Observational Study of the Entrainment-Mixing Process in Warm Convective Clouds

Frédéric Burnet Météo-France/CNRS, GAME/CNRM, Toulouse, France

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Jean-Louis Brenguier Météo-France/CNRS, GAME/CNRM, Toulouse, France

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Abstract

Thermodynamical and microphysical measurements collected in convective clouds are examined within the frame of the homogeneous/inhomogeneous mixing concept, to determine how entrainment-mixing processes affect cloud droplets, their number concentration, and their mean size. The three selected case studies—one stratocumulus layer and two cumulus clouds—exhibit very different values of the cloud updraft intensity, of the adiabatic droplet mean volume diameter, and of the saturation deficit in the environment, all three parameters that are expected to govern the microphysical response to entrainmentmixing. The results confirm that the observed microphysical features are sensitive to the droplet response time to evaporation and to the turbulent homogenization time scale, as suggested by the inhomogeneous mixing concept. They also reveal that an instrumental artifact due to the heterogeneous spatial droplet distribution may be partly responsible for the observed heterogeneous mixing features. The challenge remains, however, to understand why spatially homogeneous cloud volumes larger than the instrument resolution scale (10 m) are so rarely observed. The analysis of the buoyancy of the cloud and clear air mixtures suggests that dynamical sorting could also be efficient for the selection, among all possible mixing scenarios, of those that minimize the local buoyancy production.

Corresponding author address: Frédéric Burnet, Météo-France, CNRM/GMEI, 42 av. Coriolis, 31057 Toulouse CEDEX 01, France. Email: frederic.burnet@meteo.fr

Abstract

Thermodynamical and microphysical measurements collected in convective clouds are examined within the frame of the homogeneous/inhomogeneous mixing concept, to determine how entrainment-mixing processes affect cloud droplets, their number concentration, and their mean size. The three selected case studies—one stratocumulus layer and two cumulus clouds—exhibit very different values of the cloud updraft intensity, of the adiabatic droplet mean volume diameter, and of the saturation deficit in the environment, all three parameters that are expected to govern the microphysical response to entrainmentmixing. The results confirm that the observed microphysical features are sensitive to the droplet response time to evaporation and to the turbulent homogenization time scale, as suggested by the inhomogeneous mixing concept. They also reveal that an instrumental artifact due to the heterogeneous spatial droplet distribution may be partly responsible for the observed heterogeneous mixing features. The challenge remains, however, to understand why spatially homogeneous cloud volumes larger than the instrument resolution scale (10 m) are so rarely observed. The analysis of the buoyancy of the cloud and clear air mixtures suggests that dynamical sorting could also be efficient for the selection, among all possible mixing scenarios, of those that minimize the local buoyancy production.

Corresponding author address: Frédéric Burnet, Météo-France, CNRM/GMEI, 42 av. Coriolis, 31057 Toulouse CEDEX 01, France. Email: frederic.burnet@meteo.fr

1. Introduction

In weather forecast or climate models, liquid water clouds are primarily characterized by their liquid water content (LWC), which is driven by thermodynamics. How LWC is distributed onto droplets, the cloud droplet number concentration (CDNC), and the droplet size distribution, or their mean volume diameter (MVD), may also be of interest. The onset of precipitation for example depends on the likelihood of a few big droplets to grow enough for the collection process to become efficient at producing precipitation embryos (Pruppacher and Klett 1997). The scattering of solar radiation by cloud droplets, which is an important factor in the earth radiation budget, also depends on the droplet size distribution. The light extinction is indeed proportional to the total droplet surface (Stephens 1978). At a specified LWC, it increases when CDNC increases (Hansen and Travis 1974; Twomey 1977). Beyond LWC, CDNC that establishes a link between the aerosol, more specifically the cloud condensation nuclei (CCN), on the one hand, and precipitation efficiency or cloud optical properties on the other hand, is thus a crucial parameter to diagnose. CDNC is determined at cloud base, in convective updraft, where water vapor supersaturation activates CCN (Twomey and Warner 1967). Once CCN have been activated (within the first 50 m above cloud base in general), droplets are growing by vapor diffusion and they share the available LWC, following
i1520-0469-64-6-1995-e1
where D is the droplet diameter, A (≈4.10−10 m2 s−1) is a function of pressure and temperature, and S is the supersaturation; S = qυ/qυ(T) − 1, where qυ is the water vapor mixing ratio and qυ(T) is the saturation mixing ratio at the temperature T. Explicit calculation of the supersaturation, though, is not required to calculate the droplet spectrum at any height above cloud base because ice-free clouds are typically very close to water saturation (Brenguier 1991). Following (1), the droplet spectrum can be derived from the initial spectrum after CCN activation by a translation in the D2 scale, such that the liquid water content integrated over the spectrum is equal to the adiabatic LWC at the specified level. This adiabatic scheme provides a simple and efficient parameterization of warm convective cloud microphysics (Brenguier et al. 2000; Schüller et al. 2003, 2005).

In real clouds however convective cells entrain dryer air from the environment in isolated cumuli, or from the overlying inversion in stratocumulus clouds. Further mixing of the cloudy air with the entrained air leads to dilution of LWC and CDNC. It is not clear however how CDNC and the droplet sizes are affected by the mixing process, while this has a significant impact on the droplet spectrum (Warner 1973). In particular, there is an unexpected feature revealed by in situ measurements, namely, that most of the cloud traverses show significant variations of LWC and CDNC, while MVD appears almost constant (Paluch and Knight 1984). This ubiquitous observation motivated numerous in situ observational studies of the entrainment-mixing process in real clouds (references hereafter), as well as laboratory studies of its small scale structure (Baker et al. 1984; Malinowski et al. 1998).

Modeling of the process was also attempted, using 1D models with parameterized turbulent mixing, such as the linear eddy model (LEM), initially developed by Kerstein (1988) and further extended to cloud microphysics by Krueger et al. (1997) and Su et al. (1998). More recently, direct numerical simulations (DNSs) were performed for exploring the coupling between thermodynamics and microphysics at the microscale (Andrejczuk et al. 2004, 2006). They allow one to quantify the impact of moist processes on the microscale dynamics of mixing and the kinematics of the homogenization process. Their limited domain however prevents numerical simulation of a whole cloud and of the impact of entrainment-mixing processes on cloud microphysics. Despite these numerous observational and modeling studies, the entrainment-mixing process remains mysterious. It is not even clear if the observed features only reflect an instrumental artifact or if they are real, while they may have a significant impact on the calculation of cloud radiative properties (Chosson et al. 2007; Grabowski 2006).

This paper brings new insights with fast forward scattering spectrometer probe (Fast-FSSP) measurements of the droplet microscale spatial distribution that allow us to examine the impact of spatial heterogeneities on microphysical observations. The study relies on in situ measurements from three case studies, with very different thermodynamical and microphysical properties, to investigate (i) if the observed features can be explained entirely by an instrumental artifact, (ii) which macroscale parameters could otherwise control these features, and (iii) if the dynamics, via buoyancy production, could also play a role in the selection, among all possible mixing scenarios, of those which are actually observed.

Other processes, such as droplet scavenging by precipitating drops and radiative cooling, may also affect LWC and the droplet spectrum, but they are not considered here. The conceptual description of the entrainment-mixing process is presented in the next section. The case studies are introduced in section 3. Instrumental limitations of the microphysical measurements and their impact on the observations are examined in section 4. The possible roles of the control parameters and of the buoyancy production are discussed in section 5, and a summary is given in section 6.

2. Entrainment-mixing processes

Following the pioneering observations of Warner (1973) and Paluch and Knight (1984), Jensen and Baker (1989), Blyth (1993), Grabowski (1993), and Krueger et al. (1997) have contributed to the following conceptual picture of the entrainment-mixing process: large entraining eddies driven by the convective circulation engulf large volumes of environmental air deep into the cloud core. As turbulent mixing develops, these volumes are stretched and compressed into thinner filaments until the Kolmogorov scale is reached, where final homogenization occurs through the molecular diffusion process.

Since Paluch (1979), the mixing diagram, total water mixing ratio qt versus liquid water potential temperature θl, led to ambiguous conclusions (see section 2 of Neggers and Siebesma 2002). However, these studies have shown that mixing of cloudy air that has risen adiabatically from cloud base with environmental air entrained from a specific level well reproduces the thermodynamic composition of many cloud samples at that level. The effects of mixing on droplet spectra indeed also involve very small scale processes and require further assumptions on the kinematics of the process. This question was largely discussed in the literature (Latham and Reed 1977; Baker and Latham 1979; Baker et al. 1980; among others). Two conceptual mixing models arise from a time scale analysis involving τd ∼ −(D2/AS), the time needed for a droplet of diameter D to evaporate in a subsaturated environment, and τTL/U ∼ (L2/ε)1/3, the time needed for complete homogenization of a cloud volume of size L through the process of turbulent diffusion, where U is the relative velocity and ε is the eddy dissipation rate of turbulent kinetic energy (Baker et al. 1984). The homogeneous model shall apply when τdτT; that is, all droplets collectively contribute to the moistening of the mixture that has been first homogenized by turbulence. On the opposite, when τdτT, that is, the intertwining of clear and cloudy air filaments is much slower than the droplet response time to the saturation deficit, each droplet that is exposed to a subsaturated filament will totally evaporate within the filament, which is progressively moistened. When finally the entrained filament approaches saturation, the remaining droplets are only partly evaporated. This process is referred to as inhomogeneous mixing. Note, however, that Baker et al. (1984) also mentioned that most of the observed features could be explained by an instrumental artifact, due to the spatial heterogeneity of the counted droplets.

Former observational studies mainly support the inhomogeneous model (Hill and Choularton 1985; Paluch 1986; Bower and Choularton 1988), but later Jensen and Baker (1989) and Paluch and Baumgardner (1989) found intermediate features between the homogeneous and inhomogeneous models.

To facilitate the interpretation of the observations in this paper, two mixing diagrams will be used. Let us consider a single isobaric entrainment-mixing event between a mass fraction χ of adiabatic cloudy air at temperature Ta, and the complementary mass fraction 1 − χ of clear environmental air, at temperature Te. The cloudy air is characterized by its liquid water mixing ratio qla, while the clear air is characterized by its water vapor mixing ratio qυe, which can also be expressed as a supersaturation Se = qυe/qυs(Te) − 1. Here Se is negative because the entrained air is subsaturated; hence it shall better be referred to as a saturation deficit.

An important peculiarity of the mixing process is the possible production of negative buoyancy following droplet evaporation (Grabowski 1993). The thermodynamic diagram of Fig. 1 shows the classical variation of the virtual temperature Tυ of the final homogenized mixture as a function of χ. Among all mixtures that still contain liquid water, that is, χ > χ0, only mixtures with χ > χce remain positively buoyant with respect to the environment; αce is the critical LWC dilution ratio corresponding to the critical mixing fraction χce.

For microphysical data analysis, the mixing scenarios can then be examined with the respective reductions of droplet concentration N and droplet mean volume diameter Dυ as functions of the LWC dilution ratio, in the ND3υ diagram (Fig. 2). CDNC normalized by the adiabatic value Na is reported on the x axis, while D3υ/D3υa(z) is reported on the y axis, where Dυa(z) is the initial MVD value corresponding to the adiabatic LWC at the level z. The normalization allows one to account for small variations of the droplet sizes due to small altitude fluctuations in the collected data. Note that the Dυ/Dυa scale is indicated on the right y axis. The product of the coordinates is thus directly proportional to the LWC dilution ratio:
i1520-0469-64-6-1995-e2
which is, represented by dot–dash lines from 1 for the adiabatic value, down to 0.1, every 10%. The critical dilution ratio αce, as derived in Fig. 1, is also indicated by a thick dot–dash line.
In the homogeneous model, CDNC is diluted proportionally to the amount of entrained air:
i1520-0469-64-6-1995-e3
if we assume that the air density ρ remain constant. The liquid water mixing ratio deficit is then entirely accounted for by droplet evaporation and a decrease of the droplet sizes, so that:
i1520-0469-64-6-1995-e4
where ρw is the liquid water density. In the ND3υ diagram, homogeneous mixing is thus represented by isolines (dashed lines) corresponding to different values of the relative humidity RH, or saturation deficit S, in the entrained air. The thick isoline correspond to the RH in the environment, 30% in Fig. 2 (S = −0.7). For each S value, one can calculate the critical cloud fraction χc(S) and the critical LWC dilution ratio αc(S) at which the mixture is neutrally buoyant with respect to that premoistened entrained air. The αc(S) values are reported with diamonds in Fig. 2 for S (RH) varying from −0.7 (30%) to −0.01 (99%).

Inhomogeneous mixing is represented by a horizontal line at Dυ/Dυa = 1 (thick, short-dashed line). Note therefore that an inhomogeneous mixing process and a homogeneous mixing process with initially saturated clear air are almost superimposed.

3. Case studies

Three case studies are examined in this section. Two were documented during the Small Cumulus Microphysics Study (SCMS) experiment in 1995 in the Cape Kennedy area (Florida), with the Météo-France Merlin-IV, and one during the second Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) experiment, off the coast of California in 2001, with the National Science Foundation/National Center for Atmospheric Research (NSF/NCAR) C130 (Stevens et al. 2003). On both aircraft, the temperature was measured with a Rosemount probe. Wetting of such immersion probes in clouds with high LWC leads to a substantial underestimation of the measured temperature, typically 1°–2°C, which is roughly proportional to the LWC (Lawson and Cooper 1990). The measured Rosemount temperature Tm is then corrected from the LWC measured by a Commonwealth Scientific and Industrial Research Organisation (CSIRO) King probe as T = Tm + 0.5(LWC). The validity of this empirical correction is questionable and the resulting accuracy is not expected to be better than 1°C or so. However, for the statistical analysis that we conduct in this study, this provides a reliable way to significantly reduce errors from sensor wetting. The water vapor mixing ratio was measured with a Lyman-α hygrometer. This probe also does not perform correctly in clouds and qυ is thus derived in clouds from the corrected Rosemount temperature as qυ = qυs(T). Because of the nonlinearity between temperature and humidity, errors may also occur when the samples (10-m spatial resolution) are spatially heterogeneous, with intertwined cloudy and clear air filaments (Grabowski and Pawlowska 1993).

Table 1 lists the characteristics of the three case studies. The time scales for droplet evaporation τd and turbulent homogenization τT, are calculated at a length scale L = 10 m, which is also the resolution scale of the measurements (10-Hz data). Microphysical parameters are derived from Fast-FSSP measurements (Brenguier et al. 1998; Burnet and Brenguier 2002).

a. The 13 July 2001 DYCOMS-II case study

The stratocumulus sampled during the NCAR-C130 flight RF03 was unbroken, about 350 m thick, contained little drizzle, and was topped by a strong inversion in both liquid water potential temperature θl and total water mixing ratio qt: Δθl = 9.4 C and Δqt = −5.2 g kg−1. The data were collected along the cloud-top profiling circle flown between 1145 and 1217 UTC. The C130 was flying a series of ascents and descents from above the inversion, down to 100 m into the cloud layer. The vertical sounding of relative humidity is reported in Fig. 3a, with the cloud-base altitude and two lines for the altitude range of the observations.

Figure 4a shows the qt versus θl scatterplot: dark gray dots are the cloud samples (N > 3 cm−3), gray + symbols are for the cloud-free samples, and the black open squares correspond to cloud-edge samples, that is, clear air samples that are adjacent to a cloudy sample. Saturation curves are drawn for the mean sampling altitude (solid line) and its range (dotted lines). The mixing line connects the two black dots corresponding to the adiabatic cloud (χ = 1) and to the air from the free troposphere (χ = 0). The uncertainty bar on the cloud reference reflects an error of ±0.5°C in the estimation of the cloud-base temperature. The constant virtual temperature lines are represented for the two reference points by dashed lines. The θl and qt frequency distributions are reported at the bottom and on the left of the figure, respectively. Because the adiabatic liquid water mixing ratio in this thin cloud layer is small (0.7 g kg−1) and the saturation deficit in the free troposphere is significant (−0.7), only a limited fraction of the mixed samples still contain liquid water (χ0 = 0.84). The figure also reveals that cloud-edge samples are moister and cooler than the clear air reference for the free troposphere (χ = 0).

In the saturated area, the mixing line is almost aligned with the virtual temperature isoline for the cloud reference, while the environment is significantly warmer. It follows that cloud and clear air mixtures that still contain liquid water are negatively buoyant with respect to the overlying free tropospheric air, but neutrally buoyant with respect to the cloud layer (Fig. 5a, lower panel). This is corroborated by the statistics of vertical velocity, stratified as a function of the mixing proportion (Fig. 5a, upper panel). Except for the adiabatic cloud (χ = 1) that shows a mean updraft speed of 0.2 ± 0.8 m s−1, most of the mixture proportions exhibit vertical speed values between about −0.3 and +0.4 m s−1, with a slightly positive mean value. Slightly negative mean values are observed only for χ0 = 0.84 < χ < 0.9. These statistics support the conditioning of the entrainment interface layer between cloud top and the free troposphere by cloud detrainment as hypothesized by Gerber et al. (2005).

With an adiabatic MVD of about 15 μm and a saturation deficit of S = −0.7, the droplet response time to evaporation τd = 0.8 s, is much shorter than the turbulent homogenization time scale τT = 17 s corresponding to typical peak values of vertical velocity of 0.6 m s−1.

b. The 10 August 1995 SCMS case study

On 10 August, cumulus clouds were growing from 600 m up to 2800 m in a relatively moist environment (70% < RH < 90%), as indicated on the relative humidity profile in Fig. 3b. Between 1527 and 1548 UTC, three different convective cells were sampled by the Météo-France Merlin-IV. Starting from cloud top, the Merlin-IV made successive penetrations following each ascending cell before it collapsed: 14 cloud traverses have thus been performed between 2140 and 2470 m. The scatterplot of qt versus θl is shown in Fig. 4b. The two radio soundings of Fig. 3b are averaged to represent the environmental profile. As for the DYCOMS-II case, the statistics of θl (bottom) and qt (left) show that cloud-edge samples are slightly cooler and moister than the far environment.

The main differences with the stratocumulus case of Fig. 4a are, first that the adiabatic LWC reaches much greater values, up to 4.10 g kg−1 at 2300 m, second that the adiabatic cloudy air is positively buoyant with respect to the environment, as well as most of the possible mixtures between cloudy and clear air, namely, for χce = 0.36 < χ < 1. As a result, the statistics of vertical velocity (Fig. 5b, top) shows positive values over the whole range of mixing proportion χ, except for a few downdraft mostly found at 0 < χ < 0.3. Because of the low value of χce, very few mixtures experience negative buoyancy (Fig. 5b, bottom).

The adiabatic MVD (about 30 μm) is 2 times greater than in the stratocumulus examined above, and the saturation deficit (S = −0.2) is reduced by a factor of more than 3. This results in a droplet response time to evaporation of τd = 11.3 s, 14 times longer than in the stratocumulus case. The typical peak values of vertical velocity in the cloud cores are much greater, up to 6 m s−1, resulting in a turbulent homogenization time scale of about τT = 1.7 s.

c. The 5 August 1995 SCMS case studies

On 5 August the environment was drier than on 10 August at the level (2000 m < z < 2200 m) where penetrations were made by the Merlin-IV in the upper part of a cumulus cluster sampled between 1524 and 1555 UTC (Fig. 3c). At this level the saturation deficit dropped down to −0.7. The qt versus θl scatterplot is reported in Fig. 4c. As for the 10 August SCMS case discussed above, the adiabatic LWC reaches a high value, up to 4.03 g kg−1 at 2100 m, with a maximum MVD of more than 30 μm. The statistics of θl (lower panel) and qt (left panel) show that cloud-edge samples are slightly cooler and moister than the far environment, as in the two previous cases.

In contrast to the SCMS 10 August 1995 case, the environment is very dry. This strongly affects the buoyancy of the cloudy and clear air mixtures. While the adiabatic cloud is positively buoyant with respect to the environment at the sampling level, most of the mixtures—namely, for 0 < χ < χce = 0.71—are negatively buoyant (Fig. 5c, lower panel). This is corroborated by the observed downdraft in the statistics of vertical velocity in Fig. 5c (upper panel), with the strongest downdraft intensities around χ0 = 0.44.

The saturation deficit being more significant than in the 10 August case, the droplet response time to evaporation is reduced to τd = 3.2 s, intermediate between the two previous cases. The peak values of the updraft velocity being comparable to the 10 August case, the turbulent homogenization time scale is similar, τT = 1.7 s.

From the thermodynamical point of view, the left and lower panels in Fig. 4 show that all possible proportions of mixing between pure environmental and pure cloudy air can be observed as predicted by large eddy simulations (LESs; Neggers and Siebesma 2002). Note however that clear air samples adjacent to a cloudy sample are rarely observed at very low χ values in the DYCOMS-II case (Fig. 4a), while such samples are often observed in the SCMS cases, with χ values close to 0 (Figs. 4b,c). Indeed, traverses across cumulus clouds often exhibit sharp transitions between cloudy air and pure environmental air at the 10-m scale, while traverses across the top of a stratocumulus layer rather show smoother variations of both temperature and water mixing ratio. These features reflect the difference between the dynamics of a convective cell penetrating a dry environment and the more progressive mixing that occurs in the entrainment interface layer at the top of a stratocumulus. Such information may be of interest for modeling the dynamics of finescale structures of the mixing process.

4. Impact of the measurement technique on the observations

Microphysical observations are subject to significant uncertainties. With the Fast-FSSP, CDNC is the most accurate parameter with an accuracy of the order of ±15%. The main source of uncertainty arises from the definition of the sampling section. Droplet size measurements suffer from calibration uncertainties and ambiguities of the Mie scattering response of the instrument, resulting also in an accuracy of the order of ±15%. Overall the accuracy of the LWC, which combines uncertainties on CDNC and on droplet sizes, is not better than ±20%. A more comprehensive discussion of Fast-FSSP measurements is available in Brenguier et al. (1998) and Burnet and Brenguier (2002).

In the ND3υ diagram, the normalization parameters, which are necessary to merge cloud traverses flown at various altitudes, are a source of uncertainty. Thus Na, which represents the initial, adiabatic, CDNC value in a cloud cell may change from one cell to the other because of fluctuations of the vertical velocity at the cloud base that drives the CCN activation and/or variations of the aerosol properties along the flight path; Dυa is derived from the adiabatic LWC at the flight level, and requires an estimation of the cloud base, either directly measured, or derived from extrapolation of the LWC vertical profile. Both techniques are limited, with a resulting uncertainty in the cloud-base altitude between 50 and 100 m, which translates into about 0.1 to 0.2 g m−3 in the estimation of the adiabatic LWC. It follows that the occurrence of superadiabatic values of LWC and MVD, or of a few samples crossing the homogeneous mixing isoline characterizing the environment, does not mean that the thermodynamics of entrainment mixing shall be revised. In fact, the ND3υ diagram shall only be used for characterizing the overall scatter of the samples and how they are distributed with respect to the homogeneous mixing isolines.

As in any single particle counter, droplet counts in the Fast-FSSP must be accumulated over a specified sampling period, until there are enough particles for building a statistically significant droplet size distribution. The shortest statistically significant sample length typically ranges from 10 to 100 m. The two physical parameters that are selected here for exploring the entrainment-mixing process are in fact sensitive to the heterogeneity of the droplet spatial distribution in the measured sample. Assume for example that a sample is made of half pure cloudy air [Na; D3υa] intertwined with half diluted air, characterized by reduced values of NdNa and Dυd3D3υa. Measurements of such a sample will exhibit a concentration N = 0.5Na + 0.5Nd ≈ 0.5Na, and a MVD D3υ = (0.5NaD3υa + 0.5NdDυd3)/ND3υa. In situ measurements with a particle counter therefore tend to disguise the lowest MVD values in a spatially heterogeneous sample, when the length scale of the heterogeneities is smaller than the resolution scale of the instrument. This was mentioned by Baker et al. (1984) as a possible explanation of the unexpected contrast between large CDNC and small MVD variations in the mixed cloud samples (the inhomogeneous feature).

It is thus crucial to check if an instrumental artifact alone could be sufficient to explain the observations. For each of the case study presented above, the verification proceeds in four steps: first, a stochastic model of isobaric mixing is used to randomly produce all possible mixing proportions between adiabatic cloudy air and pure environmental air, and the results are plotted in the ND3υ diagram (Fig. 6); second, measurements of such samples with a particle counter are simulated and plotted in the same diagram (Fig. 7) to illustrate the impact of the measurement technique. In the third step, actual measurements are reported in the ND3υ diagram (Fig. 8) for comparison with the simulations, and the droplet spectral width is quantified as a function of dilution (Fig. 9). The fourth and last step relies on a unique feature of the Fast-FSSP, which provides information on the location of each counted droplet along the measured sample. This feature will be utilized to quantify the level of droplet spatial heterogeneity in the measured samples as function of the sample level of mixing inhomogeneity (Fig. 10).

a. Stochastic mixing

The model described in the appendix randomly simulates a large number of realizations of an entrainment-mixing process, during which cloudy and environmental clear air can progressively mix in random proportions, hence generating moistened clear air and diluted cloud parcels. Figure 6 shows results of three simulations for the (a) DYCOMS-II case, (b) SCMS 10 August case, and (c) SCMS 5 August case. Each simulation is made of 1000 realizations and each realization includes 50 successive mixing events. Figure 6 corroborates the assumption that ND3υ values resulting from mixing of cloudy air with pure environmental or premoistened air should fill the whole area that is prescribed by the thermodynamics. Note however that the distribution is not uniform, with more samples along the homogeneous isoline and less along the inhomogeneous one (constant MVD).

b. Simulation of droplet measurements

To illustrate the impact of accumulating droplet counts over the sample duration, we now assume that each sample generated with the stochastic model corresponds to a length of 20 cm, and that each random series of 50 successive samples represents a simulation of a 10-m sample measured with the Fast-FSSP. Droplet spectra in each series are therefore accumulated to reproduce a Fast-FSSP counting process. Figure 7, showing how the data presented in Fig. 6 would appear if they were measured with the Fast-FSSP (or any single particle counter) demonstrates that the measurement technique strongly affects the appearance of the simulated data: because each series contains a substantial proportion of clear air samples, the resulting CDNC is significantly reduced, while MVD is slightly affected and remains close to the high values of the least diluted samples.

c. Actual droplet measurements

The droplet measurements of the three case studies presented in Fig. 4 are plotted in Fig. 8. In the DYCOMS-II case, the adiabatic CDNC value Na and the adiabatic MVD value at the sampling level Dυa(z) are selected independently for each traverse to take into account fluctuations of peak CDNC and of the cloud base through the circle flown by the aircraft. The fifth cloud traverse, which was shown to be associated with a pollution plume (Twohy et al. 2005), is excluded from the analysis. Figure 8a shows statistics over the 12 remaining cloud traverses, with Na varying from 176 to 432 cm−3. If mixing was of the homogeneous type, the data should spread along the 30% (S = −0.7) relative humidity isoline (thick-dashed line in the figure). In contrast, most of the samples exhibit a constant MVD, down to a LWC dilution factor of 0.3: 75% of the samples have a D3υ ranging between 0.8 and 1.2 of the adiabatic value. With more dilution, some samples show a small MVD decrease, though D3υ/D3υa remains greater than 0.5. These features are typical of an inhomogeneous mixing process.

Cloud samples are partitioned into 11 classes of LWC dilution ratio (dot–dash lines in Fig. 8), from 0 to 1.1. The droplet spectral width is calculated in each class as the difference between the 90% and the 10% percentile of the size distribution. The resulting statistics are presented in Fig. 9a by the mean and standard deviation. There is an obvious tendency for the mean spectral width to increase, from 5 to 8 μm when the dilution ratio decreases. The increase of the spectral width due to homogeneous droplet evaporation (dotted line) is not sufficient to explain the observations. Note that the decrease of the spectral width at α < 0.2 results from the total evaporation of the smallest droplets. If the measured 10-m samples however are spatially heterogeneous, made of subsamples of more or less evaporated droplets, the measured spectral width would substantially increase. This assumption will be verified in the last step.

For the SCMS 10 August case, the three sampled cells show different CDNC peak values (Na = 250 cm−3 for the first and second, 330 cm−3 for the third one). The adiabatic MVD (about 30 μm) is 2 times greater than in the stratocumulus. Figure 8b shows the ND3υ scatterplot over the 14 cloud traverses. The droplet diameters are slightly more reduced by evaporation than in Fig. 8a. When LWC is slightly diluted (α > 0.6), data spread along the 80% RH isoline (S = −0.2), as expected for a homogeneous mixing process. At stronger LWC dilution (α < 0.3), droplet diameters are less reduced than expected, and the data spread progressively from the S = −0.2 isoline toward the S = −0.05 isoline (RH = 95%). The statistics of spectral width per class of LWC dilution ratio (Fig. 9b) shows the same tendency as for the stratocumulus case, with an increase of the spectral width, from 6 to 17 μm, when the mixing proportion decreases.

The ND3υ statistics for the SCMS 5 August case is presented in Fig. 8c. When LWC is slightly diluted (α > 0.4), most of the samples show reduced MVD values, between the 70% (S = −0.3) and the 30% (S = −0.7) relative humidity isolines. However, as in the previous SCMS case, at a stronger LWC dilution, droplet diameters are less reduced than expected, and the data spread progressively from the S = −0.3 to the S = −0.05 homogeneous mixing isolines. Finally, when α reaches values lower than 0.1, droplet sizes decrease down to Dυ/Dυa = 0.7. Figure 9c shows again a substantial increase of the spectral width from 7 μm in the undiluted cloud core up to 19 μm in the most diluted volumes.

d. Heterogeneity of the droplet spatial distribution

If the Baker et al. (1984) hypothesis is verified—namely, that accumulating droplet counts along the sample duration is the cause of the observed small MVD and large CDNC variations—then ND3υ values close to the homogeneous mixing isoline should also exhibit a more homogeneous droplet spatial distribution than ND3υ values close to the heterogeneous mixing isoline. Fast-FSSP droplet spectra cannot be processed at a spatial resolution shorter than 10 m because the number of correctly sized droplets (about 5% of the total counts) becomes too small for a statistically significant MVD computation. The data can however be used to document the statistics of the droplet spatial distribution within a 10-m sample, using the total counts, regardless of the droplet size. Each sample is subdivided into 1000 subsamples of 1-cm length. Here nj is the number of counts in the subsample j, n is its mean value over the 10-m sample, and σ2(n) the variance of counts. The Fishing test (FT) is defined as (Baker 1992; Chaumat and Brenguier 2001)
i1520-0469-64-6-1995-e5
where nt is the number of subsamples (here nt = 1000). If the droplet spatial distribution is random, droplet counts follow Poisson statistics and FT has a probability of 99.7% to be within the range ±3, as long as ntn > 100 (Baker 1992).

For both SCMS case studies, the analysis is restricted to samples with an LWC dilution ratio α < 0.4, below which the deviation from the homogeneous mixing isoline becomes significant. An equivalent homogeneous saturation deficit Seq is defined for each sample by deriving the relative humidity isoline that crosses the sample: Seq → 0 if the sample approaches the extreme inhomogeneous isoline, SeqSe if it approaches the homogeneous isoline corresponding to the pure environment.

Figure 10 summarizes the results of the Fishing test for the (a) DYCOMS-II case, (b) SCMS 10 August case, and (c) SCMS 5 August case. The three figures show that the Fishing test increases when Seq → 0, hence corroborating the hypothesis that most of the samples showing an inhomogeneous-mixing-like microphysical appearance also exhibit a heterogeneous droplet spatial distribution.

Therefore, it can be concluded that accumulating droplet counts along 10-m samples is partly responsible for the observed features. The comparison however between Fig. 7 for the simulation and Fig. 8 for actual measurements suggests that, without instrumental artifact, the actual data could not be distributed as in Fig. 6, which corresponds to equiprobable mixing proportions: for the DYCOMS-II case, the proportion of samples along the constant MVD line is much larger than in the simulation, while for both SCMS cases, the proportion of observed samples along the homogeneous isoline is much smaller. This careful examination of the instrumental limitations suggests that, beyond the artifact produced by droplet counting, there is also a physical process that limits the range of observable mixing scenarios. It also reveals that there is no universal mixing scenario, since the three cases exhibit very different features. The possible origin of these differences is now explored in the next section.

5. Discussion

a. Control parameters

The DYCOMS-II case, which is characterized by the shortest droplet response time to evaporation and the longest homogenization time scale (τd/τT = 0.05), obviously exhibits (Fig. 8a) the most inhomogeneous mixing features. In contrast, the SCMS 10 August case with τd/τT = 6.6, shows (Fig. 8b) a tendency for MVD to decrease with CDNC, though samples progressively deviate from the homogeneous mixing isoline that characterizes its environment (S = −0.2), for approaching the S = −0.05 isoline, as in the DYCOMS-II case. The SCMS 5 August case, with an intermediate value of τd/τT = 1.9, shows (Fig. 8c) the same features as the 10 August case, with quasi-homogeneous samples, along the S = −0.7 isoline, progressively moving toward the S = −0.05 isoline.

The most noticeable difference between the two SCMS cases is relative to the saturation deficit in the environment, of −0.2 on 10 August against −0.7 on 5 August. The ND3υ diagram reflects this difference, with the majority of samples distributed along the mixing isolines corresponding to the respective values of relative humidity. The most noticeable differences between the DYCOMS-II case, on the one hand, and the two SCMS cases, on the other hand, are the adiabatic droplet sizes (15 μm, against 30 μm) and the intensity of the updraft (0.6 m s−1, against 6 m s−1). The comparison, in the ND3υ diagram, of the DYCOMS-II and the SCMS 5 August cases, which both show the same saturation deficit in the environment, suggests that the ratio τd/τT actually controls the observable features of the entrainment-mixing process. These control parameters obviously play a role in the selection of the observable mixing scenarios, but they do not explain why spatially homogeneous cloud volumes larger than 10 m are so rarely observed, while the stochastic model still exhibits some homogeneous-like cloud parcels after instrumental artifacts have been corrected (Fig. 7).

b. Dynamical sorting

The main limitation of the ND3υ diagram is that it only illustrates static properties of the mixing process. The stochastic model for its part does not contain either any information about the dynamics of the mixing process, as it simulates all possible proportion of mixtures. As shown by the thermodynamic diagrams of Figs. 4 and 5, however, the three case studies are very different as far as the buoyancy of the mixed parcels is concerned. The critical LWC dilution ratio αce indicates the fraction of mixed cloud samples that are likely to experience buoyancy reversal. In the SCMS 10 August case (Fig. 4b), very diluted cloudy samples only are negatively buoyant with respect to the environment (αce = 0.26), while in the SCMS 5 August case (Fig. 4c), half of them are negatively buoyant (αce = 0.51). This is well corroborated by the statistics of vertical velocity and virtual potential temperature reported in Fig. 5. The DYCOMS-II case is conceptually different because the clear air is entrained from above, and buoyancy shall be measured with respect to the cloud layer itself. Figure 5a shows that all cloud-top mixtures containing droplets (χ > χ0) have a slightly positive buoyancy with respect to the cloud layer.

The critical values χce and αce however characterize the buoyancy of the mixed parcels with respect to the far environment and the adiabatic cloud. In fact, the dynamics of the mixing process is rather determined by local gradients of virtual temperature that generate turbulent vortices (Grabowski 1993). We shall therefore consider what the critical mixing proportion χc(S) and LWC dilution ratio αc(S) are when an adiabatic cloud volume is mixed with premoistened clear air (Se < S < 0) and buoyancy is measured with respect to that premoistened clear air. The value of αc(S) is represented on the ND3υ diagram (Figs. 8a–c) by thick diamonds for various S values, from Se to 0 (RHe and 100% relative humidity, respectively). They are remarkably consistent with the disposition of the cloud samples on the ND3υ diagram. This result suggests that dynamical sorting could be efficient for the selection, among all possible mixing scenarios, of those that minimize the local buoyancy production, a hypothesis that will have to be tested in the future with numerical simulations of the mixing process.

6. Summary

The entrainment-mixing process in convective clouds has motivated numerous experimental and numerical studies. Beyond droplet growth by vapor diffusion, it is the process that affects the most the droplet size distribution, particularly in nonprecipitating clouds. Without entrainment-mixing processes, simulated droplet spectra, at a specified altitude above cloud base, are all similar and narrower than in real clouds. Entrainment-mixing processes continuously generate a great variety of droplet trajectory and water vapor supersaturation scenarios (Cooper 1989), hence producing broad, multimodal droplet spectra. This has a significant impact on cloud radiative properties, on the formation of precipitation, and on the processing of aerosol by clouds.

Parameterizing this process in atmospheric models is nevertheless a challenge because it involves very small scales (millimeters to centimeters), at which its evolution is driven by the coupling between turbulence and droplet evaporation. A comprehensive modeling of this process would require a domain large enough to explicitly simulate the large eddies of the convection, with a resolution of millimeters to centimeters, which is beyond the performance of present computers. Compromises are possible using DNS (Andrejczuk et al. 2004, 2006) to examine the millimeter-scale coupling between dynamics and microphysics or linear eddy models (Krueger et al. 1997; Su et al. 1998) to simulate statistical properties of the physical process.

Such a very large range of scales is also an obstacle for observational studies. Most in situ measurements are performed with instrumented aircraft that are flying too rapidly for sampling the millimeter to centimeter scales, especially for temperature and humidity measurements. The Fast-FSSP was in fact developed to examine these smallest scales of cloud microphysics (Brenguier 1993; Brenguier and Chaumat 2001; Chaumat and Brenguier 2001). The dataset collected over the past 10 yr in cumulus and stratocumulus clouds has been analyzed to extract the most illustrative case studies and provide modelers with key features of the entrainment-mixing process that are crucial for constraining the simulations.

This paper summarizes the analysis of three case studies:

  • A stratocumulus layer (DYCOMS-II—13 July 2001) topped by a strong inversion: with a saturation deficit in the entrained air of S = −0.7, a droplet MVD of Dυ = 15 μm, and peak values of the vertical velocity of 0.6 m s−1, this case is characterized by a very low ratio of droplet evaporation to turbulent homogenization response times τd/τT = 0.05.

  • A series of three convective cells (SCMS—10 August 1995) growing in a moist environment: with a saturation deficit of S = −0.2, a droplet MVD of Dυ = 30 μm, and peak values of the vertical velocity of 6 m s−1, this case is characterized by a much higher ratio τd/τT = 6.6.

  • A cumulus cluster penetrating a dry layer (SCMS—5 August 1995): with a saturation deficit of S = −0.7, but a droplet MVD of Dυ = 30 μm, and updraft peak values of the vertical velocity of 6 m s−1, this case is characterized by an intermediate value of the ratio τd/τT = 1.9.

Statistics of θl and qt reveal that all possible values, between the adiabatic cloud and pure environment references, are observable and that they are distributed along the mixing line, hence justifying the use of a simplified isobaric mixing process to interpret the observations.

Fast-FSSP droplet measurements collected along 10-m samples have been projected on the ND3υ diagram that shows how a specified LWC value is partitioned into number and sizes. In this representation homogeneous mixing between pure cloudy air and pure environmental air is characterized by ND3υ values distributed along the homogeneous mixing isoline corresponding to the relative humidity in the environment. Constant MVD in contrast is the signature of an inhomogeneous mixing process (or homogeneous mixing process with saturated environmental air). The area between these two extremes characterizes partly diluted cloud volumes mixed with premoistened clear air.

The DYCOMS-II case, with the lowest τd/τT ratio, shows the most inhomogeneous features, as expected from the concepts developed by Baker et al. (1984). In contrast, the SCMS 10 August case, with the greatest τd/τT ratio, shows features intermediate between a homogeneous mixing process, when LWC is slightly diluted, approaching the inhomogeneous type, when LWC is more diluted. The same tendency is revealed by the SCMS 5 August case, despite its strong saturation deficit of S = −0.7.

The ND3υ values along the constant MVD line can be explained by the mixing of pure cloudy air with premoistened environmental air that has reached saturation. The absence of values close to the homogeneous mixing line however is unexpected, especially when considering that the θl and qt statistics show that all possible thermodynamic states of mixture exist at the vicinity of the cloud cells.

The numerical simulation of an isobaric homogeneous mixing process with a stochastic model corroborates this assessment, showing ND3υ values filling up the whole area between the inhomogeneous and the homogeneous mixing isolines. The same simulations, however, processed to replicate Fast-FSSP measurements, show features that are closer to the observations in real clouds.

This instrumental artifact only occurs if the droplet spatial distribution is heterogeneous within the sample. The Fishing test (Baker 1992) confirms that the droplet spatial distribution at the centimeter scale is highly heterogeneous in the ND3υ samples that approach the inhomogeneous mixing isoline, while the droplets are more uniformly distributed in the few samples that remain close to the homogeneous mixing isoline. A consequence of this result is that the droplet spectra in the mixed cloud volumes are not necessarily as broad as indicated in Fig. 9. The Fishing test rather suggests that the increased spectral width of the mixed samples may reflect the spatial averaging of adjacent narrow spectra with different mean diameters.

In summary, the three case studies presented here confirm that droplet spectra sampled in diluted cloud volumes show features intermediate between the two extreme scenarios and that the impact of the entrainment-mixing processes on cloud microphysics is sensitive to the respective values of the droplet response time to evaporation τd, and of the homogenization time scale τT. This study also suggests that part of the inhomogeneous-like features observed in real clouds with single particle counters may be due to an artifact of the measurement technique, which also implies that the spatial heterogeneities of the droplet distribution in most of the mixed cloud volumes have scales smaller than 10 m.

The evaporation and homogenization time scales, as well as instrumental artifacts, are not sufficient however to explain why observed samples are not distributed in the ND3υ diagram over the whole area permitted by the thermodynamics. Additional information about the dynamics of the mixing process shall be considered. In fact, the examination of the buoyancy of the mixed parcels reveals that dynamical sorting could also play a role in the selection of the mixing scenarios, namely, those that minimize the local buoyancy production at the interface between the cloud and the premoistened environment. Because local buoyancy production and the droplet response time to evaporation are closely connected through the saturation deficit in the environment, such a mechanism is compatible with the observed sensitivity of the mixing process to the evaporation time scale.

Numerical simulations of the dynamics of the mixing process shall now be performed with a spatial resolution fine enough for the homogenization time scale to be negligible, compared to the droplet response time to evaporation. They are needed to determine if the impact of the mixing process on cloud microphysics is mainly driven by the ratio of the droplet evaporation to turbulent homogenization time scales, regardless of vertical motions induced by buoyancy, or if on the opposite the turbulent production of buoyancy is the key player.

A challenge for such numerical simulation will be to replicate the typical features seen in the ND3υ diagrams of Fig. 8, with homogeneous-like mixing features at high LWC dilution ratio, progressively moving toward an inhomogeneous-like mixing process when the dilution ratio decreases. Numerous iterations between numerical modelers and experimentalists will likely be necessary before developing a complete description of this process and its parameterization.

Acknowledgments

The authors are grateful to the SCMS teams of the NCAR-C130, especially to Dr. C. Knight for the coordination of the experiment, and of the Météo-France Merlin-IV for their remarkable performance, to the Météo-France TRAMM team for data processing, and to W. W. Grabowski and the reviewers for their detailed and constructive comments. They acknowledge the contributions of the DYCOMS-II participants, and the support of Météo-France and CNRS/INSU/PATOM.

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APPENDIX

A Stochastic Model of Entrainment Mixing

This model aims at simulating random series of isobaric entrainment–mixing events, starting with a mixture of pure environmental and pure cloudy air, hence producing an elementary mixed cloud volume. Each newly formed elementary cloud volume can further mix randomly with any of the existing ones in the series. For each mixing event, both liquid water temperature Tl and total water mixing ratio qt are derived from the mixing line of Fig. 4. If the elementary cloud volume contains liquid water, droplets are homogeneously growing or evaporating until their accumulated volume is equal to the LWC prescribed by the thermodynamics (adjustment to the saturation). The scale of an elementary cloud volume is not prescribed, but the above hypothesis implies that it is small enough for the droplets to homogeneously respond to any super- or subsaturation.

The simulation starts with parcel 0 that represents the environment [Tle; qυe] and parcel 1 that represents the adiabatic cloud [Tla; qta]. The adiabatic cloud is also characterized by a monodispersed droplet spectrum at the diameter D1 = Dυa, with a concentration N1 = Na.

A mixture proportion χ2 is chosen randomly for producing parcel 2, which is characterized by [T2; qt2]. If qt2 > qυs(T2), the droplet spectrum in parcel 2 is derived from the adiabatic spectrum, that is, the concentration is N2 = χ2N1 and the diameter is calculated using
i1520-0469-64-6-1995-ea1
Parcel 3 [T3; qt3] is then obtained by mixing parcel 2 with either parcel 0 (pure environment) or parcel 1 (pure cloud). The parcel to be mixed with and the mixture proportion are chosen randomly. For the microphysics, let us assume that parcel 2 is mixed with the pure cloud. The concentration in parcel 3 is N3 = χ3N2 + (1 − χ3)N1. Its droplet spectrum is bimodal, with χ3N2 droplets of diameter (D22 + ΔD2)1/2 and (1 − χ3)N1 droplets of diameter (D21 + ΔD2)1/2. Indeed, Eq. (1) indicates that during a condensation or evaporation process, all the droplets are growing or evaporating by the same ΔD2, which is derived from
i1520-0469-64-6-1995-ea2
Iteratively, parcel k is obtained by mixing parcel k − 1 with parcel m, which is chosen randomly between parcels 0 and k − 2. The mixing proportion χk is also chosen randomly. If qtk > qυs(Tk), the droplet spectrum in parcel k is derived from the spectra of parcel m and k − 1 after adjustment to saturation:
i1520-0469-64-6-1995-ea3
where s(p) is the number of diameter values in the droplet spectrum of parcel p, Np,j is their relative concentrations, and Dp,j the diameter values. If either s(k − 1) or s(m) is null the sum is skipped. Note that this adjustment to saturation ΔD2 is very small as it only compensates the nonlinearity of the saturation mixing curve.

The use of a Lagrangian representation in the diameter scale for the droplet spectrum prevents the numerical diffusion that would necessarily affect calculations using an Eulerian droplet spectrum (also referred to as a bin distribution). The computer power however limits the number of mixing events because the maximum possible number of diameter values in the spectrum of parcel p increases as a power of p. After 50 mixing events have been simulated, the procedure is reinitialized to simulate a new mixing event series.

This model does not contain any information about the dynamics of turbulent mixing, as in the linear eddy model (Krueger et al. 1997) or DNS calculations (Andrejczuk et al. 2004). The hypothesis that a mixed parcel can further randomly mix with any of the previous ones is arbitrary and only motivated by simplification. It is nevertheless useful to see what the possibilities would be without constraint on the kinematics of the mixing process.

Fig. 1.
Fig. 1.

Thermodynamic mixing diagram of virtual temperature Tυ as a function of the mass fraction of cloudy air χ and LWC dilution ratio α. The value χ0 is the lowest cloud fraction below which all droplets are evaporated (α = 0). The values χce and αce indicate the critical cloud fraction and LWC dilution ratio below which the mixture is negatively buoyant with respect to the environment.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 2.
Fig. 2.

Microphysics mixing diagram of the MVD vs CDNC, normalized by their adiabatic values. The dot–dash lines represent the LWC dilution ratio, α = ql/qla, from 0.1 to 1, every 0.1 step. The thick line corresponds to the critical LWC dilution ratio αce. The dashed lines represent the variation of MVD and CDNC during homogeneous mixing for different values of the RH in the entrained air, from 30% to 100%, as indicated by the labels. The horizontal thick-dashed line corresponds to inhomogeneous mixing. Diamonds indicate the critical LWC dilution ratio αc(S), when buoyancy is measured with respect to premoistened entrained air with a saturation deficit S (i.e., RH), varying from −0.7 (30%) to −0.01 (99%).

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 3.
Fig. 3.

Vertical profiles of relative humidity from aircraft data: (a) DYCOMS-II RF03 flight case between 0730 and 1450 UTC, (b) 10 Aug 1995 SCMS case between 1503 and 1635 UTC, and (c) 5 Aug 1995 SCMS case between 1340 and 1555 UTC. Each gray dot is a 1-Hz clear air sample. Cloud base and the altitude range of the cloud traverses selected for this study are indicated by the horizontal bars. For the SCMS cases, solid and dashed lines are Loughman Lake soundings at (a) 1408 and 1755 UTC and (b) 1218 and 1821 UTC, respectively.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 4.
Fig. 4.

Scatterplot of qt vs θl for the three cases. Different symbols are used for cloud, clear air, and clear air samples adjacent to cloudy samples (edge samples), as indicated in the legend. Three saturation curves are drawn for the mean flight pressure altitude and its range, as indicated by the labels. Isobaric mixing between air from the environment and adiabatic cloudy air, at the mean pressure altitude, is represented by the solid straight line. The mixing line is graduated from χ = 0 to 1 every 0.1. The error bar of the adiabatic cloud reference corresponds to an error of ±0.5°C on the estimation of the cloud-base temperature. The two dashed lines are the constant virtual temperature lines corresponding to the environment and to the adiabatic cloudy air. Frequency distributions of θl and qt are shown along the two axes for cloud, clear air, and cloud-edge samples separately.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 5.
Fig. 5.

Statistics (mean value ± std dev) of vertical velocity and potential virtual temperature, as functions of the mass fraction of adiabatic cloudy air in the mixture for the three cases. Vertical dashed and dot–dash lines indicate the cloud boundary (χ0) and the critical mixing proportion (χce), respectively.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 6.
Fig. 6.

Results of the stochastic model of entrainment mixing in the ND3υ diagram for the three simulations. Each point is an elementary mixed cloud volume produced by a mixing event.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for averaged realizations. Each point is the average of 50 successive mixing events, for each of the 1000 realizations.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 8.
Fig. 8.

Scatterplot of ND3υ for in situ–measured MVD and CDNC for the three cases. Isocontours of the frequency distribution are drawn for levels corresponding to 25%, 50%, 75%, and 90% of the data.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 9.
Fig. 9.

Statistics (mean value ± std dev) of spectral width as function of the LWC dilution ratio α for the three cases. The doted line represents the expected spectral width increase for homogeneous evaporation of the adiabatic spectrum.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Fig. 10.
Fig. 10.

Fishing test as function of the equivalent homogeneous saturation deficit Seq for the three cases. (b),(c) Only data with a LWC dilution ratio α < 0.4 are shown.

Citation: Journal of the Atmospheric Sciences 64, 6; 10.1175/JAS3928.1

Table 1.

Characteristics of the case studies.

Table 1.
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  • Fig. 1.

    Thermodynamic mixing diagram of virtual temperature Tυ as a function of the mass fraction of cloudy air χ and LWC dilution ratio α. The value χ0 is the lowest cloud fraction below which all droplets are evaporated (α = 0). The values χce and αce indicate the critical cloud fraction and LWC dilution ratio below which the mixture is negatively buoyant with respect to the environment.

  • Fig. 2.

    Microphysics mixing diagram of the MVD vs CDNC, normalized by their adiabatic values. The dot–dash lines represent the LWC dilution ratio, α = ql/qla, from 0.1 to 1, every 0.1 step. The thick line corresponds to the critical LWC dilution ratio αce. The dashed lines represent the variation of MVD and CDNC during homogeneous mixing for different values of the RH in the entrained air, from 30% to 100%, as indicated by the labels. The horizontal thick-dashed line corresponds to inhomogeneous mixing. Diamonds indicate the critical LWC dilution ratio αc(S), when buoyancy is measured with respect to premoistened entrained air with a saturation deficit S (i.e., RH), varying from −0.7 (30%) to −0.01 (99%).

  • Fig. 3.

    Vertical profiles of relative humidity from aircraft data: (a) DYCOMS-II RF03 flight case between 0730 and 1450 UTC, (b) 10 Aug 1995 SCMS case between 1503 and 1635 UTC, and (c) 5 Aug 1995 SCMS case between 1340 and 1555 UTC. Each gray dot is a 1-Hz clear air sample. Cloud base and the altitude range of the cloud traverses selected for this study are indicated by the horizontal bars. For the SCMS cases, solid and dashed lines are Loughman Lake soundings at (a) 1408 and 1755 UTC and (b) 1218 and 1821 UTC, respectively.

  • Fig. 4.

    Scatterplot of qt vs θl for the three cases. Different symbols are used for cloud, clear air, and clear air samples adjacent to cloudy samples (edge samples), as indicated in the legend. Three saturation curves are drawn for the mean flight pressure altitude and its range, as indicated by the labels. Isobaric mixing between air from the environment and adiabatic cloudy air, at the mean pressure altitude, is represented by the solid straight line. The mixing line is graduated from χ = 0 to 1 every 0.1. The error bar of the adiabatic cloud reference corresponds to an error of ±0.5°C on the estimation of the cloud-base temperature. The two dashed lines are the constant virtual temperature lines corresponding to the environment and to the adiabatic cloudy air. Frequency distributions of θl and qt are shown along the two axes for cloud, clear air, and cloud-edge samples separately.

  • Fig. 5.

    Statistics (mean value ± std dev) of vertical velocity and potential virtual temperature, as functions of the mass fraction of adiabatic cloudy air in the mixture for the three cases. Vertical dashed and dot–dash lines indicate the cloud boundary (χ0) and the critical mixing proportion (χce), respectively.

  • Fig. 6.

    Results of the stochastic model of entrainment mixing in the ND3υ diagram for the three simulations. Each point is an elementary mixed cloud volume produced by a mixing event.

  • Fig. 7.

    As in Fig. 6, but for averaged realizations. Each point is the average of 50 successive mixing events, for each of the 1000 realizations.

  • Fig. 8.

    Scatterplot of ND3υ for in situ–measured MVD and CDNC for the three cases. Isocontours of the frequency distribution are drawn for levels corresponding to 25%, 50%, 75%, and 90% of the data.

  • Fig. 9.

    Statistics (mean value ± std dev) of spectral width as function of the LWC dilution ratio α for the three cases. The doted line represents the expected spectral width increase for homogeneous evaporation of the adiabatic spectrum.

  • Fig. 10.

    Fishing test as function of the equivalent homogeneous saturation deficit Seq for the three cases. (b),(c) Only data with a LWC dilution ratio α < 0.4 are shown.

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