## 1. Introduction

The climatic effects of warm clouds are characterized by their optical and microphysical properties, which are represented by parameters such as cloud droplet effective radius (CDR) and cloud optical thickness (COT). These two cloud parameters have been observed by remote sensing technique from aircraft (e.g., Nakajima et al. 1991; Asano et al. 1995; Brenguier et al. 2000) and satellite (e.g., Han et al. 1994; Nakajima and Nakajima 1995; Kawamoto et al. 2001), and there have been found characteristic relationships between CDR and COT of warm clouds.

Nakajima et al. (1991) found that the CDR and COT tend to be positively correlated when the cloud does not include drizzle-sized particles, whereas they tend to be negatively correlated when a significant concentration of drizzle particle exists in the cloud. The positive correlation for nondrizzling clouds indicates that the cloud becomes optically thicker with growth of cloud particle, whereas the negative correlation for drizzling clouds indicates that the cloud becomes optically thinner with particle growth, possibly due to the conversion process of cloud water into drizzle or rainwater. These correlation patterns can be interpreted to reflect cloud microphysical processes that take place within the clouds.

These statistical analyses have been extended into wider regions by Nakajima and Nakajima (1995), who reported that the First International Satellite Cloud Climatology Project (ISCCP) Regional Experiment (FIRE) region is dominated by negative correlations, in contrast to the Atlantic Stratocumulus Transition Experiment (ASTEX) region, where positive correlation tendencies are mainly found. These correlation statistics are shown here in Fig. 1 for the FIRE region and Fig. 2 for the ASTEX region, both cited from Nakajima and Nakajima (1995). Figures 1 and 2 show various different relationships between CDR and COT, implying more complicated microphysical mechanisms involved in the clouds over these regions. Several other observational studies have demonstrated complex positive and negative correlation patterns between CDR and COT (Han et al. 1994, 1998; Asano et al. 1995; Boers and Rotstayn 2001; Kawamoto et al. 2001; Szczodrak et al. 2001; Peng et al. 2002; Lebsock et al. 2008; Kobayashi and Masuda 2008).

Since the shortwave radiative property strongly depends on CDR and COT (e.g., Slingo 1989), clouds with positive and negative CDR–COT correlations should have largely different radiative properties. It is therefore important to understand how these correlation patterns are formed and what mechanisms are responsible for determining the CDR–COT relationships. Recent observational studies also showed that the microphysical characteristics of warm clouds significantly change from region to region associated with differing conditions of aerosol amount and static stability (e.g., Masunaga et al. 2002; Matsui et al. 2004). In this regard, it is also important to investigate how the CDR–COT relationships are influenced by these environmental factors.

For these purposes, it is useful to employ numerical cloud models. Lohmann et al. (2000) performed a numerical simulation with a general circulation model (GCM) and reported that the simulated CDR and COT are positively and negatively correlated for nonprecipitating and precipitating clouds, respectively. Their finding is consistent with previous observational studies mentioned above and implies that their GCM includes some mechanisms that may reproduce the observed CDR–COT relationships. In GCMs, however, the cloud microphysics is crudely represented by simple bulk parameterizations with coarse resolutions of several hundreds of kilometers, which are not sufficient for fully understanding the microphysical mechanisms responsible for the CDR–COT statistics.

Motivated by this difficulty with GCMs, our previous study (Suzuki et al. 2006) employed a cloud model with a more detailed representation of microphysical processes, based on a bin-type microphysics scheme that predicts the size distribution function of cloud particles in an explicit manner, and highlighted the statistics of CDR and COT obtained from the simulation with the model. The results showed that the positive and negative correlation patterns are simulated for nondrizzling and drizzling clouds, respectively, similar to the previous observational studies mentioned above, and reproduced the CDR–COT scatterplots that closely resemble the observed statistics over FIRE and ASTEX regions when pristine and polluted air conditions, respectively, are assumed.

The present study intends to provide more detailed descriptions of our previous study (Suzuki et al. 2006) using the bin-type cloud model. The main focus of this study is on fundamental mechanisms as to how these positive and negative correlation patterns are formed through microphysical processes of warm clouds. In this paper, we demonstrate that the condensation and collection particle growth processes are responsible for the positive and negative correlation patterns, respectively. We also show how the aerosol amount and dynamical conditions influence the correlation patterns based on sensitivity experiments of changing these environmental conditions, thereby providing a basis for interpreting the satellite-observed correlation statistics as shown in Figs. 1 and 2. The findings of this study may also be helpful to improve the bulk parameterizations employed in GCMs.

This paper first describes the nonhydrostatic spectral microphysics cloud model employed in the present study in section 2. The cloud model is then used for numerical simulation of warm clouds with the experimental design shown in section 3. The simulated results are analyzed with a focus on the correlation patterns between CDR and COT and their formation mechanisms in section 4. The results of the sensitivity experiments of changing the dynamical and aerosol conditions are presented in section 5. These simulation results are then used to interpret the correlation statistics observed by Nakajima and Nakajima (1995) in section 6. Section 7 summarizes the conclusions of this study.

## 2. The cloud model

The model used in this study is a nonhydrostatic spectral bin microphysics cloud model developed by Suzuki (2004) and Suzuki et al. (2006) and originally based on the Hebrew University Cloud Model (e.g., Khain and Sednev 1996; Khain et al. 1999; Khain and Pokrovsky 2004; Khain et al. 2004). This is a coupled model of the nonhydrostatic dynamical framework and the spectral bin cloud microphysics module. This model is similar to the cloud models that have recently been developed and used for various numerical simulations with detailed representations of microphysical processes (e.g., Feingold et al. 1999; Feingold and Kreidenweis 2002; Takahashi and Kawano 1998; Takahashi and Shimura 2004; Iguchi et al. 2008). The present model has also been used in a recent study of Sato et al. (2009) that implemented a Monte Carlo integration method into collection processes. The basic features of this model are described in this section.

The nonhydrostatic dynamical equations are numerically solved to predict the atmospheric density *ρ*, horizontal wind *u*, vertical wind *w*, and internal energy *e* by using the scheme developed by Stone and Norman (1992) with the addition of gravity force. This is based on an operator splitting method with an explicit time integration scheme. The Cartesian coordinate is used in the present model with a staggered grid system in which the wind components and scalar variables are located at the edge and the center of the grid point, respectively. The advection processes for density, momentum, energy, and water vapor are calculated by the Van Leer (1977) scheme.

The cloud microphysics processes involving the size distribution functions are calculated with a bin-type scheme based on Khain and Sednev (1996) and Khain et al. (1999, 2004), taking into account the microphysical processes of warm clouds (i.e., nucleation from aerosol, condensational growth, and collection processes). In the bin-type microphysics scheme, the size distribution functions of cloud particles are predicted in an explicit manner as explained below.

*S*according to the Köhler theory as follows (see Table 1 for explanations of symbols):

*r*

_{liquid}that corresponds to the aerosol particle size

*r*

_{CCN}according to the relationship obtained from the Köhler curve analysis as

*S*as

*K*(

*m*

_{1},

*m*

_{2}) describes the efficiency of collection between the particles with sizes

*m*

_{1}and

*m*

_{2}and characterizes the evolution pattern of size distribution function by collection process. There are several forms of the kernel function suggested by previous studies. In the present study, we perform experiments by assuming the form suggested by Long (1974) as

*r*

_{large}denotes the radius of larger particle, and

*υ*

_{large}and

*υ*

_{small}are the volumes of larger and smaller particles, respectively. The constants

*C*

_{1}and

*C*

_{2}are set as 9.44 × 10

^{9}cm

^{−3}s

^{−1}and 5.78 × 10

^{3}s

^{−1}, respectively, following Long (1974). Equation (4) means that the collection process has a threshold behavior. The collection process is not active when the both particles are smaller than the critical size (50

*μ*m) and discontinuously becomes active for the particles exceeding the critical size.

An example of the evolution of size distribution function due to condensation and collection processes is shown in Fig. 3 when this form of collection kernel function is used. It is illustrated that the size spectrum becomes narrow in the early stage of evolution as induced by condensation process and then produces a bimodal feature, including a drizzle mode as well as a cloud mode. This is due to the drastic increase in collection efficiency with particle size when it becomes greater than the critical radius. This bimodal behavior with the remarkable drizzle mode is likely to represent the characteristics of observed size spectra, which often tend to be bimodal. As will be demonstrated later (in section 4), the distinctive difference in evolution pattern of the size spectra between early and mature growth stages is closely related to different correlation patterns between CDR and COT.

## 3. Warm cloud simulation

*T*

_{sfc}and

*z*denote the surface temperature and the critical height of the lower unstable layer, respectively. The lapse rate Γ

_{c}_{2}is assumed to be 2 K km

^{−1}, and Γ

_{1}is changed from 7.5 to 8.5 K km

^{−1}in sensitivity experiments to examine the dynamical effects. The temperature profile given by (5) is shown in Fig. 4a for the case of

*T*

_{sfc}= 300 K,

*z*= 1 km, Γ

_{c}_{1}= 7.5 K km

^{−1}and Γ

_{2}= 2 K km

^{−1}. The vertical shear of horizontal wind is also given in the initial condition as shown in Fig. 4b. The initial humidity profile is given with the scale height of 1 km as shown in Fig. 4c.

*r*

_{0}= 0.1

*μ*m and

*f*

_{0}(

*z*) is given by an exponential decay profile as

*f*

_{sfc}and

*H*denote the surface value of

*f*

_{0}and the scale height, respectively. We assume various values of

*f*

_{sfc}ranging from

*f*

_{sfc}= 10

^{5}to 10

^{7}m

^{−3}for investigating the aerosol effect on the cloud property, and the scale height

*H*is set as 1 km. The values of

*f*

_{sfc}= 10

^{5}and 10

^{7}m

^{−3}approximately correspond to the surface aerosol number concentration of

*n*= 10

_{a}^{8}m

^{−3}and 10

^{10}m

^{−3}, respectively, and also to column aerosol particle numbers of

*N*= 10

_{a}^{11}and 10

^{13}m

^{−2}, respectively, with a 1-km scale height. A warm bubble is initially located to trigger the convection and the cloud formation as a potential temperature perturbation given by

*x*

_{0}= 9 km,

*z*

_{0}= 0.5 km,

*x*= 5 km,

_{r}*z*= 0.5 km, and Δ

_{r}*θ*= 1 K. The computational domain is set as 30 km in the horizontal and 5 km in the vertical direction, and the resolutions are 500 m in the horizontal and 50 m in the vertical. The time integration is performed with the time step of 1 s for 2 h to cover the period of formation of a single warm cloud. Although these experimental designs are simpler than those giving a forcing from a boundary layer field experiment (e.g., Bretherton et al. 1999), it is more convenient to simulate a single warm cloud in a manner as simple as possible for our purpose of obtaining the fundamental understandings of complicated microphysical processes within the cloud. More realistic experimental settings, however, are necessary in future studies for investigating how the findings of this study based on relatively simple simulations would be applied to more realistic and complicated situations.

We prepare 20 bins ranging from 0.01 to 1 *μ*m for aerosol particles and 60 bins ranging from 3 to 3000 *μ*m for liquid cloud particles. We divide the cloud size distribution into three parts (i.e., 3–50, 50–500, and 500–3000 *μ*m) and represent each range as the cloud, drizzle, and rainwater part, respectively. The mass concentration of cloud, drizzle, and rain are explicitly calculated by integrating the simulated size distribution function over these ranges.

*τ*and effective particle radius

_{c}*r*can also be explicitly calculated from the simulated size distribution function

_{e}*f*(

*x*,

*z*,

*r*) for each grid point (

*x*,

*z*) by their definitions as

*Q*

_{ext}is the extinction coefficient factor and is equal to 2 in a good approximation for cloud particles, and

*z*and

_{B}*z*denote the cloud base and top, respectively. Because the value of

_{T}*r*obtained from remote sensing represents the cloud particle size at an optical depth of 20%–40% of the total optical thickness (Nakajima and King 1990), we adopt a simulated value of

_{e}*r*at the upper 30% of

_{e}*τ*so that the calculated values can be compared with the remote sensing results.

_{c}## 4. Correlation statistics between CDR and COT

### a. Positive and negative correlation patterns

Figure 5 shows the scatterplot between CDR and COT obtained from the simulation, separately for nondrizzling (Fig. 5a) and drizzling (Fig. 5b) stages that are defined as the cloud states including the drizzle water content smaller and larger than 10^{−3} g m^{−3}, respectively. This result is obtained from the simulations with Γ_{1} = 8.0 K km^{−1} and *f*_{sfc} ranging from 7.8 × 10^{5} m^{−3} to 1.8 × 10^{6} m^{−3}, approximately corresponding to the column aerosol particle number *N _{a}* ∼ 10

^{12}m

^{−2}. Each point in the plot represents a cloud profile at a particular horizontal point and a particular time.

It is found in the figure that the COT tends to increase with increasing CDR in nondrizzling stages (Fig. 5a), leading to a positive correlation between them, whereas the COT tends to moderately decrease with increasing CDR in drizzling stages (Fig. 5b), depicting a negative correlation. These characteristics are closely similar to the observed CDR–COT statistics reported by Nakajima et al. (1991; see their Fig. 10) and those shown in Figs. 1 and 2 reported by Nakajima and Nakajima (1995), suggesting that the present model can reproduce the observed correlation patterns. This result also confirms that the positive and negative correlations relate to the presence of drizzle-sized particles as found by Nakajima et al. (1991).

### b. Microphysical interpretation

To investigate how the CDR and COT tend to change at different stages of cloud development, we show in Fig. 6 the tendencies of these quantities, which are defined as their time derivatives, as vectors projected on the CDR–COT plane. The figure shows the results from the simulations with collection as well as condensation processes (Fig. 6a) and with condensation process only (Fig. 6b), where the collection process is switched off in the model. It is shown in Fig. 6a that both COT and CDR have positive tendencies over the area of COT < 20 and CDR < 20 *μ*m, whereas the negative tendencies of COT concurrent with positive tendencies of CDR are found over the area of 5 < COT < 20 and CDR > 20 *μ*m. These characteristics confirm the notion that the CDR and COT are positively and negatively correlated for early nondrizzling stage and mature drizzling stage of cloud development, respectively. Another interesting feature found in Fig. 6a is negative tendencies of both CDR and COT over the area of COT < 5 and CDR > 15 *μ*m. These tendencies are found over later times corresponding to the decaying stage after the precipitation occurs, when the evaporation process typically take place in the model.

In the case of condensation only (Fig. 6b), the vector arrows become close to horizontal over the area of CDR > 20 *μ*m and thus the negative correlation part tends to vanish. The increasing and decreasing tendencies of both CDR and COT can also be found in this case since the condensation and evaporation processes take place for early and decaying stages, respectively, without the collection stage. It is worth noting that the CDR values are limited to smaller than about 21 *μ*m in this case, demonstrating that the particle growth is significantly suppressed compared to the cases with collection processes (Fig. 6a). These results suggest that the negative correlation pattern is formed through the collection process whereas the positive correlation pattern is formed by the condensation process.

These tendencies of CDR and COT found at specific stages of cloud development can be schematically summarized (see Fig. 7). The COT tends to increase with increasing CDR at early stage of cloud development, providing a positive correlation pattern between them as shown by the arrow A. At the drizzling growth stage shown by the arrow B, the COT tends to decrease with increasing CDR and these two quantities are negatively correlated. This branch B is typically followed by the arrow C, which represents the decaying stage of the cloud induced by the evaporation process that decreases both CDR and COT. This triangular characteristic composed of the arrows A, B, and C simply models the life cycle of a single warm cloud in terms of microphysical particle growth processes; it may be employed to interpret the basic aspects of observed CDR–COT relationships as shown in Figs. 1 and 2.

*σ*, total mass concentration

*L*, and effective radius

*r*that are defined with the size distribution function

_{e}*f*(

*r*) as

*σ*is then given from those in

*L*and

*r*as

_{e}*σ*increases (decreases) with time when the rate of change in mass concentration (

*dL*/

*dt*)/

*L*is larger (smaller) than that in effective radius (

*dr*/

_{e}*dt*)/

*r*. This leads to a positive (negative) correlation between

_{e}*σ*and

*r*.

_{e}Figure 8 shows how each term of (6) changes as a function of *r _{e}* through the evolution of size distribution function for specified values of supersaturation. It is shown in the figures that the change rate of effective radius tends to decrease with increasing

*r*up to around

_{e}*r*∼ 13–16

_{e}*μ*m because of the nature of condensation process that decelerates the particle growth, followed by an increase in the change rate with increasing

*r*due to the collection process that accelerates the particle growth. The change rate of mass concentration is found to monotonically decrease with increasing

_{e}*r*, from positive values due to condensational growth toward zero due to the mass conservation induced by the collection process.

_{e}These tendencies of the change rates of *r _{e}* and

*L*determine the change rate of

*σ*according to (6). The change rate of

*σ*decreases with increasing

*r*from positive to negative values. The positive values take place for smaller

_{e}*r*values where the change rate of

_{e}*L*is larger than that of

*r*because of the dominance of condensation process. The negative values are the result of the dominance of collection process that yields change rates of

_{e}*r*larger than those of

_{e}*L*.

Figure 8 also shows that the plot of change rates for higher values of supersaturation tend to be stretched in both horizontal and vertical axis directions, corresponding to more amplified changes in the change rates through evolution of size spectra as a function of effective radius. These results for different values of supersaturation suggest how the change rates of *r _{e}*,

*σ*, and

*L*may be influenced by environmental conditions of aerosol concentration and dynamical stability, since the supersaturation is determined by these conditions through the competing effect between the forcing due to upward air motion and the damping due to condensation. The former is determined from dynamical stability condition and the latter depends on abundance of cloud particles resulting from aerosol number concentration. The higher (lower) values of supersaturation typically tend to occur under the condition of lower (higher) aerosol number concentrations and lower (higher) dynamical stabilities. The results shown in Fig. 8 suggest that the aerosol and dynamical conditions tend to modify the amplitude of change in the change rates of

*r*,

_{e}*σ*, and

*L*although the evolution patterns of these change rates, such as the positive-to-negative switching in

*σ*, are qualitatively unchanged regardless of the environmental conditions.

Figure 9 schematically summarizes how the size distribution function expressed in mass concentration tends to evolve through these processes and how the changes in *r _{e}* and

*L*take place. When the condensation controls the particle growth, the size distribution function tends to expand in vertical dimension, corresponding to the increase in mass concentration

*L*, rather than shifting in horizontal dimension as represented by the effective radius

*r*(Fig. 9, left panel). This tendency keeps the size spectrum narrow and produces the increasing tendency of

_{e}*σ*, leading to the positive correlation between

*σ*and

*r*. When the collection process becomes active, on the contrary, the size distribution function tends to stop growing in vertical dimension and to shift in horizontal dimension, corresponding to the increase in effective radius

_{e}*r*(Fig. 9, right panel). This leads to a spectral broadening and the tendency of

_{e}*σ*then becomes negative (decreasing) in this case, providing the negative correlation between

*σ*and

*r*. The positive and negative correlation patterns between

_{e}*σ*and

*r*, thus derived from the evolution pattern of size spectra, are a fundamental cause of the correlation patterns between CDR and COT shown in Figs. 5 and 6. These results suggest that the satellite-observed positive and negative correlation patterns as shown in Figs. 1 and 2 may be correctly interpreted as being formed by condensation and collection processes through the change in size distribution function.

_{e}## 5. Dynamical and aerosol effects

The microphysical processes, which were found to form the positive and negative correlation patterns in the previous section, take place in a manner that depends on dynamical stability condition and aerosol amount. To examine how these environmental factors influence the CDR–COT correlation statistics, sensitivity experiments of changing these conditions are performed and the results are shown in this section.

### a. Adiabatic model

*q*(in g m

^{−3}) is assumed to linearly increase with height

*h*as

*λ*is a constant parameter determined from pressure and temperature and is assumed to be 2.0 × 10

^{−3}g m

^{−4}in this study. The volume-averaged particle radius

*r*is then given by

_{υ}*ρ*and

_{w}*N*denote the liquid water density and the cloud particle number concentration, respectively, and

_{c}*A*= 3

*λ*/(4

*πρ*). The effective radius

_{w}*r*is related with

_{e}*r*as

_{υ}*r*=

_{e}*k*

^{−1/3}

*r*through the parameter

_{υ}*k*that varies from about 0.67 in continental air masses to about 0.80 in maritime air masses (Pontikis and Hicks 1992; Martin et al. 1994). The value of

*k*

^{−1/3}then varies from about 1.08 to 1.14 and is assumed to be 1.1 in this study. The effective radius

*r*is thus given from

_{e}*r*as

_{υ}*h*=

*H*as

*N*is assumed to be constant. Vertical integration of (7) and (8) over cloud layer of thickness

_{c}*H*then leads to the expression for liquid water path

*W*and optical thickness

*τ*as

_{c}*r*(

_{e}*H*) and

*τ*are related through the parameters

_{c}*N*and

_{c}*H*as discussed by previous studies (e.g., Brenguier et al. 2000; Szczodrak et al. 2001). The parameter

*H*can be replaced with

*W*since these two quantities are uniquely related through (10); thus, (9) and (11) can be rewritten in terms of

*N*and

_{c}*W*as

*r*(

_{e}*H*) and

*τ*under constant values of

_{c}*N*and

_{c}*W*as

*N*is shown as solid curves in Figs. 10 and 11 and represents theoretical growth curves through the condensation process for those values of

_{c}*N*. The value of

_{c}*W*, which determines the relationship (13) shown as dotted curves in Figs. 10 and 11, represents the vertical extent that the cloud layer reaches. The theoretical curve (13) corresponding to larger value of

*W*is then expected to trace the plot for more dynamically unstable conditions.

### b. Dynamical effects

Figure 10 shows the CDR–COT scatterplots obtained from the sensitivity experiment of changing Γ_{1} as Γ_{1} = 7.5 (Fig. 10a), 8.0 (Fig. 10b), and 8.5 K km^{−1} (Fig. 10c). These results are obtained from the simulation with aerosol amount *f*_{sfc} ranging from 7.8 × 10^{5} to 1.8 × 10^{6} m^{−3} approximately corresponding to *N _{a}* ∼ 10

^{12}m

^{−2}. These plots are overlaid on the theoretical relationships between CDR and COT for specified values of

*N*and

_{c}*W*given by (12) and (13).

Figure 10 indicates that the scatterplot is confined to the area of COT < 40 under the stable condition (Fig. 10a), whereas the maximum COT reaches about 120 under the unstable condition (Fig. 10c) without significant change in CDR values. It is also found that the CDR–COT correlation patterns are not significantly changed by the change in dynamical condition. The implied values of *W* observed by these plots are about *W* < 300 g m^{−2} for the stable case (Fig. 10a) and tend to increase for more unstable cases, reaching the values larger than 1000 g m^{−2} for the most unstable case (Fig. 10c). These results suggest that the change in dynamical stability condition tends to change the amplitude of the correlation plot through the change in liquid water path *W* without significantly changing the correlation patterns between CDR and COT.

### c. Aerosol effects

Figure 11 shows the results from the sensitivity experiment of changing the aerosol amount by changing the parameter *f*_{sfc} from 10^{5} to 10^{6} m^{−3} for the “pristine” case (Fig. 11a), from 10^{6} to 10^{7} m^{−3} for the “moderate” case (Fig. 11b) and from 10^{7} to 10^{8} m^{−3} for the “polluted” case (Fig. 11c). These ranges of *f*_{sfc} values roughly correspond to surface aerosol number concentrations *n _{a}* of about 10

^{7}–10

^{8}, 10

^{8}–10

^{9}, and 10

^{9}–10

^{10}m

^{−3}, respectively, and to column aerosol particle numbers

*N*of about 10

_{a}^{10}–10

^{11}, 10

^{11}–10

^{12}, and 10

^{12}–10

^{13}m

^{−2}, respectively. The dynamical stability condition was assumed to be Γ

_{1}= 8.0 K km

^{−1}for these experiments. These figures also include the theoretical relationships between CDR and COT for specified values of

*N*and

_{c}*W*given from the adiabatic growth model [(12) and (13)].

Figure 11 indicates that the CDR–COT relationship is significantly modified with the change in aerosol amount. The least squares fitting of the relationship *r _{e}* ∝

*τ*to the scatterplots resulted in values of

_{c}^{b}*b*of −0.024, 0.0013, and 0.03 for the clean, moderate, and polluted cases, respectively, demonstrating a change in correlation pattern between CDR and COT. It is also found that the scatterplot for the pristine condition (Fig. 11a) is located around larger values of CDR and relatively smaller values of COT compared to those for the moderate and polluted conditions (Figs. 11b and 11c) that are located around smaller values of CDR and reach larger values of COT. The more polluted case (Fig. 11c) tends to have smaller CDR values than the moderate case (Fig. 11b). This systematic change in scatterplot is found to correspond to the change in number concentration

*N*implied from the adiabatic model: more polluted cases correspond to larger values of

_{c}*N*. This change in CDR–COT plot is also found to occur in accordance with a constant value of

_{c}*W*implied from the adiabatic model, which tends not to be affected by aerosol amount. These results suggest that the modification of the CDR–COT correlation plot with aerosols takes place through the change in particle number concentration

*N*without significantly changing the liquid water path

_{c}*W*.

It is worth noting that the plot shown in Fig. 11c is confined to CDR less than about 15 *μ*m and the lower edge of the plot roughly follows the condensational growth curve (12) corresponding to *N _{c}* = 300 cm

^{−3}. This feature can be understood in terms of the nonlinear behavior of Long’s collection kernel function, which includes a threshold value of particle size that distinctively divides the inactive and active states of collection process. When aerosol amount is high, more numerous cloud droplets are produced from abundant aerosols. The higher number of cloud droplets tends to consume the water vapor more efficiently through the condensation process, leading to lower values of supersaturation, which in turn slow down the condensation process according to (2). This reduction in condensational growth speed tends to confine the CDR values to those smaller than critical radius of Long’s kernel function and to prevent the collection process from becoming active.

The shape of the lower edge of the plot shown in Fig. 11c can also be interpreted with the findings of Szczodrak et al. (2001), who reported that a power-law relationship of the form *r _{e}* ∝

*τ*, with

_{c}^{b}*b*= 0.2, well explains many cases of their observed analysis over the northeastern Pacific. Since this relationship is equivalent with (12) for a constant

*N*value, their results may imply that the condensation process controls the CDR–COT statistics obtained from their analysis. There has been found, however, a much wider range of values for

_{c}*b*by Eitzen et al. (2008) using a tropical cloud observation obtained from Visible/Infrared Scanner (VIRS) onboard the Tropical Rainfall Measuring Mission (TRMM) satellite. They performed a cluster analysis for CDR and COT observed data and found several clusters with distinctive positive and negative values of

*b*as well as a cluster with

*b*close to zero. Although the clusters of positive and negative

*b*values may imply the existence of microphysical regimes controlled by condensation and collection processes, respectively, the observed variety of

*b*values suggests more complicated situations in real clouds, especially on the global scale. These global-scale observations may be explained as a complex mixtures of simple condensation and collection tendencies shown in Fig. 7 that systematically change with environmental factors, as demonstrated in this section.

## 6. Discussion

The model results shown above demonstrate how the CDR–COT correlation patterns are formed through microphysical particle growth processes and how they tend to be influenced by environmental factors of aerosol and dynamical conditions. These findings of this study provide a basis for interpreting the correlation plot observed by previous remote sensing studies as shown in Figs. 1 and 2. In this section, we discuss several aspects of the observed CDR–COT correlation plot that may be explained by the model results in terms of the microphysical processes under various environmental conditions.

The positive and negative correlations mainly found for ASTEX (Fig. 2) and FIRE (Fig. 1), respectively, suggest that the clouds over the ASTEX region observed on this occasion are mainly at the condensation regime whereas many of the clouds observed over the FIRE region are interpreted as being in the collection growth stage, according to the findings of this study. It should be noted that the observed plots shown in Figs. 1 and 2 are for specific dates and times over these regions and the microphysical regimes suggested here are also for these specific observed cases. Similar interpretations, however, can be applied to other cases given the CDR–COT correlation plots obtained from observations.

Many panels of the correlation plot shown in Fig. 1 are also found to have a small portion of positive correlation under the large part of negative correlation as typically shown in panels B3, C1, C3, C4, D2, and D3 of Fig. 1. These small parts of positive correlation appear to trace the theoretical curve (12) that corresponds to relatively small values of number concentration *N _{c}*, similar to our model results for pristine condition shown in Fig. 11a. This is possibly induced by the pristine air mass conditions over the FIRE region. The smaller values of number concentrations associated with pristine aerosol conditions may cause a large amount of drizzle particles, found as a large portion of negative correlation in many panels of Fig. 1. Our previous modeling study (Suzuki et al. 2006) also simulated a correlation pattern closely similar to those observed over the FIRE region when the pristine aerosol condition was assumed in the model. The left edge of the large negative correlation part may include the evaporation stage characterized by decreasing tendencies of both CDR and COT, which are simulated by the model as shown in Fig. 6.

The ASTEX region, on the contrary, tends to include only the positive correlation branch in many cases, as typically shown in panels A1, A2, B1, B2, C1, C2, and D1 of Fig. 2. These plots tend to cover smaller CDR values than the FIRE case and to coincide with the theoretical curve (12) corresponding to larger values of *N _{c}*, possibly because of a polluted aerosol condition over the ASTEX region. This result also confirms our previous study (Suzuki et al. 2006) that simulated the correlation plot similar to those found over the ASTEX region when turbid air conditions are assumed in the model. The polluted aerosol conditions may suppress the drizzle formation, which is shown as an absence of negative correlation part in many panels of Fig. 2. These observed characteristics with only a positive correlation branch are similar to our model result over polluted conditions (Fig. 11c), which also tends to include only a positive correlation branch.

Besides these remarkable characteristics, several interesting features are also found in these observed scatterplots. Panels D1 and D4 of Fig. 1 are found to have two branches that are likely to trace the theoretical curve [(12)] corresponding to two distinct values of number concentration *N _{c}*, implying that there exist two types of air masses with different aerosol amounts. Similar multibranch features can also be found in panels A4, B2, and B4 of Fig. 2, which also may imply the mixture of air masses characterized by different aerosol amounts. There can also be found scatterplots with different amplitudes characterized by different magnitudes of COT values. The panels A1, B1, and C1 of Fig. 1 and A1, A4, B4, and C4 of Fig. 2 tend to have smaller magnitudes compared to other plots. These differences in magnitudes can be interpreted as being induced by different dynamical conditions as shown in our model simulation (Fig. 10). Various characteristics found in observed correlation plots so far reported can thus be explained by different aerosol and dynamical conditions through their effects on microphysical particle growth processes.

## 7. Conclusions

This study investigated the microphysical mechanisms that control the CDR–COT relationships using a nonhydrostatic spectral bin microphysics cloud model. The numerical experiments were performed to form low-level warm clouds under various aerosol and dynamical stability conditions. The results showed positive and negative correlation patterns between CDR and COT for nondrizzling and drizzling stages of cloud development, similar to those reported by previous remote sensing studies. It was found that only the positive correlation pattern was simulated when the collection process was switched off in the model, whereas both the positive and negative correlation branches were reproduced when the model included the collection as well as condensation processes. The positive and negative correlations were also found to be explained by evolution patterns of the size distribution function due to the condensation and collection processes, respectively.

Sensitivity experiments of changing the aerosol and dynamical conditions were also performed. It was shown that the change in dynamical condition tends to change the amplitude of the CDR–COT scatterplot, especially in COT direction, without changing their correlation patterns, whereas the change in aerosol amount has an effect of modifying the correlation pattern between CDR and COT through changing CDR values. The former and latter changes are found to correspond to the change in liquid water path and number concentration, respectively, as implied from an adiabatic growth model. These findings provide a basis for interpreting the various characteristics found in observed CDR–COT correlation plots obtained by previous observational studies, and they may also help improve the parameterizations employed in climate models for more realistic simulations of cloud microphysical properties and their interactions with aerosols.

## Acknowledgments

The authors are grateful to Matt Lebsock and two anonymous reviewers for their valuable comments on earlier version of the manuscript. This study was supported by JAXA/EarthCARE, MEXT/VL for Climate System Diagnostics, MOE/Global Environment Research Fund B-083, JST/CREST, and MEXT/Innovative Program of Climate Change Projection for the 21st Century.

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As in Fig. 1, but for the ASTEX region (cited from Nakajima and Nakajima 1995).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

As in Fig. 1, but for the ASTEX region (cited from Nakajima and Nakajima 1995).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

As in Fig. 1, but for the ASTEX region (cited from Nakajima and Nakajima 1995).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Time evolution of size distribution function driven by the condensation and collection processes predicted by the present model for a given supersaturation value of *S* = 0.03%.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Time evolution of size distribution function driven by the condensation and collection processes predicted by the present model for a given supersaturation value of *S* = 0.03%.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Time evolution of size distribution function driven by the condensation and collection processes predicted by the present model for a given supersaturation value of *S* = 0.03%.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Initial profiles of (a) temperature, (b) horizontal wind, and (c) relative humidity assumed for the simulations.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Initial profiles of (a) temperature, (b) horizontal wind, and (c) relative humidity assumed for the simulations.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Initial profiles of (a) temperature, (b) horizontal wind, and (c) relative humidity assumed for the simulations.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot of CDR versus COT obtained from the simulation for (a) nondrizzling and (b) drizzling stages.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot of CDR versus COT obtained from the simulation for (a) nondrizzling and (b) drizzling stages.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot of CDR versus COT obtained from the simulation for (a) nondrizzling and (b) drizzling stages.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Vector charts describing the tendencies of CDR and COT obtained from the simulations (a) with both condensation and collection growth processes and (b) with condensation process only.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Vector charts describing the tendencies of CDR and COT obtained from the simulations (a) with both condensation and collection growth processes and (b) with condensation process only.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Vector charts describing the tendencies of CDR and COT obtained from the simulations (a) with both condensation and collection growth processes and (b) with condensation process only.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Schematic diagram representing the tendencies of CDR and COT at the nondrizzling (A), drizzling (B), and decaying (C) stages of a single warm cloud development, induced by condensation, collection, and evaporation processes, respectively (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Schematic diagram representing the tendencies of CDR and COT at the nondrizzling (A), drizzling (B), and decaying (C) stages of a single warm cloud development, induced by condensation, collection, and evaporation processes, respectively (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Schematic diagram representing the tendencies of CDR and COT at the nondrizzling (A), drizzling (B), and decaying (C) stages of a single warm cloud development, induced by condensation, collection, and evaporation processes, respectively (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

The normalized change rate of effective radius, mass concentration, and cross section taken for abscissa as a function of effective radius (ordinate) through the evolution of size distribution function for given supersaturation values of *S* = (a) 0.05%, (b) 0.1%, (c) 0.2%, and (d) 0.3%.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

The normalized change rate of effective radius, mass concentration, and cross section taken for abscissa as a function of effective radius (ordinate) through the evolution of size distribution function for given supersaturation values of *S* = (a) 0.05%, (b) 0.1%, (c) 0.2%, and (d) 0.3%.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

The normalized change rate of effective radius, mass concentration, and cross section taken for abscissa as a function of effective radius (ordinate) through the evolution of size distribution function for given supersaturation values of *S* = (a) 0.05%, (b) 0.1%, (c) 0.2%, and (d) 0.3%.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Schematic illustration of evolution pattern of size distribution function due to (left) condensation and (right) collection particle growth processes, showing the tendencies of the growth rate of mass concentration and effective radius that determine the sign of correlation between the effective radius and the cross section (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Schematic illustration of evolution pattern of size distribution function due to (left) condensation and (right) collection particle growth processes, showing the tendencies of the growth rate of mass concentration and effective radius that determine the sign of correlation between the effective radius and the cross section (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Schematic illustration of evolution pattern of size distribution function due to (left) condensation and (right) collection particle growth processes, showing the tendencies of the growth rate of mass concentration and effective radius that determine the sign of correlation between the effective radius and the cross section (see text for details).

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot between effective radius and optical thickness obtained from the simulation for Γ = (a) 7.5, (b) 8.0, and (c) 8.5 K km^{−1}. The theoretical relationships (12) and (13) for specified values of *N _{c}* and

*W*are also shown for reference as solid and dashed curves, respectively.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot between effective radius and optical thickness obtained from the simulation for Γ = (a) 7.5, (b) 8.0, and (c) 8.5 K km^{−1}. The theoretical relationships (12) and (13) for specified values of *N _{c}* and

*W*are also shown for reference as solid and dashed curves, respectively.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot between effective radius and optical thickness obtained from the simulation for Γ = (a) 7.5, (b) 8.0, and (c) 8.5 K km^{−1}. The theoretical relationships (12) and (13) for specified values of *N _{c}* and

*W*are also shown for reference as solid and dashed curves, respectively.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot between effective radius and optical thickness obtained from the simulation for (a) pristine, (b) moderate, and (c) polluted air conditions. The theoretical relationships (12) and (13) for specified values of *N _{c}* and

*W*are also shown for reference as solid and dashed curves, respectively.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot between effective radius and optical thickness obtained from the simulation for (a) pristine, (b) moderate, and (c) polluted air conditions. The theoretical relationships (12) and (13) for specified values of *N _{c}* and

*W*are also shown for reference as solid and dashed curves, respectively.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Scatterplot between effective radius and optical thickness obtained from the simulation for (a) pristine, (b) moderate, and (c) polluted air conditions. The theoretical relationships (12) and (13) for specified values of *N _{c}* and

*W*are also shown for reference as solid and dashed curves, respectively.

Citation: Journal of the Atmospheric Sciences 67, 4; 10.1175/2009JAS3283.1

Notations of symbols used in this paper.