1. Introduction
Shallow cumuli are the prevailing cloud type of the maritime trade wind region. Despite having diameters of less than 1 km, these small clouds play a major role in the maintenance of the tropical circulation, as they cover a large amount of the oceans (Riehl et al. 1951; Stevens 2005). They are important for the transport of heat, moisture, and momentum from the ocean surface to the free troposphere (e.g., Tiedtke 1989). At the same time, they influence boundary layer height and the vertical profiles of temperature, relative humidity, and wind (Bretherton et al. 2004).
Shallow trade wind cumuli have been studied for more than half a century (e.g., Stommel 1947; Malkus 1949; Warner 1955; Squires 1958). However, there are still numerous open questions with regard to their evolution and properties. One of them concerns the process of entraining subsaturated environmental air into the cloud and subsequent mixing with the cloudy air, which was first studied by Stommel (1947) and Squires and Warner (1957). The mixing process is described by the Damköhler number 
The mixing type is crucial for the evolution of cloud microphysical properties, dynamics, lifetime, and optical properties (Brenguier et al. 2000; Grabowski 2006) and is of importance for the first indirect aerosol effect (e.g., Pawlowska et al. 2000).
Numerous observational and numerical studies have been performed to investigate the mixing process of shallow convective clouds. Up to now, there is no consensus about the prevailing mixing type in shallow cumuli. Some observations suggest that homogeneous mixing is dominant (e.g., Jensen et al. 1985; Jensen and Baker 1989). Others indicate the inhomogeneous mixing scenario as the prevailing process (e.g., Gerber 2006; Pawlowska et al. 2000). Some studies do not clearly differentiate between the mixing processes (e.g., Gerber et al. 2008). Small et al. (2013), Burnet and Brenguier (2007), and Lehmann et al. (2009) found both homogeneous and inhomogeneous mixing in shallow cumuli depending on the cloud location. Small et al. (2013) observed that the mixing tends toward the homogeneous scenario for upper parts of the clouds and is rather inhomogeneous in the lower cloud parts. Similar results were found in the modeling studies of Jarecka et al. (2013). Burnet and Brenguier (2007) and Lehmann et al. (2009) found some indication that the mixing process was connected to the dilution and, therefore, to the evolutionary stage of the clouds.
These findings motivated the current detailed study of the mixing process in shallow trade wind cumuli as a function of the cloud evolutionary stage. Observations of Katzwinkel et al. (2014) suggest three stages in shallow cumulus evolution: “actively growing,” “decelerated,” and “dissolving.” This classification is based on updraft velocity and buoyancy in the cloud interior. Moreover, Fig. 8 of Katzwinkel et al. (2014) conceptually identifies large-scale dynamics of clouds, which are highly relevant for the microphysical response considered in this manuscript.
Data obtained during the Clouds, Aerosol, Radiation and Turbulence in the Trade Wind Regime over Barbados (CARRIBA) campaign (Siebert et al. 2013) are analyzed in this paper. CARRIBA is based on observations obtained by the helicopter-borne measurement payload Airborne Cloud Turbulence Observation System (ACTOS; Siebert et al. 2006a), plus a radiation measurements system (Werner et al. 2013, 2014). ACTOS yields high-resolution data of turbulence, thermodynamics, and cloud microphysical parameters. These high-resolution data were used to study the mixing process in shallow cumuli during different stages of their evolution. This is particularly challenging for cumuli with diameters on the order of a few hundred meters, which are common in the trade wind regime and are difficult to characterize by fast-flying research aircraft.
In section 2, a brief introduction of the CARRIBA campaign and the ACTOS payload is given. In section 3, the methods used in this study are introduced. Cloud microphysical properties, as well as an illustration of the mixing diagram with dependence on the cloud evolution, are given in section 4. This section also contains a statistical analysis of the mixing process based on mixing diagrams for clouds of different evolutionary stages. Section 5 provides a summary and the conclusions.
2. Experimental
a. The CARRIBA campaign
CARRIBA consisted of two phases, which were performed near Barbados in November 2010 and April 2011. The analysis presented here is focused on the November campaign. Characteristics of the Caribbean are the steady and uniform meteorological conditions in terms of stratification, almost constant sea surface temperature, an inversion height of approximately 3000 m, and prevailing easterly winds. All helicopter-borne measurements started with an initial profile up to approximately 2500 m followed by a number of horizontal legs in different heights under cloud-free conditions. The second half of a measurement flight was typically used to sample clouds around cloud top at relatively constant heights. However, because of changing cloud-top heights, flying at a constant height would result in a low number of sampled clouds. Therefore, the sampling altitude was adjusted, and the sampled clouds are normalized by their adiabatic value for further analysis (see section 3a). The majority of the clouds were sampled at an altitude range between 1000 and 2000 m. The cloud data were collected during horizontal flight patterns at about 100 m below the top of the cumuli. Deeper penetrations into the clouds or vertical profiling were not possible because of flight restrictions of the helicopter. The observed shallow cumuli had an average cloud-base height of approximately 500 m. Cloud droplet diameters ranged between 5 and 40 µm on days that were characterized by relatively high aerosol conditions. On days with clean conditions, often a drizzle tail could be observed with droplet sizes up to 80 µm. Also, many of the observed size distributions showed a bimodal character. Mean cloud droplet number concentrations ranged between 70 and 135 cm−3. More information about general conditions during CARRIBA can be found in Siebert et al. (2013).
b. ACTOS payload
Meteorological, turbulence, aerosol particle, and cloud microphysical parameters were sampled by instruments installed on the helicopter-borne measurement platform ACTOS (Siebert et al. 2006a, 2013). ACTOS is approximately 5 m long and has a mass of about 200 kg. It is attached to the Spectral Modular Airborne Radiation Measurement measurement system (SMART-HELIOS; Werner et al. 2013, 2014) by means of a 140-m-long rope. SMART-HELIOS, in turn, is attached to the helicopter using a 20-m-long rope. The helicopter flies with a true airspeed of about 20 m s−1, which is a compromise between a stable flight position and no influence from the rotor downwash. ACTOS is an autonomous system where data are sent to the helicopter via a telemetry connection enabling online monitoring of the data. The time for a measurement flight is limited to 2 h, and the platform can be operated up to a maximum altitude of about 3000 m.
An ultrasonic anemometer (Solent HS, Gill Instruments Ltd., United Kingdom) measured the three-dimensional wind vector. The data were corrected for attitude and platform motion by means of a GPS-aided inertial navigation system. In-cloud temperature measurements were carried out with an ultrafast thermometer (UFT) with an accuracy of 0.2 K (Haman et al. 1997). The device is based on a resistance wire with a diameter of 2.5 µm. A shielding rod in front of the wire prevents droplet impacts on the sensor, which yields reliable temperature measurements within clouds (e.g., Siebert et al. 2006a). To measure absolute air humidity, an open-path infrared absorption hygrometer (LI-7500, LI-COR Inc., United States) and a dewpoint hygrometer (TP 3-S, MeteoLabor, Switzerland) were used. All instruments have a temporal resolution of at least 100 Hz, which yields a spatial resolution down to the decimeter scale. More detailed information on these instruments can be found in Siebert et al. (2006a). Measurements of cloud condensation nuclei (CCN) were obtained using a mini-CCN counter (Roberts and Nenes 2005) operated at a supersaturation of 0.26% with an uncertainty of about 10%, which is mainly limited by the counting statistics.
3. Methods
a. Estimating adiabatic cloud properties
At cloud base, a certain number of aerosol particles will be activated to cloud droplets. During adiabatic lifting of the cloudy air parcel, no further activation occurs, and the droplet number concentration remains constant at its adiabatic value 
Because no measurements with ACTOS at cloud base are available, 
As a next step, we estimate the error of LWCad resulting from an error of 50 m in cloud-base estimation. For simplicity, we estimate this error for only one selected measurement height of 1.5 km, which is a typical height for our cloud observations. With 
This discussion of the accuracy of the measurements is important for the following analysis of the mixing diagrams.
b. Classification of shallow cumuli with respect to evolution stage
A typical feature of the trade wind area is the concurrent presence of clouds at different stages of evolution. Actively growing clouds can be observed next to dissolving ones. This is caused by the permanent triggering of development of shallow cumuli under almost-constant thermodynamic and dynamic conditions. The three cloud life cycle stages (actively growing, decelerated, and dissolving) following Katzwinkel et al. (2014) are defined by the 90th percentiles of vertical wind velocity w and buoyancy acceleration B in the cloud interior. As in Katzwinkel et al. (2014), the cloud interior is defined as the region with LWC > 0.2 g m−3. The actively growing clouds are characterized by positive buoyancy and updrafts (
This classification is based on the measured vertical velocity and the calculated buoyancy 
A total of 177 active, 91 decelerated, and 32 dissolving clouds were classified. The disproportion results from the flight strategy aiming at well-developed shallow cumuli, as well as from the strict criteria for individual clouds. Some of the dissolving clouds might have too-low LWC or might be too narrow in diameter. The analysis of this work focuses mainly on the actively growing and the dissolving clouds, as the decelerated cloud stage can be seen as an intermediate state between these two cloud stages.
The classification has some limits because of the fact that most of the measurements are performed about 100 m below cloud top. It might occur that the top of an actively growing cloud was sampled, but this region might belong to an overturning eddy, and therefore the cloud would be falsely classified as a dissolving cloud.
c. The mixing diagram
d. Statistical mixing analysis
To classify the mixing process of all considered clouds, two methods are applied. The first is based on the mixing diagrams described in the previous subsection. The spread in the droplet size 
To estimate a theoretical maximum value of the 
4. Data analysis
a. Cloud stage classification
In Fig. 1, examples are given for clouds from the actively growing and the dissolving cloud stage. The red lines indicate the cloud interior (LWC > 0.2 g m−3). Figures 1a and 1b show time series (distance) of LWC for each cloud. The actively growing cloud reveals large values of LWC with mean values of approximately 1 g m−3, whereas the dissolving cloud shows low values of LWC ≈ 0.3 g m−3. The edges of the clouds are characterized by low LWC because of the definition of cloud edges, and they vary in extent. In Figs. 1c and 1d the time series of w are shown for each cloud type. The active cloud has an updraft in the interior region with maximum values of about 3 m s−1. The dissolving cloud has negative vertical velocities of about −2 m s−1 in its interior. Time series of B are shown in Figs. 1e and 1f, with the actively growing cloud being positively buoyant with maximum values around 0.01 m s−2, whereas the dissolving cloud is negatively buoyant with a mean of B ≈ −0.005 m s−2 in the interior.
Horizontal cross section through single clouds showing (a),(b) liquid water content; (c),(d) vertical velocity; and (e),(f) buoyancy for actively growing and dissolving clouds. Red lines mark the cloud interior region, and x is the horizontal distance during the helicopter flight.
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
In Fig. 2, the statistics of the 90th percentiles of w and B in the cloud interior region are shown for all classified clouds. Each box represents 50% of the data and indicates the median value by a horizontal line. Whiskers denote the 12.5th and 87.5th percentiles and, thus, represent 75% of the data. Actively growing clouds have high vertical velocities with a median value of about 2 m s−1. They also show a large spread in w where 50% of the data lie in a range of approximately 1.3–3.1 m s−1. The dissolving clouds are characterized by a median w of around −0.5 m s−1 and a small spread where 50% of the data are in a range from −0.2 to −0.8 m s−1.
Box-and-whisker plots of 90th percentile of (a) vertical wind velocity and (b) buoyancy acceleration. The box represents, as in all box-and-whisker plots in this work, 50% of the data and contains the median value. The whiskers mark the 12.5th and 87.5th percentile, thus representing 75% of the data.
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
Buoyancy acceleration is positive for the active clouds with a median value of 
b. Cloud microphysical properties
Figure 3 shows the variability of 
Box-and-whisker plots of (a) normalized cloud droplet number concentration, (b) normalized mean volume diameter, and (c) normalized liquid water content for actively growing and dissolving clouds. Open boxes represent cloud-interior data, and gray-shaded boxes represent cloud-edge data.
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
The behavior of 
These findings can already be used to get a first idea of the mixing processes occurring in these clouds: Burnet and Brenguier (2007) analyzed data from the Small Cumulus Microphysics Study (SCMS) experiment and found that these clouds showed features of both homogeneous and inhomogeneous mixing. They observed homogeneous mixing when the cloud LWC was slightly diluted and inhomogeneous mixing when the LWC was more diluted. Therefore, it is likely that the dissolving clouds mix rather inhomogeneously, whereas the actively growing clouds tend to mix more homogeneously.
The differences in 
In Fig. 3 the mean droplet size is analyzed by using height-independent normalized values. As a next step, the droplet size distribution is investigated for the active and dissolving cloud stages by looking at the mean droplet size 
Profile of (a) mean droplet diameter, (b) mean spectral width, (c) spectral dispersion, and (d) the difference between 
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
When analyzing the cloud droplet size distribution of the observed clouds, three different types of size distributions could be found. Some clouds showed monomodal distributions with only one distinct peak at small droplet sizes in the range between 10 and 20 µm. Other clouds showed a bimodal structure in their droplet size distributions, with a small mode peaking at about 10 µm. To measure the significance of the individual modes, Gaussian fits are applied, and the absolute number of droplets 
Example of cloud droplet size distributions for (a) a monomodal distribution, (b) a bimodal distribution with nonsignificant contribution of droplets formed by secondary activation, and (c) a bimodal distribution with a significant small mode formed by secondary activation. The solid red lines represent the fitted distribution, and the dashed red lines represent the individual modes. The ratio of the number of droplets in the small mode to the number of droplets in the main mode is given for the bimodal distributions.
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
c. Mixing diagram analysis
An example of a mixing diagram is shown in Fig. 6 for the actively growing and the dissolving cloud stages, where each diagram contains data from one individual cloud. The colors indicate the mixing fraction χ of entrained air, with 
Mixing diagrams of (a)–(c) actively growing clouds and (d)–(f) dissolving clouds. Solid lines starting at [1, 1] represent the variation of 
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
Figures 6a–c show three examples of actively growing clouds, which are not significantly affected by secondary activation, according to our definition (section 4b). Figures 6a and 6b show a clustering of data points around the saturation line for 99% RH, indicating homogeneous mixing, but at a higher RH than the environment (RH = 80%). Data points with small χ are derived from the interior of the cloud and also show the largest values of 
Figures 6d–f show mixing diagrams for dissolving clouds, where data points with the lowest χ belong to the largest 
Based on Fig. 6, the hypothesis is made that there is a trend for a transition from homogeneous toward inhomogeneous mixing as the clouds evolve from the actively growing to the dissipating stage. In actively growing clouds the droplet size and droplet number concentration decrease as a result of homogeneous mixing. When the cloud reaches the dissipating stage, the droplet size deviates significantly from its adiabatic value because of the reduction in size when the cloud was still at the actively growing stage. Because of inhomogeneous mixing, the cloud droplet size remains then rather constant, but the droplet number concentration decreases strongly in the dissolving clouds, and the data points are located almost horizontally at lower values of 
d. Statistical mixing analysis
The spread in the cubed droplet size 
PDF of the spread in droplet mean volume diameter for (a) actively growing clouds and (b) dissolving clouds, which show no signs of significant secondary activation. The median value is given in each panel.
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
In Fig. 8, the analysis of the 
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
The actively growing clouds with purely monomodal size distributions show a relatively large
To summarize, we found that when only taking into account clouds with purely monomodal droplet size distributions, the actively growing clouds have a larger decrease in diameter as the number concentration decreases compared with the dissolving clouds. It is concluded that the homogeneity of mixing decreases with cloud stage from active to dissolving, even if actively growing clouds are not close to pure homogeneous mixing (i.e., α ≈ 1.66). Also, part of the deviation from perfect homogeneity may likely be due to more complicated factors, such as secondary activation, which tend to mask the signature of homogeneity.
5. Discussion
The mixing diagrams presented in section 4 show features that cannot be explained using the theory of homogeneous and inhomogeneous mixing. The existence of data points with relatively constant 
a. Secondary activation
Secondary activation of CCN is a well-known process that can explain a significant proportion of small droplets that is sometimes observed at higher altitudes than the cloud base (Baker and Latham 1979; Paluch and Knight 1984; Austin et al. 1985; Hill and Choularton 1985; Stith and Politovich 1989; Paluch and Baumgardner 1989; Pontikis and Hicks 1993; Brenguier and Chaumat 2001). Also, in large-eddy simulation (LES) studies, this phenomenon was observed to produce small droplets in the mixing zone well above cloud base (Slawinska et al. 2012). The occurrence of secondary activation seems a plausible explanation because of the weak condensational sinks due to low cloud droplet number concentrations of 100–200 cm−3 observed during CARRIBA. There are different possible mechanisms to obtain [
For the discussion of secondary activation, the evolution of supersaturation, described by 
In the case of homogeneous mixing, the environmental and cloudy air mix instantaneously. Because of the reduction in droplet size and number concentration, the adjusted 
During inhomogeneous mixing, a large portion of the environmental air that is mixed into the cloud remains separated from the cloudy air for the time span of the turbulent mixing time scale 
Figure 4d shows a profile of the difference between the mean volume diameter 
Krueger (2008) studied the entrainment mixing process using an Explicit Mixing Parcel Model, which calculates the growth of thousands of individual droplets based on each droplet’s local environment. By entraining CCN with a number concentration identical to the CCN concentration at cloud base, as well as subsequent lifting of the mixed parcel, similar results were obtained as in Figs. 6d and 6e. In their case [Fig. 12 in Krueger (2008)] the normalized droplet size also decreased to about 0.4, whereas the droplet number was close to that expected in an adiabatic parcel.
Even without mixing, secondary activation would be possible if there were some larger cloud droplets that can serve as collector droplets in the cloud. By collision and coalescence with smaller cloud droplets, N would be reduced, which would again lead to an increase in 
b. Humid shells
Another explanation of the observed large changes in LWC and N but nearly constant 
Figure 9 shows a cross section of w (black line) and the absolute humidity a (blue line) in representative clouds at the actively growing and dissolving stage. The cloud in terms of LWC is shown as a gray area, and the edges of each cloud are marked by a solid vertical line. It is quite obvious that, for each cloud, a region directly adjacent to the cloud edges is found, where a is still relatively high compared to environmental values. That is, the clouds are surrounded by a humid shell with a horizontal dimension on the order of a few tens of meters.
Cross section through (a) an actively growing cloud and (b) a dissolving cloud showing LWC (gray bars), vertical velocity (black line), and absolute humidity (blue line). Solid vertical lines mark the edges of the cloud and the dashed lines mark the edges of the humid shell. The light-gray-shaded rectangles represent the region of the humid shell.
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
As a first step, we are interested in the general existence of such a humid shell for the different cloud stages. Therefore, the following simple criteria are applied to identify a humid shell: the shell has to be positioned in the downdraft area outside the cloud (LWC < 0.02 g m−3), and a has to be larger than the median value of a in the nearby cloud-free environment. In Fig. 9, this humid shell region is shaded light gray in the graphics, and its end is marked by a dashed vertical line. The horizontal extent of these shells is highly variable and ranges from 20 to 1000 m and also differs for downshear and upshear sides of the cloud.
PDF of 
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
Closely linked to humid shells are what are commonly referred to as subsiding shells (Telford and Warner 1962). Heus and Jonker (2008) analyzed subsiding shells around shallow cumuli using LES, and Katzwinkel et al. (2014) observed such shells around shallow cumuli sampled during CARRIBA. Both found that the horizontal extension of these subsiding shells increases with increasing cloud age.
Box-and-whisker plot of the width of the humid shell around the clouds obtained for actively growing and dissolving clouds.
Citation: Journal of the Atmospheric Sciences 72, 4; 10.1175/JAS-D-14-0230.1
As the humid shells grow to the length of 
It is often stated that it is impossible to distinguish between homogeneous and inhomogeneous mixing if the environmental air is close to saturation. The classical picture of inhomogeneous mixing is entrainment of unmodified environmental air and subsequent complete evaporation of portions of the cloud, while the cloud core is preserved, until the mixture of clear but humidified air is able to mix thoroughly with the cloudy air. The result is simple dilution on the mixing diagram. The humid shell can be viewed as the product of that initial stage of inhomogeneous mixing, and the subsequent mixing of humid-shell air with the cloud can be viewed as the final stage of inhomogeneous mixing. Therefore, we consider the humid shell as part of the inhomogeneous mixing process. This is consistent with the schematic of Fig. 8 in Katzwinkel et al. (2014), where the evolution of the different cloud regions over time is illustrated.
6. Summary
The mixing process of environmental air into shallow trade wind cumuli is analyzed based on data sampled during the CARRIBA campaign in November 2010 near Barbados. The analysis is based on helicopter-borne measurements performed with the payload ACTOS with a spatial resolution in the decimeter range. The investigated clouds were classified into three stages of their life cycle according to criteria based on vertical velocity w and buoyancy acceleration B in the cloud interior (Katzwinkel et al. 2014). A dataset of 177 actively growing, 91 decelerating, and 32 dissolving clouds has been identified, and this study focuses on the actively growing and dissolving stage. This work presents the first detailed analysis of cloud microphysical response to entrainment as a function of cloud evolutionary stage.
It is found that the cloud droplet number concentration N and cloud liquid water content (LWC) decrease strongly during the transition from the active to the dissolving stage. This decrease occurs in the cloud interior as well as in the edge region of the clouds. The mean volume diameter 
Signs of secondary droplet activation are found in the droplet size distributions for about 46% of the observed clouds. It is found that there is a significant tendency from homogeneous mixing of actively growing clouds to more inhomogeneous mixing with increasing cloud life time, when only looking at clouds with purely monomodal size distributions. This is derived from mixing diagrams of individual clouds, as well as from two statistical analyses. These findings are supported by previous observations of shear-induced turbulence at the edges of actively growing clouds (e.g., Siebert et al. 2006b). This increased local turbulence results in a reduced mixing time scale, which favors homogeneous mixing. Also, indications of an active collision–coalescence process were found in some clouds. Both secondary activation and collision–coalescence make the interpretation of mixing diagrams more complex. The additional droplet source/sink leads under certain conditions to somewhat ambiguous results, and further investigation of this important problem is needed.
Acknowledgments
The authors thank the German Research Foundation Deutsche Forschungsgemeinschaft (DFG) for funding this project (SI 1534/3–1 and WE 1900/18-1). We like to thank Heike Wex for providing data from the mini-CCN counter. We are grateful to Christoph Klaus and Dieter Schell from enviscope GmbH, as well as Thomas Conrath from TROPOS, for their professional technical support. We also thank our pilots Alwin Vollmar and Milos Kapetanovic for great helicopter flights. Moreover, we are thankful to Paul Archer and to National Helicopters in Canada for providing the helicopter service.
REFERENCES
Ackerman, B., 1958: Turbulence around tropical cumuli. J. Meteor., 15, 69–74, doi:10.1175/1520-0469(1958)015<0069:TATC>2.0.CO;2.
Arabas, S., H. Pawlowska, and W. W. Grabowski, 2009: Effective radius and droplet spectral width from in-situ aircraft observations in trade-wind cumuli during RICO. Geophys. Res. Lett.,36, L11803, doi:10.1029/2009GL038257.
Austin, P. H., M. B. Baker, A. M. Blyth, and J. B. Jensen, 1985: Small-scale variability in warm continental cumulus clouds. J. Atmos. Sci., 42, 1123–1138, doi:10.1175/1520-0469(1985)042<1123:SSVIWC>2.0.CO;2.
Baker, M. B., and J. Latham, 1979: The evolution of droplet spectra and the rate of production of embryonic raindrops in small cumulus clouds. J. Atmos. Sci., 36, 1612–1615, doi:10.1175/1520-0469(1979)036<1612:TEODSA>2.0.CO;2.
Baker, M. B., R. G. Corbin, and J. Latham, 1980: The influence of entrainment on the evolution of cloud droplet spectra. I: A model of inhomogeneous mixing. Quart. J. Roy. Meteor. Soc., 106, 581–598, doi:10.1002/qj.49710644914.
Baker, M. B., R. E. Breidenthal, T. W. Choularton, and J. Latham, 1984: The effects of turbulent mixing in clouds. J. Atmos. Sci., 41, 299–304, doi:10.1175/1520-0469(1984)041<0299:TEOTMI>2.0.CO;2.
Blyth, A. M., and J. Latham, 1990: Airborne studies of the altitudinal variability of the microphysical structure of small, ice-free, montanan cumulus clouds. Quart. J. Roy. Meteor. Soc., 116, 1405–1423, doi:10.1002/qj.49711649608.
Brenguier, J.-L., and L. Chaumat, 2001: Droplet spectra broadening in cumulus clouds. Part I: Broadening in adiabatic cores. J. Atmos. Sci., 58, 628–641, doi:10.1175/1520-0469(2001)058<0628:DSBICC>2.0.CO;2.
Brenguier, J.-L., H. Pawlowska, L. Schüller, R. Preusker, J. Fischer, and Y. Fouquart, 2000: Radiative properties of boundary layer clouds: Droplet effective radius versus number concentration. J. Atmos. Sci., 57, 803–821, doi:10.1175/1520-0469(2000)057<0803:RPOBLC>2.0.CO;2.
Bretherton, C. S., J. R. McCaa, and H. Grenier, 2004: A new parametrization for shallow cumulus convection and its application to marine subtropical cloud-topped boundary layers. Part I: Description and 1D results. Mon. Wea. Rev., 132, 864–882, doi:10.1175/1520-0493(2004)132<0864:ANPFSC>2.0.CO;2.
Burnet, F., and J.-L. Brenguier, 2007: Observational study of the entrainment-mixing process in warm convective clouds. J. Atmos. Sci., 64, 1995–2011, doi:10.1175/JAS3928.1.
Chuang, P. Y., E. W. Saw, J. D. Small, R. A. Shaw, C. M. Sipperley, G. A. Payne, and W. Bachalo, 2008: Airborne phase Doppler interferometry for cloud microphysical measurements. Aerosol Sci. Technol., 42, 685–703, doi:10.1080/02786820802232956.
Cooper, W. A., 1989: Effects of variable droplet growth histories on droplet size distributions. Part I: Theory. J. Atmos. Sci., 46, 1301–1311, doi:10.1175/1520-0469(1989)046<1301:EOVDGH>2.0.CO;2.
Ditas, F., 2014: Microphysical properties of aerosol particles in the trade wind regime and their influence on the number concentration of activated particles in trade wind cumulus clouds. Ph.D. thesis, University of Leipzig, 113 pp.
Feingold, G., L. A. Remer, J. Ramaprasad, and Y. J. Kaufman, 2001: Analysis of smoke impact on clouds in Brazilian biomass burning regions: An extension of Twomey’s approach. J. Geophys. Res., 106, 22 907–22 922, doi:10.1029/2001JD000732.
Gerber, H., 2006: Entrainment, mixing, and microphysics in RICO cumulus. 12th Conf. on Cloud Physics, Madison, WI, Amer. Meteor. Soc., 14.2A. [Available online at https://ams.confex.com/ams/pdfpapers/109676.pdf.]
Gerber, H., G. M. Frick, J. B. Jensen, and J. G. Hudson, 2008: Entrainment, mixing, and microphysics in trade-wind cumulus. J. Meteor. Soc. Japan, 86A, 87–106, doi:10.2151/jmsj.86A.87.
Grabowski, W., 2006: Indirect impact of atmospheric aerosols in idealized simulations of convective–radiative quasi equilibrium. J. Climate, 19, 4664–4682, doi:10.1175/JCLI3857.1.
Haman, K. E., A. Makulski, S. P. Malinowski, and R. Busen, 1997: A new ultrafast thermometer for airborne measurements in clouds. J. Atmos. Oceanic Technol., 14, 217–227, doi:10.1175/1520-0426(1997)014<0217:ANUTFA>2.0.CO;2.
Heus, T., and H. J. J. Jonker, 2008: Subsiding shells around shallow cumulus clouds. J. Atmos. Sci., 65, 1003–1018, doi:10.1175/2007JAS2322.1.
Hill, T. A., and T. W. Choularton, 1985: An airborne study of the microphysical structure of cumulus clouds. Quart. J. Roy. Meteor. Soc., 111, 517–544, doi:10.1002/qj.49711146813.
Jarecka, D., W. W. Grabowski, H. Morrison, and H. Pawlowska, 2013: Homogeneity of the subgrid-scale turbulent mixing in large-eddy simulation of shallow convection. J. Atmos. Sci.,70, 2751–2767, doi:10.1175/JAS-D-13-042.1.
Jensen, J. B., and M. B. Baker, 1989: A simple model for droplet spectra evolution during turbulent mixing. J. Atmos. Sci., 46, 2812–2829, doi:10.1175/1520-0469(1989)046<2812:ASMODS>2.0.CO;2.
Jensen, J. B., P. H. Austin, M. B. Baker, and A. M. Blyth, 1985: Turbulent mixing, spectral evolution and dynamics in a warm cumulus cloud. J. Atmos. Sci., 42, 173–192, doi:10.1175/1520-0469(1985)042<0173:TMSEAD>2.0.CO;2.
Kain, J. S., and J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization. J. Atmos. Sci., 47, 2784–2802, doi:10.1175/1520-0469(1990)047<2784:AODEPM>2.0.CO;2.
Katzwinkel, J., H. Siebert, T. Heus, and R. A. Shaw, 2014: Measurements of turbulent mixing and subsiding shells in trade wind cumuli. J. Atmos. Sci.,71, 2810–2822, doi:10.1175/JAS-D-13-0222.1.
Korolev, A. V., and I. P. Mazin, 2003: Supersaturation of water vapor in clouds. J. Atmos. Sci., 60, 2957–2974, doi:10.1175/1520-0469(2003)060<2957:SOWVIC>2.0.CO;2.
Krueger, S., 2008: Fine-scale modeling of entrainment and mixing of cloudy and clear air. Extended Abstract, 15th Int. Conf. on Clouds and Precipitation, Cancun, Mexico, Int. Commission on Clouds and Precipitation, P1.10. [Available online at http://cabernet.atmosfcu.unam.mx/ICCP-2008/abstracts/Program_on_line/Poster_01/Krueger_extended.pdf.]
Kumar, B., J. Schumacher, and R. A. Shaw, 2013: Cloud microphysical effects of turbulent mixing and entrainment. Theor. Comput. Fluid Dyn., 27, 361–376, doi:10.1007/s00162-012-0272-z.
Laird, N. F., 2005: Humidity halos surrounding small cumulus clouds in a tropical environment. J. Atmos. Sci., 62, 3420–3425, doi:10.1175/JAS3538.1.
Lehmann, K., H. Siebert, and R. A. Shaw, 2009: Homogeneous and inhomogeneous mixing in cumulus clouds: Dependence on local turbulence structure. J. Atmos. Sci., 66, 3641–3659, doi:10.1175/2009JAS3012.1.
Lu, M.-L., J. Wang, A. Freedman, H. H. Jonsson, R. C. Flagan, R. A. McClatchey, and J. H. Seinfeld, 2003: Analysis of humidity halos around trade wind cumulus clouds. J. Atmos. Sci., 60, 1041–1059, doi:10.1175/1520-0469(2003)60<1041:AOHHAT>2.0.CO;2.
Malkus, J. S., 1949: Effects of wind shear on some aspects of convection. Eos, Trans. Amer. Geophys. Union, 30, 19–25, doi:10.1029/TR030i001p00019.
Nuijens, L., I. Serikov, L. Hirsch, K. Lonitz, and B. Stevens, 2014: The distribution and variability of low-level cloud in the North Atlantic trades. Quart. J. Roy. Meteor. Soc., 140, 2364–2374, doi:10.1002/qj.2307.
Paluch, I. R., and C. A. Knight, 1984: Mixing and the evolution of cloud droplet size spectra in a vigorous continental cumulus. J. Atmos. Sci., 41, 1801–1815, doi:10.1175/1520-0469(1984)041<1801:MATEOC>2.0.CO;2.
Paluch, I. R., and D. G. Baumgardner, 1989: Entrainment and fine-scale mixing in a continental convective cloud. J. Atmos. Sci., 46, 261–278, doi:10.1175/1520-0469(1989)046<0261:EAFSMI>2.0.CO;2.
Pawlowska, H., J.-L. Brenguier, and F. Burnet, 2000: Microphysical properties of stratocumulus clouds. Atmos. Res., 55, 15–33, doi:10.1016/S0169-8095(00)00054-5.
Perry, K. D., and P. V. Hobbs, 1996: Influences of isolated clouds on the humidity of their surroundings. J. Atmos. Sci., 53, 159–174, doi:10.1175/1520-0469(1996)053<0159:IOICCO>2.0.CO;2.
Pontikis, C., and E. Hicks, 1993: Droplet activation as related to entrainment and mixing in warm tropical maritime clouds. J. Atmos. Sci., 50, 1888–1896, doi:10.1175/1520-0469(1993)050<1888:DAARTE>2.0.CO;2.
Riehl, H., T. Yeh, J. S. Malkus, and N. la Seur, 1951: The north-east trade of the Pacific Ocean. Quart. J. Roy. Meteor. Soc., 77, 598–626, doi:10.1002/qj.49707733405.
Roberts, G. C., and A. Nenes, 2005: A continuous-flow streamwise thermal-gradient CCN chamber for atmospheric measurements. Aerosol Sci. Technol., 39, 206–221, doi:10.1080/027868290913988.
Rogers, R. R., and M. K. Yau, 1989: A Short Course in Cloud Physics. 3rd ed. Pergamon Press, 293 pp.
Siebert, H., H. Franke, K. Lehmann, R. Maser, E. W. Saw, D. Schell, R. A. Shaw, and M. Wendisch, 2006a: Probing finescale dynamics and microphysics of clouds with helicopter-borne measurements. Bull. Amer. Meteor. Soc., 87, 1727–1738, doi:10.1175/BAMS-87-12-1727.
Siebert, H., K. Lehmann, M. Wendisch, and R. Shaw, 2006b: Small-scale turbulence in clouds. 12th Conf. on Cloud Physics, Madison, WI, Amer. Meteor. Soc., 12.1. [Available online at https://ams.confex.com/ams/Madison2006/techprogram/paper_112465.htm.]
Siebert, H., and Coauthors, 2013: The fine-scale structure of the trade wind cumuli over Barbados—An introduction to the CARRIBA project. Atmos. Chem. Phys., 13, 10 061–10 077, doi:10.5194/acp-13-10061-2013.
Slawinska, J., W. Grabowski, H. Pawlowska, and H. Morrison, 2012: Droplet activation and mixing in large-eddy simulation of a shallow cumulus field. J. Atmos. Sci., 69, 444–462, doi:10.1175/JAS-D-11-054.1.
Small, J. D., P. Y. Chuang, and H. H. Jonsson, 2013: Microphysical imprint of entrainment in warm cumulus. Tellus, 65B, 19 922, doi:10.3402/tellusb.v65i0.19922.
Squires, P., 1958: Penetrative downdraughts in cumuli. Tellus, 10A, 381–389, doi:10.1111/j.2153-3490.1958.tb02025.x.
Squires, P., and J. Warner, 1957: Some measurement in the orographic cloud of the island of Hawaii and in trade wind cumuli. Tellus, 9A, 475–494, doi:10.1111/j.2153-3490.1957.tb01909.x.
Stevens, B., 2005: Atmospheric moist convection. Annu. Rev. Earth Planet. Sci., 33, 605–643, doi:10.1146/annurev.earth.33.092203.122658.
Stith, J., and M. Politovich, 1989: Observations of the effects of entrainment and mixing on the droplet size spectra in a small cumulus. J. Atmos. Sci., 46, 908–919, doi:10.1175/1520-0469(1989)046<0908:OOTEOE>2.0.CO;2.
Stommel, H., 1947: Entrainment of air into a cumulus cloud. J. Meteor., 4, 91–94, doi:10.1175/1520-0469(1947)004<0091:EOAIAC>2.0.CO;2.
Su, C.-W., K. Krueger, P. A. McMurtry, and P. H. Austin, 1998: Linear eddy modeling of droplet spectral evolution during entrainment and mixing in cumulus clouds. Atmos. Res., 47–48, 41–58, doi:10.1016/S0169-8095(98)00039-8.
Telford, J. W., and J. Warner, 1962: On the measurement from an aircraft of bouyancy and vertical air velocity in cloud. J. Atmos. Sci., 19, 415–423, doi:10.1175/1520-0469(1962)019<0415:OTMFAA>2.0.CO;2.
Telford, J. W., and S. K. Chai, 1980: A new aspect of condensation theory. Pure Appl. Geophys., 118, 720–742, doi:10.1007/BF01593025.
Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 1779–1800, doi:10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.
Topping, D., P. Connolly, and G. McFiggans, 2013: Cloud droplet number enhanced by co-condensation of organic vapours. Nature Geosci.,6, 443–446, doi:10.1038/ngeo1809.
Warner, J., 1955: The water content of cumuliform clouds. Tellus, 7A, 449–457, doi:10.1111/j.2153-3490.1955.tb01183.x.
Warner, J., 1969: The microstructure of cumulus cloud. Part I: General features of the droplet spectrum. J. Atmos. Sci.,26, 1049–1059, doi:10.1175/1520-0469(1969)026<1049:TMOCCP>2.0.CO;2.
Wendisch, M., and J.-L. Brenguier, 2013: Airborne Measurements for Environmental Research: Methods and Instruments. Wiley, 641 pp.
Werner, F., H. Siebert, P. Pilewskie, T. Schmeissner, R. A. Shaw, and M. Wendisch, 2013: New airborne retrieval approach for trade wind cumulus properties under overlying cirrus. J. Geophys. Res. Atmos., 118, 3634–3649, doi:10.1002/jgrd.50334.
Werner, F., and Coauthors, 2014: Twomey effect observed from collocated microphysical and remote sensing measurements over shallow cumulus. J. Geophys. Res. Atmos., 119, 1534–1545, doi:10.1002/2013JD020131.
