During September–October 2014, the German HALO (High Altitude and Long Range Research Aircraft) performed a total of 96 h of research flights over the Amazon. The 14 flights were part of the ACRIDICON–CHUVA campaign (Machado et al., 2014; Wendisch et al., 2016) that took place in cooperation with the second intensive operation period (IOP2) of the GoAmazon2014/5 experiment (Martin et al., 2016). Here we focus on cloud profiling sections during six flights that occurred in different regions in the Amazon (Fig. 1). The research flights of ACRIDICON–CHUVA were named chronologically from AC07 to AC20; the six flights selected (AC07, AC09, AC12, AC13, AC18, and AC19) accumulated 16.8 h of data (in or out of clouds), of which 50 min were inside the lower 6 km of the clouds. We concentrate primarily on the first 6 km for the DSD analysis in order to capture both warm-phase characteristics and early mixed-layer formation. There were other flights with cloud penetrations, but they are not considered in this study because of higher aerosol variability below clouds. The flights chosen for analysis presented relatively low aerosol variability, meaning that the clouds probed in the same flight were likely subject to similar aerosol conditions. The time frame of the campaign corresponds to the local dry-to-wet season transition, when biomass burning is active in the southern Amazon (Artaxo et al., 2002; Andreae et al., 2015). For clarity, the flights of interest are renamed in this study according to the region probed. Flight AC19 will be referred as M1 (Maritime 1), flights AC09 and AC18 as RA1 and RA2 (Remote Amazon 1 and Remote Amazon 2, respectively), and flights AC07, AC12, and AC13 as AD1, AD2, and AD3 (Arc of Deforestation 1, Arc of Deforestation 2, and Arc of Deforestation 3, respectively). Those definitions are listed in Table 1.

The flight paths followed a regular three-stage pattern: (i) sampling of the air below clouds for aerosol characterization, (ii) measurements of DSDs at cloud base, and (iii) sampling of growing convective cloud tops (Braga et al., 2017; Wendisch et al., 2016). The latter step was deployed as follows. After the cloud base penetration, the aircraft performed several penetrations in vertical steps of several hundred meters. In each step, the aircraft penetrated the cloud tops available, thus avoiding precipitation from above. In this way, different clouds can be penetrated in the same altitude level and the vertical steps followed the growing cumuli field overall. Surface and thermodynamic conditions were different for the various flights (see Figs. 1 and 3) with high contrasts in the north–south direction. Logging, agriculture, and livestock activity management involves burning extended vegetated areas in the region, which emits large quantities of particles that serve as CCN in the atmosphere (Artaxo et al., 2002; Roberts et al., 2003). Because of this, this region is known as the “arc of deforestation,” and its thermodynamic properties tend toward pasture-like characteristics. The energy partitioning over pasture-like areas is different compared to regions over the rainforest (Fisch et al., 2004), favoring sensible heat flux and higher cloud base heights (see Table 1). Contrasting with the arc of deforestation, the region named Remote Amazon in this study has much lower background aerosol concentrations, producing cleaner clouds. Clouds over the Atlantic Ocean developed under cleaner conditions as compared to the continental counterparts, and also had lower cloud bases (Table 1).

The cloud profiling missions were mostly characterized by cumulus fields, with some developed convection in two flights over the arc of deforestation (Fig. 2d and f). For flight AD1 some precipitation-sized droplets were observed (not shown); the clouds sampled during AD2 and AD3 presented almost no droplets with D > 100 µm. The precipitation during AD1 might be explained by the lower aerosol particle number concentrations compared to flights AD2 and AD3, later start time of the profile, and the presence of deep convection nearby (Table 1 and Fig. 2).

The results to be presented here are based on five sensors carried by HALO. A comprehensive description of the airborne instrumentation introduced below can be found in Wendisch and Brenguier (2013). Aerosol particle number concentrations (NCN) were measured using a butanol-based condensation particle counter (CPC). The flow rate was set to 0.6 L min-1, with a nominal cutoff particle size of 10 nm. NCCN at a given supersaturation (S, averaging 0.48 % ± 0.033 % for the data used here, with 10 % error) was measured using a cloud condensation nuclei counter (CCN-200; Roberts and Nenes, 2005). This instrument contains two columns and was connected to two different inlet systems for aerosol sampling: the HALO Submicrometer Aerosol Inlet (HASI) for the aerosol particles and the Counterflow Virtual Impactor (CVI) inlet to sample cloud droplets, evaporate the cloud water, and analyze the residual particles. The aerosol measurements reported in this study refer to the HASI inlet.

Cloud DSDs were measured using a cloud droplet probe (CDP; Lance et al., 2010; Molleker et al., 2014) that is part of the cloud combination probe (CCP). The CCP also contained a grayscale cloud imaging probe (CIPgs; Korolev, 2007), but we focus on CDP measurements in which D < 50 µm. The intent is to focus on cloud droplet growth processes and bringing the analysis closer to modeling scenarios. Additionally, the percentage of data with significant liquid water content (LWC) for D > 50 µm is relatively small. The number of data with LWCD>50 > 0.1 g m-3 is only 12 % of the number of DSDs with LWCD<50 > 0.1 g m-3, meaning that drizzle and precipitation are relatively infrequent in the dataset. This observation combined with the possibility of higher uncertainty (especially on the lower CIPgs bins) when combining two different instruments with distinct measurement principles further justifies the exclusive focus on CDP. The CDP counted and sized the droplets based on their forward scattering characteristics, sorting them into 15 droplet size bins between 3 and 50 µm. The sample volume had an optical cross section of 0.278 mm2 (±15 %). Uncertainties in the cross-section area, the sampling volume, and counting statistics were the major sources of uncertainty for the DSD measurements (Weigel et al., 2016). According to Molleker et al. (2014), the CDP uncertainty is about 10 %. Additionally, Braga et al. (2017) performed an intercomparison between HALO probes, as well as hot-wire measurements, and concluded that they agree well within instrumental uncertainties. We excluded all cloud DSDs with droplet number concentrations (Nd) less than 1 cm-3 from further analysis.

The DSDs measured using the CDP were fitted to gamma distributions (Eq. 1) by matching the zeroth, second, and third moments. These moments were chosen in order to favor the study of the DSD properties of interest to this study (i.e., droplet number concentration, LWC, and effective diameter), but they also coincide with the properties usually predicted using bulk microphysics models (zeroth and third moments in two-moment schemes). The complete gamma function is used to be consistent with modeling scenarios, in which the gamma parameters are calculated by μ=6G-3+1+8G2(1-G)Λ=(μ+3)M2M3N0=Λμ+1M0Γ(μ+1), where Mp is the pth moment of the DSD. The symbol G is a nondimensional ratio, given as follows: G=M23M32M0. The three parameters N0, μ, and Λ define the gamma distribution in Eq. (1); they are used to construct the phase space described in the next section. Previous studies comparing the complete and incomplete (or truncated) gamma fits suggest that, while there are differences in the resulting parameters, the relation between them remains similar. The first indication of that comes from the study of Ulbrich (1985) that analyzed the relation between rainfall DSD moments in the empirical form Mp = αMqβ, where p and q are the two distinct moment orders and α and β are fit parameters. The author notes that β is relatively insensitive to DSD truncation, meaning that the relation between the moments remains similar while their overall values change. Brandes et al. (2003) also note that the μ–Λ relation introduced by Zhang et al. (2001) is relatively insensitive to DSD truncation.

In order to confirm that both the complete and incomplete gamma fits result in similar correlations among the DSD parameters, a method similar to the one presented in Vivekanandan et al. (2004) was applied. This method aims to find the incomplete gamma parameters by using an iterative method to adjust μ and Λ. Here the moments of order zero, two, and three will be used instead of two, four, and six as in Vivekanandan et al. (2004). The first step is to calculate the ratio G using the incomplete gamma as Ginc=[γΛDmax,μ+3-γΛDmin,μ+3]3γΛDmax,μ+1-γΛDmin,μ+1γΛDmax,μ+4-γΛDmin,μ+42, where γ is the incomplete gamma function and ΛDmin and ΛDmax are its integration range. Dmin and Dmax were calculated from the measured DSDs as the lowest and highest diameter bins associated to drop concentrations higher than 10-6 cm-3 (considering both CDP and CIPgs for testing purposes). This ratio is found to be greater than or equal to its counterpart in Eq. (5). Therefore, μ and Λ should be lowered until Ginc is sufficiently close to G (the threshold of 0.001 is used here). For that purpose, μ was lowered in 0.01 steps, where the respective Λ is calculated using Eq. (3) until Ginc - G ≤ 0.001. When this condition is met, N0 is recalculated from Eq. (4).

For the incomplete gamma fit, the adjustment described above was needed in 64 % of the data used here. In the other cases, the ratios G and Ginc were similar and resulted in the same gamma parameters. For the 64% of the dataset, median relative differences in log⁡(N0), μ, and Λ ranged from 10 to 20 % towards higher N0 and lower μ and Λ. However, regardless of their different values, the relation among the gamma parameters remains unchanged in the incomplete fit because they are based on the same underlying equations. As will be shown later, the main interest of this study is in the relation among the gamma parameters and not in their values themselves. Therefore, we will focus on the complete gamma, noting that the results can be slightly shifted if truncation were to be considered.

The DSD bulk properties, such as droplet number concentration (Nd), LWC, effective droplet diameter (Deff), and relative dispersion (ε), can be derived from the gamma parameters N0, μ, and Λ by taking into account the complete gamma function integral properties. In the units considered here, the equations are given by Nd=∫0∞N(D)dD=N0Γ(μ+1)Λμ+1LWC=10-9π6ρw∫0∞N(D)D3dD=10-9π6ρwN0Γ(μ+4)Λμ+4Deff=∫0∞N(D)D3dD∫0∞N(D)D2dD=μ+3Λε=σDg=1μ+1, where ρw = 1000 g m-3 represents the density of liquid water and σ and Dg are the DSD standard deviation and mean geometric diameter, respectively. Nd, LWC, and Deff are given in per cubic centimeter, gram per cubic meter, and micrometer, respectively. Given the choice of the conserved moments, they exactly match the respective characteristics of the observed DSDs. The parameter ε is described in detail in Tas et al. (2015). The relative dispersion of the gamma DSD may differ from the observations, given the differences between the parameterized and observed DSDs. However, our measurements show that the gamma and observed ε values are closely related by εGamma = 0.95εObserved (R2 = 0.93), showing that the gamma DSDs are slightly narrower on average. We focus on ε as obtained using the gamma parameters and do not use subscripts.

Cloud hydrometeor sphericity was analyzed with the NIXE-CAPS probe (New Ice eXpEriment – Cloud and Aerosol Particle Spectrometer; Luebke et al., 2016; Costa et al., 2017). NIXE-CAPS also contains two instruments, a CIPgs as the CCP and the CAS-Depol for particle measurements in the size range 0.6 to 50 µm. The sizing principle of CAS-Depol is similar to the CDP; the difference is the particle probing: while CAS-Depol has an inlet tube (optimized with respect to shattering), CDP is equipped with an open path inlet. In addition to the sizing, CAS-Depol is equipped with a detector to discriminate between spherical and aspherical particles by measuring the change of the polarized components of the incident light. Spherical particles do not strongly alter the polarization state, in contrast to nonspherical ice crystals. The cloud particle phase of the whole cloud particle size spectrum was analyzed from the combination of phase determination in the size ranges < 50 µm (from the CAS-Depol polarization signal) and > 50 µm (from visual inspection of the CIPgs images) (for details, see Costa et al., 2017). Here, the phase states are defined as follows: “Sph (liquid)” stands for many only spherical (D < 50 µm) and predominantly spherical (D > 50 µm) hydrometeors, “Asph small (mixed phase)” for many predominantly spherical (D < 50 µm) and only aspherical (D > 50 µm) hydrometeors, “Asph large (ice)” for only very few aspherical (D < 50 µm) and only aspherical (D > 50 µm) hydrometeors. The NIXE-CAPS classification is a separate analysis and will not be considered as a filter to apply the gamma fits to the CDP measurements. The CDP data fits are primarily focused on the warm phase and the transition to the mixed layer, where liquid droplets predominate.

Meteorological conditions, including three-dimensional (3-D) winds, were obtained with the Basic HALO Measurement and Sensor System (BAHAMAS) located at the nose of the aircraft (Wendisch et al., 2016). The wind components were calibrated according to Mallaun et al. (2015), with an uncertainty of 0.2 and 0.3 m s-1 for the horizontal and vertical directions, respectively. All probes were synchronized with BAHAMAS and operated at a frequency of 1 Hz. All HALO instruments are listed in Wendisch et al. (2016).

The gamma fit parameters can be plotted in a 3-D subspace where each parameter (N0, μ, and Λ) represents one dimension. Each point in this 3-D gamma phase space is defined by one (N0, μ, and Λ) triplet and thus represents one fitted DSD. This space includes all possible combinations of gamma parameters of the theoretical variability in the DSDs.

The 3-D gamma phase space is illustrated in Fig. 3. There are two points in this figure defined by two location vectors P1 and P2, each one representing a fit to a specific DSD (see the insert in the left side of Fig. 3) at different times (t1 and t2 for t2 > t1). If we consider that P1 and P2 represent the same population of droplets evolving in time (i.e., a Lagrangian case), we can link the two points by a displacement vector P = P2 - P1, which can be associated with a pseudo-force F (blue arrow in Fig. 3). We use the term pseudo-force in order to illustrate that the growth processes produce displacements in the phase space. Alternatively, displacements in the phase space can also be understood as phase state transitions, in which case each phase-state is related to a DSD. The pseudo-force F can be decomposed into two components, one related to condensational growth and the other to the collision–coalescence (collection) process. The respective pseudo-forces are illustrated as Fcd and Fcl in Fig. 3, respectively. This approach can be applied to multiple points, defining a trajectory through the phase space (gray dotted line). The change of the DSD results in modified gamma parameters, which determine the trajectory through the gamma phase space. The direction and speed of the displacements forming the trajectory are determined by the direction and intensity of the underlying physical processes that modify the DSD (condensation and collection). These pseudo-forces are defined by properties such as the initial DSD, CCN, updraft speed, and supersaturation. Of course, this generalization considers only condensation and collision–coalescence. The pseudo-forces can be represented with more sophistication in models, including the several processes involved in DSD changes, such as evaporation, turbulence, melting from the layer above, breakup, sedimentation, etc. Therefore, these two processes can be replaced by a number of pseudo-forces as a function of the level of sophistication of the model. We should remember that this approach does not consider contributions from other levels because advection is not directly addressed. To describe the whole process of DSD evolution during the entire cloud life cycle, the contribution from other layers should be considered.

The direction of the Fcd pseudo-force in Fig. 3 represents the transition of the DSD during the condensation process, which favors high values of μ and slightly increasing Λ. This induces both the narrowing of and a slight increase in the effective droplet diameter (see equations in Sect. 2.2) of the DSD, which is expected from conventional condensation growth theory. Because of the DSD narrowing, the intercept parameter (N0) is also reduced. Condensational growth may cause a broadening of the DSD in specific situations such as at the cloud base of polluted systems. However, this is an exception and most of the time condensational growth leads to DSD narrowing. The collision–coalescence pseudo-force acts in a significantly different way in the phase space. From theory and precise numerical simulations that solve the stochastic collection equation, it is known that this process leads to DSD broadening (given the collection of small droplets and breakup of bigger ones) and faster droplet growth in size (compared to condensation). In the gamma phase space, it should be reflected in lower values of Λ and μ, the former decreasing at a faster pace. The intercept parameter N0 can remain relatively constant or increase because the effects of increased mean diameter and DSD broadening balance each other. If N0 remains constant, lower values of Λ and μ result in reduced droplet number concentration, which is consistent with theory (see Fig. 7).

To confirm the overall directions of the pseudo-forces and the characteristics of the gamma phase space, we performed some calculations with the Lagrangian model described in Feingold et al. (1999) – see their Section 3c and references therein. Basically, the model solves CCN activation (only at cloud base), condensation and collision–coalescence growth, and the effects of giant CCN on the DSDs (the latter process was turned off in our runs). The DSDs are sorted into 35 mass-doubling bins from ∼ 3 µm to ∼ 9 mm; thus, the condensation and collision–coalescence processes are not parameterized as in bulk approaches. The model was initialized with conditions that mimic flight RA1 (Table 1, Fig. 4). By performing two runs, one with exclusively condensational growth and the other with both growth processes, it was possible to isolate their effects on the DSDs. In the run with both processes active, by the time the collection was significant the droplets were big enough (Deff > 25 µm) to grow very slowly by condensation.

From the Lagrangian model runs it was possible to calculate the direction of the displacements caused by condensation and collision–coalescence growth in spherical coordinates. For this first introduction of the phase space, we will focus on the elevation angle φ (from the plane N0 × μ to the Λ axis) and azimuth angle θ (calculated from the N0 to the μ axes), when N0, μ, and Λ are in logarithm (base 10) scale (as in Figs. 5–8). The angles vary depending on the relative values of N0, μ, and Λ, but the following numbers are provided as a first estimate. For condensational growth, φ averaged 0.26 ∘ and θ averaged 179.6∘, while they were -4.23 and -13.7 ∘ for collision–coalescence, respectively. Note that the angles have opposite signs for the two processes and their overall direction is the same as exemplified in Fig. 3. The direction of the displacements remains consistent even when other moments are chosen to fit the gamma DSDs. For more details on the model runs, refer to the Supplement.

Note that, given the relation between the gamma parameters, the phase space is non-orthogonal and it is not trivial to mathematically represent the pseudo-forces. The mathematical treatment of such forces is beyond the scope of this paper, which intends to illustrate microphysical processes in the phase space. But this aspect should be considered in potential future implementations of this methodology in practical applications.

In Sect. 3.2, we show gamma parameters fitted to real DSD observations. As it is not feasible to follow fixed populations of droplets in a Lagrangian way with an aircraft, the evolutions we analyze in the gamma space are not strictly over time. As a compromise, we use the altitude above cloud base (H) of the measurements instead of time evolution, given the conditions of the measurements and our data handling. The cloud profiling missions were planned to capture growing convective elements before reaching their mature state, which is the reason why they usually started at around 12:00 LT (local time). Additionally, we only consider DSD measurements in which updraft speed w > 0 in order to focus on the ascending part of the growing clouds.

Another point to take into consideration are the ellipsoids discussed in McFarquhar et al. (2015). Basically, by considering the instrument and gamma fitting uncertainties, it is possible to define volumes (with ellipsoid shapes) rather than individual points in the gamma phase space. Inside each ellipsoid, all DSDs are equally realizable and therefore the movements within it have no particular physical meaning and are statistically the same. In this study, however, we estimate that the results evolve beyond individual ellipsoids and the patterns are associated with physical processes. The results shown in the next sections will not consider the ellipsoid approach, but the points shown can be considered to be the central points of such volumes.

The HALO flights are classified according to the region they covered and the respective aerosol and CCN number concentrations (Table 1). Note the close link between region of the measurements and the aerosol concentrations. From the most pristine clouds at the coast to the most polluted cases in the arc of deforestation, there is a 10-fold increase in NCN. Remote regions in the Amazon have aerosol particle concentrations slightly higher than over the coast, which is one of the reasons for the term “Green Ocean” used for the unpolluted Amazon regions (Williams et al., 2002). Flights AD1, AD2, and AD3 present flight paths progressively shifted to the south, which are accompanied by increasing values of NCN and NCCN. The farther away the flights take place from the forest, and consequently closer to developed regions, the higher the pollution levels.

Cloud profiles started at the end of the morning or beginning of the afternoon. The flights were specifically planned for this time period because the convective systems are usually in their developing stages at this time. The freezing level varied between 4500 and 5000 m, while cloud base altitudes were more variable (500 to 2000 m), which resulted from the regional meteorological conditions (Fig. 4), and which affects the characteristics of the cloud layers. Clouds in the arc of deforestation grow from drier air, given the diminished evapotranspiration rate, and form higher in the atmosphere. As a result, there are thinner warm layers in the polluted clouds, which reduces the time available for droplets to grow by collision–coalescence. Flight RA2 was characterized by a just slightly higher depth of the warm layer compared to the polluted clouds, partly due to the lower altitude of the freezing level. Nevertheless, cleaner clouds can present warm layers 1000 m thicker than clouds affected by pollution.

The vertical profile of the relative humidity (RH) should also be taken into account when comparing clouds formed over different regions. Figure 4b shows that all clouds measured formed in a surrounding environment with a RH between 60 and 90 % for their lower 2500 m layer, with RH being higher for forested areas compared to the arc of deforestation. For 2500 m and above, there was a significant drying of the atmosphere for flights M1, RA2, and AD2. It is not clear if the other flights presented similar behavior given the relatively low data coverage for this layer. Regardless, surrounding dry air can significantly enhance the entrainment mixing process (Korolev et al., 2016). As pointed out by Freud et al. (2008), the mixing in Amazonian convective clouds (and also in other regions – Freud et al., 2011) tends toward the extreme inhomogeneous mixing case, in which the effective droplet diameter Deff presents almost no sensitivity to the entrainment. Our result largely corroborates this finding (see Fig. 11). It should be pointed out, however, that the recent studies by Korolev et al. (2016) and Pinsky et al. (2016a, b) show that homogeneous and inhomogeneous mixing can be indistinguishable depending on meteorological conditions and DSD characteristics when considering the time-dependent characteristics of the entrainment process. Mixing processes may have an impact on the shape of the DSDs measured, thus affecting displacements in the gamma phase space. The specific type of mixing responsible for it, however, is beyond the scope of this work.

In this study, we use the gamma phase space as a means to study DSD variability. As described in Sect. 2.3, this space is obtained when the DSD measurements are fitted to Eq. (1), and N0, μ, and Λ are used as the dimensions of the 3-D subspace. In this space, each point represents one DSD. As the different DSDs were obtained close to the cloud top at the time of the cloud development, the ensemble of positions in the gamma phase space can be hypothesized as the evolution of the DSDs of a typical cloud through stages of its life cycle. The sequential connection of points (here we use cubic spline fits for illustrating purposes) can be considered as trajectories describing multiple processes responsible for the DSD variability observed. The advantage of using this space is that this variability can be readily observed and compared between different cloud life cycles with different properties. Given the relations between gamma parameters and DSD properties (Sect. 2.2), the variability in all cloud microphysical properties can also be inferred from the points in the trajectories. We limited the analysis regarding cloud DSDs and the gamma phase space to the regions in which w > 0 in order to capture the developing parts of the growing convective elements.

Figures 5 to 7 show the gamma phase space for all profiles considered in this study, grouped by region. The coloring represents the altitude above cloud base (H), with the 1 Hz measurements shown as small markers. Bigger markers represent averages at every 200 m vertical interval with available information. Curves (or trajectories) represent cubic spline fits to the averaged points. At first glance, it is possible to see stronger differences between the trajectories in the different regions, while internal variations are much weaker. Aerosol concentrations seem to be a key factor controlling warm-phase properties in the Amazon; thus, the internal similarities can be attributed to similar pollution conditions. Conversely, differences between the regions stem from the different weights of growth processes. Pristine clouds, like the ones found over the remote Amazon and the coast of the Atlantic Ocean, are characterized by faster droplet growth with altitude associated with enhanced collisional growth. In the gamma space, this is seen as diagonally tilted trajectories in Figs. 5 and 6, contrasting with the more vertical trajectories found in polluted clouds (Fig. 7).

The differences in the DSD variability in each region highlight the relation of growth processes and trajectories in the gamma phase space. From the theory described in Sect. 2.3, it is expected that collisional growth results in diagonal trajectories where the droplets get progressively bigger with DSD broadening. Pristine clouds over the coast and remote Amazon show such tilting (Figs. 5 and 6), indicating that this process is effective in these systems. The more vertically oriented trajectories of polluted clouds (Fig. 7) show that there is a different balance between condensational and collisional growth. In terms of the gamma phase space characteristics, this can be understood as weaker Fcl as a result of smaller droplets and narrower DSDs. This highlights that the interaction between aerosols and collisional growth occurs mainly through changes in the initial DSD (i.e., P1 in Fig. 3). For each point in the gamma phase space the collisional pseudo-forces have different intensities and directions, suggesting that a vector field can be constructed. This could only be achieved with idealized model experiments, however, where the updraft speeds can also be prescribed.

Condensational growth can also be illustrated by some points in Figs. 6 and 7. Under polluted conditions, this type of growth is expected to be dominant close to cloud base where the droplets are too small to trigger collision–coalescence. In Fig. 7, this is seen in the first two or three points in the trajectories (dark blue colors), where the points evolve to higher μ values with altitude. This results in DSD narrowing and almost opposite displacement in the gamma space compared to collisional growth. This trend is shifted when the altitude at which collection processes start to become relevant is reached. Another example of condensational growth can be seen in Fig. 6 at 3000 m. At this point, which is close to the freezing level, there is a sudden increase in the updrafts (see Tables 2 and 3) and consequently increased condensation rates. The rapid increase in condensational growth, with no significant changes in collision–coalescence, tilts the trajectories to a direction similar to that observed close to cloud base in polluted systems. The displacement is closer to the horizontal direction (i.e., the plane N0 × μ), because droplets grow concomitantly by collision–coalescence in the cleaner clouds.

The magnitude of the condensational pseudo-force (Fcd in Fig. 3) also depends on the initial DSD characteristics (P1). Condensational growth rates are inversely proportional to droplet size, meaning that they get weaker higher in the cloud. The different dependences of Fcd and Fcl on P1 and their balance throughout the warm-phase life cycle ultimately define the cloud trajectory in the phase space. If they can be mapped with sufficient resolution, covering different updraft and supersaturation conditions, trajectories may be forecast from a single DSD at cloud base and the evolving thermodynamic conditions. Aerosols are a key aspect in this regard because they significantly change the cloud base DSD in the gamma space (Figs. 5–7) and also affect cloud thermodynamics, impacting condensation rates and consequently latent heat release. Note that clouds subject to similar aerosol conditions have similarities in their trajectories represented by small variability along the trajectories of the respective flights (Figs. 6 and 7).

The Fcd and Fcl tabulation over the gamma space can potentially be achieved with the help of Lagrangian large-eddy-simulation bin-microphysics models that precisely solve the condensation and collection equations for varying input DSDs and updraft conditions. Initial DSDs can be obtained from observations and analytical considerations. For instance, Pinsky et al. (2012) show an analytical way to obtain the maximum supersaturation (which is usually a few meters above cloud base) and the relative droplet concentration. If Deff behaves adiabatically (Freud et al., 2008; Freud et al., 2011) and is linearly correlated to the mean volumetric diameter (Freud and Rosenfeld, 2012), it is possible to estimate the initial DSD based on gamma DSD equations and adiabatic theory given that the aerosol population is known. The advantage of such an approach is that all DSD characteristics, most notably the shape, would be realistically represented and there would be no need for fixing or diagnosing (Thompson et al., 2004; Milbrandt and Yau, 2005; Formenton et al., 2013a, b) gamma parameters for various hydrometeor types – which works for specific applications but may be lacking the physical representation of the processes. This study focuses on introducing the gamma phase space and its characteristics, and further work is needed if new parameterizations are to be developed.

In this section, we focus on flights RA1 and AD2 in order to study the differences between natural and human-affected clouds in the gamma space. Figure 8 shows the trajectories of the clouds measured during these flights, where the points related to the averaged DSDs are numbered and the corresponding properties are shown in Tables 2 and 3. The numbers start at 1 close to cloud base and grow with altitude (“p” stands for polluted, while “c” is for clean). Also presented in Tables 2 and 3 are the adiabatic fractions that correspond to the ratio between the observed and adiabatic LWC. Some observed DSDs and their corresponding gamma DSDs are shown in Fig. 9, highlighting different growth processes.

It is clear from Fig. 8 that clean and polluted clouds cover different regions of the gamma phase space. Nevertheless, it is possible to see that the trajectories can evolve almost in parallel depending on the dominant growth process. Polluted clouds have wider DSDs at cloud base because of the tail to lower diameters (Fig. 9), which brings down the value of μ (see Eq. 9). Given the lower droplet size (Table 3), condensation is efficient and the trajectories evolve in the overall direction of Fcd illustrated in Fig. 3. From points 1p to 2p, Nd and LWC are approximately doubled. Condensational growth seems to be the dominant growth process in the polluted clouds up to the point 3p, corresponding to a cloud depth of 600 m. A similar layer does not exist in cleaner clouds, where there are enough big droplets to readily activate the collision–coalescence growth. Collisional growth dominates the DSD shape evolution between points 1c and 6c for flight RA1 and between 4p and 7p for AD2. Note that the trajectories are almost parallel in this region. Condensation is still active in this period given the increasing LWC, but collision–coalescence has a comparatively bigger impact on the overall DSD shape. Both sections of the trajectories represent 1400 m thick layers, but droplet growth and DSD broadening is more efficient in the cleaner clouds (Fig. 9). This explains the pronounced tilting of its trajectory, consistent with a stronger Fcl pseudo-force.

Eventually, the trajectories reach a point close to the 0 ∘C isotherm where the updrafts are enhanced given the continued latent heat release. This w-enhanced layer can be several hundred meters thick and culminates in narrower DSDs. This is exemplified between points 7c and 9c and between 8p and 10p. Although droplets still grow by collision–coalescence, the enhanced updrafts increase condensational growth sufficiently to produce observable effects on the DSDs. Both trajectories evolve in the condensational growth direction, but with slightly different tilting. The tilting is less pronounced in the cleaner clouds given the stronger Fcl component. The way in which the DSDs evolve in this region is important for the mixed-phase initiation, given that both primary and secondary ice generation depend on the characteristics of the liquid droplets. The different properties of the polluted and clean DSDs (see Tables 2 and 3, Figs. 8 and 9) indicate that ice formation may follow distinct pathways.

Previous studies suggest that droplets bigger than 23 µm at concentrations higher than 1 cm-3 favor secondary ice generation, which was identified as the main mechanism for cloud glaciation (Mossop, 1978; Saunders and Hosseini, 2001; Heymsfield and Willis, 2014; Lawson et al., 2015). In order to visualize these conditions in the gamma phase space, it is interesting to consider constant Nd surfaces. These surfaces are defined when Nd is fixed in Eq. (6), resulting in a relation of the form Λ = f(N0μ) when inverted. Examples are shown in Fig. 10, where Nd = {10, 100, 1000} cm-3 (axes are rotated for clarity). The surfaces are evidently parallel and are stacked in relatively close proximity (on the scale used here). The trajectories evolve through the surfaces depending on their Nd, where polluted clouds tend toward higher droplet concentration (i.e., closer to the red surface). These surfaces can be used to delimit specific regions of interest. Additionally, further DSD properties can be analyzed along these surfaces. Figure 10 highlights the region of 23 µm < Deff < 50 µm with black lines along the surface of Nd = 10 cm-3. The purple ellipse represents an estimate of the size of the ellipsoids introduced by McFarquhar et al. (2015). For the calculation of the ellipse, we considered that the 10 % error in the CDP measurements translates to 10 % error in Nd, Deff, and ε. Under those conditions, the size of the ellipsoids can be represented by the purple ellipse in Fig. 10. Note that its dimension in the normal direction from the Nd surfaces is very small (1/100 of the distance between the surfaces shown); that is why we only show an ellipse rather than an ellipsoid. This simple estimate is meant to find the order of magnitude of the ellipsoids of McFarquhar et al. (2015) and it shows that the trajectories are not just random displacement inside one ellipsoid (the trajectories evolve beyond them).

Regarding cloud DSDs (drizzle droplets are not analyzed here, although they also contribute to ice formation), the region delimited by the black lines for the different surfaces of constant Nd can be interpreted as the most favorable for secondary ice generation, thus indicating a quick glaciation process. Note that the trajectory of the cleaner clouds enters this region while in the w-enhanced layer mentioned previously, which corresponds to the transition to temperatures below 0 ∘C. Polluted clouds are able to produce high droplet number concentrations, but their smaller droplet size means that they are out of the delimited region. More details about the transition to the mixed phase are given in the next section.

The observation of constant Nd surfaces poses an interesting question for parameterizations. In existing two-moment schemes, both Nd and LWC are predicted. For each pair of such properties, it is possible to define two surfaces (with constant Nd and LWC) based on Eqs. (6) and (7). These surfaces intersect, defining a curve where both properties are conserved. In this curve, the mean volumetric diameter (proportional to the ratio between LWC and Nd) is also constant. Based on the limited information provided by the model (only two moments for three gamma parameters), this curve represents the infinite DSD solutions for the undetermined equation system. A good parameterization scheme should be able to choose one of the DSDs that best fits observations. Given the undetermined equation system, other considerations have to be made.

One parameter that varies along the infinite DSD solution curve is the relative dispersion ε. If ε can be constrained in the model, it should be possible to obtain the full gamma DSD – which is the point in the intersection curve that presents the given ε. The advantage of relying on ε is that it has low variability between clean and polluted clouds and its average is almost constant with altitude. Tas et al. (2015) studied the relative dispersion parameter in detail, noting that averaged values for ε were independent of Nd, LWC, or height, but its variability is significantly lower for the most adiabatic portions of the cloud (notably its updraft core). For precise parameterizations, ε variability should be taken into account in regions with relatively low Nd and LWC, but averaged values may be considered for the updraft cores. Our observations show that ε is slightly higher in polluted Amazonian clouds compared to the ones measured over remote regions mainly because of their reduced droplet size (Tables 2 and 3). This can be considered to produce slight corrections to ε based on CCN number concentrations.

The gamma phase space provides an insightful way to study the formation of the mixed phase by providing the history of the warm-phase development as a trajectory. Liquid cloud droplet properties are important for the glaciation process because they determine the probability of contacting INPs and the conditions for secondary generation. As shown in the previous sections, different aerosol and thermodynamic conditions alter warm-phase characteristics and can thus impact the early formation of ice in the clouds.

Figure 11 shows vertical profiles of Nd, LWC, Deff, and ε for clouds subject to background and polluted conditions (flights RA1 and AD2, respectively). It shows the different microphysical properties (1 Hz) of the clouds associated with the trajectories presented in Figs. 8 to 10 (w > 0). It shows that droplet concentrations are much higher in polluted clouds, which are not depleted with altitude as much as cleaner clouds (Fig. 11a and b). The lower effective diameter for clouds over the arc of deforestation may contribute to enhanced evaporation, leading to lower adiabatic fractions. As commented in the previous section, ε shows small variations between the flights and does not change much with altitude.

The properties of the DSDs around the 0 ∘C level in Fig. 11 are a significant feature regarding the mixed-phase formation. Note that cleaner clouds have a sudden change in behavior right above the freezing level. At this point, there is a fast decrease in LWC, with higher variability in both Deff and ε. This suggests that ice processes have been triggered, disrupting the smooth evolution observed in the warm phase. In polluted clouds, this transition takes place at considerably different DSD properties. Averaged Nd reaches values above 1000 cm-3 (compared to 50 cm-3 in cleaner clouds) with very strong updrafts, bringing LWC closer to adiabaticity. However, no significant variability was observed for Deff, suggesting that most of the water is still in a condensed state.

In order to further detail the characteristics of the hydrometeors in the transition from warm to mixed phase, we analyzed the sphericity criteria obtained with the NIXE-CAPS probe (Costa et al., 2017). The methodology developed by Costa et al. (2017) indicates whether each individual 1 Hz measurement contained some aspherical hydrometeors or not. This criterion can be used to indicate whether the hydrometeors are liquid (spherical), mixed (spherical and aspherical), or frozen (aspherical). By combining all measurements for clouds over the remote Amazon (RA1 and RA2) and the arc of deforestation (AD1, AD2, and AD3), we obtained the results shown in Fig. 12.

The classifications shown in Fig. 12 separate the volumes probed as containing only spherical hydrometeors (Sph (liquid)) or if there are also aspherical particles too. In that case, the data are further divided into containing both small (D < 50 µm) spherical and large aspherical particles (D > 50 µm) – Asph small (mixed phase) – or if there are only large (D > 50 µm) aspherical particles – Asph large (ice). It is possible to observe that close to cloud base most of the hydrometeors were detected as spherical for both regions, which is expected given that it is the warmest layer of the cloud. However, higher in the clouds the distribution of the classifications becomes different. The number of measurements with aspherical particles increases relatively fast for the cleaner clouds, being higher than 90 % at the layer around 0 ∘C. For polluted clouds, however, almost half of the measurements contained exclusively spherical hydrometeors at this level. Exclusively spherical hydrometeors persisted with a frequency of ∼ 20 % down to temperatures of -15 ∘C. This is in line with previous studies that found supercooled droplets high into continental convective systems (Rosenfeld and Woodley, 2000; Rosenfeld et al., 2008). Our results show that the persistence of supercooled droplets in Amazonian clouds is more likely under polluted conditions.

The characteristics of the cloud warm layer determine the properties of the liquid DSDs close to the 0 ∘C level and should have a determining role in the glaciation initiation. Our measurements show that clean clouds can produce droplets roughly twice the size of the ones found in polluted systems at this layer, at 95 % lower droplet concentrations (Tables 2 and 3). Bigger droplets are not only more likely to interact with INPs and glaciate by immersion or contact freezing, but may also trigger a cascading effect through secondary ice generation (Heymsfield and Willis, 2014; Lawson et al., 2015). This process is able to quickly glaciate the cloud, which fits the results shown in Fig. 12. Beyond the DSD bulk properties, the gamma phase space can also provide more information regarding the kind of DSD that enables or inhibits the glaciation process. In the present study, we have only a few examples to compare warm- and mixed-phase characteristics, but it is clear that it is possible to correlate some regions of the phase space with the characteristics of the ice initiation. Detailed model experiments would greatly enrich this discussion by providing control over the liquid DSD properties and the resulting formation of the mixed layer. More specifically, it would be invaluable to study the impacts of the properties of DSDs at cloud base and at the 0 ∘C isotherm on the primary and secondary ice production.