The Ice Particle and Aggregate Simulator (IPAS). Part III: Verification and Analysis of Ice–Aggregate and Aggregate–Aggregate Collection for Microphysical Parameterization

Vanessa M. Przybylo aUniversity at Albany, State University of New York, Albany, New York

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Kara J. Sulia aUniversity at Albany, State University of New York, Albany, New York

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Zachary J. Lebo bUniversity of Wyoming, Laramie, Wyoming

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Carl G. Schmitt cNCAR, Boulder, Colorado

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Abstract

The Ice Particle and Aggregate Simulator (IPAS) is used to theoretically represent the aggregation process of ice crystals. Aggregates have a variety of formations based on initial ice particle size, shape, and falling orientation, all of which influence water phase partitioning. Aggregate dimensional properties and density changes are calculated for monomer–monomer (MON–MON), monomer–aggregate (MON–AGG), and aggregate–aggregate (AGG–AGG) collection to be used by ice-microphysical models for improvement in aggregation parameterizations. Aggregates are chosen from a database of 9 744 000 preformed combinations to be further collected (see ). AGG–AGG collection results in more extreme and a smaller range of aggregate aspect ratios than MON–AGG collection. A majority of aggregates are closer to prolate than oblate spheroids, regardless of collection type, except for quasi-horizontally oriented particles that have extreme aspect ratios to begin with. MON–AGG collection frequently results in an increase in density upon collection, whereas MON–MON and AGG–AGG collection almost always result in particle density decreases, with extreme reductions near 99% for MON–MON collection. MON–MON collection results in the greatest decreases in density but then quickly becomes unaffected by the addition of more monomers due to inherent size differences between monomers and aggregates. Finally, a holistic analysis to in situ observations of cloud particle images is presented. IPAS 2D aspect ratios surround a median value of 0.6 and closely follow that of previous studies while varying by no more than ≈12% on average from observed aggregates.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Schmitt’s current affiliation: University of Alaska, Fairbanks, Fairbanks, Alaska.

Corresponding author: Vanessa Przybylo, vprzybylo@albany.edu

Abstract

The Ice Particle and Aggregate Simulator (IPAS) is used to theoretically represent the aggregation process of ice crystals. Aggregates have a variety of formations based on initial ice particle size, shape, and falling orientation, all of which influence water phase partitioning. Aggregate dimensional properties and density changes are calculated for monomer–monomer (MON–MON), monomer–aggregate (MON–AGG), and aggregate–aggregate (AGG–AGG) collection to be used by ice-microphysical models for improvement in aggregation parameterizations. Aggregates are chosen from a database of 9 744 000 preformed combinations to be further collected (see ). AGG–AGG collection results in more extreme and a smaller range of aggregate aspect ratios than MON–AGG collection. A majority of aggregates are closer to prolate than oblate spheroids, regardless of collection type, except for quasi-horizontally oriented particles that have extreme aspect ratios to begin with. MON–AGG collection frequently results in an increase in density upon collection, whereas MON–MON and AGG–AGG collection almost always result in particle density decreases, with extreme reductions near 99% for MON–MON collection. MON–MON collection results in the greatest decreases in density but then quickly becomes unaffected by the addition of more monomers due to inherent size differences between monomers and aggregates. Finally, a holistic analysis to in situ observations of cloud particle images is presented. IPAS 2D aspect ratios surround a median value of 0.6 and closely follow that of previous studies while varying by no more than ≈12% on average from observed aggregates.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Schmitt’s current affiliation: University of Alaska, Fairbanks, Fairbanks, Alaska.

Corresponding author: Vanessa Przybylo, vprzybylo@albany.edu

1. Introduction

Microphysical parameterizations are incapable of resolving every individual particle within a cloud due to computational and theoretical limits; instead, they parameterize the particle size distribution analytically, or rely upon statistical inference from a population of particles either from observations or direct numerical simulations at high resolution. Even if individual particles could be extensively resolved, there are inherent uncertainties in formation (e.g., nucleation, diffusion) and evolution (e.g., deposition, riming, melting) that are amplified for frozen hydrometeors that have varied shapes, densities, and orientations. In addition, particle misrepresentation at the microscale can be propagated upscale through thermodynamic and dynamic interactions (e.g., diffusion, evaporation, turbulence, sedimentation).

Heterogeneity in ice particle type or habit can lead to forecast error through the inability to predict or empirically derive snow precipitation rates solely using bulk characteristics. This study aims to explore uncertainties attributable to the aggregation process by statistically quantifying aggregate particle characteristics, namely, aspect ratio (shape), and density changes from the collection of a wide range of platelike and columnar ice crystals. This is accomplished on a statistical basis using detailed Monte Carlo simulations reliant on collections of polygons that mimic the hexagonal structure of ice crystals.

Aggregation is generally thought to be the dominant growth mechanism in producing large ice particles and is enhanced in regions of relatively high ice concentrations (Field 1999; Lawson and O’Connor 2006; Ovchinnikov et al. 2011; Lawson 2011), dendritic growth zones (−17° and −13°C), and near the melting level (Pruppacher and Klett 1997; Field 1999). Aggregation is also present in a multitude of systems, which makes it complex to model due to abundant microscale fluctuations of environmental properties that alter ice particle growth. To address these variations in a broad sense, many ice microphysical schemes use empirically derived relationships from in situ observations. For example, ice particle mass derived from probe measurements of projected area can be related to particle maximum dimension through a power law, which also takes into account the nonsphericity of ice particles (Locatelli and Hobbs 1974; Heymsfield and Kajikawa 1987; Mitchell et al. 1990; Brown 1995; Mitchell 1996; Woods et al. 2007; Morrison and Milbrandt 2015; and others). These models require individualized coefficients for each particle type for a given size range and are taken from either cloud imaging probes or more subjective surface-based methods, which fail to capture particle dimensions near the point of origination (e.g., photographs of fallen snow particles on a petri dish; Mitchell et al. 1990). Even at cloud level, these coefficients are fixed regardless of ice crystal density and shape evolution and are taken from a population of particles in one localized region or cloud type (e.g., midlatitude cirrus in the Cascade Mountains; Locatelli and Hobbs 1974). The nonlinearity of particle growth with size from increased vapor flux is undermined when the coefficients of empirical relationships are fixed; however, advancements are being made (Eidhammer et al. 2017). This is especially important in stratiform regions where there can be a steady supply of small crystals available for aggregation (Houze et al. 1993).

Korolev et al. (2000) found that only 3% of particles (out of 95 655) in Arctic clouds from 0° to −45°C are pristine (defined as faceted ice crystals) taken from cloud particle imagery. In addition, most ice particles in frontal systems are “irregular” and likely a mixture of complex polycrystals and aggregated particles (Korolev et al. 2000). Diagnostic mass and fall speed relationships with respect to size do not appropriately capture hydrometeor morphology, including the evolution of particle shape during both vapor diffusional growth and continuous collection events. Further, these mD parameterizations assume that all aggregates have the same mass for a given diameter using a priori assumptions via coefficient magnitudes, which limits differences in particle terminal velocities and diminishes particle heterogeneity within a size regime (Passarelli and Srivastava 1979). This work aims to eliminate these confinements by expanding potential aggregate characteristics through variation in particle density, aspect ratio, radius (calculated from two axes), and collection type.

Some recent models use spheroids to encapsulate a broader range of ice particles that may have taken on complex shapes from convoluted attachment trajectories in turbulent environments (Harrington et al. 2013; Jensen et al. 2017; Sulia et al. 2021). Simulations in this study similarly circumscribe aggregates using three-dimensional ellipsoids for generality and amenability into bulk models that evolve two primary spheroid axes as particles grow or decay (e.g., Sulia et al. 2021). This approach is advantageous for transitioning from ice to snow with avoidance of harsh autoconversion thresholds where ice is arbitrarily transferred to snow based on bulk mass or size quantities. Similar to the riming scheme of Morrison and Milbrandt (2015), this work smoothly and explicitly evolves aggregate properties (e.g., axis lengths or aspect ratio, density, and radius) with avoidance of traditional thresholding techniques that neglect sensitivity of process rates to the environment. By predicting aggregate properties instead of prescribing them, models can utilize environmental information to better represent microphysical processes, which no longer need ad hoc conversion parameterizations between categories. This is especially important for aggregates as their mass and size can quickly alter precipitation rates at the surface, diminish small particles through collection, and redistribute mass and number quantities aloft (Karrer et al. 2020; Dunnavan 2021).

To quantitatively specify the range of typical characteristics of aggregates, the Ice Particle and Aggregate Simulator (IPAS) is used (Schmitt and Heymsfield 2010, 2014). IPAS is a theoretical framework that mimics simplified laboratory investigations to perform sensitivity tests, visualize, and better understand growth via collection. IPAS collects any number of solid hexagonal prisms that are simulated to represent plates, columns, or isometric particles. A detailed background description on monomer–monomer (MON–MON) and monomer–aggregate collection (MON–AGG) in IPAS can be found in Parts I and II (Przybylo et al. 2019, 2022b) and bulk implementation and testing of which can be found in Sulia et al. (2021).

IPAS has the capability of generating an immense number of aggregates from predefined monomer properties and acts as an invaluable test bed to analyze trends in aggregate characteristics. A database of aggregates was created and analyzed as a function of monomer attributes in Part II (Przybylo et al. 2022b). Now, this database is used in further collection between a single monomer and an aggregate from the database (MON–AGG collection) or two aggregates from the database (AGG–AGG collection). A detailed methodology including imposed criteria for MON–AGG collection is found in section 2. Section 3 then analyzes aggregate parameters after further collection. To ensure confidence that IPAS sufficiently reproduces bulk aggregate properties, IPAS is validated against microphysical aircraft probe imagery in section 4. Table 1 includes a reference for variables included throughout the following discussions.

Table 1

Variables used in IPAS simulations and in the results section are listed below.

Table 1

2. Methodology

Methodologies for monomer–aggregate (MON–AGG) and aggregate–aggregate (AGG–AGG) collection extend from that of MON–MON collection as outlined in Przybylo et al. (2019). Note that for MON–AGG and AGG–AGG collection, one or two aggregates must already exist. The creation of a new aggregate for both MON–AGG and AGG–AGG collection requires individual construction of each aggregate, which, as the aggregates grow can become computationally intensive for each collection event. Thus, a database of 9 744 000 aggregates with monomer counts ranging from 2 to 30 was created (Part II, Przybylo et al. 2022b). IPAS-generated aggregates were formed from identical monomer properties (e.g., the same size and aspect ratio); however, formation variability in the amount of overlap, orientation, and number of monomers (nm), quickly diversified aggregate arrangements. Monomer aspect ratios (ϕm), are defined as the half-length of the prism or rectangular face (cm) to the half-length of the basal or hexagonal face (am, ϕm=cm/am) from ϕm = 0.01 (platelike crystals) to ϕm =100.0 (needlelike crystals). Twenty monomer aspect ratios (ϕm = 0.01 to ϕm = 100.0) and 28 monomer radii (rm = 1.0 to rm =1000.0) were initialized, where units are arbitrary in IPAS, yet thought of as micrometers in this instance. Formed aggregates are composed of 2–30 monomers specified by these ϕmrm criteria. The creation of a database allows for direct and prompt aggregate acquisition for further collection and allows for control on aggregate shape and size, while allowing for diversity in geometry and monomer count. Database creation, postprocessing calculations, and average characteristics are described in Part II (Przybylo et al. 2022b).

After formation, the database is truncated for aggregates ≤ 5 mm in radius to eliminate unrealistic growth and for comparison to cloud particle imagers that have a finite resolution and upper size range limit. The database is then split into 400 bins or ranges based on the aggregate aspect ratio (ϕca) and radius (ra) each with an equal number of aggregates. Aggregate aspect ratios are calculated as the semiminor axis (c) divided by the semimajor axis (a) from a minimum volume ellipsoid fit around each aggregate. We calculate ra based on whether the aggregate is oblate (ab)<(bc):ra=a2×c3, or prolate (ab)>(bc):ra=c2×a3, where a, b, and c are the three radii of the fit ellipsoid in descending size order. Note that the aggregate semiaxis lengths are from an encompassing ellipsoid whereas the monomer radii measure along the basal and prism faces of the crystal. Because “platelike” and “columnar” descriptions do not apply to aggregates, all aggregates have aspect ratios that range between 0.0 ≤ ϕca ≤ 1.0, where ϕca = 1.0 are isometric aggregates. To capture the sensitivity to orientation in IPAS, two orientations were tested for the monomers within the aggregates and the aggregates themselves; random: tumbling as if in a turbulent environment; and quasi horizontal: oriented so that the projected area from overhead is maximized and the major axis dimension is near-parallel to the xy plane (refer to Parts I and II, Przybylo et al. 2019, 2022b). After binning the database, each bin contains 9 857 (random) and 10 750 (quasi-horizontal) aggregates, respectively, and are sampled 300 times to be collected with a single monomer (MON–AGG collection) or another aggregate (AGG–AGG collection) following methods outlined in the following sections 2a and 2b for methods and criteria, respectively. In other words, two particles are collected 300 times to create statistically significant conclusions for the evolution of aggregate properties.

Przybylo et al. (2019) demonstrated that 300 newly formed aggregates for MON–MON collection was sufficient in reproducing ϕca trends and in limiting variability between simulations. Brief testing up to 1000 aggregates for MON–AGG and AGG–AGG collection provided no additional conclusions nor generated any discrepancies for all aggregate properties. To test the robustness of the final statistics that emerge from the aggregates comprising the database, two independent simulations were run, wherein 300 aggregates were formed (MON–AGG) for each simulation. While IPAS is heavily randomized, the correlations (R) between the two simulations for ra, ϕca, and the semimajor axis length from a minimum-volume ellipsoid (a) were R = 0.992, 0.935, and 0.998, respectively, concluding that high consistency is achievable using a Monte Carlo approach from 300 aggregates for all collection types.

a. Collection methodology

The following outlines collection methods for MON–AGG and AGG–AGG collection, which follow the same principle methodology:

  1. To begin a collection event, two particles are selected by loading the vertices [dimensions of nm by 12 (x, y, z) coordinates] from the configuration and orientation at the time of database storage. The location of these vertices, which vary with each particle, determines the size of the encompassing domain and the spatial relation between particles. Both particles are then recentered based on their center of mass.

  2. A random point in the xy plane is chosen from a bounding rectangle1 around each aggregating particle (either a monomer and an aggregate or two aggregates from the database).

  3. The second particle is then moved atop the first particle such that the two chosen points from each bounding rectangle align in the xy plane, creating overlap diversification. Note that the aggregates may not be touching in the xy plane at this stage since the randomly chosen points within the bounding rectangles may not be located on the particles themselves. Subsequent steps certify that the particles connect in the x and y dimensions.

  4. It is ensured that the two particles are not lodged within one another, but that one is entirely above the other in the z direction. If the particles are indeed lodged, the second initialized particle is moved up a minimum distance so that all vertices are above the first initialized particle.

  5. A Python package from Shapely,2 which focuses on spatial analysis in two dimensions, is then iteratively used between the xz and yz planes to determine the closest points between the two particles.

  6. Both particles are “attached” in x, y, and the larger of the two distances in z from either xz or yz. This ensures that the particles attach with minimal lodging.

  7. The above steps are repeated 300 times for each of the 400 bins, for two orientations (random and quasi horizontally), for MON–AGG and AGG–AGG aggregation, totaling 480 000 individual aggregation events. Associated geometrical computations are performed for each event.

b. Collection criteria

There are a wide range of monomer aspect ratios and sizes simulated by IPAS (many orders of magnitude); thus, the monomer to be collected should lie within realistic bounds as compared to the monomers comprising the collecting aggregate, but with ample variability. Therefore, in IPAS, the size and shape of the monomers that make up the aggregate are used to guide the allowable range of monomer size (rm) and aspect ratio (ϕm) to collect for MON–AGG collection.

After one aggregate is selected from the database, the shape and size of the monomers that comprise this aggregate are determined; recall that all monomers that comprise each aggregate in the database are identical. An array of 20 logarithmically spaced aspect ratios is then created, uniformly centered around the aspect ratio of the monomer within the aggregate, extending one order of magnitude in both directions. One of these 20 monomer aspect ratios is then randomly chosen from the uniform distribution to be collected. Should the randomly selected ϕm be ≤0.01 or ≥70, it is reset to 0.01 or 70, respectively. Ice crystals with ϕm outside these thresholds are rarely found in nature and create visually unrealistic aggregates in IPAS. In addition, this choice is guided by 29 060 airborne images of pristine columns that never grew past ϕm = 50 [dataset taken from classification methods in Przybylo et al. (2022a)]. To allow for model-generated crystals that may grow more extreme than ϕm = 50, ϕm is truncated at 70. In IPAS, monomers used in collection only exceed ϕm = 0.01 or ϕm = 70% 10% of the time, respectively. An aggregate made up of plates can collect a column and vice versa if ϕm within the aggregate is between 0.1 and 10.0. A similar method is used to bound rm for collection, except a logarithmically spaced array is split into 20 values centered around the monomer radii within the aggregate that extends in both directions by a factor of 2. Should the randomly chosen rm be <1 μm or >1000 μm, it is reset to <1 or 1000 μm, respectively. Monomer radii are smaller than <1 μm 2% of the time and never exceed 1000 μm based on sampling from an uniform distribution.

Example aggregates formed from MON–AGG collection (left) and AGG–AGG collection (right) are shown in Fig. 1 for both orientations with particles distinguished by color (red and black). Values for the complexity of each particle are located next to each aggregate and are discussed in section 3a. For the quasi-horizontal orientation, viewing angle is in the xz plane (y orientation) for MON–AGG collection and in the xy plane (z orientation) for AGG–AGG collection for different viewing perspectives.

Fig. 1.
Fig. 1.

Theoretical aggregates from (left) MON–AGG collection and (right) AGG–AGG collection for the random and quasi-horizontal orientations. Corresponding complexity values are located next to each aggregate. Viewing angle is in the xz plane (y orientation) for MON–AGG collection and in the xy plane (z orientation) for AGG–AGG collection for different viewing perspectives.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0180.1

While the monomers comprising a single aggregate are identical, the monomers within both aggregates (AGG–AGG collection) can vary across shapes and sizes as long as the aggregates themselves are pulled from within the same bin or ϕca and ra ranges within the database. Considering a thick cloud or a cloud deck with multiple layers (e.g., seeder–feeder effect) it is often found that the small crystals forming in the upper part of the cloud begin to form an aggregate that then further grows by collection of monomers with vastly different (or similar) aspect ratios at lower layers creating a mixture of habits within the collection of aggregates. In addition, particles with slow fall speeds can grow by depositional growth to relatively large sizes before aggregation becomes the dominant process; therefore, it is not unrealistic for aggregates to have very different monomer sizes; however, collection of unrealistically different aggregate and monomer sizes are avoided through restrictions of the collected particle to one order of magnitude greater than or less than the collector particle for both aspect ratio and size.

3. Results

After splitting the database into 20 × 20 or 400 equal-count bins based on aspect ratio (ϕca = c/a) and ra, 300 aggregates are formed from each bin for MON–AGG and AGG–AGG collection. From the collections, aggregate attributes are analyzed in terms of shape, complexity, area ratio, and density.

a. Aggregate aspect ratio evolution

Figure 2 shows aspect ratio evolution for MON–AGG and AGG–AGG collection (Fig. 2, blue and red, respectively) for both random (top) and quasi-horizontal (bottom) orientations. Since two aspect ratio measures are required to specify an ellipsoid shape, aggregate aspect ratios are calculated as ϕba = b/a (Fig. 2, x axis) and ϕca = c/a (Fig. 2, y axis), where a, b, and c are semiaxis lengths in descending order from a minimum volume ellipsoid fit around each aggregate; hence, ϕca will always be <ϕba, or “more extreme,” unless spherical or prolate. A kernel density estimate is used to smooth the distribution of aggregate aspect ratios across each population of 300 aggregates within the multivariate domain of ϕba (x axis) and ϕca (y axis). The contribution from each of the 300 aggregates is projected out from a single point into the surrounding phase space for inference on the probability density between ϕba and ϕca for each collection type, which is useful for predictions if specific data points are not available (Fig. 2, contours). The most frequent values for ϕba and ϕca are plotted in white and red dots in Fig. 2 for MON–AGG and AGG–AGG collection for both orientations, respectively. General statistics on the modes of the kernel density estimates across all ϕca bins are also displayed in Table 2. Every other ϕca bin is plotted in Fig. 2 starting with the most extreme aggregates (top left) to the most spherical (bottom right); titles represent ϕca bin edges after splitting the database so that each bin has the same number of aggregates. The middle ϕra bin (from 20) is chosen for all ϕca, which varies from 164 to 213 μm. Aggregates are randomly chosen for MON–AGG and AGG–AGG collection within each bin; therefore, the sampled ϕca may not reach the values at the bin edges. In Fig. 2, the entire bin range is sampled for ϕca except for the first and last; however, bin ranges are sufficiently small to limit substantial variability in aggregate characteristics should the extremes of the bin range be neglected. While the relative evolution across aspect ratios for random (top) versus quasi-horizontal (bottom) orientations provide insight, caution should be exercised when directly comparing ϕca ranges between orientations as they can vary by up to an order of magnitude (Fig. 2, titles).

Fig. 2.
Fig. 2.

Evolution of aggregate aspect ratio distributions (ϕba, x axis) and (ϕca, y axis) for MON–AGG (blue) and AGG–AGG (red) collection. Each aggregate is formed 300 times for the (top) random orientation and (bottom) quasi-horizontal orientation to create a multivariate kernel density estimate (contoured). Distribution modes are shown in white and red dots for MON–AGG and AGG–AGG collection types, respectively. ϕca ranges from the database before collection are listed in the title going from (top left) most extreme to (bottom right) most spherical. Percentage oblate values (out of 300) are marked in the upper-left corner of each subplot for AGG–AGG collection (red) and MON–AGG collection (blue). If a majority of the aggregates are defined as oblate (prolate) spheroids the subplot is shaded to the right (left), with color representing collection type. If both collection types produce the same majority shape (oblate or prolate), the appropriate region is shaded purple.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0180.1

Table 2

Minimum and maximum modes of the distributions in Fig. 2 along with the difference (total range) across all ϕca bins for MON–AGG and AGG–AGG collection for the random orientation (top) and quasi-horizontal orientation (bottom). Both aspect ratio calculations are included under ϕba and ϕca.

Table 2

Recall from Part II (Przybylo et al. 2022b) that a maximum of 30 monomers comprise each aggregate in the database, and so a maximum of 31 (60) monomers comprise each aggregate following MON–AGG (AGG–AGG) collection. After collection, no aggregate becomes completely spherical (ϕca = ϕba = 1.0) for nm ≤ 60 (AGG–AGG collection, red) nor nm ≤ 31 (MON–AGG collection, blue) (Fig. 2). For extreme ϕca bins (top-left plots), the random orientation leads to broader distribution shapes for both collection types due to unbounded collection where monomers are not required to fall flat. In addition, outer edge collection is more common when randomly tumbling, leading to greater aggregate diversity through an assortment of formations. Differences between orientations (Fig. 2, top versus bottom) are not pronounced for most ϕca bins that encompass similar ranges (bottom-right plots) because the aggregates in the quasi-horizontal case start to mimic the random case after a few monomers, depending on ϕm and how much tilt is needed to maximize monomer 2D projected area (refer to Fig. 1 in Part I, Przybylo et al. 2019). As ϕca approaches 1.0, ϕm is generally less extreme as well, requiring significant tilting in all directions from the isometric monomers to maximize the projected area.

Quasi-horizontally oriented monomers and aggregates create more extreme aggregate aspect ratios than randomly oriented when ϕca ≪ 1.0. The most extreme mode is 0.02 for both ϕba and ϕca for the quasi-horizontal orientation for MON–AGG and AGG–AGG collection (Table 2, bottom). The least extreme modes (most spherical aggregates, ϕba ≥ 0.85) occur for calculations using the middle semiaxis length for MON–AGG collection in both orientations for the least extreme ϕca bins. Based on the monomer criteria for MON–AGG collection, assuming rm is relatively small compared to ra, the monomer could fit into gaps within the aggregate and would not substantially alter ϕca or ϕba, leading to the least extreme newly formed aspect ratios, especially as nm increases. Orientation becomes less significant for both collection types as the ϕca bin ranges increase toward ≈1.0 since the monomers within the aggregates for the quasi-horizontal orientation are stacked in all directions, the aggregate is already near spherical, and nm is relatively high (Part II, Figs. 2c–d, Przybylo et al. 2022b).

MON–AGG collection (blue) reaches near-isometric aggregates and starts at more extreme ϕba and ϕca values compared to AGG–AGG collection (red), especially for the quasi-horizontal orientation, so the range in modes across all bins is greater for MON–AGG collection than AGG–AGG collection (Table 2, “difference”). For the random orientation, AGG–AGG collection results in the smallest modal variation across all ϕca bins (0.15 for ϕca and 0.12 for ϕba, Table 2). Since the aggregates are required to have similar sizes and shapes and are inherently larger than a single monomer, the distributions are broad but modal values are more consistent when averaging across all attachment possibilities when randomly tumbling.

A bimodal distribution for ϕba is present for the quasi-horizontal orientation for bin ranges between ϕca ≈ 0.1 and ϕca ≈ 0.3 (Fig. 2, bottom). In this range, there are two distinct types of aggregate formations. The first is a very large number of small, near-isometric particles attaching in a vertical chain that fall into ϕca ranges between 0.101 and 0.287 (i.e., the aggregate major axis is long from the vertical chaining, while the aggregate minor axis is short from small monomer radii). Given the small radii, the b axis length (width of the aggregate) is minimal, leading to high frequencies of ϕba near 0.2. The second type of formation is very few yet long columns that attach parallel to each other (again with a thickness that falls into the ϕca range between 0.101 and 0.287), which creates a long major axis and short aggregate width or b axis. As nm increases for large monomer radii and less extreme ϕca bins, there is a greater chance in expanding the aggregate width and the bimodality becomes unnoticeable (Fig. 2, bottom). Consulting Figs. 2d and 2f from Part II (Przybylo et al. 2022b), the most extreme ϕca bins are composed of very thin needles with approximately 10 monomers collecting on average, which leads to a wide variety in ϕba as monomers rotate and collect in all directions in the xy plane; therefore, a unimodal distribution is evident for the first two ϕca bins.

b. Aggregate ellipsoid shape

As previously stated, ice-microphysical models often use reduced-density spheres or spheroids as proxies for aggregate representation in a bulk sense (e.g., Thompson et al. 2008; Morrison and Milbrandt 2015; Jensen et al. 2017). Similarly, ellipsoids are fit around IPAS-generated aggregates and categorized into prolate and oblate spheroids for comparison to other modeling studies and for understanding general aggregate characteristics before implementing into bulk models. Figure 2 displays the percentage of oblate aggregates (out of 300) marked in the upper-left corner of each subplot for MON–AGG (blue) and AGG–AGG (red). Regions within the diagonals represent prolate (left region) and oblate (right region) aggregates (Fig. 2). See Fig. 5 in Part II (Przybylo et al. 2022b) for aid in the interpretation of aggregate shapes in Fig. 2. For each subplot, the shading represents if a majority of the aggregates were oblate (shaded to the right) or prolate (shaded to the left) with color representing collection type. Purple shaded regions represent a majority of aggregates as either both oblate or both prolate from both collection types (i.e., red and blue overlap), which happens a majority of the time, especially for prolate aggregates (left shading).

For reference, past literature often refers to aggregates as oblate spheroids (e.g., Kennedy and Rutledge 2010) with aspect ratios of 0.6 (Hogan et al. 2012) from projections of aggregates < 1 mm aloft (Korolev and Isaac 2003) and at the surface (Garrett et al. 2015). In contrast, these detailed simulations of the aggregation process suggest aggregates are primarily prolate, except for extreme aggregates (ϕca < 0.158) that collect in a quasi-horizontal orientation; note that the aggregate aspect ratio bin ranges for the only three majority oblate distributions in the quasi-horizontal orientation are more extreme than even the most extreme aspect ratio bin for the random orientation. From Fig. 2d in Part II (Przybylo et al. 2022b), the most extreme bins in the quasi-horizontal orientation hold aggregates composed of a relatively low number of monomers, so they spread more than stack, favorably leading to a majority of oblate aggregates. It should be emphasized that the most extreme bin ranges in the quasi-horizontal orientation are considered uncommon on a broad scale; therefore, for most purposes, the collection of two particles in IPAS routinely produce prolate aggregates consistent with other modeling studies (e.g., Jiang et al. 2017, 2019; Dunnavan et al. 2019).

c. Change in density of snow

Ice microphysical models that evolve particle shape also benefit from prognostically evolving particle density. Density is a critical component in parameterizing snow collection as it is used in fall speed calculations and can delineate between precipitating and nonprecipitating particles. While aggregate axis lengths are easily computed in IPAS, aggregate density is unable to be analytically solved due to the lack of mass. Since a typical mass/volume density computation is unattainable, the relative density change from aggregation (Δρ) is determined as monomers continuously attach in the simulator. First, the volume of the hexagonal prism(s) [Vp, Eq. (1)] of the two collecting particles (any collection type) is explicitly calculated and divided by an equivalent-volume ellipsoid [Ve, Eq. (2)] that is fit to each particle using the smallest encompassing volume defined by
Vp=332am2×cm×nm,
Ve=4*π3ae×be×ce.
This unitless volume ratio (Vpi/Vei, initial) is equivalent to a density ratio with the premise that the mass from the hexagons that make up the aggregate is equal to the mass of the ellipsoid. Every time a new monomer or aggregate is added to the preexisting particle, a new volume ratio is calculated (Vpf/Vef, final). The final and initial volume ratios are subtracted to obtain the change in density. Finally, the change is normalized by the initial ratio (Vpi/Vei) for a unitless relative change (Δρ):
Δρ=(Vpf/VefVpi/Vei)Vpi/Vei,
where subscripts p, m, e, i, and f represent “polygon,” “monomer,” “ellipsoid,” “initial,” and “final,” respectively. If the volume of the polygon is being calculated for a monomer, in the case of MON–MON or MON–AGG collection, nm is 1. This methodology for computing the change in density is beneficial for implementation into bulk models due to the avoidance of direct unit mapping between arbitrary units in IPAS and dimensional units in bulk models such as micrometers. Usage of IPAS-derived density changes requires preexisting magnitudes for ϕm and rm (MON–MON collection), and additionally, ϕca, ra for MON–AGG and AGG–AGG collection.

For MON–MON collection, the two monomers to collect have the same rm and ϕm so the fit ellipsoid [Ve, Eq. (2)] surrounds equivalent points; therefore, the initial volume ratio is the same for both monomers. For MON–AGG collection, the initial volume ratio of the monomer is subtracted from the final aggregate volume ratio, and for AGG–AGG collection the average of the two initial volume ratios is subtracted from the final aggregate volumetric ratio.

Figure 3 shows the change in density from IPAS as a function of orientation and collection type; MON–MON collection is included for a well-rounded analysis across all collection types. MON–MON collection is initialized by evenly incrementing 28 radii logarithmically from rm = 1.0 to 1000 μm and 20 aspect ratios from ϕm = 0.01 to 100.0. Three hundred aggregates were formed per radius–aspect ratio pairing (each bin) for all collections resulting in 168 000 MON–MON aggregates and 120 000 MON–AGG and AGG–AGG formations, all included in Fig. 3. Positive (negative) values indicate an increase (decrease) in density for each collection event, with a minimum value near 1.0 or near 100% decrease after collection.

Fig. 3.
Fig. 3.

The relative change in density as a function of the (left) random orientation and (right) quasi-horizontal orientation for MON–MON (green), MON–AGG (orange), and AGG–AGG (blue). All aggregate data are plotted for each collection with a sample size of 168 000 for MON–MON collection and 120 000 for MON–AGG and AGG–AGG collection. A density change of −1.0 represents a 100% decrease in density from the initial particle. Both a histogram and a smoothed kernel density estimate are plotted.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0180.1

MON–MON collection results in the greatest reduction in density with multiple modes present in the quasi-horizontal case (Fig. 3, green, right). Of importance are the 560 discrete ϕmrm pairings that are collected 300 times each. Subtracting multiple identical volumetric ratios (same monomer properties) from the volumetric ratio of the aggregate produces high frequencies of the same density changes given that the aggregate formations are relatively similar. The most frequent decrease in density for MON–MON collection is >90% for the random orientation; frequencies are less for the quasi-horizontal orientation at the most extreme density decreases (Fig. 3, green). In the quasi-horizontal orientation (Fig. 3, right), especially for ϕm ≪ 1.0 where the minor axis (along the vertical dimension) of the ellipsoid is relatively small, plates, given their basal face expanding in both x and y directions, will result in a high volumetric ratio (aggregate volume relative to encompassing ellipsoid volume closer to 1). In contrast, columns with a prism face expanding in only x or y directions, will result in lower volumetric ratio given the holes that result from the likely collection with offset (cross formation rather than stacked parallel). These geometric differences result in a separation between the density change of plates and columns and is apparent in the distribution bimodality for quasi-horizontally oriented monomers, with the collection of plates most often decreasing their density by ≈25% and columns near 100% (Fig. 3, green, right); note that the initial volumetric (density) ratio of the single monomer is nearly 1. Both extreme plates and columns generate a high frequency of exceptionally low densities from MON–MON collection for the random orientation; unlike quasi-horizontal simulations where contact angle is minimal resulting in minor variation in the z direction, the random orientation simulations allow particles, particularly platelike particles, to collect with higher contact angles, expanding the ranges in the z direction and increasing the likelihood that the aggregates would contain empty space when fit with the ellipsoid, reducing the aggregate density relative to the initial monomer density. The fit ellipsoid around a single monomer is always smaller than the fit ellipsoid after collection for the random orientation so the density change is always negative (Fig. 3, green, left).

For MON–AGG and AGG–AGG collection, the volume of the ellipsoid and aggregate changes based on the shape and number of crystals within the aggregate(s). MON–AGG collection results in the greatest increase in density at the highest frequency out of all collection types for both orientations (Fig. 3, orange). Collection of a single monomer with an aggregate minimally varies the fit-ellipsoid volume while the aggregate volume increases. In other words, the aggregate can become denser if the monomer fills in gaps within the aggregate or attaches within the perimeter of the aggregate in all dimensions. Aggregate density is more likely to increase from MON–AGG collection in the random orientation compared to the quasi-horizontal orientation due to more vacant area; monomers that are quasi-horizontally oriented stack on top of each other, spreading out horizontally more than vertically (for extreme ϕm) making it less likely to fill in any gaps due to a small ellipsoid minor dimension. Recall that the monomer criteria for MON–AGG collection has a generous spread of two orders of magnitude around rm and ϕm such that MON–AGG collection can result in greater decreases in density than increases upon collection. However, most often, density increases <10% from the aggregate volume increasing more than the ellipsoid with minimal aggregate expansion (Fig. 3, orange).

AGG–AGG collection produces a Gaussian distribution and most frequently reduces the aggregate density by ≈30% (Fig. 3, blue). The collection of a wide variety of aggregate shapes and sizes results in a normal distribution for the density change unlike MON–MON collection with discrete monomer properties that have repetitive volumetric ratios. Since both aggregates are coming from the same bin range for ϕca and ra, it is unlikely for them to collect tightly together without expanding the encompassing ellipsoid around both aggregates (see Fig. 1, right, for illustration).

Figure 4 shows the relative change in density (y axis) from 300 collections for the random orientation (left) and quasi-horizontal orientation (right) as a function of monomer aspect ratio (ϕm, x axis) and the number of monomers per aggregate (nm, shaded dots). In this simulation, all monomers are identical and keep collecting from MON–MON (nm = 2) to MON–AGG (nm > 2). The greatest density changes occur for the collection of two monomers, primarily for the random orientation due to expansion of the encompassing ellipsoid with very thin monomers within (Fig. 4, dark blue). The collection of two monomers in a random orientation with ϕm = 0.01 and ϕm = 100.0 results in a 96% and 99% decrease in particle density, respectively (Fig. 4, left) due to the likelihood of a larger contact angle resulting in an aggregate with more vacant space, as described above. The effects on the change in density are diminished as monomers are progressively collected since there are so many gaps already present by fitting a large ellipsoid to the aggregate; adding one more monomer does not greatly contribute to density fluctuations after nm = 10. While the range in the monomer aspect ratios in Fig. 4 are extreme and rarely observed, the density change is significant for two monomers even as ϕm approaches 1.0. When ϕm is 1.0, density changes are much less extreme as monomers collect from nm = 2 to nm =9 due to the compact nature of the monomers (Fig. 4, middle). For the quasi-horizontal orientation, the sensitivity to monomer volume is evident: plates inherently have more volume across their basal face than columns along their prism face when falling flat; thus, the decrease in density is not nearly as significant across ϕm < 1.0 (Fig. 4, right). After nm ≈ 4 or 5, the average change in density is <20% for each additional monomer, regardless of orientation for MON–AGG collection.

Fig. 4.
Fig. 4.

The mean percentage decrease in density (y axis) from 300 collections as a function of ϕm (x axis, 0.01 < ϕm < 100.0) for rm = 10 μm. All monomers are identical within the aggregate. Colors represent the number of monomers in the aggregate after collection for the (left) random orientation and (right) quasi-horizontal orientation.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0180.1

Density change values theoretically determined by IPAS are used by bulk models to evolve particle density from the ice to snow categories (e.g., Sulia et al. 2021). While it is understood that aggregating dissimilar particle shapes with intricate features will result in differing density changes, additional work is necessary in both the modeling and observational realms to quantify such affects in a bulk approach. Smooth density transitions as more particles attach to the aggregate is an improvement over past schemes that use constant density assumptions (e.g., Reisner and Rasmussen 1998; Gilmore et al. 2004; Straka and Mansell 2005; Thompson et al. 2008; Jensen et al. 2017; and others).

4. Verification of IPAS

Before IPAS can be synthesized with bulk models to smooth the evolution from ice to snow, there must be confidence that IPAS aggregates are capturing bulk properties of observed aggregates. Since the wide acceptance and ever increasing use of optical array probes beginning in the early 1970s with the Particle Measuring Systems 2DC (Knollenberg 1972), documentation of the microstructural composition of individual particles (Korolev and Isaac 2003; Baum et al. 2005) has shed light on particle scattering properties and satellite retrievals, such as effective particle size, growth rates, and terminal fall velocity (Korolev et al. 2000; Heymsfield et al. 2002, 2004; Heymsfield and Westbrook 2010). The SPEC Inc. CPI probe (see Lawson and O’Connor 2006) is used for verification due to the fine-scale resolution: 2.3-μm pixel size, 256 levels of gray at each pixel, and detectable particle size of 15–2300 μm. The difference in resolution is easily noticeable compared to other optical array probes (OAPs) (see Lawson et al. 2006, their Fig. 1; Lawson and O’Connor 2006, their Figs. 4–6 and 8).

The CPI operation relies on a square photo-detector array, meaning that an entire image is captured instantaneously as a two-dimensional “snapshot” when the device is triggered. Subsequently, the measurement is less sensitive to distortion effects, but discontinuous. While the CPI is unreliable for particle size distributions due to limitations in detected size range and slow sampling speeds compared to other OAPs, only the imagery itself is used. Furthermore, while it is possible for shattered particles to enter the sample volume (McFarquhar et al. 2013), artifacts are minimized by only using single, in-focus particles per frame, which is quality controlled during preprocessing such that no more than 5% of the particle is cutoff or intersecting the border with respect to the perimeter of the particle.

A collection of 110 920 aggregates are gathered from seven field campaigns (multiple intensive operation periods each) including the 2000 Department of Energy Atmospheric Radiation Measurement Intensive Operating Period (ARM IOP) at the Southern Great Plains site, 2002 Cirrus Regional Study of Tropical Anvils and Cirrus Layers–Florida-Area Cirrus Experiment (CRYSTAL-FACE), 2003 Alliance Icing Research Study II (AIRS-II), 2004 Midlatitude Cirrus Experiment (MidCiX), 2007 Ice in Clouds Experiment-Layers (ICE-L), 2015 The Olympic Mountains Experiment (OLYMPEX), and 2004 Mixed Phase Arctic Cloud Experiment (MPACE). Culling aggregate images from these campaigns is accomplished through a machine learning approach, the detailed methodology is discussed in Przybylo et al. (2022a). The following sections discuss three parameters that are used for direct unitless comparison between IPAS and CPI imagery: aggregate complexity, aspect ratio, and area ratio. Note that while IPAS aggregates are represented from the vertices of each hexagonal prism and converted to polygonal objects before calculations, variables are parsed from the pixels that each crystal occupies from the CPI images. Despite intrinsic differences, the methodologies are consistent. Moreover, since all three parameters are ratios of particle characteristics, magnitudes between IPAS and CPI aggregates can be systematically evaluated without concern of unit discrepancies.

a. Complexity

Complexity, first discussed in Schmitt and Heymsfield (2010) with references to IPAS in Przybylo et al. (2022b, Part II) is a useful parameter in determining the degree of aggregation and has been shown to clearly separate aggregates from single particles (monomers) (Schmitt and Heymsfield 2010). Complexity is calculated as
C=(Ac×Ap)P2,
C=10×(0.1C),
where Ac is the area of the circumscribed particle in the xy plane, Ap is the particle projected area in the xy plane, and P is the perimeter of the exterior polygon after collection. Values closer to C = 1.0 are more complex; see Part II (Przybylo et al. 2022b) for a detailed description on how each variable influences C. An illustration of the collection geometries and associated complexities are presented in Fig. 1 for theoretical aggregates from MON–AGG collection (left) and AGG–AGG collection (right) for both orientations. Complexity is generally higher for more monomers (high nm and P) and columns with ϕm ≫ 1.0 (low Ap).

Complexity is a valuable parameter for verification as it utilizes and consolidates multiple parameters into one comprehensive value. Figure 5a shows normalized probability distributions of complexity magnitudes between aggregates within the IPAS database (3 527 442, for nm ≤ 30, blue) and all aggregates gathered from the CPI probe (110 920, red). The database of IPAS aggregates is truncated to only include aggregates that consist of identical, randomly oriented monomers with ϕm ≤ 70 and rm ≤ 5 mm. This truncation is guided by 29 060 CPI observations of pristine columns that never grew past ϕm = 50 but extended to ϕm = 70 to allow for model-generated crystals that may grow more extreme. The dataset is taken from classification methods in Przybylo et al. (2022a). IPAS aggregates are distinguished by the maximum number of monomers per aggregate [nm ≤ 30 (solid line), nm ≤ 20 (dashed line), nm ≤ 10 (dash–dotted line), and nm ≤ 5 (dotted line), Fig. 5a]. A primary and secondary mode is present for all IPAS distributions (blue) due to the distinct regimes of aggregates composed of plates versus columns (Fig. 5a). The modes for nm ≤ 30 (solid line) are C = 0.87 for plates (ϕm < 1.0) and C = 0.96 for columns (1.0 < ϕm ≤ 50). Subsequent modes for the primary distribution peak (mostly aggregates composed of plates) for nm ≤ 20, nm ≤ 10, and nm ≤ 5 are C = 0.83, C = 0.72, and C = 0.62, respectively (Fig. 5a). Complexity distribution modes continuously decrease as nm decreases, primarily due to a reduction in the perimeter of the aggregate. Additionally, the distribution becomes more disperse for fewer number of monomers per aggregate; C is driven by the amount of overlap between monomers for low nm (especially plates), which can vary considerably compared to the relatively small perimeter.

Fig. 5.
Fig. 5.

(top) Normalized probability distributions of aggregate (a) complexity, (b) aspect ratio, and (c) area ratio for IPAS (blue) and CPI (red) aggregates. IPAS aggregates are plotted with respect to nm ≤ 30 (solid lines), nm ≤ 20 (dashed lines), nm ≤ 10 (dash–dotted lines), and nm ≤ 5 (dotted line). IPAS 2D aspect ratios [ϕ2D, (b), light blue] are calculated from a projected ellipse surrounding the aggregate in a random orientation and IPAS three-dimensional aspect ratios [ϕca, (b), dark blue] are calculated from a fit ellipsoid (both minor and major semiaxes). (bottom) Bulk statistics for (d) complexity, (e) aspect ratio, and (f) area ratio are shown for all aggregates up to 30 monomers with rm ≤ 5 mm with each box plot corresponding to a different range of monomer attributes within the IPAS-formed aggregates (y axis). From top to bottom, each box plot in (d)–(f) consists of 3 681 556 (dark blue), 3 010 409 (medium blue), 436 612 (light blue), and 110 920 (CPI, red) aggregates, respectively. All calculations are independent of viewing angle in a random orientation.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0180.1

Compared to IPAS, the mode of CPI C values is C = 0.76, which falls closest to IPAS aggregates with nm ≤ 10 (red, Fig. 5a). Visual inspection of the CPI database generally conforms to semidiscernible monomers with nm ≤ 10 per aggregate. It should be noted that there is a wide variety of aggregate formations present in the CPI database, which include the collection of plates, columns, bullet rosettes, or combinations thereof. The collection of bullet rosettes and columns generate the highest C values, but are more infrequent compared to the collection of compact irregular particles with lower C. Predefined initializations for IPAS aggregates cover more monomers (high nm) with high P (i.e., columns and needles) compared to relatively compact CPI aggregates with monomers that merge and deform from vapor deposition and sintering.3 Figure 5a demonstrates that there is no one assemblage of IPAS monomers to precisely match C magnitudes to observed aggregates, but instead, there are multiple collections with varied nm that resemble CPI imagery.

It cannot be assumed that all aggregates in nature are formed from the collection of identical monomers, one at a time; thus, AGG–AGG collection is also considered between IPAS and CPI aggregates. AGG–AGG collection results in a mode of C = 0.92 from a histogram of 120 000 aggregates (not shown). From Fig. 2c, left, in Part II (Przybylo et al. 2022b), there is, on average, greater than 10 monomers per bin. Therefore, upon collection, P can greatly increase from the compounding effect on nm, which is most profound for a large number of extreme columns. Even though Ac tends to increase to encompass the broad collection from randomly oriented aggregates that tend to collect via outer-edge attachment, C is highly dependent on the magnification of P. While AGG–AGG collection is undoubtedly important in the creation of large aggregates, the maximum size range of the CPI probe (2.5 mm) limits further verification to larger aggregates in IPAS.

b. Aspect ratio

While complexity is a useful parameter for broad associations between IPAS and CPI aggregates, this work serves to provide quantitative estimates of aggregate aspect ratios for bulk models that prognose the growth of the major and minor axes of spheroids. Before implementing aggregate properties from IPAS into bulk models, there must be confidence that aspect ratios from IPAS aggregates are resemblant of observed aggregates. Due to the fact that CPI imagery is two-dimensional and may not capture the most extreme aspect ratios (i.e., the major axis is “hidden” from the field of view), the sampling of 110 920 aggregates reassures that bulk statistics are robust. It is assumed that particles enter the probe inlet in a random orientation from self-induced turbulence, especially for small particle diameters. Since the CPI probe images particles up to 2.5 mm in diameter, it is also assumed that there is no preferential orientation of larger particles due to resistance of nonuniform flow from particle mass. For a more direct comparison to two-dimensional imagery, 2D aspect ratios are similarly computed in IPAS. Both the monomers and aggregates are prescribed with a random orientation before a minimum volume ellipse is fit to each aggregate. The primary axes of the ellipse are then divided such that the two-dimensional aspect ratio (ϕ2D) ranges from 0 < ϕ2D ≤ 1.0. ϕ2D is averaged from three viewing angles separated by 36° of rotation to mimic the Multi-Angle Snowflake Camera (MASC; Garrett et al. 2012); however, the extra computations made negligible difference on distribution shape; therefore, only one viewing angle is considered. The same fit-ellipse function is used for all aspect ratio calculations by converting binary thresholded CPI images to a polygon using the Python Shapely package for manipulation and analysis of planar geometric objects.

Figure 5b shows normalized probability distributions for IPAS 2D aspect ratios (ϕ2D, light blue), 3D aspect ratios (ϕca, dark blue), and CPI aggregate aspect ratios (red) for nm ≤ 30 (solid line), nm ≤ 20 (dashed line), and nm ≤ 10 (dash–dotted line). Naturally, ϕ2D is less extreme than ϕca (on average by ≈0.1) since ϕca takes into account the true minimum and maximum extent of the aggregate. As nm increases and the aggregates become less eccentric, the modes of both aspect ratio calculations also increase to a maximum of ϕ2D = 0.59 for nm ≤ 30, but fail to reach the approximate modal value of 0.66 for CPI aggregates.

The tails of the monomer aspect ratio distributions initialized in IPAS could be considered unnatural or unlikely to occur on a broad scale. While it is impossible to measure each monomer aspect ratio within the CPI aggregate, it is very unusual to have monomers collecting at the end of the ϕm spectrum in IPAS (e.g., ϕm = 0.01 and ϕm = 100.0). For modeling purposes, these extreme monomer aspect ratios need to be considered, while for verification purposes, they can artificially skew aggregate aspect ratios to be too extreme even when only considering up to ϕm = 50. Naturally formed aggregates have less rigidity in particle edges, are more isometric, and exhibit greater diversity in particle habit attributable to sedimentation and growth in regions of varied temperature and saturation. The relatively narrower spread in IPAS aspect ratios (standard deviation of 0.13 for ϕ2D compared to 0.16 for the CPI distribution) is likely a consequence of the limited habits that are modeled, each with similar features that only differ by scale. It is important to note that the database of CPI aggregates does not include rimed particles, which would skew aspect ratios toward unity and increase particle density. Due to the fact that MON–AGG collection results in aspect ratios that are more extreme than observed aggregates, AGG–AGG collection is not presented in Fig. 5b as the collection of two aggregates tends to create more extreme aspect ratios and more complex aggregates than MON–AGG collection, outside of very eccentric aggregates to begin with (Fig. 2, top).

c. Area ratio

To justify why aspect ratio deviations exist between IPAS aggregates and observed aggregates, a third parameter is considered: roundness or area ratio (Ar), defined as the ratio of the particle projected area (Ap) to the area of the smallest circle that will cover a two-dimensional image of the particle (Ac) in the xy plane (Ar=Ap/Ac). Ar lies in the range 0.0 < Ar ≤ 1.0; smaller values represent increased spatial dispersion, gaps, or lacunarity taken in two dimensions. Branched and columnar ice crystals tend to have relatively low Ar, whereas compact or quasi-spherical ice particles have relatively high Ar, near unity. Figure 5c shows Ar for CPI aggregates (red) and IPAS aggregates (blue), again for nm ≤ 30 (solid line), nm ≤ 20 (dashed line), and nm ≤ 10 (dash–dotted line) for 0.01 ≤ ϕm ≤ 50.0. Interestingly, Ar distributions are consistent for IPAS aggregates regardless of the number of monomers included for analysis and despite monomer projected area variations given the random orientation. As monomers begin to chain together, there are large gaps present between the aggregate and encompassing circle such that the area of additional monomers becomes nearly insignificant toward Ar alterations, on average. Furthermore, if a monomer is collected within the perimeter of an aggregate composed of plates or is parallel to the field of view, the area ratio will not change.

Two distinct modes are displayed in Fig. 5c for IPAS aggregates (blue): Ar = 0.02 for aggregates composed of columns and Ar = 0.29 for aggregates composed of plates. For columns and needles, Ap is primarily attributable to the surface area of the major axis, which is insignificant compared to the broad surface area across the basal face for plates. Recall that the major axis of columns is longer than the major axis of plates given the same mass, so columns are able to encompass a larger area (large Ac) for the same nm, yet Ap remains much smaller, leading to extremely low Ar. The secondary, higher mode mostly encompasses aggregates composed of monomers with ϕm < 1.0.

In comparison to IPAS aggregates, CPI aggregates have an area ratio of Ar = 0.44, which is approximately 50% higher than IPAS-formed aggregates composed of plates (Fig. 5c). Analysis of Ar confirms the compactness and sphericity of CPI aggregates that leads to higher aspect ratios, on average, compared to IPAS aggregates. Figure 5 (bottom row) shows bulk statistics of (Fig. 5d) complexity, (Fig. 5e) aspect ratio, and (Fig. 5f) area ratio for CPI aggregates (red) and IPAS aggregates (blue) for 0.01 ≤ ϕm ≤ 50.0 and ra ≤ 5 mm. Box plots are displayed for different ranges of monomer attributes within IPAS-generated aggregates (y axis) to draw comparisons across more readily observed monomer ranges. As the range in ϕm converges toward 1.0, distributions for complexity, aspect ratio, and area ratio better align with CPI aggregates (Figs. 5d–f), which proves that IPAS initializations encompass an extreme yet sufficiently broad spectrum of monomers.

For computational efficiency, it is assumed that IPAS monomers collect as long as they intersect, meaning there is no minimum surface area required for attachment. As such, edge-to-edge collection between IPAS monomers is much more prevalent, although less realistic, leading to generally higher complexity and more extreme aspect ratios and area ratios compared to CPI aggregates, especially as monomers randomly tumble (Fig. 5, bottom row). In addition, the extreme monomer shapes that are included in all distributions in Fig. 5, top row, are mostly used to encompass a broad spectrum of possible collections for modeling purposes, but skew IPAS aggregates toward more extreme and complex aggregate formations. While complexity and area ratio are useful measures toward IPAS verification, recall that only aspect ratios and density change calculations in IPAS are to be used by bulk models to smoothly evolve particle shape; aspect ratios vary by ≈12% between the mean distribution values in CPI (15–2300 μm) and IPAS (nm ≤ 30, ϕ2D, all ϕm) aggregates, which gives sufficient confidence that bulk characteristics are similar between theoretical and observed aggregates. A more thorough analysis of the CPI geometries could be a topic of future work to discern the causality for parameter deviations, but dissecting the unimodal nature at this moment is outside the purpose of this paper, which is to introduce the methodology of MON–AGG and AGG–AGG collection in IPAS.

d. Comparison to other studies

There is often a mutual reliance between theoretical and observational efforts to fill in data gaps, examine and evaluate parameter sensitivities, and verify model simulations. Remote sensing initiatives have helped validate and extend current understanding of particle shape, which is valuable in refining regional and global climate models for nonsphericity and particle irregularities that influence microphysical and radiative processes. Hogan et al. (2012) found that using horizontally oriented oblate spheroids with an aspect ratio of 0.6 (compared to spheres) significantly improved the forward calculation of radar reflectivity (with use of aircraft observed size distributions) compared to coincident observations from ground-based radar, although there is evidence that deviations from 0.6 could result in better agreement in certain geographic regions. Aspect ratios were determined from 2D projections of ice particles from the 2DC probe in a midlatitude ice clouds but improvements were not universally validated across all aircraft tracks (Hogan et al. 2012). Moreover, mixed-phase clouds were not analyzed but can be dominated by deposition instead of aggregation, which can lead to more extreme particles such as in the dendritic growth zone. In addition, during the StormVEx field project, Marchand et al. (2013) observed a small number of events were dominated by columnar or prolate particles; therefore, a general assumption that oblate particles characterize all sample volumes is not sensible.

Reinking et al. (2002) explicitly differentiates particle type (e.g., drizzle and cloud droplets, plates, columns, and irregular ice particles such as graupel and aggregates) using a slant quasi-linear DR4 at 45° assuming particle bulk densities. Consequently, Matrosov (2015) found that radar-based retrievals of particle aspect ratios from mountain winter storms are generally 0.5 ± 0.2, dependent on elevation angle, particle orientation, size, and mass–dimensional relationships chosen a priori. Additionally, the methods of both Reinking et al. (2002) and Matrosov (2015) are sensitive in regions of light drizzle or low reflectivity clouds. Given uncertainty estimates of 0.2, pristine crystals with extreme aspect ratios (e.g., dendrites) should be cautiously evaluated.

Using the surface-based MASC instrument, Garrett et al. (2015) found the median aspect ratio of 73 000 imaged aggregates to be 0.60 (averaged across three viewing angles) for particles up to 14 mm in diameter with a median of 3.3 mm. Korolev and Isaac (2003) found similar aspect ratios between 0.6 and 0.8 (based on temperature) from 897 033 CPI images between 70 and 1000 μm in winter midlatitude and polar stratiform clouds during three field projects. The aggregation modeling work by Westbrook et al. (2004) also finds aspect ratios in the range 0.6–0.65 independent of initial habit (rods or columns and bullet rosettes shown in their Fig. 2). Contrary to the aforementioned studies, Jensen et al. (2017) assumes a constant yet more extreme characteristic aggregate aspect ratio of 0.2 and a density of 50 kg m−3 for all newly formed aggregates, which could have significant influence on snowfall quantitative precipitation estimates given nonlinear vapor deposition with extreme particle shape.

IPAS 2D aspect ratios surround a bulk value of 0.6 and closely follow that of previous findings that use remote radar retrievals, in situ imagery, and some modeling efforts; however, there must be consideration on calculation discrepancies. IPAS true or “real” aspect ratios are calculated using all three ellipsoid dimensions, contrasting imagers that retrieve two-dimensional snapshots. Projection uncertainties hinder orientation predictability as 2D observations are distorted away from the true 3D aggregate structures (Dunnavan et al. 2019). Particles will tend to look more circular, on average, since the primary axes rarely will be aligned perpendicular to the viewing direction, leading to mean 3D aspect ratios less than 0.6. From Jiang et al. (2017), their Table 1, projected 2D aspect ratios from predefined ellipsoids of varied prolate and oblate spheroids from various distribution shapes loosely range between 0.5 and 0.7 for a random orientation (oblates trend higher than prolates), which are very consistent with 2D aspect ratios taken from the database of IPAS aggregates (Fig. 5b, light blue). Also from Jiang et al. (2017), random orientations of 0.33 oblate spheroids yield a mean 2D projected aspect ratio of 0.6, which conforms with ϕca calculations in Fig. 5b when extrapolating for a reduced number of monomers per aggregate that resemble single spheroids modeled by Jiang et al. (2017).

5. Discussion

In addition to design limitations of IPAS discussed in Part II (Przybylo et al. 2022b), such as lack of particle branching, multihabit representations beyond primary particle types, fluid-dynamical flow, a collection kernel, and appropriate collection constraints (e.g., forcing a certain amount of overlap for collection), there are additional confinements to consider that impact verification. IPAS aggregates are only compared to CPI particles in a random orientation, which undermines the fact that the maximum dimension tends to lie closer to the horizontal than the vertical at the surface, although orientation also depends on local aerodynamic conditions (Hogan et al. 2012; Xie et al. 2012; Garrett et al. 2015). This orientation is opted as aircraft observations are unable to capture particles in their prevailing orientation due to horizontal aircraft motion along with turbulence and flow disturbance from the aircraft fuselage. Furthermore, IPAS ranges for the number of monomers per aggregate (2 ≤ nm ≤ 30), monomer aspect ratios (0.01 ≤ ϕm ≤ 100.0), monomer radii (1.0 ≤ rm ≤ 1000.0), and aggregate radius (ra ≤ 5 mm) are somewhat ad hoc and discrete even though they are guided by observations. Premature truncation of the database of IPAS aggregates in the case of ra ≤ 5 mm, or overly expansive ranges of monomer attributes, have the potential to alter bulk characteristics used in verification. It is critical to note that the IPAS aggregate database on which these analyses are performed was constructed to encompass as many possible aggregate scenarios as possible, including unlikely scenarios, but those that may be encountered in rare instances in bulk model simulations. These data likely skew results to more extreme values, especially when compared to frequently observed CPI aggregates.

While the CPI probe imagery is taken to be “truth,” there are limitations of the probe itself. A restriction is imposed on the percent of the particle perimeter allowed to intersect the border of the image; there are cases where an aggregate with intricate branches or a bullet rosette is largely cutoff but since the aggregate perimeter is large, the cutoff percentage is less than 5%. Even though a <5% cutoff criteria is strict given aircraft motion, some aggregates are sampled that are not entirely in frame, which alters all parameter calculations. Calculating cutoff with respect to the perimeter of the border instead of particle perimeter tends to be less reliable with no bearing on particle features and overcompensates by discarding too many particles (e.g., if a bullet rosette is mostly in frame except for one branch).

Furthermore, the CPI probe size range (≤2300 μm) largely limits analysis of aggregate properties at larger sizes, particularly relevant for IPAS simulations of aggregates with a larger number of monomers. Supplementary use of multiple probes that image larger aggregates would be disputable for resolution-dependent calculations such as perimeter and area; therefore, due to the detailed analysis of aggregate properties formed in IPAS, only one high-resolution probe is considered. While the CPI probe resolution is sufficient in capturing particle properties at a fine scale, there are quality and consistency issues with background noise between field campaigns. Moreover, regions of low particle density or high transparency are omitted in parameter calculations if they fall outside of the darker bounding perimeter.

The CPI database is truncated to neglect rimed particles but other thermodynamic processes that alter particle shape morphology such as deposition and sublimation are inherently incorporated yet not represented in IPAS. A fully dynamical model that includes vapor, liquid, and ice quantities would be needed to deduce particle shape and density alterations from thermodynamic processes, which is outside the scope of this work but reconciled through conjunction with present dynamical microphysical parameterizations that represent aggregates as spheroids.

6. Conclusions

Bulk ice-microphysical models parameterize the dynamic evolution of ice particles from advection, collection, and sedimentation (among other mass-altering processes) from cloud top to surface. Frozen hydrometeors have a multitude of shapes and sizes that need to be accounted for as their influence on the distribution of mass within mixed-phased clouds can be significant. A statistical model that simulates the aggregation process of pristine plates and columns is used to quantitatively evolve particle shape and density from monomer to aggregate sizes. MON–AGG and AGG–AGG collection is analyzed from a database of 9 744 000 aggregates from the collection of monomers in a random and quasi-horizontal orientation. Naturally, quasi-horizontally oriented monomers create more extreme aggregate aspect ratios for the most extreme bin ranges; however, orientation is insignificant in terms of aspect ratio evolution if the collecting aggregate aspect ratio is ⪆5 for both MON–AGG and AGG–AGG collection. Neither MON–AGG nor AGG–AGG collection results in spherical aggregates; ϕca is most isometric for MON–AGG collection for the quasi-horizontal orientation at 0.76.

Accurate shape evolution from ice to snow must be accompanied by proper density changes as ice particles collect and redistribute mass and number quantities. Erfani and Mitchell (2016) show at least an order of magnitude spread in estimated mass for any particular aggregate size between −20° and −40°C (their Fig. 8). Particle habit, collection type, and the number of monomers per aggregate all influence the magnitude in which particle mass and density change. MON–MON collection results in the most extreme decrease in density from identical monomers that fill a larger encompassing area after collection yet a smaller polygonal area—notably for columns. MON–AGG collection can both increase and decrease aggregate density depending on whether the monomer fills in aggregate gaps or collects along the aggregate perimeter. AGG–AGG collection generally decreases particle density from expansion of the encompassing ellipsoid. IPAS acts to clarify and hone density magnitudes as a function of multiple independent variables for more accurate fall speed estimates.

IPAS aggregates are validated against 110 920 CPI images of aggregates in terms of complexity, 2D aspect ratio, and area ratio. Generally, observed aggregates are more compact and circular since IPAS aggregates encompass a wide range of monomer aspect ratios and number of monomers that collect with no enforcement on the amount of overlap needed for attachment. While there are differences between IPAS and CPI aggregates, bulk statistics on particle aspect ratios prove to be adequately similar for usage in bulk ice-microphysical schemes.

1

A bounding rectangle surrounds the minimum and maximum x and y points of the aggregate.

3

Sintering is the bonding between particles from adhesive forces with increasing temperatures near 0°C (Kuroiwa 1961).

4

Depolarization ratio is the ratio between the perpendicular component and the parallel component of power received after a signal is transmitted from a radar.

Acknowledgments.

V. Przybylo, K. Sulia, C. Schmitt, and Z. Lebo would like to thank the Department of Energy for support under DOE Grant DE-SC0016354. K. Sulia is additionally supported through an appointment under the SUNY 2020 Initiative. The authors would also like to thank the ASRC Extreme Collaboration, Innovation, and Technology (xCITE) Laboratory for IPAS development support. The authors are grateful for the efforts of the instrument personnel and aircraft flight crew in collection of the campaign data in addition to the funding sources.

Data availability statement.

The database of aggregates generated and used for collection are not publicly available due to size limitations. However, the source code relating to the specific version of this work can be found at https://doi.org/10.5281/zenodo.4749478. The latest version of IPAS has been made open source at https://github.com/vprzybylo/IPAS and can be installed using the pip package manager for Python available at https://pypi.org/project/ipas/.

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Save
  • Baum, B. A., A. J. Heymsfield, P. Yang, and S. T. Bedka, 2005: Bulk scattering properties for the remote sensing of ice clouds. Part I: Microphysical data and models. J. Appl. Meteor., 44, 8851895, https://doi.org/10.1175/JAM2308.1.

    • Search Google Scholar
    • Export Citation
  • Brown, P. R. A., 1995: Improved measurement of the ice water content in cirrus using a total water probe. J. Atmos. Oceanic Technol., 12, 410414, https://doi.org/10.1175/1520-0426(1995)012<0410:IMOTIW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dunnavan, E. L., 2021: How snow aggregate ellipsoid shape and orientation variability affects fall speed and self-aggregation rates. J. Atmos. Sci., 78, 5173, https://doi.org/10.1175/JAS-D-20-0128.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dunnavan, E. L., Z. Jiang, J. Y. Harrington, J. Verlinde, K. Fitch, and T. J. Garrett, 2019: The shape and density evolution of snow aggregates. J. Atmos. Sci., 76, 39193940, https://doi.org/10.1175/JAS-D-19-0066.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Eidhammer, T., H. Morrison, D. Mitchell, A. Gettelman, and E. Erfani, 2017: Improvements in global climate model microphysics using a consistent representation of ice particle properties. J. Climate, 30, 609629, https://doi.org/10.1175/JCLI-D-16-0050.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Erfani, E., and D. Mitchell, 2016: Developing and bounding ice particle mass- and area-dimension expressions for use in atmospheric models and remote sensing. Atmos. Chem. Phys., 16, 43794400, https://doi.org/10.5194/acp-16-4379-2016.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Field, P. R., 1999: Aircraft observations of ice crystal evolution in an altostratus cloud. J. Atmos. Sci., 56, 19251941, https://doi.org/10.1175/1520-0469(1999)056<1925:AOOICE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, T. J., C. Fallgatter, K. Shkurko, and D. Howlett, 2012: Fall speed measurement and high resolution multi-angle photography of hydrometeors in freefall. Atmos. Meas. Tech., 5, 26252633, https://doi.org/10.5194/amt-5-2625-2012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Garrett, T. J., S. E. Yuter, C. Fallgatter, K. Shkurko, S. R. Rhodes, and J. L. Endries, 2015: Orientations and aspect ratios of falling snow. Geophys. Res. Lett., 42, 46174622, https://doi.org/10.1002/2015GL064040.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gilmore, M., J. M. Straka, and E. N. Rasmussen, 2004: Precipitation uncertainty due to variations in precipitation particle parameters within a simple microphysics scheme. Mon. Wea. Rev., 132, 26102627, https://doi.org/10.1175/MWR2810.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harrington, J. Y., K. J. Sulia, and H. Morrison, 2013: A method for adaptive habit prediction in bulk microphysical models. Part I: Theoretical development. J. Atmos. Sci., 70, 349364, https://doi.org/10.1175/JAS-D-12-040.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., and M. Kajikawa, 1987: An improved approach to calculating terminal velocities of plate-like crystals and graupel. J. Atmos. Sci., 44, 10881099, https://doi.org/10.1175/1520-0469(1987)044<1088:AIATCT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., and C. D. Westbrook, 2010: Advances in the estimation of ice particle fall speeds using laboratory and field measurements. J. Atmos. Sci., 67, 24692482, https://doi.org/10.1175/2010JAS3379.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., A. Bansemer, P. R. Field, S. L. Durden, J. L. Smith, J. E. Dye, W. Hall, and C. A. Grainger, 2002: Observations and parameterizations of particle size distributions in deep tropical cirrus and stratiform precipitating clouds: Results from in situ observations in TRMM field campaigns. J. Atmos. Sci., 59, 34573491, https://doi.org/10.1175/1520-0469(2002)059<3457:OAPOPS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., A. Bansemer, C. Schmitt, C. Twohy, and M. Poellot, 2004: Effective ice particle densities derived from aircraft data. J. Atmos. Sci., 61, 9821003, https://doi.org/10.1175/1520-0469(2004)061<0982:EIPDDF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., L. Tian, P. Brown, C. Westbrook, A. Heymsfield, and J. Eastment, 2012: Radar scattering from ice aggregates using the horizontally aligned oblate spheroid approximation. J. Appl. Meteor. Climatol., 51, 655671, https://doi.org/10.1175/JAMC-D-11-074.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Houze, R. A., Jr., W. Schmid, R. G. Fovell, and H.-H. Schiesser, 1993: Hailstorms in Switzerland: Left movers, right movers, and false hooks. Mon. Wea. Rev., 121, 33453370, https://doi.org/10.1175/1520-0493(1993)121<3345:HISLMR>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jensen, A. A., J. Y. Harrington, H. Morrison, and J. A. Milbrandt, 2017: Predicting ice shape evolution in a bulk microphysics model. J. Atmos. Sci., 74, 20812104, https://doi.org/10.1175/JAS-D-16-0350.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jiang, Z., M. Oue, J. Verlinde, and E. E. Clothiaux, 2017: What can we conclude about the real aspect ratios of ice particle aggregates from two-dimensional images? J. Appl. Meteor. Climatol., 56, 725734, https://doi.org/10.1175/JAMC-D-16-0248.1.

    • Crossref
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  • Fig. 1.

    Theoretical aggregates from (left) MON–AGG collection and (right) AGG–AGG collection for the random and quasi-horizontal orientations. Corresponding complexity values are located next to each aggregate. Viewing angle is in the xz plane (y orientation) for MON–AGG collection and in the xy plane (z orientation) for AGG–AGG collection for different viewing perspectives.

  • Fig. 2.

    Evolution of aggregate aspect ratio distributions (ϕba, x axis) and (ϕca, y axis) for MON–AGG (blue) and AGG–AGG (red) collection. Each aggregate is formed 300 times for the (top) random orientation and (bottom) quasi-horizontal orientation to create a multivariate kernel density estimate (contoured). Distribution modes are shown in white and red dots for MON–AGG and AGG–AGG collection types, respectively. ϕca ranges from the database before collection are listed in the title going from (top left) most extreme to (bottom right) most spherical. Percentage oblate values (out of 300) are marked in the upper-left corner of each subplot for AGG–AGG collection (red) and MON–AGG collection (blue). If a majority of the aggregates are defined as oblate (prolate) spheroids the subplot is shaded to the right (left), with color representing collection type. If both collection types produce the same majority shape (oblate or prolate), the appropriate region is shaded purple.

  • Fig. 3.

    The relative change in density as a function of the (left) random orientation and (right) quasi-horizontal orientation for MON–MON (green), MON–AGG (orange), and AGG–AGG (blue). All aggregate data are plotted for each collection with a sample size of 168 000 for MON–MON collection and 120 000 for MON–AGG and AGG–AGG collection. A density change of −1.0 represents a 100% decrease in density from the initial particle. Both a histogram and a smoothed kernel density estimate are plotted.

  • Fig. 4.

    The mean percentage decrease in density (y axis) from 300 collections as a function of ϕm (x axis, 0.01 < ϕm < 100.0) for rm = 10 μm. All monomers are identical within the aggregate. Colors represent the number of monomers in the aggregate after collection for the (left) random orientation and (right) quasi-horizontal orientation.

  • Fig. 5.

    (top) Normalized probability distributions of aggregate (a) complexity, (b) aspect ratio, and (c) area ratio for IPAS (blue) and CPI (red) aggregates. IPAS aggregates are plotted with respect to nm ≤ 30 (solid lines), nm ≤ 20 (dashed lines), nm ≤ 10 (dash–dotted lines), and nm ≤ 5 (dotted line). IPAS 2D aspect ratios [ϕ2D, (b), light blue] are calculated from a projected ellipse surrounding the aggregate in a random orientation and IPAS three-dimensional aspect ratios [ϕca, (b), dark blue] are calculated from a fit ellipsoid (both minor and major semiaxes). (bottom) Bulk statistics for (d) complexity, (e) aspect ratio, and (f) area ratio are shown for all aggregates up to 30 monomers with rm ≤ 5 mm with each box plot corresponding to a different range of monomer attributes within the IPAS-formed aggregates (y axis). From top to bottom, each box plot in (d)–(f) consists of 3 681 556 (dark blue), 3 010 409 (medium blue), 436 612 (light blue), and 110 920 (CPI, red) aggregates, respectively. All calculations are independent of viewing angle in a random orientation.