1. Introduction
One of the challenges in cloud physics is to understand the fast formation of rain in warm clouds. Radar measurements by Laird et al. (2000) in Florida and by Szumowski et al. (1997) offshore the island of Hawaii indicate that warm shallow cumulus clouds can produce rain in 15–20 min; that is, cloud droplets grow from micrometer to millimeter size within tens of minutes. Jonas (1996) showed that for a supersaturation of 0.2% a cloud droplet with 10-μm radius needs about 20 min to grow to a size of 20 μm through condensation. To further grow to drizzle size (∼100 μm) through the coalescence process in a cloud with a liquid water content (LWC) of 1 g m−3 requires another hour. Clearly these calculations are in disagreement with the aforementioned observations. A closely related challenge is the discrepancy between the observed and modeled width of the cloud droplet size distribution. Yum and Hudson (2005) compared the width of the size distribution measured with a forward scattering spectrometer probe (FSSP) in several stratocumulus and cumulus clouds with results from their adiabatic parcel model, which calculates the condensation growth of droplets in a predetermined updraft. They noted large discrepancies in magnitude between observed and modeled widths of the size distribution. Part of the observed broadness of the size distribution might be caused by the measurement instrument itself (Cerni 1983; Schmidt et al. 2004). However, even if this artificial broadening is taken into account, the narrowest droplet size distributions found in nonprecipitating cumulus clouds are still broader than reference spectra derived from calculations assuming pure adiabatic growth (Brenguier and Chaumat 2001). Generally it is found from observations that the width of the size distribution stays nearly constant or even increases with height and with the median diameter, whereas pure condensational growth yields a narrowing of the number size distribution with height.
Several processes originating from the turbulent nature of the velocity, temperature, and water vapor field have been proposed to influence not only the growth of single droplets but also the macroscopic evolution of cumulus clouds (Shaw 2003). For instance, the collision kernel in turbulent air can be enhanced compared to a laminar flow (e.g., Wang et al. 2006; Pinsky and Khain 2004). Fluctuations of the supersaturation field were proposed to contribute to the width of the droplet size distribution (Stepanov 1975; Korolev and Isaac 2000; Vaillancourt et al. 2002), but different conclusions regarding the effectiveness in producing large droplets were drawn.
Another process of potential importance for the evolution of the cloud droplet size distribution and formation of larger droplets is the entrainment and consecutive mixing of subsaturated, droplet-free air into the cloud and its effect on the droplet number and size. It is generally assumed that subsaturated ambient air is entrained in blobs with a typical length scale lE. The entrained air is turbulently mixed with the cloudy air, such that smaller and smaller filaments of subsatured ambient and cloudy are produced (e.g., Baker et al. 1984; Brenguier 1993). Finally, at length scales close to the Kolmogorov microscale (≈1 mm for atmospheric conditions), gradients in the fields of temperature and water vapor are homogenized by molecular diffusion.
Baker et al. (1980) suggested that the relative magnitudes of the time scales for turbulent mixing and the time scale for droplet evaporation determine the characteristics of the entrainment/mixing process. If the mixing proceeds rapidly, then the fields of water vapor and temperature will essentially be the same for all droplets. Thus, all droplets will experience nearly the same subsaturation and will shrink toward smaller sizes until equilibrium is reached. Consequently, the droplet size distribution is shifted toward smaller diameters. This type of mixing is termed homogeneous. If, in contrast, the mixing proceeds slowly and the borders between cloudy and subsaturated ambient air exist long enough, individual droplets adjacent to the borders will completely evaporate, while other droplets that are surrounded by saturated cloudy air remain unaffected. In this case, the droplet size distribution will not be shifted toward smaller sizes, but the droplet number will be diminished by the complete evaporation of single droplets. This is referred to as inhomogeneous mixing.
If a diluted cloud parcel experiences an updraft, it may experience higher supersaturations than its neighboring undiluted cloud parcels because of the diminished number of droplets that compete for the available water vapor. Thus, the droplets may grow to larger than average sizes, especially when previous mixing was inhomogeneous. Numerical studies that attempt to represent entrainment and turbulent mixing have suggested that this mechanism operates in clouds (e.g., Baker and Latham 1979; Telford and Chai 1980; Lasher-Trapp et al. 2005; Andrejczuk et al. 2006; Krueger et al. 2006). They also suggest that further broadening toward smaller droplet diameters can occur when fresh cloud condensation nuclei (CCN) entrained with the subsaturated air are activated in the raising air parcel, known as secondary activation. Finally, whether a dilution manifests itself in a reduction of droplet size or droplet concentration or both has important consequences for the cloud’s radiative properties (e.g., Brenguier et al. 2000; Grabowski 2006; Jeffery 2007) and thus should be considered when detecting and assessing the first indirect aerosol effect (e.g., Pawlowska et al. 2000).
Many observations and modeling studies have aimed at understanding the entrainment/mixing process since the pioneering work of Latham and Reed (1977) and Baker et al. (1980) on homogeneous and inhomogeneous mixing. Some of them point to the inhomogeneous scenario (e.g., Gerber 2006; Burnet and Brenguier 2007; Pawlowska et al. 2000). Others suggest that the observations are closer to the homogeneous mixing scenario (Jensen et al. 1985; Jensen and Baker 1989). Recently, Burnet and Brenguier (2007) analyzed observations and suggested that a homogeneously mixed cloud will appear inhomogeneously mixed because of an instrumental artifact caused by spatial averaging of the Fast-FSSP measurements over spatially inhomogeneous parts (with respect to droplet number concentration and droplet size) of the cloud. The authors suggested further studies of this problem and explicitly pointed to the need to include finescale measurements of the dynamic fields in the analysis.
The entrainment and mixing process is important for the evolution of a cloud and its microphysical and optical parameters, but until now it has remained unclear whether the mixing is predominantly homogeneous, inhomogeneous, or between the two cases, and what the controlling factors are and how they interact. We address these questions here by reconsidering the time and length scales relevant to the problem. We then explore implications with observations of the entrainment/mixing process in shallow cumulus clouds made with the helicopter-borne instrument platform known as the Airborne Cloud Turbulence Observation System (ACTOS). The use of this platform enables the analysis of the cloud microphysical, thermodynamic, and dynamic parameters on the decimeter scale. The microphysical characteristics of the mixing process are analyzed using a so-called mixing diagram introduced by Brenguier et al. (2000). Additionally, the thermodynamic and dynamic properties of the cloudy and subsaturated entrained air that govern the mixing process are analyzed on the meter scale up to the cloud scale.
2. The concept of mixing: Time and length scales
a. Homogeneous versus inhomogeneous mixing
How does the entrainment of subsaturated air into a cloud influence its microphysical properties? Two observations provide some appreciation of the complexity of this question. First, mixing begins with the engulfment of dry air by “large eddies” followed by the breakup into progressively smaller eddies until the gradients created by the nonlinear cascade become sharp enough to result in their dissipation through molecular diffusion (e.g., Broadwell and Breidenthal 1982). The key point here is that there is a continuous range of eddy or fluctuation scales between the large eddy and dissipation scales and that cloud droplets are inherently coupled to these multiscale fluctuating fields through liquid–vapor phase changes and the associated energy transports. Second, when dry air is mixed into cloudy air the resulting cloud liquid water content is reduced, but how this reduction in liquid water content manifests itself in the droplet size distribution is not unique. Keeping in mind that LWC = (π/6)ρwnd Dυ3, where ρw is the density of liquid water, nd is the droplet number density, and Dυ is the droplet mean volume diameter, equivalent reductions of LWC can manifest themselves in quite distinct droplet number–size distributions. How nd and Dυ vary depends on the detailed interaction of cloud droplets with their turbulent environment during the mixing process.
The time scales for mixing and evaporation will be discussed in detail later in this section, but first we consider hypothetical limiting microphysical scenarios for homogeneous and inhomogeneous mixing. Motivated by the relationship LWC ∝ nd Dυ3 discussed earlier, Brenguier and Burnet (1996), Pawlowska et al. (2000), and Burnet and Brenguier (2007) characterized the microphysical properties of the mixing by means of a diagram in which the abscissa is given by the droplet number concentration normalized by its adiabatic value (nd /nd,a) and the ordinate is given by the cube of the mean volume diameter normalized by its adiabatic values [(Dυ /Dυ,a)3]. By choosing these coordinates, variations of the cloud base height and the sampling altitude are accounted for by the normalization with the adiabatic value. An example of this mixing diagram is displayed in Fig. 1.
If the mixing is inhomogeneous, Dυ stays constant during dilution while nd is diminished. Thus, the measured [(nd /nd,a); (Dυ /Dυ,a)3] values will form a horizontal line at (Dυ /Dυ,a)3 ∼ 1 (thick horizontal dashed line in Fig. 1). During homogeneous mixing, the [(nd /nd,a); (Dυ /Dυ,a)3] values will lie along the solid black mixing lines, which are here shown for four exemplary values of the ambient saturation ratio S. The mixing lines for homogeneous mixing are calculated as follows. During homogeneous mixing, nd is diminished, and droplets evaporate because of the saturation deficit (1 − S) of the mixture. The saturation deficit dependent on the fraction of (subsaturated) entrained air can be calculated from the wet equivalent potential temperature θq and the total water mixing ratio Q, which are conservative variables and mix linearly, and which are known for the pure cloudy and ambient air. The partial evaporation of droplets will shift the droplet size distribution to smaller sizes (smaller Dυ) until the mixture is in equilibrium again (S = 1). The homogeneous mixing lines are obtained by varying the fraction of entrained air and calculating the saturation deficit of the mixture and the corresponding shrinking of the droplets to account for that saturation deficit.
For values of S close to 1, the inhomogeneous mixing lines become nearly indistinguishable from the inhomogeneous mixing line (e.g., Gerber et al. 2008).
b. Evaporation and mixing scales: The transition length scale
To determine whether mixing tends to be homogeneous or inhomogeneous, it is necessary to quantify the evaporation and mixing time scales for realistic ranges of cloud microphysical, thermodynamic, and turbulence properties. In this subsection we argue that the typical treatment of the two time scales misses some important aspects of the problem. We then use the revised approach in the data analysis and interpretation subsequently in the paper.
In reality, Eqs. (3) and (4) constitute a coupled system of differential equations. The two time scales τe and τp come from assumptions of constant S and constant
Irrespective of what specific value one chooses to use for the entrainment length scale, we argue that characterizing the mixing process with a single Damköhler number is ambiguous. In reality the situation is much more complex: the classical picture of the turbulent energy cascade, which has been shown to be valid in homogeneous cloud regions (e.g., MacPherson and Isaac 1977; Smith and Jonas 1995; Siebert et al. 2006b), is one of a continuous spectrum of turbulent eddy sizes. For cumulus clouds this ranges from the energy injection scale, typically hundreds of meters, down to scales at which energy is dissipated by viscosity, typically on the order of one millimeter. The mixing of temperature and vapor fields occurs over a similar range of scales, with the dissipation scales being those at which temperature and water vapor gradients are smoothed out by analogous molecular diffusion processes. Equation (5) is obtained from such inertial-range scaling arguments; it follows, therefore, that τmix varies by over a factor of 103 within that range even for constant ε. Consequently, instead of considering only a single mixing time or length scale, we recognize that there is a continuum of mixing time scales, and likewise a continuous range of Damköhler numbers. The dependence of the Damköhler number on the length scale of the considered eddy is displayed in Fig. 3. The full range of eddy sizes within the inertial subrange is plotted, from the largest typical energy injection scales to typical viscous dissipation scales. The different line styles indicate some typical values of the energy dissipation rate, subsaturation of the entrained air, and droplet diameter and concentration in the unmixed cloud.
Therefore, in any given cloud the mixing could be taking place inhomogeneously at some scales and homogeneously at other scales. For example, the results for τreact summarized in Fig. 2 for typical cloud microphysical properties do not exceed 10 s; thus, it is safe to conclude that mixing at scales greater than several meters is always inhomogeneous (i.e., Da ≫ 1), especially for the low dissipation rates common in weakly turbulent clouds. At centimeter scales, however, there exist typical microphysical conditions for which the mixing would be homogeneous (Da ≪ 1). (This concept can be thought of as an extension of the idea that a flow is characterized by two Damköhler numbers for the large and the dissipating eddies, respectively; Libby and Williams 1994, 52–57). As a result of this continuous range of scales, we alter the question from “At what scale does mixing occur?” to “At what scale does mixing make the transition from inhomogeneous to homogeneous?”.
Following Eq. (6), the energy dissipation rate, the adiabatic droplet size and concentration, and the saturation ratio of the entrained air are required for a calculation of l*. Siebert et al. (2006b) derived energy dissipation rates in cumulus clouds from balloon-borne measurements with a spatial resolution of about 8 m. The authors showed that these local energy dissipation rates (ετ, where τ indicates the time period for which ε is calculated) vary considerably within the cloud and that their probability density function (PDF) can be approximated by a lognormal distribution (see, e.g., Kolmogorov 1962). It follows that the mixing time scales and the transition length scales are not universal for a cloud but rather have to be calculated locally.
Droplet sedimentation may also influence the microphysical response to mixing because it allows droplets to move across fluid streamlines (e.g., Jensen and Baker 1989; Vaillancourt et al. 2002). This sedimentation-induced “mixing” can be thought to occur on a time scale of τsed ≈ l /uT, where l is the length scale of interest and uT is the droplet terminal speed. Droplets with diameters between 10 and 15 μm, such as are typical in small continental cumulus, fall at less than 1 cm s−1. To understand whether and on what scales gravitational sedimentation can be as efficient in mixing as turbulence, we set τsed = τmix. Specifically, for the case of high turbulence (ε = 10−2 m2 s−3), τsed = τmix at l = 10−4 m; that is, turbulent mixing is faster than gravitational sedimentation for all length scales within the inertial subrange. For weaker turbulence (ε = 10−4 m2 s−3), τsed equals τmix at about 1 cm. Thus, for the droplet sizes and terminal fall velocities under consideration in this paper, we find that only for length scales at the bottom of the inertial range can sedimentation be faster than turbulent mixing. We conclude that although the effect of gravitational sedimentation can be of importance in general (e.g., for large droplets and very weak turbulence), because of the dominating role of turbulent mixing for the conditions under consideration in this paper we limit our analysis to the turbulent mixing time scale only.
3. Experiment
The helicopter-borne instrument payload ACTOS was used to measure turbulence and cloud microphysical properties in shallow boundary layer clouds. The general system is described in detail by Siebert et al. (2003), and a discussion of the new helicopter-borne version is given by Siebert et al. (2006a); therefore, only a brief overview of the instrumentation used for this analysis and the general setup is given here.
ACTOS is equipped with an ultrasonic anemometer of type Solent HS (Gill Institute, Lymington, UK) to measure the three-dimensional wind vector with a frequency of 100 Hz. Because these measurements refer to a platform-fixed coordinate system, they have to be corrected for the payload motion and attitude, which are measured by a combination of a differential global positioning system (GPS) and inertial sensors. High-resolution measurements of the temperature are obtained from an ultrafast thermometer (UFT; Haman 1992; Haman et al. 1997). The saturation ratio S is measured with a capacitance hygrometer (HMP243, manufactured by Vaisala Oy, Helsinki, Finland). The absolute uncertainty of these measurements are given by the manufacturer as 1% at values of S below 0.9 and more than 2% at values of S above 0.9. The response time of the sensor is several seconds. During cloud transects wetting of the sensor housing might occur, and the measurements are then biased until the condensate is evaporated. Thus, there is no reliable measurement of relative humidity (RH) in the vicinity of the clouds, and the RH profile measured at the start of each flight is used to determine the RH at the respective altitude.
The liquid water content is measured with the airborne particle volume monitor (PVM-100A; Gerber et al. 1994). Single droplet measurements are performed with the modified fast forward scattering spectrometer probe (M-Fast-FSSP; Schmidt et al. 2004), which is a special version of the Fast-FSSP (Brenguier 1993; Brenguier et al. 1998). From the measurements of the M-Fast-FSSP, the droplet number size distribution dnd /dD, the mean volume diameter Dυ, the droplet number concentration nd, and the LWC can be inferred.
ACTOS is attached to the helicopter (Bell Long Ranger) by means of a 140-m-long rope. For typical true air speed (TAS) between 15 and 20 m s−1 the measurements are unaffected by the helicopter’s downwash and the payload is stable (Siebert et al. 2006a).
In this paper, data from the two first helicopter-borne experiments in Koblenz/Winningen, Germany, in 2005 and 2006 are presented. Every measurement flight started with a vertical profile under cloud-free conditions followed by horizontal flight legs flown at approximately constant altitudes. Horizontal transects through individual cumulus clouds are analyzed with respect to the entrainment/mixing process. During all flights, ACTOS was dipped into the clouds from above while the helicopter remained outside of the clouds. Thus, all measurements were made within ∼50–100 m from cloud top.
4. Data analysis and results
During the field experiment in Koblenz/Winningen in April 2005, 10 helicopter flights in warm cumulus humilis and mediocris were performed. Out of these 10 flights, three cloud transects representative of typical conditions have been selected. In addition, a fourth transect from a 2006 field experiment at the same location is presented.
The flight dates, flight leg number and pressure level, adiabatic droplet concentration and droplet diameter, and the ambient saturation ratio for the four cases are given in Table 1.
a. Uncertainties and biases of mixing diagrams
Quantifying the mixing using mixing diagrams depends on knowledge of the adiabatic droplet diameter and concentration. To obtain the value of the adiabatic volume diameter, the cloud base height for each case was determined from the video camera installed on ACTOS and compared with the cloud base determined by fitting the peak LWC values with a linear profile. We estimate the uncertainty of the cloud base to be ±100 m. Given the cloud base, the adiabatic liquid water content was calculated, and the highest measured droplet concentration of the entire flight leg is taken to be the adiabatic value. Using the profile of the adiabatic LWC and the maximum droplet concentration, we calculate the adiabatic mean volume diameter. The uncertainty of the cloud base translates into an uncertainty of the mean volume diameter of less than 1 μm at the observation altitude. This uncertainty in the derived adiabatic mean volume diameter is a systematic error; that is, it causes all points in the mixing diagram to be equally shifted along the vertical axis. However, it would not change the distribution of points relative to each other.
By assuming the maximum measured droplet concentration to be the adiabatic value, we assume that no cloud droplet activation took place above cloud base. This assumption is supported by the measurements of the droplet size distribution, which do not exhibit a second droplet mode indicative of secondary activation.
The measured values of the normalized droplet concentration in a mixing diagram can be affected by several uncertainties and biases, which will be discussed here. First, the maximum of the measured droplet concentrations throughout the flight is taken to be an estimate of the adiabatic value. Since we cannot be certain that these maximal measured droplet concentrations are truly measured in an undiluted cloud parcel, the adiabatic droplet concentration could be higher than our estimate, which would shift all data points equally toward smaller values of nd /nd,a. Second, it was found that even in undiluted, adiabatic cloud parcels, droplet concentrations can vary by ±20% (Jensen et al. 1985). The absolute uncertainty of this random error is marked by an error bar in the mixing diagram.
Third, in a recent paper Burnet and Brenguier (2007) suggested that averaging droplet counts over heterogeneous cloud regions would bias the retrieved mixing diagrams to indicate inhomogeneous mixing. Likewise, averaging the droplet concentration over partly cloudy, partly cloud-free air would underestimate the measured droplet concentration, shifting nd /nd,a values to the left and thus artificially forming an inhomogeneous mixing line. To ensure that our analysis is not biased by sampling partly cloudy, partly cloud-free air in the 1-s (∼15 m) interval over which nd and Dυ are averaged, we analyzed the droplet interarrival time distribution. For a single-particle counter like the M-Fast FSSP, the measured droplet interarrival time distribution for droplets that are distributed with perfect randomness follows an exponential distribution (e.g., Brenguier et al. 1993; Shaw et al. 2002). For this type of distribution, the standard deviation of the droplet interarrival times σ equals the mean value μ. (Beyer 1991, p. 488). Any physical process that leads to positive spatial correlations in droplet positions will result in σ/μ > 1; for example, inertial clustering in unmixed cloud regions can cause small deviations (Lehmann et al. 2007). However, if a 1-s measurement interval is affected by cloud holes, we expect σ/μ ≫ 1. We have calculated σ/μ for all four measurement cases and found that at cloud edges and inhomogeneous cloud regions, it reaches values up to 10–100. To be conservative, a threshold of σ/μ = 2 was chosen to exclude measurement intervals that are possibly affected by heterogeneities and cloud holes.
b. Case I
In Fig. 4-I, a transect through cumulus mediocris clouds measured on 24 April 2005 is shown. During this passage, the flight level was approximately constant at a pressure level of 875 hPa (∼1100 m AGL). Figure 4-I(a) shows the time series of the droplet number concentration, reaching a maximum value of 870 cm−3. In Fig. 4-I(b), the median droplet diameter (D50), along with the fifth and 95th percentiles (D5, D95), is depicted. The median diameter ranges from 7 μm in more diluted cloud parts (with respect to droplet number concentration) to 12 μm in regions where the number concentration is largest. The low-pass filtered vertical wind velocity is shown in Fig. 4-I(c). The regions with the highest number concentration at about 37 509–37 519 s are accompanied by rising air motions with w ≈ 1.7 m s−1, whereas descending air with w ≈ −1.3 m s−1 can be found in proximate regions with decreased droplet concentration. The updrafts are well correlated with temperature, which ranges from 5.8° to 6.8°C. The local energy dissipation rates displayed in Fig. 4-I(d) confirm that the ascending cloud is a region of intense turbulence compared to the surroundings, with ετ up to 10−2 m2 s−3 at the edges of the updraft region and values between 10−4 and 10−3 m2 s−3 for most of the more diluted cloud regions and the surrounding cloud-free areas. The close correlation of LWC, w, and T in the cloud core indicates it is actively growing, whereas in the outer parts, where this close correlation is lost, the cloud seems to be dissipating.
The values of (nd /nd,a) and (Dυ /Dυ,a)3 averaged over 1-s intervals are plotted on a mixing diagram in Fig. 5a. The subset of [(nd /nd,a); (Dυ /Dυ,a)3] values marked by black circles closely follows the homogeneous mixing line for entrained air with a saturation ratio between 0.90 and 0.95. The reasoning for coloring this subset differently is based on a distinct state of the flow for this subset, which will be further addressed in section 4g. For (Dυ /Dυ,a)3 < 0.6, the close correlation between the data and the homogeneous mixing lines is lost, and it is not obvious whether mixing is homogeneous or inhomogeneous.
From Fig. 5a it can be concluded that the mixing is close to the homogeneous scenario in some regions of the cloud. This is corroborated by the probability density function for droplet size, PDF(D), which is shown in Fig. 6a. The PDF is calculated for the areas labeled A and B in Fig. 4a, where B is a region that is more diluted compared to region A. Clearly, the whole size distribution taken in the more diluted region B is shifted toward smaller sizes. The fact that the size distribution as a whole is shifted toward smaller sizes indicates that evaporation has acted and that the smaller mean volume diameter in this region cannot be attributed to secondary activation.
c. Case II
In the following, a case study with mixing properties similar to those of case I is presented. The data stem from the second helicopter-based campaign in Winningen in 2006. During this flight, cumuli mediocris were sampled. Figure 4-II(a)–(d) depict, respectively, nd and LWC; D5, D50, and D95; w; and ετ. For this flight, fast temperature measurements were not available, so the temperature trace is not shown. For this section of the cloud, the flight level was constant within 150 m.
By comparing Figs. 4-II(a) and 4-II(b), one can find regions of the cloud where Dυ and nd are highly correlated (e.g., between 35 600 and 35 700 s UTC). The mixing diagram is shown in Fig. 5b. The distribution of [(nd /nd,a); (Dυ /Dυ,a)3] values has a clearly evident kink at values of (nd /nd,a) ∼ 0.6. The scatter of the data indicates inhomogeneous mixing below this value, where the points lie nearly along a horizontal line (gray circles), and homogeneous mixing for higher values of (nd /nd,a) (black circles). As in case I, the different coloring of a subset of the [(nd /nd,a); (Dυ /Dυ,a)3] values is based on the fact that this data stems from a region with different flow properties, further discussed in section 4g.
In Fig. 6b, two size distributions for regions A and B in Fig. 4-II(a) are depicted. The size distribution for the more diluted region (A) is clearly shifted toward smaller droplet sizes compared to the size distribution for region B. For regions A and B, the altitude is constant within 10 m; therefore, it can be concluded that the shift of the droplet size PDF toward smaller sizes is caused by homogeneous mixing of subsaturated air and subsequent evaporation of droplets.
d. Case III
About 80 m above the flight leg introduced in case I, and about 15 min later, another group of clouds was sampled that did not exhibit a well-defined cloud core. Figure 4-III gives an overview of the microphysical parameters and dynamic conditions for this flight leg, which was measured at a constant pressure level of 865 hPa (∼1180 m). The droplet number concentration nd is smaller compared to case I, with maximum values of 600 cm−3. LWC correlates well with nd, with maximum values of 0.2 g cm−3, indicating that the sampled clouds are more diluted compared to the case I. The median diameter is about 10 μm [Fig. 4-III(b)]. In cloudy regions, w is mostly positive [see Fig. 4-III(c)], with small maximum values below +1 m s−1. The temperature varies from 5.2° to 5.9°C, and generally LWC, w, and T are much less correlated than in case I. From Fig. 4-III(d) it can be seen that the cloud section is less turbulent compared to the first two examples, with maximum values of ετ ∼ 10−3 m2 s−3. For this cloud there is no significant trend for ετ to be higher in cloudy regions than in the ambient dry air, suggesting that the cloud is dissipating.
The corresponding mixing diagram is shown in Fig. 5c. Both droplet diameter and concentration are below the adiabatic values for that altitude. With increasing dilution, that is, toward smaller (nd /nd,a) values, (Dυ /Dυ,a)3 decreases only slightly. The scatter of the [(nd /nd,a); (Dυ /Dυ,a)3] values indicates that the mixing is close to the inhomogeneous scenario. The fact that the mean volume diameter of all samples is much smaller than the adiabatic value suggests that homogeneous mixing acted before to make the clouds as dilute as they are, but further dilution is via inhomogeneous mixing. This is again corroborated by the droplet size PDFs shown in Fig. 6c, which reveal that for the regions where the droplet concentration is diminished (curve B), the mode diameter is only slightly shifted toward smaller diameters compared to the region of relatively high concentration (curve A).
e. Case IV
As a fourth example, another time series is presented in Fig. 4-IV. During that day, the bulk microphysical parameters were similar to the previous cases, with maximum values of the droplet number concentration up to ∼600 cm−3 and maximum values of LWC of 0.6 g m−3 [Fig. 4, case IV(a)]. The median droplet diameter was around 13 μm. The vertical velocity in Fig. 4-IV(c) reveals sharp fluctuations between up- and downdrafts with values between +3.1 m s−1 and −1.4 m s−1. The time series of temperature [Fig. 4-IV(c)] is hardly correlated with w and LWC, implying that this cloud is aged and not actively growing. Because of the fluctuating vertical velocity, local values of energy dissipation rate [Fig. 4-IV(c)] reveal maxima up to 10−2 m2 s−3 but, similar to w, ετ is marked by strong fluctuations even within the clouds. In cloud-free regions, ετ drops to ∼10−5 m2 s−3.
The mixing diagram for case IV is shown in Fig. 5d. For ratios (nd /nd,a) > 0.6, the data points scatter around a horizontal line; that is, (Dυ /Dυ,a)3 is constant during dilution, which corresponds to the extremely inhomogeneous mixing scenario. For smaller values of (nd /nd,a), (Dυ /Dυ,a)3 increases with decreasing (nd /nd,a), and for the most diluted parts of the cloud even exceeds the adiabatic value. In Fig. 6d, two PDFs of the droplet size are shown, one sampled in the rather undiluted region marked as A in Fig. 4-IV(a), the other sampled in region B, which is more representative for the diluted regions of the leg. Note that the size distribution for region A is averaged over a 5-s interval, whereas the size distribution for region B is averaged over 15 s. Obviously the values of (Dυ /Dυ,a)3 > 1 in Fig. 5d correspond to a shift of the droplet size PDF to larger diameters, which is opposite to the effect observed in cases I and II and will be discussed in more detail later.
It is obvious that the mixing diagrams for cases I and II and cases III and IV differ significantly, although their microphysical characteristics (e.g., mean droplet number and diameter) are similar. In the two former cases, the mixing is rather homogeneous, with small droplets in the most diluted cloud regions, whereas in the latter two cases it is extremely inhomogeneous, and in case IV the largest droplets are found in the most diluted cloud parcels. What causes the difference in the mixing characteristics of the four cases? An apparent difference between the four clouds is their degree of dilution.
There are several reasons to believe that the cloud presented in cases I and II are younger and less diluted than the cloud in cases III and IV. For example, it can be seen in Fig. 4-Ia that the regions with the highest LWC are well correlated with positive vertical velocity, temperature and highest energy dissipation rates. This indicates that the cloud is in an actively growing state. On the other hand, for the cloud presented in cases III and IV, the close relationship among LWC, w, T, and ετ is lost. Even though the energy dissipation rate for case IV is quite high, it seems that a well-defined updraft core no longer exists. In the following section, we attempt to corroborate these findings through a thermodynamic analysis that quantifies the nature and degree of cloud dilution.
f. Analysis of dilution—Paluch diagrams
Entrainment and its thermodynamic consequences can be observed using conserved-variable diagrams. In the following, Paluch diagrams for cases I, III, and IV are presented. Because there is no reliable fast temperature measurement for the flights presented in case II, no Paluch diagram is shown for this flight. Detailed information on how these diagrams are computed and on their interpretation can be found in Paluch (1979), Betts (1982), Taylor and Baker (1991), and Heus et al. (2008) and references therein.
In Fig. 7 the Paluch diagram for case I is depicted. The thin black solid line indicates the ambient profile measured during the final descent of the flight. The arrows mark the [θq, Q] values measured during the cloud passage shown in Fig. 4-I, where black upward-pointing arrows denote raising air motion and gray downward-pointing arrows indicate negative vertical velocity, and where the length of the arrows is proportional to the vertical velocity. A least squares fit indicates the source level of the air to be around 960 hPa, whereas the cloud base was as 914 hPa.
As expected, the highest positive velocities are found in regions with high [θq; Q] values, and small negative velocities are found almost exclusively in more diluted regions, with lower [θq; Q] values. The fact that there are some [θq; Q] values in the cloud that are identical with the ambient conditions in about 940 hPa shows that at least some regions of the cloud have risen adiabatically more than 500 m with very little dilution by the surrounding air. In contrast, the Paluch diagrams for cases III and IV (Figs. 8 and 9) indicate that these clouds are highly diluted, with many values of [θq; Q] close to the saturation line (i.e., the clouds are close to complete evaporation). In these regions of high dilution, the vertical velocity is mainly negative; however, there are a few samples in case IV (Fig. 8) where [θq; Q] is close to the saturation line (i.e., highly diluted), but with positive vertical velocities. For case IV, the least squares fit was derived from the entire flight leg and points to a source region at 987 hPa.
In conclusion, the analysis of the Paluch diagrams corroborates the finding from the previous section that the cloud in case I is “younger” and significantly less diluted than the clouds in case III and IV.
g. Transition length scale
In the following, we will examine the transition length scale introduced in section 2 for the four cases. As mentioned before, all cases are quite similar with respect to their microphysical properties. Moreover, all presented cases have similar mean energy dissipation rates as well, being 8 × 10−4, 6 × 10−4, 4 × 10−4 and 2 × 10−3 m2 s−3, respectively. Figure 10 shows the time series of the transition length scale [Eq. (6)] calculated using nd,a, 
Figures 10c and 10d correspond to cases III and IV resembling inhomogeneous mixing. In Fig. 10c, the local transition length scales are small, with maximal values smaller than 5 cm. The low values of l* suggest a vanishing range of scales over which homogeneous mixing can occur and are consistent with the observations of inhomogeneous mixing for this case. However, the average transition length scale for case IV depicted in Fig. 10d is comparable to the values derived for cases I and II. Because of the generally high values of the transition length scale, in this case we would expect the mixing to be homogeneous; however, this is in disagreement with the mixing diagram for this case (Fig. 5d). We try to further elucidate this disparity in the following section. Nevertheless, for the region of elevated transition length scale around 46 527 s (marked by a black bar) we find
The main finding of the analysis of the transition length scale is the presence of regions with elevated energy dissipation rates and transition length scales within a single cloud. Within these regions, we find the mixing to be homogeneous.
5. Discussion and conclusions
In this paper, we analyze the entrainment/mixing characteristics of shallow cumulus clouds, one of the key processes in the evolution of droplet size distribution. Mixing has been suggested to affect clouds’ radiative properties and to play a role in the formation of warm rain.
First, we introduced a new conceptual framework of a transition length scale that separates the inertial subrange into a range of length scales for which inhomogeneous mixing dominates and a range for which homogeneous mixing is prevalent. In the second part of the paper, four case studies were presented and their mixing characteristics were analyzed using the new theoretical framework.
A major conclusion of this work is that a description of the mixing process by a single Damköhler number is not sufficient because it is based on the assumption that mixing is determined by a single length scale. Following the concepts of classical turbulence theory, a continuum of length scales and thus mixing time scales exists during subsequent mixing after dry air has been entrained into the cloud at the entrainment length scale. Therefore, we suggest defining a transition length scale above which the mixing is inhomogeneous and below which the mixing is homogeneous. If the transition length scale lies within the inertial subrange, both homogeneous and inhomogeneous mixing occur. A small transition length scale yields a high probability for the bulk mixing characteristics to appear primarily inhomogeneous, whereas a large transition length scale implies a high probability for homogeneous mixing to occur.
In addition to the new concept of a transition length scale, we have attempted to clarify the connection between the droplet evaporation time and the phase relaxation time. While these are usually calculated by assuming either the saturation ratio or the integral radius to be constant, they are in fact connected through a coupled differential equation system that describes the adjustment of the saturation ratio to an evaporating droplet population, and vice versa. The time scale that describes this coupled differential equation system can differ significantly from the traditionally defined droplet evaporation time and the phase relaxation time.
Generally, the four case studies presented in this paper indicate that both homogeneous and inhomogeneous mixing occur and can be observed in cumulus clouds. Specifically, we found both mixing scenarios to coexist even within the same cloud.
In a recent paper, Burnet and Brenguier (2007) suggested that measurements rarely resemble homogeneous mixing because the spatial averaging of the droplet-sizing instrument will bias the measurements of an actually homogeneous mixed cloud to indicate inhomogeneous mixing. In contrast to these measurements, we found regions in cumulus clouds where the microphysical measurements clearly indicate mixing close to the homogeneous scenario. However, these regions seem to be limited to the more turbulent cloud core. By analyzing the droplet interarrival time distribution and excluding samples with apparent cloud holes, we ensured that this result is not an artifact introduced by spatial averaging.
In clouds that were highly diluted and close to dissipation, we found that the mixing is inhomogeneous. We note, however, that entrainment and mixing of ambient air with a saturation ratio close to 1 leads to a reduction of the droplet concentration with little change in droplet size and therefore would also lead to a distribution of [(nd /nd,a); (Dυ /Dυ,a)3] values that is nearly indistinguishable from the inhomogeneous mixing line. The fact that the [(nd /nd,a); (Dυ /Dυ,a)3] values form an inhomogeneous mixing line at a smaller than adiabatic mean volume diameter suggests that homogeneous mixing had acted at some point before. The identification of cloud samples where a reduction of droplet number was not accompanied by a reduction in droplet size is important in the context of precipitation formation in warm clouds. Recent results from numerical simulations of Lasher-Trapp et al. (2005) showed that entrainment and mixing contributes to the broadening of the droplet size distribution toward smaller and larger sizes, and that larger than adiabatic droplets are formed when an inhomogeneously mixed parcel is subject to an ascent. Similarly, simulations by Krueger et al. (2006) using an explicit mixing parcel model indicate that ascending mixed parcels can produce larger droplets than during an adiabatic ascent. These large droplets are possible precipitation embryos. The resemblance between Fig. 5 of Krueger et al. (2006) and Fig. 5d in this paper is striking. However, we cannot rule out that other mechanisms acted to produce the large droplets in the most diluted cloud regions of case IV. Among the possible mechanisms is enhanced condensational growth through radiative cooling at cloud top; however, because of the limited residence time of cloud droplets at cloud top this is not thought to be significant in cumulus clouds (Harrington et al. 2000). Another possible mechanism is the growth of the largest droplets through collision and coalescence; however, for the droplet sizes measured in case IV the collision efficiency is thought to be small.
By utilizing ACTOS’ ability to measure local dynamical properties, we have tested the theoretical concept introduced in section 2b and calculated local transition length scales. For the two cases I and II, we find that the regions that exhibit homogeneous mixing correlate with regions of increased transition length scale, supporting our concept. These regions coincide with the turbulent cloud core. For clouds that do not possess a defined updraft core (cases III and IV) and are more aged and diluted, we find the mixing to be inhomogeneous. However, especially in case IV, the values of calculated local transition length scale are comparable to the cases that mix homogeneously, challenging our concept. We interpret this discrepancy as follows: The logic of the mixing diagram analysis and the calculation of transition length scales follow from the idea that an initially adiabatic cloud mixes with environmental air and the microphysical properties evolve homogeneously or inhomogeneously, depending on the thermodynamic and dynamic conditions before the mixing event. As mixing progresses, however, the cloud becomes diluted and the cloud conditions evolve, possibly in such a way that the nature of the mixing itself is altered. The further the mixing proceeds, the more difficult it may be, based on in situ measurements alone, to understand the current state of the cloud in terms of a simple view of homogeneous or inhomogeneous mixing. Taken together, the data profiles (Fig. 4), mixing diagrams (Fig. 5), and Paluch analyses (Figs. 7 –9) suggest that the clouds sampled in cases I and II were not yet strongly diluted, at least in parts, and a reasonably clear interpretation of the microphysical and turbulent environment emerges. The clouds sampled in cases III and IV, however, were likely highly diluted and the situation is therefore more complex. How might the early mixing cause the cloud to evolve in such a way that would explain the observed patterns in these two cases? We can suggest the following plausible changes as mixing progresses:
The value of ε decreases as the cloud becomes less buoyant because of entrainment. We should note that there is evidence that initial entrainment may in fact lead to local increases in ε near cloud edges, but ultimately the dissipation rate will decrease. A decrease in ε, all else being equal, leads to a decrease of l* and therefore tends to favor inhomogeneous mixing (see Fig. 3).
The saturation ratio S of the local environment outside the cloud will increase, possibly approaching unity. This leads to large possible increases in τe, and therefore either increases l* or, more likely, leads to a situation in which τp is the dominant (meaning smaller) reaction time scale (see Fig. 2).
The value of
D likely decreases, albeit perhaps only slightly, depending on the initial form of mixing. This tends to increase τp, which is likely the relevant reaction time scale because of humidification as just described, and thereby favors an increased role for homogeneous mixing.The value of nd decreases as a result of dilution and the possible complete evaporation of some droplets. This tends to increase τp, again assumed to be the reaction time scale, and therefore to favor homogeneous mixing.
At least two of the four effects of mixing tend to favor subsequent homogeneous mixing as the cloud evolves; one of them favors subsequent inhomogeneous mixing. Without more detailed calculations it is difficult to determine which effects will dominate. The presumed tendency to evolve to a state of homogeneous mixing occurs simultaneously with the vanishing distinction between homogeneous and inhomogeneous mixing. This can be seen in Figs. 1 and 5: as S → 1 the idealized homogeneous mixing curves approach the inhomogeneous mixing curve. The limit, of course, is that entrainment of saturated air (S = 1) results in dilution and therefore reduction of nd, but no evaporation and decrease in
Another limitation of our analysis is the inability to measure the saturation ratio in the cloud vicinity with a sufficient spatial resolution and precision. Instead, we use the vertical profile of S measured at the beginning of each flight. It is likely that this profile, taken some distance away from the analyzed clouds, does not exactly represent the air surrounding the cloud that is being entrained. Often, cumulus clouds are surrounded by thin shells of air with increased saturation ratio, also referred to as humidity halos (e.g., Lu et al. 2003; Heus and Jonker 2008). This might significantly affect the derived time scales and thus the transition length scale, especially for the more aged clouds presented in case IV.
Acknowledgments
During the review process of this paper, our colleague and first author Katrin Lehmann died in a tragic hiking accident. This paper, which is an extension of Katrin’s Ph.D. work, will therefore represent one of her major scientific achievements. We have lost a wonderful scientist, colleague, and friend, and we dedicate this paper to her memory.
Part of this work was funded by the Deutsche Forschungsgemeinschaft (DFG Projekt WE 1900/7-3). Participation by R. Shaw was supported by the Alexander von Humboldt Foundation and by U.S. National Science Foundation Grant ATM0535488. We thank Rotorflug GmbH and their pilots for great helicopter flights, and enviscope GmbH for their support during the field campaigns. We are grateful to S. Schmidt and M. Wendisch for their work on developing the M-Fast-FSSP. We thankfully acknowledge comments from three anonymous reviewers who helped improve the manuscript.
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Example of a mixing diagram. The solid lines shown the dependence of (Dυ /Dυ,a)3 on (nd /nd,a) during homogeneous mixing. Because the homogeneous mixing lines depend on the saturation deficit of the ambient air, four exemplary curves are shown, each corresponding to different values of S.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
Temporal evolution of the relative droplet size (D/D0, gray curve) and saturation ratio S (black curve) for a coupled microphysical–thermodynamic system. The droplet evaporation time and the phase relaxation time are indicated by the dotted and solid vertical lines, respectively.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
Damköhler number as a function of the entrainment length scale, for four exemplary values of energy dissipation, saturation ratio, droplet concentration, and size. The transition length scale at Da = 1 is indicated for one of the examples. All eddies with characteristic sizes to the right of l* will mix inhomogeneously, whereas all with sizes smaller than l* will mix homogeneously.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
For cases I, II, III, and IV, (a) droplet number density (black line, left scale) and LWC (gray line, right scale) with 1-s resolution, (b) mean volume diameter with 1-s resolution, (c) vertical wind velocity (black line, left scale) and temperature (gray line, right scale) at 0.1-s resolution, and (d) local energy dissipation rates. For the regions marked A and B, PDFs of the droplet size are calculated and displayed in Fig. 6-I. Times are displayed as seconds of the day in UTC.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
Mixing diagrams for cases (a) I, (b) II, (c) III, and (d) IV. The solid lines show values of (nd /nd,a) and (Dυ /Dυ,a)3 calculated for homogeneous mixing of environmental air with different saturation ratios S following the procedure described in section 2a. The subsets of the data marked by black circles in (a),(b), and (d) stem from a specific cloud region with high transition length scale, as discussed in section 4g. The uncertainty of the adiabatic droplet concentration is marked by an error bar in the lower right corner of each mixing diagram.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
Example droplet number size distributions expressed as PDFs for the regions marked as A and B in time series of droplet concentration for cases I, II, III, and IV. The number of droplets that passed the center of the sampling cross section and that were therefore used to calculate the size distribution are indicated in each panel.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
Paluch diagram constructed for case I. The thin solid black curve shows values of θq and Q measured during the descent. Upward-pointing black arrows represent in-cloud measurements of θq and Q with positive w, whereas downward-pointing gray arrows mark in-cloud measurements with negative w. The length of the arrows is proportional to the vertical wind speed. Gray dots are isopleths of virtual potential temperature, denoting isolines of density, where the respective labels can be found as gray numbers at the bottom of the panel. The dashed line indicates the saturation line (i.e., all samples below that line are subsaturated). The solid straight gray line indicates a least squares fit to the in-cloud data. The data is processed with 0.1-s resolution.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
As in Fig. 7, but for case III.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
As in Fig. 7, but for case IV.
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
Time series of local transition length scales for the four cases previously presented. The black curves indicate local transition length scales calculated using the ambient saturation value, the adiabatic droplet concentration and size, and local energy dissipation rates. The thick bars denote regions with high transition length scale (peak values l* > 0.1 m).
Citation: Journal of the Atmospheric Sciences 66, 12; 10.1175/2009JAS3012.1
Flight dates and adiabatic droplet concentration (na) and size (Da), saturation ratio of the ambient air (S), flight level (pobs), and cloud base pressure (pCB) for the four different cases analyzed in this paper.