2.1 Cloud probes

The 2012 DC3 experiment investigated the impacts of deep midlatitude continental convective clouds on upper tropospheric chemistry and composition in the US Midwest (Barth et al., 2015). The National Science Foundation (NSF)/National Center for Atmospheric Research (NCAR) Gulfstream V (GV), the National Aeronautics and Space Administration (NASA) DC-8, and the Deutsches Zentrum für Luft- und Raumfahrt (DLR) Falcon aircraft were deployed during DC3.

In this study, in situ measurements were acquired from the GV equipped with a Stratton Park Engineering Company Inc. (SPEC) 3V-CPI instrument, a cloud droplet probe (CDP, manufactured by Droplet Measurement Technologies, DMT), and a specially modified Particle Measuring Systems (PMS) optical array probe (2DC), which uses high-speed electronics and a 64-element 25 µm-resolution diode array in order to shadow particles at the sampling speeds of the GV. The 3V-CPI instrument provided high-resolution (i.e., 2.3 µm) images of ice crystals with sizes up to ∼2300 µm at 400 frames per second. The CDP determined the number distribution function N(D_max) and total concentration (N_CDP) of ice crystals with a D_max between 2 and 50 µm from the amount of light forward scattered. The 2DC optical array probe measured N(D_max) for $D_{max}<\mathrm{1550}$ µm. Other details on the GV instrumentation are provided in Stith et al. (2014).

Figure 1(a) Example CPI images of ice crystals observed at $T=-\mathrm{58.16}$ ^∘C (altitude of 12.11 km) between 22:12:13 and 22:12:19 UTC, and (b) example CPI images of ice crystals observed at $T=-\mathrm{57.72}$ ^∘C (altitude of 12.03 km) between 22:21:02 and 22:22:14 UTC. The 200 µm scale bar is embedded in each figure.

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The shattering of large cloud particles on the shrouds, tips, or inlets of cloud probes can cause artificial increases in in situ measured concentrations of small particles. Thus, the impacts of shattering must be prevented or removed (Field et al., 2003, 2006; McFarquhar et al., 2007; Korolev et al., 2011; Lawson, 2011; Jackson and McFarquhar, 2014; Jackson et al., 2014; Korolev and Field, 2015). The CDP used during DC3 did not have a shroud; thus, shattering is not expected to be substantial (Stith et al., 2014). Anti-shattering tips (Korolev et al., 2011) were installed on the 2DC, and post-processing methods of removing particles with small interarrival times (Field et al., 2003, 2006) were applied. 2DC measurements of only particles with $D_{max}>\mathrm{100}$ µm were used in this study due to a poorly defined depth of field for smaller particles (e.g., McFarquhar et al., 2017). Thus, the CDP (N_CDP) and 2DC (N_2DC>100) concentrations are used to identify the presence of small ($D_{max}<\mathrm{50}$ µm) and large ($D_{max}>\mathrm{100}$ µm) particles, respectively. Information about particle shape was obtained from the CPI component of the 3V-CPI. But, only CPI images with focus >20 were used (McFarquhar et al., 2013), and analysis of multiple particles on the same frame was not performed as Um and McFarquhar (2011) suggested that they might be shattered artifacts.

Table 1Segregated time periods of the 6 June flight and contributions (%) of crystal habit to the total number (total projected area) of ice crystals for the given time period. The average and standard deviation of temperature (T), altitude, and maximum dimension (D_max) of ice crystals determined from CPI images are also listed for the given time period.

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2.2 Anvil observations

During the 6 June 2012 flight, the GV sampled the upper anvils of two storms that developed in eastern Colorado between 22:10:00 and 22:30:00 UTC near the CSU-CHILL radar (40.45^∘ N and 104.64^∘ W). The two storms were aligned north–south, separated by ∼70 km, and had similar size and intensity based on the next generation weather radar (NEXRAD) images (see Figs. 1–3 of Stith et al., 2014). Examples of CPI crystal images sampled during this flight are shown in Fig. 1. They were mainly (>93 % by number) single frozen droplets with quasi-circular shapes and their aggregates (i.e., FDAs). This is consistent with the analysis of Stith et al. (2014), who showed that these upper anvil regions were primarily composed of frozen droplets with differing degrees of aggregation, with FDAs being most frequent in the center and lower regions of the upper anvil. More details about these two storms are discussed in Stith et al. (2014, 2016). The GV flew two constant altitude runs during this period, at altitudes and temperatures of 12.0 to 12.4 km and −61 to −55 ^∘C, respectively (see Fig. 2). This period is further segregated into three periods: (1) 22:11:30–22:14:55 UTC, (2) 22:19:00–22:23:40 UTC, and (3) 22:23:55–22:27:50 UTC (Table 1 and Fig. 2). These periods are selected in order to separate measurements in the north and south anvils and at different temperatures. During time periods 1 and 2, the same anvil of the south storm was sampled with a ∼4 min interval between the samples at two different temperatures ($T\sim -\mathrm{68.4}$ and −57.5 ^∘C, respectively), whereas period 3 sampled the north anvil at the higher temperature of −56.6 ^∘C. In the next section, the frequency at which single frozen droplets and FDAs were observed is determined using a technique derived to identify the frozen droplet elements within FDAs.

3.1 Ice crystal habit classification

Based on CPI crystal images obtained in tropical ice clouds, Um and McFarquhar (2009) developed a classification scheme to sort crystals into eleven habits: small, medium, and large quasi-spheres, columns, plates, bullet rosettes, aggregates of columns, aggregates of plates, aggregates of bullet rosettes, capped columns, and unclassified. To represent other crystal habits commonly found in midlatitude and Arctic clouds, the capability of sorting into more habits (i.e., dendrite, needle, aggregates of needles, and FDAs) has been added to the scheme (McFarquhar et al., 2017). Thus, this habit classification scheme now sorts crystals into 15 different categories in a quasi-automatic manner that requires some manual intervention.

Figure 2(a) Temperature and altitude of the NSF/NCAR GV aircraft as a function of time on the 6 June 2012 flight and (b)  the 1 s average concentration of measured CDP and 2DC_>100. Determined habit fraction (c) by number and (d) by projected area as a function of time. Habit categories for panels (c) and (d) are shown using the colored legend under the figure . Habits are sorted into 15 categories: frozen droplet aggregates (FDAs), small quasi-sphere (SQS), medium quasi-sphere (MQS), large-quasi sphere (LQS), plate, aggregates of plates, bullet rosette, aggregates of bullet rosettes, column, aggregates of column, needle, aggregates of needles, dendrite, capped column, and unclassified as shown in Fig. 2. For simplicity single frozen drops (FD) denoted in this figure includes SQS, MQS, and LQS, while other habits except FDAs and UC are indicated as “Else” in this figure. Time periods 1, 2, and 3 are shaded gray in each panel.

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In this study, ice crystals classified as small (SQS), medium (MQS), and large quasi-spheres (LQS) are regarded as single frozen droplets. The FDAs that occur near anvil tops are often classified as bullet rosettes, aggregates of bullet rosettes, or unclassified from the automated part of the algorithm; therefore, an additional manual check was necessary to confirm whether or not these crystals were FDAs. To be classified as FDAs, there must be at least two quasi-circular frozen droplets as elements. Habits that frequently occurred during the 6 June 2012 flight were single frozen droplets (i.e., SQS, MSQ, and LQS) and their aggregates (i.e., FDAs), whereas very few pristine shape crystals, such as plates, columns, and bullet rosettes, were observed (see Table 1 and Fig. 3).

Figure 3Contributions of ice crystal habits by number (red) and by projected area (blue) during (a) all periods, (b) period 1, (c) period 2, and (d) period 3. The total number of samples is indicated in each panel. Acronyms for crystal habits are indicated in the caption of Fig. 2.

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3.2 Technique to identify element frozen droplets

The circle Hough transform (CHT, Duda and Hart, 1972) detects circular objects in digital images and is one of many feature-extracting techniques that use the Hough transform (Hough, 1962). Several variants of the Hough transform have been developed, such as, the fast Hough transform (Li et al., 1986), two-stage CHT (Yuen et al., 1990), space saving approach CHT (Albanesi and Ferretti, 1990), and the phase-coding method (Atherton and Kerbyson, 1999). These techniques have been used to detect natural particles with circular shapes in digital images, such as circular nanoparticles in transmission electron microscopy (TEM) images (Bescond et al., 2014; Mirzaei and Rafsanjani, 2017).

Figure 4CPI images of frozen droplet aggregates (FDAs, left image of each column) and those with determined element frozen droplets (red circle, right image of each column). The 46 µm scale bar is shown in each FDA image.

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Prior studies have used such techniques to identify the elemental or primary particles within black carbon aggregates (e.g., Bescond et al., 2014; China et al., 2013), most of which are circular. Similar techniques can be applied to the FDAs observed near the tops of anvil clouds assuming the element frozen droplets have spherical shapes. The biggest difference between TEM and CPI images is that the quality of TEM images is, in general, better than that of CPI images. A CPI image has an inhomogeneous background and debris or noise, such as impulse noise (i.e., salt-and-pepper noise), which causes lower quality images. Thus, additional image-quality control was required before applying the CHT technique to the images. This was accomplished in a number of steps. First, a median filter that is a nonlinear digital filtering technique to remove noise is applied to the CPI images classified as FDAs. The 256-level gray-scale CPI images are then converted to binary images based on the average intensity of pixels to further remove background noise and debris. Figure 4 shows example images of CPI FDAs. Two different CHT techniques, the two-stage CHT and phase-coding method, are then applied to the images to detect element circles (i.e., frozen droplets) as shown by the red circles in Figs. 4 and 5. Two different techniques are used because the performance of each technique varies depending on the CPI image being classified. The technique used for the subsequent analysis is chosen as that for which the projected area of the FDAs determined for the element frozen droplets identified by CHT technique (i.e., area determined by red lines in Fig. 5) best matches that for the original CPI image (i.e., area enclosed by green line in Fig. 5). However, the performance of both techniques is quite similar. For example, the phase-coding technique shows closer agreement with the imaged area for the FDAs shown in the top row of Fig. 5, while the two-stage CHT shows closer agreement for the FDAs shown in the bottom row. Although the phase-coding method provided marginally better results for ∼54 % cases, the differences in projected area determined by the two techniques were within 9.7 % for all FDAs. Thus, since the performance of both methods in replicating the determined area of the CPI images is similar, there should be minimal bias in the determined results. The number and size of the element frozen droplets within the FDAs were then determined automatically. The relative positions (i.e., x and y coordinates) of the element frozen droplets were also identified.

Figure 5Element frozen droplets (red circles) determined using the phase-coding (left column) and two-stage CHT (middle column) techniques. Examples of two different FDAs are shown in the top and bottom rows, respectively. Original CPI images of FDAs are shown in the right column along with the 46 µm scale bar. The detected boundary (green lines) of the FDAs are shown on panels in the left and middle columns.

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4.1 Frequency of occurrence of ice crystal habits

Figure 3 shows the normalized contribution of each habit to the total number (red) and to the total projected area (blue) of measured ice crystals during the three different time periods and integrated over the entire time period. For all time periods, single frozen droplets represented the dominant habit by number, whereas FDAs were dominant by projected area (see also Table 1). The fraction (by number) of single frozen droplets was 73.0 % (84.1 %; 70.6 %; 71.2 %) for all periods (period 1; period 2; period 3), whereas the area fraction of FDAs was 46.3 % (27.8 %; 49.6 %; 47.5 %). The fraction of well-defined pristine ice crystals, such as plates and columns, was less than 0.04 % by number and 0.12 % by area for all time periods, whereas unclassified crystals represented 6.1 % (3.9 %; 5.3 %; 7.5 %) by number for all periods (period 1; period 2; period 3) and 13.5 % (6.5 %; 10.9 %; 16.9 %) by area. These fractions of unclassified crystals were lower than those obtained from anvil cloud in the tropics (Um and McFarquhar, 2009) that showed more than 22 % and 37 % contributions by number and area, respectively. The presence of small crystals with relatively simple habit distributions shown in this study indicates that the anvil clouds were sampled in an early stage of development, which was verified using radar observations (Stith et al., 2014, 2016).

The average D_max for all crystal habits was 80.7±45.4 µm for all periods, with the average D_max of 68.4±37.1 µm during period 1 at $T=-\mathrm{60.0}\pm \mathrm{1.2}$ ^∘C smaller than those of 72.4±42.9 and 84.4±48.8 µm during periods 2 and 3 at $T=-\mathrm{57.5}\pm \mathrm{0.3}$ ^∘C and $T=-\mathrm{56.6}\pm \mathrm{0.5}$ ^∘C, respectively (Table 1). Table 1 also shows that the contributions of single frozen droplets and FDAs sampled in the south and north anvil in periods 2 and 3 are similar, whereas a much higher fraction of small frozen droplets was revealed in the south anvil during period 1 at a slightly lower temperature. For example, the fraction (by number) of single frozen droplets was 84.1 % in period 1, and 70.6 % and 71.2 % in periods 2 and 3, respectively. The fraction of FDAs in period 1 was 12.1 % and 27.8 % by number and projected area, respectively, which was substantially lower than those sampled in periods 2 and 3 (Table 1). Figure 2 shows that the fraction of single frozen droplets, in general, decreased as the GV penetrated into the center of anvil cloud (i.e., center of each gray shaded area shown on panels in Fig. 2), and then increased as it approached the cloud edge (i.e., both sides of each gray shaded area shown on panels in Fig. 2) for all periods. This variation in the relative occurrence of small and large crystals is more distinctly seen by comparing the N_CDP and N_2CD>100 shown in Fig. 2.

In summary, single frozen droplets and their aggregates dominated the upper anvil clouds sampled in situ, with the relative frequency of occurrence of single frozen droplets and FDAs dependent on temperature and position within the anvil, consistent with the conceptual model proposed by Stith et al. (2014, Fig. 12) and further detailed in Stith et al. (2016, Fig. 9).

4.2 Morphology of single frozen droplets and FDAs

Among the 4667 CPI images of FDAs, the CHT technique succeeded in identifying element frozen droplets for 4356 FDAs, whereas it failed for 311 FDAs (6.66 %). The number, size, and 2-D position of the element frozen droplets within the FDAs were thus determined automatically. Figure 6a shows the frequency distribution of the number of element frozen droplets within FDAs. The average number of frozen droplets within FDAs is 4.7±5.0 and the FDAs with two element frozen droplets are dominant with a frequency of occurrence of 42.0 %. This occurrence frequency gradually decreases with the number of element frozen droplets. The average and standard deviation of the diameter of the determined element frozen droplets (blue) are shown as a function of the number of element frozen droplets (Fig. 6b). The average and standard deviation of D_max of the single frozen droplets (red) are also shown for comparison. Considering plausible errors ($\sim \pm \mathrm{4.6}$ µm) in the identifying algorithm and the 2.3 µm CPI resolution, the average and standard deviation of the diameter of the element frozen droplets (31.79±7.12 µm) are comparable with those of the D_max of single frozen droplets (34.03±6.22 µm). The CPI errors in sizing particles vary with focus (Connolly et al., 2007) and can be larger than those considered in this study. The quasi-spherical shapes, non-pristine shapes, and similarity of single and element droplet sizes indicate that diffusional growth was likely not effective for the anvils sampled, and the large ice crystals (i.e., FDAs) grew mainly through aggregation. They also indicate that the sampled anvil clouds are in an early stage of development as verified by radar observations. The more complex structure of FDAs with an increase in the number of elements may cause errors in the estimated size of the element frozen droplets. For example, the increase in the determined diameter of element frozen droplets as the number of element frozen droplets increases from two to five in Fig. 6b may not be a physical effect, but rather caused by uncertainty in the methodology. The sizes of the element frozen droplets here (31.79±7.12 µm) are larger than those (15–20 µm) noted by Gayet et al. (2012) and those (∼10 µm) measured during laboratory experiments (Pedernera and Ávila, 2018). It is hard to determine the extent to which differences in methodology as opposed to physical differences in droplet sizes caused these differences because Gayet et al. (2012) did not specify how they determined frozen droplet size. However, despite the unavoidable errors in identifying the element frozen droplets due to the quality of the CPI images, the differences are large enough to suggest physical differences.

Figure 6(a) Normalized frequency of occurrence of the number of component frozen droplets within 4356 FDAs analyzed. (b) The average and standard deviation of the diameter of frozen droplets as a function of number of component frozen droplets within FDAs (blue). The average (red solid line) and standard deviation (red dash line) of the D_max of single frozen droplets are also shown.

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4.3 Three-dimensional representations of FDAs and comparison with black carbon aggregates

To determine microphysical (e.g., fall velocity) and scattering (e.g., asymmetry parameter) properties of cloud particles required for models, idealized 3-D models of the crystals are needed. However, cloud particle images recorded by cloud probes are silhouettes (i.e., 2-D images) of 3-D cloud particles (Nousiainen and McFarquhar, 2004). Retrieving the 3-D shapes of cloud particles based on the recorded silhouettes is difficult, especially for non-spherical ice crystals that have non-pristine shapes. It is easier to reconstruct 3-D shapes of well-defined pristine crystals, such as columns and plates. For example, an iterative approach to retrieve the 3-D shapes of bullet rosette crystals was developed (Um and McFarquhar, 2007). Assuming that the element crystals all had the same shape (e.g., plates), the 3-D shapes of more complex crystal aggregates (e.g., aggregates of plates) have also been reconstructed from crystal silhouettes (Um and McFarquhar, 2009).

FDAs consist of at least two element frozen droplets whose shapes are assumed to be spheres even though the elements are in fact quasi-spherical, meaning they have some departures from a spherical shape. The number, size, and 2-D position of the element frozen droplets within the FDAs were determined from the CPI images as explained in Sect. 3.2. Using this information, the 3-D shapes of FDAs are reconstructed for the given 2-D silhouette (i.e., CPI image) with the following assumptions:

• element frozen droplets of FDAs are spheres;

• there is no overlap between the elements of the frozen droplets; and

• the maximum number of contacting points of an element frozen droplet with other frozen droplets is two.

Since the relative positions (i.e., x and y coordinates) and sizes of the element frozen droplets are known, the reconstruction problem becomes one of stacking spheres with varying combinations of a vertical (z) coordinate. The abovementioned assumptions reduce the number of possible 3-D realizations of FDAs for a given 2-D projection so that a maximum number of ${\mathrm{2}}^{n_{\mathrm{p}}-\mathrm{2}}$ 3-D realizations is possible, where n_p is the number of element frozen droplets. For example, for FDAs with five element frozen droplets (i.e., n_p=5), a maximum number of eight different 3-D realizations is possible, while a maximum number of 256 3-D realizations is possible for FDAs with 10 elements. Figure 7 shows six different example 3-D realizations of the FDA shown in the top-left panel of Fig. 4. Since this FDA has 12 element frozen droplets, theoretically a total of 1024 3-D realizations are possible. However, FDAs such as this with a large number of element frozen droplets usually have notably fewer 3-D realizations due to the abovementioned assumptions, in particular due to the no overlap assumption.

As the number of element frozen droplets increases, the number of possible 3-D realizations also increases. For FDAs with 20 element frozen droplets, a maximum number of 262 144 different 3-D realizations is possible. Considering all 262 144 3-D realizations of FDAs is impractical for calculations of single-scattering properties. Thus, parameters that characterize the 3-D shapes of particles and link the 3-D shapes to scattering properties are required. Um and McFarquhar (2009) used several parameters, such as the aggregation index (AI), the area ratio, and the normalized projected area, to characterize the 3-D shape and to link to the scattering properties of aggregates of plate crystals. The motivation for the use of the AI is that the asymmetry parameter (g) of aggregates of plates was previously seen to increase with AI (Um and McFarquhar, 2009). In this study, a similar approach is adapted and the AI is defined as

where D_ij is the distance between the center of frozen droplet i and that of frozen droplet j and n_p≥3. Physically, AI represents the ratio of the sum of the distances between the centers of all frozen droplets compared to that when they all lie on a straight line. Thus, when all frozen droplets lie on a straight line AI=1, whereas a smaller AI implies a more compact shape. The AI is calculated for every 3-D realization. Thus, there are 262 144 AIs for FDAs with 20 element frozen droplets. Since the 3-D shape complexity of a particle (i.e., AI) and its g was previously shown to have a positive relationship for aggregates of plate (Um and McFarquhar, 2009), the maximum, minimum, and average AI are calculated for a given FDA. The calculated maximum, minimum, and average AI of FDAs as a function of the number of element frozen droplets (for n_p>2) are shown in Fig. 8. The AI of FDAs decreases with n_p, which indicates that larger FDAs with more element frozen droplets have more compact shapes. Figure 9 shows that the AI of FDAs decreases slightly with increasing temperature. Stith et al. (2014) showed that FDAs were frequent in the lower regions (i.e., higher temperatures) of the upper anvil.

Figure 7Six different examples of 3-D representations of FDA. Each image has the same projected area (gray) as the CPI image shown in Fig. 4 (top-left image).

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Figure 8The calculated maximum (blue), minimum (green), and average (red) aggregation index (AI) of FDAs (circles) as a function of the number of element frozen droplets. Best-fit lines are shown using solid lines.

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A fractal dimension (D_f) has been widely used to describe the 3-D shape or compactness of black carbon (soot) aggregates (e.g., Bescond et al., 2014; China et al., 2013) and can be represented using the following scaling law:

where R_g, r_p, and k_f are the radius of gyration, average radius of elements frozen droplets, and fractal prefactor (or structural coefficient) (Mandelbrot, 1982), respectively. The radius of gyration is represented as

where m_i is the mass of ith element and x_i is the distance between element i and the center of mass of the FDA. The radius of gyration is an overall cluster radius. The chain-like shapes of FDAs observed near the tops of anvil clouds (e.g., Fig. 1) show similarity with those of black carbon (BC) aggregates (see Figs. 6–8 in Lewis et al., 2009). Thus, identifying fractal dimensions of FDAs and comparing them against those of BC aggregates is of great interest to calculate the scattering properties as a function of shape. In this study, possible 3-D realizations of FDAs are represented using both AI and D_f with subsequent comparison to the shape of BC aggregates.

Figure 9The average aggregation index (AI, red circles in Fig. 8) as a function of temperature (blue circles). The mean and standard deviation of AI for six temperature ranges are indicated by the red circles and vertical bars, respectively.

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To provide an overview of the shapes of the FDAs, the D_f, k_f, R_g, and r_p are calculated for three 3-D realizations of a FDA that represent the maximum, minimum, and average AI. For each 3-D realization R_g∕r_p is plotted against n_p in Fig. 10. Then, the D_f and k_f are determined by fitting to Eq. (2). For a given D_f, k_f represents the degree of compactness, with a smaller k_f indicating less packing density (Lewis et al., 2009). The FDAs with the maximum, minimum, and average R_g∕r_p have a D_f (k_f) of 1.2083 (2.0998), 1.4329 (2.3864), and 1.4124 (2.0412), respectively. A smaller D_f indicates a more linear shape with 1.0 indicating a perfectly linear shape; the compactness also increases with D_f. Thus, FDAs with the maximum R_g∕r_p have more linear (chain-like) shapes, while FDAs with the minimum R_g∕r_p have more compact shapes.

Figure 10Relationships between the ratio of the radius of gyration (R_g) to the average diameter of element frozen droplets (r_p) and the number of element frozen droplets (n_p). Maximum (blue), minimum (green), and average (red) R_g∕r_p are shown using circles, and corresponding best-fit lines are plotted using solid lines. The calculated fractal dimension (D_f) and the fractal prefactor (k_f) of FDAs with maximum, minimum, and average R_g∕r_p are embedded. The D_f and k_f of ambient (black) and denuded (yellow) black carbon aggregates determined in China et al. (2013) are indicated along with those commonly determined (purple) in several studies.

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Figure 11Relationship between aggregation index (AI) and the ratio of radius of gyration to the average radius of elements (R_g∕r_p) of FDAs. The number of element frozen droplets (no. elements) and number of all possible 3-D realizations (no. 3-D realizations) are indicated. The best fit and correlation coefficient (r) for the relationship between the AI and R_g∕r_p are shown in red. Corresponding CPI images of FDAs are also embedded in each panel.

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The D_f=1.8 and k_f=1.35 for BC aggregates determined in previous studies (Meakin, 1983; Kolb et al., 1983; Sorensen and Robert, 1997; Lattuada et al., 2003; Pierce et al., 2006; Heinson et al., 2012; Heinson and Chakrabarty, 2016) are shown in the purple dashed line in Fig. 10 for comparison. The D_f=1.85 and k_f=1.46 determined for ambient BC aggregates (black dashed line) and D_f=1.53 and k_f=2.40 for denuded BC aggregates (yellow dashed line) sampled from the Las Conchas fire (New Mexico, 2011) (China et al., 2013) are also shown in Fig. 10. The calculated fractal dimensions (1.20–1.43) of FDAs are smaller than those of BC aggregates (1.53–1.85), which indicates that FDAs have more linear branched shapes compared with the compact shapes of BC aggregates. Previous studies have shown that the D_f of fresh BC aggregates is between 1.6 and 1.8 (e.g., Chakrabarty et al., 2006; Köylü et al., 1995; Sorensen, 2001; Pierce et al., 2006; Heinson et al., 2012) and becomes larger than 1.8 in aged BC aggregates (Lewis et al., 2009). It was also shown that the scattering properties of BC aggregates depended heavily on their D_f (e.g., Sorensen, 2001; Liu et al., 2008). Thus, it is required to characterize the 3-D shapes of FDAs for the accurate calculation of radiative properties.

There is a fundamental difference in the nature of the variables AI and D_f that describe the 3-D shapes of aggregates. The former is a 3-D shape indicator of individual aggregates, whereas the latter is an indicator for a group of aggregates. Each parameter has advantages and disadvantages. For example, AI is useful to describe the 3-D shape of individual aggregates and their scattering properties. As the D_f is intended to describe a group of aggregates, a statistically significant number of samples is required to determine meaningful values and they should not be used to describe individual aggregates. Though AI and D_f cannot be compared, a comparison between AI and R_g∕r_p is possible. Comparisons between AI and R_g∕r_p of FDAs with 10 (left) and 20 (right) element frozen droplets are shown in Fig. 11. The CPI images of FDAs are embedded in each panel. For the FDA with 10 elements a total 127 3-D realizations are possible, whereas a total 65 535 3-D realizations are possible for the FDA with 20 elements (Fig. 11). The best fits illustrated in Fig. 11 show that AI and R_g∕r_p of FDAs are positively correlated with high correlation coefficients. For all FDAs with n_p >2, an average r of 0.942±0.094 is revealed. It indicates that the AI and R_g∕r_p of aggregates are highly correlated and both parameters can be used to describe individual and/or a group of aggregates for the calculations of microphysical and radiative properties.