3.2.1. Theoretical Formulation
 Except for the microphysical relationship between N and rv, the partition between homogeneous and inhomogeneous entrainment-mixing processes has been often based on two characteristic time scales, evaporation time τevap and turbulent mixing time τmix, since the pioneering work by Baker and Latham  and Baker et al. . The evaporation time expresses the time needed for a population of droplets with a mean radius rm to evaporate in a subsaturated environment and is given by
where A is a function of air pressure and temperature and s the supersaturation [e.g., Rogers and Yau, 1989] (see Appendix B for details). The negative sign is introduced to denote the fact that s is negative in a subsaturated environment. The turbulent mixing time represents the time needed for complete homogenization of a volume of a linear size L through the process of turbulent diffusion and is given by
where ɛ is the eddy dissipation rate of turbulent kinetic energy [Baker et al., 1984; Burnet and Brenguier, 2007; Wyngaard, 2010]. The occurrence of homogeneous or inhomogeneous mixing can be discerned from the ratio of the two characteristic time scales defined as the Damköhler number (Da) or its reciprocal [Siebert et al., 2006; Burnet and Brenguier, 2007; Jeffery, 2007; Andrejczuk et al., 2009]:
Despite its popularity, this approach suffers from a serious shortcoming: The value of L used in the calculations of τmix and Da is ambiguous [Lehmann et al., 2009]. Major progress was achieved lately by Lehmann et al.  to overcome this difficulty by introducing the concept of transition length (L*) as the value of L that corresponds to the unit Da. The substitution of Da = 1 and equation (2) into equation (3) leads to the expression for the transition length:
where reaction time τreact is defined as either the time when the droplets have completely evaporated or the time at which the relative humidity has reached 99.5% (s > −0.005). Note that τreact is used instead of τevap to relax the limited assumption that the supersaturation s is a constant for the calculation of τevap. See Appendix B for details about the calculation of τreact. Lehmann et al.  argued that if L* falls within the turbulent inertial subrange, after a blob of size LE of subsaturated air is entrained into a cloud, all eddies of size L in the range of L* < L < LE will experience inhomogeneous entrainment-mixing processes, whereas eddies smaller than L* will mix homogeneously.
 Following Lehmann et al.  and recognizing that the lower end of the turbulent inertial subrange is represented by the Kolmogorov length scale η, here we further introduce a new dimensionless number, called the scale number (NL) and defined as the ratio of L* to η:
where η is given by
and ν is the kinematic viscosity [Wyngaard, 2010] (see Appendix B for details).
 Similar to L*, NL is a dynamical measure of the degree of the homogeneous or inhomogeneous entrainment-mixing process, but accounts for both L* and η: the larger the NL, the stronger the homogenous entrainment-mixing process and the weaker the inhomogeneous entrainment-mixing process.
3.2.2. Leg-Averaged Scale Number
 To examine the applicability of this idea and make a connection to the microphysical results presented in section 3.1, we have computed L*, η, and NL for all the legs. In the calculation, the air with temperature and moisture in the environment just above the cloud tops (Table 1) is assumed to be entrained into the clouds. Basically, these values are based on the vertical profiles of temperature, relative humidity, and water vapor mixing ratio along the first vertical penetrations of clouds at the beginning of flights. A similar method was used by Lehmann et al.  in their cumulus study. If such a penetration is not available, as, for example, in the 19 March 2000 case, a penetration in the middle of the flight is used.
Table 1. Air Temperature, Relative Humidity and Water Vapor Mixing Ratio Above the Cloud Top in the Five Cases
|Case||Type||Temperature (°C)||Relative Humidity (%)||Water Vapor Mixing Ratio (g kg−1)|
|3 March 2000||Non-drizzling||0.5||60.8||2.9|
|17 March 2000||Drizzling||−5.0||61.6||2.5|
|18 March 2000||Drizzling||−3.1||38.1||1.7|
|19 March 2000||Non-drizzling||2.3||42.4||2.2|
|21 March 2000||Drizzling||−0.9||26.7||1.5|
 Figure 5 shows the NL for all the legs, where the blue, red, and black colors represent the extreme inhomogeneous mixing, inhomogeneous mixing with subsequent ascent, and homogeneous mixing as identified by the microphysical analysis, respectively.
 The only leg (Leg 2 of the 17 March 2000 case) dominated by the homogeneous entrainment-mixing process according to the microphysical analysis conspicuously had the largest NL of 53.7, while other legs affected by the inhomogeneous entrainment-mixing process had smaller NL. Lehmann et al.  pointed out that the homogeneous mixing process was more likely to occur in the vicinity of the cloud core of cumulus, where vertical velocity, temperature, ɛ, N, and LWC were positively correlated, while the inhomogeneous entrainment-mixing mechanism dominated in the more-diluted cloud regions. The five cases presented here did not have such cloud cores; positive correlations among vertical velocity, temperature, ɛ, N, and LWC were not identified, implying that these clouds were aged and not actively growing. However, the comparison between Leg 2, affected by the homogeneous entrainment-mixing process in the 17 March 2000 case, and Leg 1, dominated by the inhomogeneous entrainment-mixing process in the same case, could still provide some hints to further examine the causes of the two mechanisms (Figure 6). The reason for choosing Leg 1 of the 17 March 2000 case representing the inhomogeneous entrainment-mixing mechanism was that these two legs were in the same case and were expected to have a similar environment. Thus, we can focus on the effect of entrainment mixing on the microphysical relationships to the greatest extent.
Figure 6. (a) Temporal variations of vertical velocity (w), (b) dissipation rate (ɛ), (c) number concentration (N), (d) liquid-water content (LWC), (e) scale number (NL) along Leg 1 and Leg 2 of the 17 March 2000 Case. The dots represent the 1 Hz data.
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 It is evident that the vertical velocity was mainly positive along Leg 2 but negative along Leg 1. Moreover, the ɛ, N, LWC, and NL were much larger along Leg 2 than along Leg 1. Therefore, compared with Leg 1, Leg 2 was closer to the “cloud core” with less dilution and Leg 1 had experienced extensive dilution. This was consistent with the aforementioned conclusion given by Lehmann et al. . In addition, if a blob with size LE was assumed to be entrained from above the cloud top and moved downward, the parcel would be stretched into smaller blobs. The more the blob moved downward, the smaller the size that would be obtained [Krueger, 1993]. The height of Leg 2 was lower than that of Leg 1; therefore Leg 2 could have smaller blobs, increasing the probability of their sizes being located in the range from η to L*. This could be another reason for the homogeneous entrainment-mixing process along Leg 2.
 Except for the two legs in the 17 March 2000 case, which had the largest and smallest NL (Figure 5), the correspondence between the analyses of microphysical relationship and scale number was not that obvious in their relations to entrainment-mixing types along the legs with intermediate values of NL; a careful comparison of the results for the two legs dominantly affected by the inhomogeneous mixing with subsequent ascent to other legs suggests that these two legs shared similar scale numbers with some other legs affected by the extreme inhomogeneous entrainment-mixing mechanism. Although the samples here are too limited to be conclusive, this phenomenon should be emphasized and warrants substantiation. The reason for such a lack of unique one-to-one correspondences between scale number and microphysical relationships could be that dynamical analysis through scale number here or Damköhler number [Siebert et al., 2006; Burnet and Brenguier, 2007; Jeffery, 2007; Andrejczuk et al., 2009] or transition length [Lehmann et al., 2009] in previous studies can distinguish homogeneous and inhomogeneous entrainment-mixing processes, but cannot further distinguish the two types of inhomogeneous mixing processes: extreme scenario and inhomogeneous mixing with subsequent ascent. Thus such a lack of unique one-to-one correspondences between microphysical and dynamical properties highlights the importance of combined microphysical and dynamical analyses in the investigation of turbulent entrainment-mixing processes.
3.2.3. A Probabilistic View of the Scale Number
 Lehmann et al.  obtained a transition scale L* that starts to favor homogeneous mixing at approximately 10 cm, which is close to leg-averaged value L* = 7.4 cm for Leg 2 of the 17 March 2000 case; L* of ∼10 cm means NL of ∼50 for a typical value of η ∼2 mm. Except for the leg-averaged NL, the further examination of spatial distribution of NL is necessary because NL along one leg is not uniform. The spatial distribution of NL along one leg is expected to be responsible for the simultaneous occurrence of different types of entrainment-mixing mechanisms, as mentioned in section 3.1. For example, Figure 7 shows the probability density function of NL (1 Hz) along Leg 3 of the 3 March 2000 case; it is clear that most NL values were in the range of 5–20, corresponding to the dominance of the extreme inhomogeneous entrainment-mixing mechanism; however, there were still some values larger than 50 or even 150, giving a sign of the occurrence of homogeneous entrainment-mixing process, which was supported by the weak positive correlation between rv and N for N < ∼100 m along this leg (Figure 2c). Similarly, for Leg 2 of the 17 March 2000 case, although it was dominantly affected by the homogeneous entrainment-mixing mechanism, the negative correlation of rv versus N for N > ∼200 cm−3 (Figure 4) showed the occurrence of the inhomogeneous entrainment-mixing process, which could be related to the existence of small NL (Figure 8). The wider distribution of NL from smaller to larger than 50 was a common phenomenon; only Leg 1 in the 17 March 2000 case had maximum NL smaller than 50. The wider distribution of NL could be further augmented by Figure 9 in that the standard deviation of NL generally increased with its mean for all the legs in general. Therefore, even a dominance of one specific entrainment-mixing mechanism could not completely rule out the occurrence of other mechanisms.