain insight into the behavior of schemes presently used in global models. It should nevertheless be noted that the SCM introduced here does not intend to simulate detailed structure of clouds but is used only as a simple theoretical interpretive tool of the results of the full three-dimensional NICAM-SPRINTARS model.
5.1 The model
 The SCM framework employed here is an extended version of the zero-dimensional box model of Wood et al. . The model is characterized by two prognostic variables, i.e. mixing ratios of cloud water (qc) and rain water (qr) for representing the liquid precipitation process, as in the original NICAM-SPRINTARS model that is based on the single-moment cloud microphysics scheme of Grabowski, . These two variables are allowed to have vertical profiles and to evolve with time in the SCM.
 The model is governed by the prognostic equations for qc and qr as
 The notations of the symbols are given in Table 1, and the detailed description of the SCM is provided in Appendix A.
Table 1. Notations of symbols in the single-column model
cloud water mixing ratio
rain water mixing ratio
cloud droplet number concentration
rain particle number concentration
aerosol number concentration
column aerosol particle number
adiabatic water mixing ratio
replenishment time constant
auto-conversion time constant
accretion time constant
terminal fall velocity of rain water
liquid water density
 The cloud water qc is generated by the condensation process given by the first term of (1), which replenishes the cloud water to the adiabatic value ρqadb over the relaxation time scale of τrep. The adiabatic value ρqadb is determined from the assumed profiles of pressure and temperature as described in Appendix A. The cloud water thus produced is depleted by the cloud-to-rain water conversion process represented by the second term of (1) that occurs over the precipitation time scale of τp. The precipitation process is represented as combined contributions from the auto-conversion and accretion processes as
where the time scales of the auto-conversion (τaut) and accretion (τacc) processes are parameterized in terms of cloud properties (i.e. the mixing ratio ρqc and the number concentration Nc) and rain properties (i.e. the mixing ratio ρqr and the number concentration Nr), respectively, as detailed in Appendix A. The dependency of the auto-conversion time constant τaut on droplet number concentration Nc provides a pathway through which the second aerosol indirect effect is represented in the same way as in NICAM-SPRINTARS when the linkage of Nc to the aerosol index is incorporated as detailed in Appendix A. The rain water qr thus produced by the auto-conversion and accretion processes is also depleted by the sedimentation process according to the terminal fall velocity Vt as represented by the second term of (2). Other details of the SCM are described in Appendix A and the values of parameters therein are summarized in Table 2.
Table 2. Default parameter values assumed in the single-column model
 The SCM is numerically solved starting from the initial condition of ρqc(z) = 0 and ρqr(z) = 0 for the entire cloud layer to seek for a steady-state solution under different assumed values of the aerosol index. The vertical domain size of the model is assumed to be 2 km, and the vertical resolution and the time increment are set as 20 m and 1 sec, respectively. The replenishment time constant τrep is assumed as τrep = 1800sec. The atmospheric profile is assumed to be hydrostatic with a temperature lapse rate of 7Kkm-1, a surface pressure of 1000 hPa and a surface temperature of 285 K.
5.2 Radar reflectivity profile
 We use the SCM to examine the major discrepancy found between the NICAM-SPRINTARS-simulated and the satellite-observed radar reflectivity profiles shown in Fig. 3. For this purpose, the radar simulator of Okamoto et al. , analogous to QuickBeam, is adapted to the SCM output to simulate the radar reflectivity profile. The layered in-cloud optical depth is also computed to recreate the SCM equivalent of Fig. 3.
 Figure 5 shows the results from the SCM simulations for three different values of AI (=0.01, 0.1 and 1.0) in comparison to the average results from the original NICAM-SPRINTARS simulations (Fig. 5e) and A-Train observations (Fig. 5f). The results of Fig. 5a represent the SCM run with assumptions of parameter values that are same as those in the original NICAM-SPRINTARS model based on the Berry auto-conversion parameterization [Berry, 1968] given by (A3). These results resemble the NICAM-SPRINTARS results shown in Fig. 5e, and validate the use of the SCM for examining the behavior of the parameterizations. It is shown in Fig. 5a that the radar reflectivity increases downward reaching around Ze = 5 − 10dB and producing substantial rainfall even when AI = 1.0. This confirms the interpretation that rain formation occurs more rapidly in the model than observed and suggests that the model-observation difference is likely to be attributed to microphysics parameterizations.
 To investigate how the model assumptions within the present formulation of the microphysics scheme influence the reflectivity profile, we perform sensitivity experiments of changing the parameter c1, c2 or c3 in the auto-conversion formula (A3) that determines how the auto-conversion time scale τaut varies with cloud water content ρqc and cloud droplet number concentration Nc. Changing either of these parameters corresponds to a change in time scales of auto-conversion and is then expected to change the rain formation rate. Shown in Figs. 5b, 5c and 5d are the results from such experiments where c1, c2 and c3 are respectively changed by 0.1 (Fig. 5b), 10 (Fig. 5c) and 10 (Fig. 5d) times their default values in NICAM-SPRINTARS. These parameter changes correspond to longer time scales of auto-conversion than its default. The results indeed show that the reflectivity profiles are generally shifted to smaller Ze values due to these changes, illustrating a “slow-down" of rain formation associated with the overall increase in the auto-conversion time scale. These changes in reflectivity profiles, however, do not mean the improvement of the model discrepancy from observations because the model reflectivity profiles for a given change of c1, c2 or c3 do not cover the observed range of variation with respect to the AI range shown in Fig. 5f. In order for the model to reproduce the observed variation, the auto-conversion time scale needs to vary more widely for the given change of the aerosol index from 0.01 to 1.0.
 To identify the time scale range required to reproduce the observation, the SCM is also run with the Kessler auto-conversion scheme [Kessler, 1969] that assumes a constant value of the auto-conversion time constant (τaut = const.). Figure 6 shows the results from such computations for various specified values of the time constant. In Fig. 6, we show the ranges of radar reflectivity profile for specified values of the auto-conversion time constant as obtained when considering the uncertainty of model thermodynamical assumptions in the SCM. For this purpose, we posed the ranges of surface temperature from 280 K to 290 K, lapse rate from 6.0 Kkm-1 to 8.0 Kkm-1, and the replenish time scale τrep from 900 sec to 3600 sec. The ranges of the reflectivity profiles thus obtained for each specified value of τaut are shown as the shaded areas in Fig. 6. Fig. 6 illustrates how different values of the auto-conversion time constant approximately correspond to different profiles of radar reflectivity. This correspondence also provides reference values of the auto-conversion time scale in the context of the optical depth-radar reflectivity relationship. Such a correspondence could be understood by the notion that the relationship of radar reflectivity relative to cloud optical thickness represents the drop collection process and the slope in the optical depth-reflectivity diagram is a gross measure of the water conversion efficiency [Suzuki et al., 2010] whose inverse is the conversion time scale.
 The comparison of Fig. 6 with Fig. 5f implies that the time constant τaut needs to vary over roughly two to three orders of magnitude, from about 103sec to about 105 − 106sec, so as to cover the observed range of variation in reflectivity profile. The simulated range in Fig. 5 for a given set of (c1,c2,c3) values, when compared to Fig. 6, turns out to cover approximately only one-order of magnitude variation of the time scale, and is much narrower than implied from the observation. This narrower range of variation can be understood in terms of formulation of the particular parameterization schemes employed in the SCM (and in the original NICAM-SPRINTARS model) as discussed below in next section.
 The model biases found in the results above are further explored in an attempt to systematically characterize and constrain the microphysics parameterizations with satellite observations. To this end, we introduce a generic theoretical framework that simply quantifies the aerosol-cloud-precipitation interaction and provides a context in which the role of model microphysics parameterizations in representing the second aerosol indirect effect is identified.
 The interrelationship among aerosol, cloud and precipitation, particularly in terms of the linkage of the rain formation time scale to the aerosol index analyzed above, is characterized by three different relationships;
The aerosol index (AI) is related to the aerosol particle number concentration Na as proposed by Nakajima et al.  as
where γ depends on aerosol species and size spectrum.
ii.The aerosol particle number concentration Na is then related to the cloud droplet number concentration Nc through the nucleation process according to
where the parameter k depends on airmass conditions including aerosol's hygroscopicity and size spectrum.
iii.The cloud droplet number concentration Nc comes into play in the cloud-to-precipitation process according to its influence on the auto-conversion rate. This effect is typically parameterized as dependency of the auto-conversion time constant τaut on Nc as
 Combining the relationships (3), (4) and (5) leads to the relationship of τaut with the aerosol index as
 The relationship (6) is a quantitative statement of the second aerosol indirect effect in terms of the aerosol index, which is directly obtained from two aerosol satellite observables (i.e. the optical thickness τa and the Ångström Exponent α). The strength of the second indirect effect is then characterized by the parameter μ, which is determined as a simple product of the parameters k, β and γ. Each of these three parameters characterizes a fundamental relationship (3), (4) and (5), respectively, that represents different component of the aerosol-cloud-precipitation interaction. The overall relationship (6) is thus obtained as a result from the linkage of these component processes.
 Each of these component parameters (k, β and γ) has respectively been studied by previous investigators and has its own uncertainty. These uncertainties are then combined to produce overall uncertainty of the parameter μ that characterizes the strength of the second aerosol indirect effect.
Nakajima et al.  found γ ~ 0.87 by a fitting to the results from their global aerosol retrieval. The retrieval essentially determines the two peak volumes of the assumed bi-modal log-normal size spectrum from the two channels radiances [Higurashi and Nakajima, 1999], and estimates the τa and α to derive the aerosol index (τaα). The two peak volume values can also be used to estimate the column aerosol particle number, which is then compared to the aerosol index in Nakajima et al.  to obtain their globally averaged relationship under the assumption of the bi-modal log-normal size spectrum. This globally-averaged relationship is shown in Fig. 7 in comparison to the relationship obtained from the NICAM-SPRINTARS global simulation results. The simulated scatter plot suggests that the power law function of the form of (3) well describes the relationship, and a fitting to the simulated results gives a global average value of γ ~ 0.76. The comparison of Fig. 7 shows that the simulated relationship is fairly close to that derived from satellite observations by Nakajima et al. .
 The value of k has been studied over decades since the pioneering work of Twomey  who suggested a value of k = 0.8. Kaufman et al.  assembled in-situ observations and proposed k ≈ 0.7. More recently, Nakajima et al.  found k = 0.5 based on their global satellite retrieval of cloud and aerosol parameters. These values found in the literature have been summarized by recent studies such as McComisky and Feingold  and Nakajima and Schulz . In the former, the relationships of Nc to the aerosol index are summarized as kγ ~ 0.26 − 1.0. The latter surveyed the past studies and summarized the range of k as k ~ 0.2 − 0.7. The assumed relationship in the NICAM-SPRINTARS model given by (A4) translates to a range of k ~ 0.5 − 0.8 with a global average of k ~ 0.7.
 The value of β varies depending on the auto-conversion parameterization scheme. The Berry scheme [Berry, 1968] employed in NICAM-SPRINTARS assumes β = 1.0, meaning that the time scale linearly depends on Nc. The popular examples for non-linear scheme in the literature are Tripoli and Cotton , Khairoutdinov and Kogan  and Beheng , which respectively parameterize τaut as
 These existing schemes suggest that β appears to have a range of β ~ 0.33 − 3.3.
 Table 3 summarizes the values of the parameters γ, k, β and μ reported in the literature and derived in the present study. As shown in Table 3, the ranges of γ, k and β in the literature lead to a range of μ ~ 0.05 − 3.3, which appears to represent the current uncertainty in understanding the second aerosol indirect effect.
Table 3. Summary of parameters fundamental to the aerosol-cloud-precipitation interaction.
 The narrower range of reflectivity variation in the NICAM-SPRINTARS model (Figs. 3d–f and Fig. 5e) and in the SCM (Figs. 5a–d) compared to satellite observations (Figs. 3a–c and Fig. 5f) can be understood in the context of the relationships of (3), (4), (5) and (6) and their parameterizations in the model. According to the Berry auto-conversion scheme adopted in NICAM-SPRINTARS, τaut ∝ Nc i.e. β = 1. The Nc is then related to Na through (A4) that is translated to the approximate range of k ~ 0.5 − 0.9 with an average value of k ~ 0.7. The Na is further linked to AI as (3) with a global average of γ ~ 0.76 in the NICAM-SPRINTARS model as shown in Fig. 7. These model assumptions lead to the relationship of the form of (6) with approximate value of μ ~ 0.53. This dependency of τaut on AI appears to explain the simulated variation of the reflectivity profile corresponding to only approximately one-order change of τaut for the two-order change of AI from 0.01 to 1.0 (Figs. 5a–e).
 The observed variation in reflectivity profile (Fig. 5f), when compared to the reference values of τaut in Fig. 6, suggests that the two-to-three-order in magnitude variation of the time constant needs to occur to obtain the observed change for the given aerosol change of over two orders of magnitude (AI = 0.01-1.0). This implies that the time constant τaut should be scaled by AI approximately as τaut ∝ (AI)μ with μ ~ 1.0 − 1.5. Since the parameter μ characterizes the strength of the second indirect effect, this model-observation discrepancy in μ implies that the model representation of the second indirect effect is fundamentally biased toward a much weaker response than reality due to the model assumptions that determine the value of μ.
 This argument suggests that the joint CloudSat and MODIS analysis as in the form of Figs. 3a–c and Fig. 5f, when combined with the theoretical relations used to form the SCM, provides observational constraint on the value of μ. The fact that the μ value in the NICAM-SPRINTARS model (μ ~ 0.53) is much smaller than inferred from observation (μ ~ 1.0 − 1.5) implies that the auto-conversion power exponent β should be significantly larger than 1 since the values of γ and k assumed in the model tend to be in the upper portion of the reasonable ranges. The observational information about μ thus offers a specific hint at appropriate values of β, given the fact that γ and k are better constrained than β.
 To examine the effect of changing the β values, we also perform a sensitivity experiment using the SCM to replace the Berry auto-conversion scheme with that of Khairoutdinov and Kogan  ((7); β = 1.79) or Beheng  ((8); β = 3.3) that has larger value of β than Berry . Figure 8 shows the results with uncertainty ranges given from the range of model thermodynamical assumptions described above, and illustrates how the variations of the reflectivity profile with regard to aerosol change widen and are closer to the observation (Figs. 3a–c and Fig. 5f). Figure 8 also shows that the variation in range tends to be wider in the case of Beheng (Fig. 8b) than in the case of Khairoutdinov-Kogan (Fig. 8a), reflecting the difference in β between these two schemes. It is worth noting that the observed variation with regard to aerosol change (Fig. 5f) appears to fall between these two model cases. This is consistent with the fact that the value of μ ~ 1.0 − 1.5 inferred from observations with assumptions of k ~ 0.7 and γ ~ 0.76 leads to a rough estimate of the “observation-implied" value of β ~ 1.9 − 2.8, which lies between β = 1.79 [Khairoutdinov and Kogan, 2000] and β = 3.3 [Beheng, 1994]. These results provide a hint at how the observational analysis could be used to improve the model microphysics parameterization. It is nevertheless important to note that the “observation-implied" values of μ ~ 1.0 − 1.5 and β ~ 1.9 − 2.8 should be taken as highly approximate values based on order-of-magnitude estimates with the aid of a simple model. The simple model even contains some uncertainties of thermodynamic parameters that lead to the non-negligible range of the auto-conversion time scale as shown above. More detailed analysis of the uncertainties will be necessary in future studies to better quantify the sensitivity parameters and their uncertainty ranges.
 The insight into the auto-conversion process obtained in this study should also be extended into more complete understanding of the second indirect effect that is typically defined as the cloud water change resulting from the precipitation change due to perturbed aerosols. In this regard, it is worth noting that the relative contribution from auto-conversion and accretion processes has an impact on the magnitude of the cloud water change in response to aerosols [e.g., Posselt and Lohmann, 2009; Wang et al., 2012], although the accretion process itself typically does not have a direct dependency on cloud droplet number concentration and thus on aerosols. Further studies investigating the effect of such a coupling of auto-conversion and accretion on the precipitation formation process are necessary to better understand the second indirect effect, where the analysis approach described in this study can be applied to more detailed examinations of both the auto-conversion and accretion processes and their parameterizations.
 This study examines the aerosol effect on the warm rain formation process using multi-sensor satellite observations and an aerosol-coupled global cloud-resolving model (GCRM). Methodologies developed to examine microphysical processes of warm clouds are applied to joint CloudSat and MODIS satellite observations to reveal how the rain formation characteristics vary for different aerosol conditions over global ocean. The results show process-level signatures of aerosol suppression of liquid precipitation on the global scale. The corresponding statistics are also constructed from the GCRM to illustrate that the simulated reduction of rain production for a given aerosol increase is much smaller than that observed due to the model deficiency of representing the water conversion process. These results complement the previous study of Suzuki et al.  to provide more complete picture of how the NICAM-SPRINTARS model represents the aerosol-cloud-precipitation interaction, and point to a fundamental model bias in the cloud-to-precipitation process and its influence on representation of aerosol effect on precipitation.
 Analysis of a simplified single-column model (SCM) that contains the key microphysical processes adopted in the GCRM is also performed to characterize the behavior of the parameterization scheme and to explore parameter ranges that will lead to model improvement. The SCM results are analyzed in the form of the radar reflectivity profile as a function of in-cloud optical depth in a manner comparable to satellite observations. The different reflectivity profiles in such a depiction are found to correspond to different time scale values of the cloud-to-rain water conversion process. This correspondence is then used to infer the range of variation of the time scale that is required to represent the observed variation of the radar reflectivity profile for a given perturbation of aerosol amount. The “observation-implied" range of the time scale is found to be substantially wider than that of the time scale in the model that is parameterized to vary in response to aerosol change. This can explain the model-observation discrepancy in the aerosol effect on the rain formation. It is then argued that the model bias is attributed to the formulation of the auto-conversion scheme that appears to have weaker dependency of the time scale on cloud droplet number concentration than reality. This is also supported by additional experiments with the SCM replacing the auto-conversion scheme with those having stronger dependency of the time scale on the number concentration, which showed wider range of reflectivity variations closer to those observed.
 The importance of the auto-conversion parameterization is also demonstrated in the context of the overall relationship among aerosol, cloud and precipitation that characterizes the strength of the second aerosol indirect effect. The relationship is established in a generic form that combines fundamental relationships reported in the literature describing different components of the aerosol-cloud-precipitation interaction. The overall relationship is represented as a linkage of the water conversion time scale to the aerosol index by way of connections between the “intermediate" parameters, i.e. aerosol and cloud particle number concentrations. The general formulation simply reveals how the uncertainties in each component relationship reported by past studies are combined to produce overall uncertainty in understanding and representing the second indirect effect in the models. The auto-conversion parameterization is identified as the chief source of uncertainty and is constrained by the model-observation comparison analyses described in the present paper.
Appendix A: Description of the single-column model
 The formulations and assumptions of the single-column model used in this study are described in detail here. The governing equations of the model are the prognostic equations for cloud water (qc) and rain water (qr) mixing ratios that are again written as
 The notations of symbols and the values of parameters adopted in the present SCM are summarized in Tables 1 and 2, respectively.
 The first term on the right hand side of (A1) represents the condensation process as a replenishment to the assumed adiabatic value ρqadb over the relaxation time scale of τrep. The adiabatic profile ρqadb(z) is determined by assumed profiles of pressure p and temperature T as
where Γ(p,T) is a thermodynamic function of p and T that represents the increase in liquid water content for a pure adiabatic parcel ascent, and the factor fad(z) represents the sub-adiabaticity that takes into account the effect of air-parcel dilution due to the lateral and cloud-top entrainments. We adopt the functional form for fad(z) proposed by Wood et al.  as
 The second term in (A1) represents the cloud-to-rain water conversion process that occurs at characteristic time scale of τp. This process consists of the auto-conversion and accretion processes that have the time constant of τaut and τacc, respectively, as [Suzuki and Stephens, 2009]
 The auto-conversion process is parameterized in terms of the cloud water content ρqc and the cloud particle number concentration Nc according to the scheme of Berry  as in NICAM-SPRINTARS as
where c1, c2 and c3 are preset parameters whose default values are given in Table 2, and σ(Nc) is a function of Nc given as
 (A3) means that the auto-conversion time scale linearly depends on Nc.
 This dependency provides a pathway through which aerosols influence the precipitation process when Nc is related to aerosol amount. Following the original NICAM-SPRINTARS model, we relate Nc with Na via the empirical relationship as [Suzuki et al., 2004]
where Nc,max denotes the assumed maximum value of Nc and is set as Nc,max = 400cm− 3.
 The aerosol number concentration Na is further related to the aerosol index (AI) as argued by Nakajima et al.  who proposed the relationship between the columnar number concentration na and the aerosol index as
where n0 = 4.57 × 1012m− 2 and γ = 0.87. This relationship also approximately applies to the NICAM-SPRINTARS results as shown in Fig. 7 with n0 = 3.73 × 1012m− 2 and γ = 0.76, which are obtained from the least square fitting to the simulated results of na and AI. Assuming the exponentially decaying vertical profile of the aerosol number concentration as Na(z) = Nsfcexp(−z/za), the above relationship (A5) is reduced to
 We employ this relationship with the assumption of za = 1km in the present SCM.
 The accretion rate is parameterized as a function of the rain water content ρqr and the rain particle number concentration Nr in the same manner as in NICAM-SPRINTARS. Since the microphysics scheme used here [Grabowski, 1998] is a single-moment scheme, the number concentration Nr is diagnosed from ρqr through an assumption of the size distribution function of rain particle n(D) with regard to particle diameter D. We assume the exponential function as n(D) = N0exp(−λD) with fixed value of N0 = 107m− 4 following the assumption in NICAM-SPRINTARS and in Grabowski . The number concentration Nr is determined from ρqr through λ.
 The accretion time constant τacc is then written as [Suzuki and Stephens, 2009]
where the constant c4 is determined from the assumption of the terminal fall velocity and the collection efficiency of Grabowski  and is given in Table 2. The auto-conversion and accretion processes act as a sink for cloud water and a source for rain water as described in (A1) and (A2).
 The second term on the right hand side of (A2) represents the sedimentation process by which the rain water falls down through the cloud layer according to the terminal fall velocity Vt. Vt is determined as the mass-weighted average of the single-particle terminal velocity vt(D) = cDd (c = 130.0m1/2sec− 1, d = 0.5) over the assumed size spectrum n(D) = N0exp(−λD) as [Grabowski, 1998]
where c5 is a preset constant parameter and its value is derived from the assumptions in (Grabowski, 1998) as c5 = 188.0m1/2sec− 1 (also given in Table 2). The advection scheme of Van Leer  is used to solve the sedimentation process.
 This study was supported by National Aeronautics and Space Administration (NASA) grant NNN13D771T and NNN13D968T, and carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. K.S. is grateful to J. Haynes and H. Okamoto for their providing a code of radar signal simulator.
In the Acknowledgments of the originally published article, two incorrect grant numbers were used. The grant numbers as they should be corrected are as follows: Grant number “NNX07AR11G” should be “NNN13D771T”. Grant number “NNX09AJ45G” should be “NNN13D968T”.
Acknowledgments. This study was supported by National Aeronautics and Space Administration (NASA) grant NNN13D771T and NNN13D968T, and carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. K.S. is grateful to J. Haynes and H. Okamoto for their providing a code of radar signal simulator.