The Ice Particle and Aggregate Simulator (IPAS). Part II: Analysis of a Database of Theoretical Aggregates 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

Bulk ice-microphysical models parameterize the dynamic evolution of ice particles from advection, collection, and sedimentation through a cloud layer to the surface. Frozen hydrometeors can grow to acquire a multitude of shapes and sizes, which influence the distribution of mass within cloud systems. Aggregates, defined herein as the collection of ice particles, have a variety of formations based on initial ice particle size, shape, falling orientation, and the number of particles that collect. This work focuses on using the Ice Particle and Aggregate Simulator (IPAS) as a statistical tool to repetitively collect ice crystals of identical properties to derive bulk aggregate characteristics. A database of 9 744 000 aggregates is generated with resulting properties analyzed. After 150 single ice crystals (monomers) collect, the most extreme aggregate aspect ratio calculations asymptote toward ϕca=(c/a)0.75 and ϕca ≈ 0.50 for aggregates composed of quasi-horizontally oriented and randomly oriented monomers, respectively. The results presented are largely consistent with both a previous theoretical study and estimates derived from ground-based observations from two different geographic locations. Particle falling orientation highly influences newly formed aggregate aspect ratios from the collection of particles with extreme aspect ratios; quasi-horizontally oriented particles can produce aggregate aspect ratios an order of magnitude more extreme than randomly oriented particles but can also produce near-spherical aggregates as the number of monomers comprising the aggregate reach approximately 100. Finally, a majority of collections result in aggregates that are closer to prolate than oblate spheroids.

© 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

Bulk ice-microphysical models parameterize the dynamic evolution of ice particles from advection, collection, and sedimentation through a cloud layer to the surface. Frozen hydrometeors can grow to acquire a multitude of shapes and sizes, which influence the distribution of mass within cloud systems. Aggregates, defined herein as the collection of ice particles, have a variety of formations based on initial ice particle size, shape, falling orientation, and the number of particles that collect. This work focuses on using the Ice Particle and Aggregate Simulator (IPAS) as a statistical tool to repetitively collect ice crystals of identical properties to derive bulk aggregate characteristics. A database of 9 744 000 aggregates is generated with resulting properties analyzed. After 150 single ice crystals (monomers) collect, the most extreme aggregate aspect ratio calculations asymptote toward ϕca=(c/a)0.75 and ϕca ≈ 0.50 for aggregates composed of quasi-horizontally oriented and randomly oriented monomers, respectively. The results presented are largely consistent with both a previous theoretical study and estimates derived from ground-based observations from two different geographic locations. Particle falling orientation highly influences newly formed aggregate aspect ratios from the collection of particles with extreme aspect ratios; quasi-horizontally oriented particles can produce aggregate aspect ratios an order of magnitude more extreme than randomly oriented particles but can also produce near-spherical aggregates as the number of monomers comprising the aggregate reach approximately 100. Finally, a majority of collections result in aggregates that are closer to prolate than oblate spheroids.

© 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

The process of aggregation, or the collection of one or more ice crystals, deemed snow in this study, is a critical component in ice-microphysical parameterizations within cloud-resolving models due to the inherent size and mass of aggregates. A diverse range of sizes and shapes (habits) form in nature based on environmental growth conditions and advective transport strength, which leads to varied sedimentation velocities and easily attained precipitation-sized particles (Heymsfield et al. 2002; Schmitt et al. 2019). In general, bulk ice-microphysical schemes simplify the multitude of features that aggregates can acquire. For example, the prediction of snow mass can be constrained by a priori empirical relationships (e.g., Straka and Mansell 2005; Thompson et al. 2008; Morrison and Milbrandt 2015), which only considers the maximum dimension of the particle and smooths the fractal nature of aggregates. In this approach, while mass is conserved in the transition between frozen categories (e.g., ice, snow, rime, graupel), particle fall speed and maximum dimension may not evolve in a natural manner from artificial thresholding (e.g., the transition from ice to snow when the maximum diameter is ≥200 μm; Thompson et al. 2008), which can potentially lead to ambiguous spatial distributions of precipitation, especially in nonstatic environments where particle properties are dynamically changing (Morrison et al. 2015). More recent models using both a bin (McSnow; Brdar and Seifert 2018) and bulk (P3; Morrison and Milbrandt 2015) approach have made progress on smoothly simulating the aggregation and riming process by adding more prognostic variables, such as rime mass and volume and the number of monomers that comprise an aggregate, to alleviate the ad hoc conversion parameters for traditionally defined particle categories; however, this comes at a cost of computational expense required to resolve a four-dimensional particle distribution on a three-dimensional grid (McSnow) or the inherent lack of ability to simulate drastically different ice particle populations at the same spatial coordinate at some time (P3).

Because aggregation is inherently related to the evolution of ice and vapor quantities through both particle collection and vapor depositional growth, accurate representation in cloud models is critical, especially within precipitating systems. Modeled particle size distributions (PSDs) in ice-containing clouds are highly dependent on aggregates because number concentrations and particle maximum dimensions are quickly altered through the loss of single ice crystals to aggregates. As small ice particles are collected by larger particles, a bimodal distribution can result in a clear delineation between a diminishing number of small particles and a growing number of large particles (e.g., Cotton et al. 2013, their Fig. 3). A natural transition of frozen mass from the ice to snow category resulting from aggregation where aggregate size, shape, density, and terminal fall speed is evolved smoothly is essential for accurate predictions of hydrometeor type, distribution of mass within a system, and cloud lifetime.

Ice microphysical schemes have commonly used separate mass–dimensional relationships based on habit composition to diagnose bulk microphysical quantities (e.g., Harrington et al. 1995; Meyers et al. 1997; Heymsfield et al. 2002; Woods et al. 2007), but this approach can be cumbersome, not realistic near size ranges at the boundaries of observations, and does not allow for the nonlinear evolution of particle shape with growth. In addition, outside of Woods et al. (2007), the empirically derived coefficients are not modified from initial declarations based on habit, have finite size ranges (e.g., Locatelli and Hobbs 1974; Mitchell et al. 1990; Mitchell 1996), and thus, are unable to capture the detailed growth (mass and size) and advective history of these particles [see Sulia and Harrington (2011) and Sulia et al. (2013) for details on this importance]. Such shortcomings intrinsically neglect the multitude of particle shapes that can form and aggregate in nature. Few observational studies have been performed to link power-law relationships with empirically derived constants to multiparticle aggregates that encompass nonpristine habits with wide-ranging sizes. Aggregate characteristics are particularly difficult to predict, validate, and extend past a given size range for derived relationships; thus, a variety of aggregation models have been used to elucidate aggregate characteristics through theorized statistical modeling based on observations (Westbrook et al. 2004; Hashino and Tripoli 2011a,b; Leinonen and Szyrmer 2015; Dunnavan et al. 2019; Przybylo et al. 2019).

2. IPAS

The Ice Particle and Aggregate Simulator (IPAS; Schmitt and Heymsfield 2010, 2014) is a theoretical framework that mimics laboratory investigations in a statistical sense to retrieve, visualize, and better understand growth via collection. A detailed description of IPAS can be found in Przybylo et al. (2019, 2021). IPAS joins hexagonal prisms that are modifiable in shape and size yet solid in structure in a three-dimensional architecture. IPAS can create an innumerable number of theoretical aggregates, solely limited by computational expense, which provides sufficient confidence that calculated bulk parameters capture the most representative aggregate properties when verified against thousands of naturally occurring aggregates from airborne probe images (Part III, Przybylo et al. 2022). Of particular interest is the influence of monomer (single-crystal) habit on aggregation and how these initializations scale with an increasing number of monomers per aggregate. The scalability of IPAS is explored to achieve a well-rounded and representative dataset to be used by bulk models for parameterization of the aggregation process.

The primary motivation of using IPAS is to determine and evolve aggregate aspect ratios, which is adaptable to any habit through the use of reduced density oblate and prolate spheroids as proxies for particle shape. Numerical models that predict and evolve aggregate properties are only as accurate as the aggregate initializations for size, shape, number, and mass quantities. IPAS is used to derive aggregate parameters from a predefined number of simulations with fixed monomer properties and acts to hone and bound aggregate dimensional characteristics offline before cloud-resolving models apply further processes (e.g., deposition, sublimation, melting). This paper primarily focuses on the collection of a single monomer by an aggregate (MON–AGG collection) as monomer–monomer (MON–MON) collection is described in detail in Przybylo et al. (2019), bulk testing of which can be found in Sulia et al. (2021). Part III of this work (Przybylo et al. 2022) considers verification and analysis of aggregate–aggregate collection.

a. Aggregate database

For MON–AGG collection, an aggregate must already exist; methodologies for MON–AGG collection extend from MON–MON collection as outlined in Przybylo et al. (2019). The creation of a new aggregate for MON–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 has been created. In addition to easy aggregate acquisition for further collection, this methodology allows for control of aggregate shape and size, while allowing for diversity in geometry and monomer count. Database initialization and collection methodologies are described in sections 2b and 2c, respectively. Then postprocessing calculations and average database characteristics are analyzed in sections 2d and 2e. Aggregates saved in the database are later used to collect with a single monomer (MON–AGG collection) or another aggregate (AGG–AGG collection; Part III of this work, Przybylo et al. 2022). Aggregate properties are then examined as a function of the number of monomers that comprise the aggregate and the shape of these monomers in section 3a. Subsequent verification to past theoretical studies in section 4 is performed. This investigation concludes with a discussion on model caveats in section 5 and remarks on aggregation parameterization in bulk microphysical models in section 6.

b. Initialization

The aggregates in this study are composed of a maximum of 30 monomers to encompass a broad yet computationally justifiable range of observed crystal sizes in addition to consideration of other theoretical studies (e.g., Westbrook et al. 2008, their Fig. 13). To create the database of aggregates, monomer radii were increased across three orders of magnitude from 1 to 1000 μm logarithmically (28 values); each size assumed a constant monomer density from an equivalent volume sphere that is independent of shape. For every radius, 20 different monomer shapes were initialized by logarithmically increasing the aspect ratio (ϕm), defined as the ratio of the prism or rectangular face (cm) to the basal or hexagonal face (am, ϕm=cm/am) from ϕm = 0.01 (platelike crystals) to ϕm = 100.0 (needlelike crystals). While these monomer aspect ratios encompass a scale that pushes the limits of readily observed monomers, bulk microphysical models sometimes grow extreme instances of particles (such as dendrites with ϕm = 0.01), and therefore, IPAS needs to simulate such instances. Three hundred aggregates were formed and saved for every aspect ratio–radius pairing and for every monomer addition (up to 30, starting at 2) to create multiple configurations of aggregates from varied orientations and overlap with the same monomer properties. Three hundred is a flexible initialization yet deemed to be statistically reliable in reproducing aggregate characteristics (see Part I, Przybylo et al. 2019). This results in a total of 8700 aggregates per monomer aspect ratio–radius pairing, totaling 4 872 000 aggregates in the database for two falling orientations simulated (9 744 000 aggregates total). Each aggregate carries eight attributes: the vertices of the hexagonal prisms to account for their configuration, three radii from a minimum volume ellipsoid circumscribing the aggregate, a calculation for the complexity of the aggregate, the number of monomers within the aggregate (nm), the monomer aspect ratio (ϕm), and the monomer radius (rm) of the crystals that comprise the aggregate. While the monomers that comprise the aggregates in the database for a given rmϕm pair are identical, the resulting aggregates are not given the geometrical diversity of monomer incorporation (discussed in section 2c). Table 1 includes a reference of variables included throughout the following discussions.

Table 1

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

Table 1

Note that, in actuality, two databases are created to account for particle falling orientation (Table 2). Past studies have indicated that particle falling orientation greatly affects scattering properties and radar retrievals (Noel and Sassen 1983; Matrosov et al. 2005; Matrosov 2007; Garrett et al. 2015). Particle orientation is important because the horizontal cross section of ice particles affects fall speed through drag. Moreover, shear and deceleration of air in and around airborne probes randomly orient particles, especially ≤50 μm, due to the turbulent environment (Korolev and Isaac 2003). In addition, particles may fall in spiral patterns should their center of gravity be offset from the geometrical center (Kajikawa 1985). 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.

Table 2

Statistics on the aggregate database for both orientations.

Table 2

c. Collection methodology

Aggregates were all formed in the same manner outside of orientation adjustments: A monomer is initialized at the origin of a domain with 12 vertices that create a hexagonal prism based on the given radius. The predefined aspect ratio determines whether this so-called seed crystal is a plate (ϕm < 1.0), a column (ϕm > 1.0), or isometric (ϕm = 1.0). The seed crystal is held steady at an orientation that produces the maximum projected area from overhead for the quasi-horizontal orientation, which is determined by reorienting the crystal around the horizontal axes every 0.01° between 0° and 90° so that the orientation with the maximum projected area is selected (albeit with a tolerance inherent to the step size). For the random orientation, a single rotation is chosen from a series of randomly generated Euler angles.1 A second crystal of identical size and shape but different orientation is then moved to a randomly generated point on the seed crystal in the xy plane. The crystals are then merged vertically to the closest set of points in the xz and yz dimensions. In the quasi-horizontal orientation, the aggregate is then reoriented with maximized area from overhead following an identical process to reorientation for each monomer. This process repeats for every additional monomer. See section 5 for a discussion on the simplified collection methodology in IPAS.

Resultant theoretical aggregates from IPAS are displayed in Fig. 1. On the left, aggregates are composed of plates, and on the right, columns, all with a quasi-horizontal orientation. Monomer aspect ratios closer to ϕm = 1.0 represent hexagons with similar basal and prism face lengths, are thicker, and their maximized projected area results in a slight tilt relative to the xy plane (see Przybylo et al. 2019, their Fig. 1). Due to this tilting, as more monomers are added to the aggregate, collection begins to randomize (rather than stack) from monomers sloped in all directions (Fig. 1, middle to bottom). For extreme monomer aspect ratios (ϕm ≪ 1.0 or ϕm ≫ 1.0), this rotation effect is visually unnoticeable for the quasi-horizontal orientation up to 50 monomers due to minimal tilting (Fig. 1, top). Random rotation around the z axis is constrained to a range between 0° and 90° instead of between 0° and 180° to limit how quickly the aggregates in the quasi-horizontal orientation collect in all directions. A quasi-horizontal orientation is first applied to each particle to be collected then to the newly formed aggregate after collection; therefore, upon the collection of each subsequent monomer, the aggregate is likely to be in a different orientation such that each monomer within the aggregate is no longer oriented with a maximum projected area (Fig. 1).

Fig. 1.
Fig. 1.

Theoretical aggregates from the collection of identical monomers with both the monomers and resulting aggregates quasi-horizontally oriented. Corresponding ϕm values for the aggregate are listed above the aggregates for (left) ϕm < 1.0 and (right) ϕm ≥ 1.0 from (top) most extreme to (bottom) least extreme.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0179.1

d. Postprocessing

After database creation and aggregate storage, final aggregate properties are calculated to mitigate unnecessary data transfer. Two aggregate aspect ratios are calculated for every aggregate, ϕca, defined as the semiminor axis (c) divided by the semimajor axis (a) from a minimum volume ellipsoid2 that encompasses the aggregate. Since two aspect ratio measures are required to specify an ellipsoid shape, aggregate aspect ratios are calculated as ϕba = b/a and ϕca = c/a 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. ϕca is considered the true aspect ratio due to the fact that it embodies all aggregate extremes. 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 ≤ ϕcaϕba ≤ 1.0, where ϕca = 1.0 are isometric aggregates. Using the largest and smallest semiaxis lengths from the minimum volume ellipsoid produces the most extreme aspect ratio calculation regardless of orientation. Additionally, an equivalent volume spherical radius (ra) is calculated to have a comparable size measurement across all database aggregates. ra is calculated 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. Only two radii are used in the calculation to be amenable to bulk models that use spheroids to mimic the primary3 shapes of ice and snow.

After calculating ϕca and ra, the database is divided into 20 equal-count bins (i.e., all bins contain the same number of aggregates) based on the range of calculated ϕca. Each bin is then further split into 20 equal-count bins based on ra for a total of 400 bins. Note that in some cases, aggregates grew past realistic sizes that can be verified by aircraft imagery with increasing nm (e.g., 20-mm ra); hence, any aggregates with radii exceeding 5 mm are excluded from these analyses to focus on cloud-level aggregate sizes. Statistics on the number of aggregates for each orientation after truncating and dividing into equal-count bins are shown in Table 2. Seventy-eight percent (86%) of the database is less than 5 mm for the random (quasi-horizontal) orientation, which is 3 820 668 (4 214 075) aggregates. A sensitivity test was performed to assess the influence that larger aggregates had on database statistics but trends prove to be extremely consistent when aggregates of all sizes are included. Now, at this point, the original database of aggregates created and defined according to their monomer rm and ϕm have been remapped according to the resulting aggregate properties, ra and ϕca. This partitioning is critical as the 8700 aggregates formed for each monomer rmϕm pair can have highly variable final aggregate properties: Remapping the database according to aggregate properties is useful for analysis of aggregate evolution after further collection and for the creation of discrete intervals for bulk model usage via lookup tables.

e. Analysis

The above outlines the methodology for building the millions of aggregates stored in the database. A variety of parameters useful in determining particle characteristics in bulk models are calculated using IPAS. An analysis of the resulting characteristics of these formations is explored, gleaning insight on the patterns that emerge in aggregate formation as a function of size, shape, monomer count, and falling orientation.

Figure 2 shows aggregate properties with respect to the average monomer radius (Figs. 2a,b) and the average number of monomers (Figs. 2c,d) across all aggregates within the database for ra < 5 mm. Figures 2e and 2f show the primary monomer habit within each bin, calculated as the percentage of aggregates made up of monomers with ϕm < 1.0 (plates); a majority of plates (columns) per bin are shown in red (blue) (e.g., a value of 20% is such that 20% of the aggregates within that bin are plates, and 80% are columns). Note that, unlike monomers, aggregates do not fit cleanly into a plate or column category. Rather than using two lengths across each monomer face for monomer aspect ratio calculations, it is more accurate to consider all three dimensions from an ellipsoid when defining aggregate aspect ratios. The x axes (y axes) in Fig. 2 represents the edges of all 20 consecutive ϕca (ra) bins with an equal number of aggregates in each from the database; recall that the bin edges are determined according to the actual properties of the aggregates formed to populate the database, and are not diagnosed a priori, which explains their nonuniformity. Every rectangle in Figs. 2a–h represents one bin. Notice that the ranges of the bins vary; some ra bins span larger radii values than others to reach the same aggregate count as bins that contain a higher concentration of aggregates all with similar ϕca and ra. While the radii bin edges (y axes) are similar in range for both the random orientation (Fig. 2, left) and quasi-horizontal orientation (Fig. 2, right), the aggregate aspect ratio lower limit (x axis) is an order of magnitude smaller for the quasi-horizontal case when ϕca ≪ 1.0. Quasi-horizontally oriented monomers with extreme aspect ratios (ϕm ≪ 1.0 or ϕm ≫ 1.0) span a wider area in the horizontal plane than the vertical plane due to less thickness, which consistently produces the most extreme aggregate aspect ratios (e.g., Fig. 1, top). For this reason, caution should be used for analysis between orientations as the ranges for ϕca can considerably vary when ϕca ≪ 1.0.

Fig. 2.
Fig. 2.

Database characteristics truncated at 5 mm for (left) randomly oriented monomers and aggregates and (right) quasi-horizontally oriented monomers and aggregates. Aggregate aspect ratio ranges (ϕca) are shown on the x axis and aggregate radii (ra) are shown on the y axis for (a),(b) the average monomer radius, (c),(d) the number of monomers within the aggregate (nm), (e),(f) the majority shape of the monomers within the aggregates, and (g),(h) aggregate complexity.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0179.1

For every monomer radius in IPAS, the major and minor axes scale to maintain a constant volume while the aspect ratio varies between ϕm = 0.01 and 100.0 to achieve the same mass. IPAS is fundamentally designed to initialize monomers with equivalent volumes instead of equivalent major axis lengths, which would create the same probability of attachment along the major axis. By holding monomer volumes constant and by assuming constant densities across particle types, aggregate characteristics are compared from monomers of the same mass, independent of shape. Plates are primarily two-dimensional in nature with a broad basal face, while columns are longer in one dimension; given the same equivalent volume radius (e.g., rm = 1000 μm), the major axis of a column is 4.6 times longer than the major axis of a plate between the most extreme monomer aspect ratios of ϕm = 0.01 and 100.0. This means that for aggregates primarily composed of columns (e.g., extreme ϕca in Figs. 2e,f), fewer monomers are required to reach the same ra values (Figs. 2c,d). Notice that in all locations where database bins primarily consist of columns (blue shading in Figs. 2e,f), especially for the random orientation, nm is lower for the same ra magnitude. Aggregates mostly composed of long needles quickly grow two ellipsoid axes after two monomers attach (Fig. 2d). The most extreme ϕca ranges for the quasi-horizontal orientation are an order of magnitude more extreme than the random orientation: as more needlelike monomers collect with their major dimensions along the horizontal, they can continuously spread in the xy plane. Regions with a higher number of aggregates composed of plates require more monomers to reach the same ra since the particle volume is distributed more evenly in two directions, not controlled by a relatively longer single axis for columns (Fig. 2d).

From Figs. 2a and 2b, the monomer radius, rm (shading), is typically one order of magnitude smaller than the aggregate radius, ra (y axis), and larger monomers result in larger aggregates. Since all monomer aspect ratios are initialized such that rm is constant (equivalent volume independent of shape), there is minimal variation in rm across all ϕca bins (Figs. 2a,b). For random orientations (Fig. 2a), ra is directly proportional to rm and almost exactly a factor of 10rm; more variability is shown in the quasi-horizontal orientation (Fig. 2b) where slightly larger monomers are needed to reach the same ra as in the random orientation. In other words, the quasi-horizontal orientation produces slightly smaller aggregates for the same size monomers compared to the random orientation as ϕca approaches 1.0.

While rm is fairly consistent between orientations, the average number of monomers (nm) that make up the aggregates shows greater variation. For random orientations, Fig. 2c shows more spherical aggregates are generally composed of more monomers; higher nm provides more monomers available to fill in any gaps in the aggregates where monomers are evenly distributed in all directions. Figures 2a and 2b show that within an order of magnitude for ra, rm increases, but also, to reach the largest of sizes within this ra range, nm increases (Figs. 2c,d). This is especially prevalent for abrupt increases in nm where rm and ra have not increased to the next order of magnitude and the aggregates are composed of a relatively higher number of plates. The cyclical trends in nm are an artifact of monomer initializations and database discretization, and is not correlated with any physical mechanism during collection. All trends are smoothed in the random case since there is a wider variety of possible configurations with no constraints. For example, fewer randomly oriented plates need to stack to reach larger ra magnitudes since they can perpendicularly attach basal to prism face to increase aggregate sizes; columns are no longer solely responsible for producing extreme aggregates.

Complexity

Pristine ice crystals grow in heterogeneous conditions from turbulent environments within mixed-phase clouds. Variety in crystal size and shape leads to complex aggregate configurations based on temperature and cloud saturation. Similarly, IPAS simulates aggregate diversity through randomized collection methods despite the attachment of identical monomers within the database. To quantify the fractal appearance of atmospheric ice particles, a parameter that includes multiple aggregate properties, namely, complexity, is defined in Schmitt and Heymsfield (2014). Ice particle complexity can be used to quantify the transition from single to complex particle shapes based on different environmental conditions (Schmitt et al. 2016). IPAS-generated aggregates carry all necessary information to compute complexity following Schmitt and Heymsfield (2014), their Eqs. (1) and (2):
C=(Ac×Ap)P2,
C=10×(0.1C),
where Ac is the area of the circle that 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. An illustration of the collection geometries and associated complexities are presented in Fig. 3 for theoretical aggregates within the database for the random orientation (top) and quasi-horizontal orientation (bottom). As C approaches 1.0 (C′ approaches 0), the perimeter of the aggregate is extensive (many monomers), while the polygonal and circular encompassing area are rather compact. Values fall on a continuous scale but are not exclusive to a particular range; C values from database aggregates range between 0.103 and 0.997 for the quasi-horizontal orientation and 0.087–0.994 for the random orientation. Negative C values, while rare, are possible when C′ is >0.1. In these instances, it is most common for a few platelike monomers (low P) with a quasi-horizontal orientation (large Ap) to collect in a spread-out fashion (large Ac). Platelike crystals lead to lower C values compared to columns given the same initial mass and degree of aspect ratio deviation from ϕm = 1.0; plates have a smaller perimeter and larger area, particularly when oriented horizontally. There is a greater dependence on P than Ap and Ac in Eq. (1), so the highest complexities are evident by a large number of needles that all collect toward their centers (Fig. 3).
Fig. 3.
Fig. 3.

A sampling of IPAS-generated theoretical aggregates in the database with corresponding complexity values for the (top) random orientation and (bottom) quasi-horizontal orientation. Higher values are more complex. View is in the xy plane.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0179.1

Figures 2g and 2h show the average complexity per bin (shaded) within the database for the (i) random orientation and the (ii) quasi-horizontal orientation as a function of aggregate aspect ratio (x axis) and aggregate radius (y axis). A comparison of Figs. 2g and 2h with Figs. 2c–f illustrates that nm and the average monomer shape (ϕm) that comprises the aggregate quickly take precedence in governing C: As nm increases, the aggregate perimeter becomes convoluted and intricate, leading to greater particle complexity despite orientation (Figs. 2g,h,c,d). In addition, if the monomers attach mostly on top of each other (high overlap), P can increase faster than Ac and Ap, especially for aggregates composed of randomly tumbled columns, leading to higher C from a large dependence on P. ϕm illustrates an especially strong effect on C in the quasi-horizontal orientation in Fig. 2h at extreme ϕca because this region is primarily composed of columns (Fig. 2d). While columns are also largely present at extreme ϕca in the random orientation, there are relatively more aggregates composed of plates (larger areas) but in fewer numbers per aggregate (smaller perimeter), which leads to lower C values. The range of C values, on average in Figs. 2g and 2h, is larger for the quasi-horizontal orientation than the random orientation, largely influenced by the number and majority of columns that comprise the aggregates within each bin. An increase in rm does not have a large influence on C since all variables equivalently scale within Eq. (1), which results in constant C given the same ϕm for a single monomer (no collection).

3. Aggregate ellipsoid shape

Complexity is a useful parameter to describe the fractal nature of aggregates, and can also serve to distinguish monomer crystals from aggregates, and can be similarly applied to observations of particles for direct comparison. There are many additional metrics that can be used to not only compare to observations (e.g., aspect ratio), but are also amenable to implementation into bulk microphysical parameterizations. At present, the complex geometries of simulated particles such as those in IPAS cannot be simulated in numerical parameterizations, particularly in bulk models. Instead, 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). As such, ellipsoids fit around IPAS-generated aggregates are categorized into prolate and oblate spheroids for comparison to other modeling studies and for understanding general aggregate characteristics before integrating into bulk models. Figure 4 shows a schematic of an oblate spheroid (a = b > c) and prolate spheroid (a > b = c), which are two specific instances of the ellipsoids (a > b > c) used to encompass IPAS-generated aggregates. Figure 4 (middle panel) divides aspect ratios into prolate and oblate regions with ϕba on the x axis and ϕca on the y axis. When ϕba = 1.0 (y axis, a = b), the ellipsoid is oblate and when ϕba = ϕca (b = c), the ellipsoid is prolate. If the aggregate fit-ellipsoid b-axis length is closer to the a-axis length, the aggregate is categorized as oblate (vertical hatched region, Fig. 4) and if the b-axis length is closer to the c-axis length, the aggregate is categorized as prolate (horizontal hatched region, Fig. 4). Figure 4 is to be used as aid in the interpretation of Fig. 5.

Fig. 4.
Fig. 4.

Schematic of (right) oblate and (left) prolate ellipsoids used to encompass IPAS aggregates. Aggregates are prolate when ϕba = ϕca (diagonal line) and oblate when ϕba = 1.0 (vertical line, right). The region in between is split into aggregates with aspect ratios that would be either closer to prolate (horizontal hatching) or oblate (vertical hatching). This schematic aids the interpretation of Fig. 5.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0179.1

Fig. 5.
Fig. 5.

Evolution of ϕba (x axis) and ϕca (y axis) for (top to bottom) different monomer aspect ratios (ϕm) and (left to right) number of monomers (nm). Each aggregate is formed 300 times every time a new monomer is added for the random orientation (red) and quasi-horizontal orientation (blue) to create a multivariate kernel density estimate (contoured). Distribution modes are shown in red and white dots for each orientation. rm = 10 μm for all monomers that make up the aggregates. Percentage oblate values (out of 300) are marked in the upper-left corner of each subplot for randomly oriented monomers (red) and quasi-horizontally oriented monomers (blue). If a majority of the aggregates are defined as oblate (prolate) spheroids (percentage values > 50%) the subplot is shaded on the right (left) based on the orientation. Purple shading represents a majority prolate aggregates, where both orientations are shaded on top of each other.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0179.1

Identical monomer collection

To compare with other modeling studies and to ensure a foundational understanding of aggregate aspect ratio trends across a range of monomer sizes and shapes, 300 identical monomers were collected for every combination of predefined monomer attributes. Figure 5 shows the resulting aggregate aspect ratio evolution as a function of the number of identical monomers (nm = 2, 3, 10, 50, 100, and 150, left to right) and monomer aspect ratio (ϕm = cm/am = 0.01, 0.10, 0.50, 1.0, 2.0, 10.0 and 100.0, top to bottom) for randomly (red) and quasi-horizontally (blue) oriented monomers and aggregates, for rm = 10 μm. No aggregates are truncated at ra = 5 mm for the remainder of this study, and while 150 monomers are unlikely to collect in nature, the goal is to broadly understand aggregate characteristics upon continuous collection. 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, which is useful for predictions if specific data points are not available (Fig. 5, contours). The primary peak, or mode of each distribution, is marked with white and red dots for the quasi-horizontal and random orientations, respectively. The percentage of IPAS-formed aggregates that are closer to oblate spheroids than prolate spheroids are located in the upper left corner of each subplot, again for the random orientation (red) and quasi-horizontal orientation (blue). For reference, within each grid cell, the layout is identical to the schematic in Fig. 4 for ellipsoid shape interpretation: regions within the diagonals represent prolate (left region) and oblate (right region) aggregates. 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 orientation. Purple shaded regions represent a majority prolate aggregates for both orientations (i.e., red and blue overlap). The random orientation (red) almost always produces a majority of particles that are closer to prolate spheroids than oblate spheroids; thus, there is only one subplot with red shading on the right (Fig. 5, ϕm = 100.0, nm = 2). Randomly oriented particles can quickly create elongated aggregates in the vertical dimension, which increases the likelihood of producing prolate aggregates. When nm = 2 for thin columns, particle shape is largely balanced between oblates and prolates based on whether the columns are parallel or perpendicular to each other, independent of falling orientation.

Plates are defined with an oblate shape; thus, when more quasi-horizontally oriented plates attach, there is a tendency for the encompassing ellipsoid to also take on an oblate shape (Fig. 5, top, blue shading on the right). Extreme plates (ϕm = 0.01) always produce a majority of oblate aggregates up to nm = 150 (top row, Fig. 5). For ϕm = 0.10, by nm = 50, enough monomers are stacked onto the reoriented aggregate that a majority of the 300 reoriented aggregates resemble prolate spheroids. At the other end of extreme monomer aspect ratios, a majority of aggregates composed of quasi-horizontally oriented columns with ϕm = 10.0 are closer to oblate spheroids from nm = 3 to 50 (Fig. 5, bottom, blue shading on the right). Recall that due to predefined equivalent volume initializations, columns have a longer major axis length than plates; the major axis of a column with ϕm = 10.0 is 2.1 times longer than the major axis of a plate with ϕm = 0.10; thus, more monomers are required to stack (nm > 50) to offset the width of the aggregates for a majority of prolate aggregates (Fig. 5, top and bottom, fourth column). Similar to monomers with ϕm = 0.01, thin monomers with ϕm = 100.0 always produce a majority of oblate aggregates through nm = 150 (Fig. 5, top and bottom row). Even though columns are defined as prolates, due to randomized rotation around the z axis, the resulting aggregates take on a variety of broad yet flat shapes, especially after nm = 2. Monomers in the middle of the ϕm spectrum (isometric monomers) are thicker, so subsequent tilting for the maximum projected area in the quasi-horizontal orientation produces a majority of prolate aggregates (shading on the left in the middle rows of Fig. 5).

A wide variety of aggregate aspect ratio distribution shapes are present in Fig. 5 due to variety in monomer attributes (ϕm, nm, and falling orientation). Vertically oriented distributions, as seen for ϕm = 0.01 and ϕm = 0.10 for nm = 2 (Fig. 5, top left, red contours), have a larger range in ellipsoid c-axis lengths (semiminor axis) than b-axis lengths (middle radius). The collection of two plates that are randomly oriented (red) can greatly increase the encompassing aggregate ellipsoid minor axis length since each monomer extends in two directions, and thus, has a low probability of intersecting center to center as two columns might. However, the middle semiaxis (b) has less variation and is driven by the inherent width of the plate. For quasi-horizontally oriented plates that attach basal face to basal face, b (driven by the amount of overlap between plates) varies more than c (minimal stacking), which creates more variability for ϕba and a large difference in modes between orientations for ϕm = 0.10 and nm = 2 (Fig. 5, top left). On the contrary, considering the one-dimensional nature of the prism face of columns, ϕca and ϕba will greatly vary if the columns that comprise the aggregate are oriented parallel versus perpendicular to one another (broad range in ϕba). For nm = 2, regardless of how two extreme columns collect, only two ellipsoid axes can substantially increase, depicted by a small range in ϕca from minimal stacking (bottom left, Fig. 5). Once nm ≥ 3, all three ellipsoid axes can vary for the random orientation, illustrated by the broadened range for ϕca in Fig. 5 (bottom row). In general, there is more variability in aggregate aspect ratio distributions from differing monomer aspect ratios than from the number of monomers within the aggregates, especially for the earliest stages of aggregation (Fig. 5), which is also found in Dunnavan et al. (2019) and Karrer et al. (2020). By nm = 50, there is minimal diversity in distribution shapes for both orientations regardless of ϕm due to the intrinsic size of an aggregate compared to a monomer. This result is critical, and suggests that the resulting shape of the aggregate becomes independent of monomer shape once the number of monomers that comprise the aggregate approaches nm = 50. Figure 6 hones in on the number of monomers per aggregate required to stabilize aggregate aspect ratios, with the best fit line from the modes of each aggregate aspect ratio distribution (i.e., dots in Fig. 5) for ϕca (top) and ϕba (bottom) with respect to the number of monomers that comprise the aggregate (x axis). Each color represents a different monomer aspect ratio (ϕm = 0.01–100.0) (all identical within an aggregate) for the random orientation (dashed lines) and quasi-horizontal orientation (solid lines). Polynomial regression fit coefficients as a function of nm, ϕm, and orientation for ϕca and ϕba can be found in Tables 4 and 5, respectively. By nm ≈ 30, both ϕca (Fig. 6, top) and ϕba (bottom) have nearly asymptoted for all monomer aspect ratios in the random orientation and all monomers toward the middle of the ϕm spectrum for the quasi-horizontal orientation. With the exception of aggregates composed of extreme monomers (e.g., ϕm = 0.01, 10.0 and 100.0), quasi-horizontally oriented monomers (Fig. 6, solid lines) result in more spherical aggregates than randomly oriented monomers (Fig. 6, dashed lines) by nm ≈ 20 for ϕca (Fig. 6, top). Quasi-horizontally oriented monomers always produce more spherical aggregates than randomly oriented monomers for ϕba (Fig. 6, bottom). Quasi-horizontally oriented monomers initially create compact aggregates with limited gaps; new additions are collected surrounding a tightly packed aggregate, which conforms aggregate aspect ratios to be more isometric. On the contrary, randomly oriented monomers collect with nonuniform rotation angles, which quickly disperse monomers toward outer collection sites as monomers chain together within the aggregate. Preferential collection along protruding portions of randomly oriented monomers elongates surface area along an aggregate major axis, leading to a feedback for more collection along the major axis. Randomly oriented monomers tend to create more extreme aggregates as monomers can be placed farther from the center of mass with each new addition given that each aggregate occupies a geometrically broader region.

Fig. 6.
Fig. 6.

Aggregate aspect ratio evolution as a function of the number of monomers that comprise the aggregate (x axis) for (top) ϕca and (bottom) ϕba. Best-fit lines from the modes of each kernel density estimate (e.g., dots in Fig. 5) are plotted for the random orientation (dashed lines) and the quasi-horizontal orientation (solid lines). Monomer aspect ratios (ϕm) are distinguished by different colors. Aspect ratios closer to ϕca(ba) = 0.0 represent more extreme aggregates, and values closer to ϕca(ba) = 1.0 represent more spherical aggregates.

Citation: Journal of the Atmospheric Sciences 79, 6; 10.1175/JAS-D-21-0179.1

Quasi-horizontally oriented extreme columns and plates (ϕm = 0.01 and 100.0) produce the most extreme ϕca values (Fig. 6, top) from minimal particle height or tilting upon collection. Contrarily, the most isometric monomers (and those next closest to isometric—ϕm = 0.5 and 2.0) trend toward the most isometric (spherical) aggregates throughout the entire aggregation process from ϕca calculations for the quasi-horizontal orientation (Fig. 6, top, solid lines). However, there is no clear delineation for which ϕm produces the most extreme (or isometric) aggregates for the random orientation as ϕca and ϕba quickly overlap (Fig. 6, dashed lines). As nm increases, built-in randomizations in IPAS quickly complicate any aspect ratio trends evident between aggregates composed of different monomer shapes. For precise quantitative analysis, Table 3 displays both the mode of each kernel density estimate (out of 300 aggregates) and the mean for ϕca and ϕba after nm = 100.0 for monomer aspect ratios between ϕm = 0.10 and 10.0 (as seen in Figs. 5 and 6) for the random orientation (left) and quasi-horizontal orientation (right). Values on the left of each column represent IPAS aspect ratios, whereas values on the right are discussed in IPAS verification in section 4. Additionally, as a loose reference point to confirm consistency in aggregate ellipsoid shapes and aspect ratios, data from a Multi-Angle Snowflake Camera (MASC) assumed in a random orientation for any given number of monomers is included from Dunnavan et al. (2019) (Table 3, top two rows). Table 3 reiterates what is displayed in Fig. 5 (second to last column) and Fig. 6 for nm = 100 [extreme monomers in IPAS are excluded due to lack of equivalent comparisons in Dunnavan et al. (2019)]: the random orientation produces more extreme aggregate aspect ratios than the quasi-horizontal orientation, there is minimal delineation (outside of the most extreme monomers for the quasi-horizontal orientation) for which monomer aspect ratios result in the most extreme aggregate aspect ratios, and ϕca (ϕba) modal values asymptote toward 0.52 (0.68) on average for randomly oriented aggregates by nm = 100. Hence, it can be surmised from this analysis that monomer aspect ratio is important for computations of vapor diffusional growth and the initial onset of aggregation, and become less important for continuous collection except for extreme aspect ratios when the falling orientation is quasi horizontal, such as that which could be expected for a dendritic growth regime.

Table 3

Modal and mean ϕba and ϕca values from a kernel density estimate of 300 aggregates (separated by a comma, respectively) as a function of identical ϕm (rows) and orientation (random orientation, left and quasi-horizontal orientation, right) for IPAS-generated aggregates. Additionally, mean ϕba and ϕca values are displayed from Monte Carlo simulations in Dunnavan et al. (2019) (D19) for nm = 100 and observed Multi-Angle Snowflake Camera (MASC; top row) imagery taken from Dunnavan et al. (2019) (assumed to have a random orientation and variable number of monomers).

Table 3
Table 4

Fourth-order polynomial regressions fit to the modes of ϕca for each kernel density estimate (Fig. 5) as a function of nm and constant ϕm. The equations follow the lines in Fig. 6 for both the random orientation (top) and quasi-horizontal orientation (bottom).

Table 4
Table 5

Fourth-order polynomial regressions fit to the modes of ϕba for each kernel density estimate (Fig. 5) as a function of nm and constant ϕm. The equations follow the lines in Fig. 6 for both the random orientation (top) and quasi-horizontal orientation (bottom).

Table 5

4. Verification of IPAS

Due to the idealized nature of IPAS, simulated aggregates must be verified to observed ice crystals, which is critical in understanding and identifying any uncertainties that were not considered during methodology formulation. A full analysis with detailed comparisons to microphysical observations is found in a companion paper (Part III, Przybylo et al. 2022). The work presented here focuses on IPAS as an idealized statistical tool, where aggregate sensitivities to ice crystal shape, number, and orientation are explored. IPAS, is not the first of its kind in the theoretical simulation of ice particles and aggregates, and so analysis alongside other peer-reviewed studies is necessary to compare model conclusions, whether they agree or diverge.

Dunnavan et al. (2019) investigates aggregate aspect ratio distributions using a Monte Carlo approach from the collection of spheroidal monomers with varied aspect ratios. Figure 5 is similarly generated to Fig. 5 in Dunnavan et al. (2019) but with extensions for ϕm, nm, and orientation (note the different ordering for each monomer aspect ratio). After nm = 10, all cases have distributions that are consistent with Dunnavan et al. (2019). An overview and comparison of modal and mean quantities from IPAS (truncated) and Dunnavan et al. (2019) is found in Table 3, including relations between ϕm, ϕca, and ϕba. Excluding ϕm = 10.0 due to the elongated nature of columns given an equivalent volume, ϕca (ϕba) varies by no more than 14% (11%) for the random orientation between IPAS and Dunnavan et al. (2019) modes. A comparison to the quasi-horizontal orientation in Dunnavan et al. (2019) is limited by available and equivalent data, but the collection of plates (columns) in IPAS with ϕm = 0.10 (ϕm = 10.0) creates more isometric aggregates by 0.06 (0.02) for ϕca than in Dunnavan et al. (2019) based on the modes of the distributions. Dunnavan et al. (2019) enforces that dipoles or lattice sites only be available when not surrounded by other occupied dipoles, which intrinsically creates more spread between monomers, and could reduce ϕca. Additionally, IPAS monomers are initialized as hexagonal prisms to mimic the stacking of water molecules in layers of hexagonal rings; thus, IPAS-generated aggregates are generally more extreme in both ϕba and ϕca than spheroidal aggregate representations in Dunnavan et al. (2019). Last, the timing in the transition of a distribution of particles from a majority oblate to prolate ellipsoids for quasi-horizontally oriented monomers occurs much faster in Dunnavan et al. (2019), likely due to monomer representation and attachment methodological differences. Note that the intent is not to mimic the results of Dunnavan et al. (2019), especially since differences in fit-ellipsoid algorithms could lead to deviations in aggregate characteristics, but instead, to demonstrate a different approach and perspective in the creation of aggregates based on varied monomer properties. All in all, there is general agreement in final aggregate aspect ratio compared to Dunnavan et al. (2019) for both orientations, all aspect ratios, and after collection of 100 monomers.

Ellipsoid shape comparison

Beyond the above comparison to Dunnavan et al. (2019), there have been diverging conclusions as to the likely spheroidal shape of aggregates. Jiang et al. (2017) found that it is unlikely that aggregates are well characterized as oblate spheroids with the commonly used aspect ratio of 0.6. Furthermore, Jiang et al. (2019) found that aggregates rarely looked like oblate spheroids even for the aggregation of horizontally oriented planar crystals taken from surface-based MASC imagery. Further, Dunnavan et al. (2019) states, “The consistency of the 0.6 aspect ratio value derived from in situ observations and that used in homogeneous oblate spheroid radar approximations should be viewed as strictly coincidental since aggregates are rarely oblate.” In accordance with these statements, IPAS-generated aggregates also prove to be mostly prolate for randomly oriented monomers. Excluding IPAS monomers with ϕm = 0.01 and 100.0 for analysis of the most abundant and realistically observed monomers, quasi-horizontally oriented monomers also exhibit a majority prolate shape (Fig. 5, shading to the left) for ϕm = 0.5, 1.0, and 2.0 for all nm, and for ϕm = 0.1 and 10.0 when nm ≥ 50. This is not by chance; as nm increases, monomers stack more than spread out horizontally. For isometric monomers that largely tilt to maximize area and quickly begin to collect randomly, there is more surface area for attachment along the major axis, increasing the probability of extension along the major axis, favorably leading to prolate aggregates regardless of nm.

Contrarily, extreme theoretical aggregates in IPAS show that a majority of aggregates for the collection of quasi-horizontally plates with ϕm = 0.01 (for all nm) and ϕm = 0.10 (up to nm = 10) are oblate (i.e., early stages of aggregation, Fig. 5, top). The percentage of oblate particles for ϕm = 0.10 quickly decreases as nm increases: when nm = 2, all aggregates are oblate, whereas by nm = 10 there is a more balanced amount of oblate and prolate aggregates formed out of 300 (57% oblate). Similarly, all quasi-horizontally oriented columns with ϕm = 100.0 and most with ϕm = 10.0 produce slightly more oblate aggregates (crossed formations that spread in the horizontal plane), although with less distinction than planar monomers during early aggregation due to the variations between the direction of the one-dimensional prism face of columns. It should be acknowledged that IPAS methods are highly idealized in that most monomers and aggregates will not consistently fall with major axes perpendicular to the flow due to canting, turbulence, electrical forces, and rotational variations around the aggregate center of mass. Moreover, the monomer properties that create oblate-like aggregates are typically more extreme than what is readily observed within large sample volumes of in situ observations, therefore microphysical quantities should not be greatly altered by shape morphology unless there is an abundant prediction of quasi-horizontally collecting plates/dendrites within the dendritic growth zone or stacking needles at temperatures colder than −40°C (Bailey and Hallett 2009).

5. Discussion

There are notable regions for improvement in IPAS, such as adding more crystal habits (e.g., dendrites, bullet rosettes, irregulars) and allowing for the collection of multiple habits within each aggregate in the database, which is relevant for calculating parameters that require analytic, habit-dependent, measurements, such as area, perimeter, or volumetric calculations. Westbrook et al. (2004), Leinonen and Szyrmer (2015), and extensions thereof (e.g., Leinonen et al. 2017; Seifert et al. 2019; Karrer et al. 2020) include a variety of realistic monomer structures such as hexagonal plates, dendrites, stellars, columns, and needles for scattering calculations, some of which include rimed particles. The use of complex particle shapes is advantageous for calculating individual terminal fall speed relationships with ice particle drag coefficients and for prognosing rime fraction as hydrometeors sediment. In IPAS, ellipsoids surround the primary and pristine habits of plates and columns for amenability into bulk models that evolve particle shape via two axis lengths without the use of mass–dimensional or area–dimensional relationships, which can have multiple coefficients for similar ice particle types and assume that all snow particles have the same mass for a given diameter (Sulia et al. 2021). The basis of this study aims to gather geometrical data from idealized aggregates (no riming) for utilization in microphysical models that apply further dynamical processes to the aggregates such as riming, deposition, and temperature-dependent collection.

Monomer sizes, shapes, and orientations are extremely influential in collection geometries regardless of the dynamical system in which these particles reside; however, the intent is not to model intricate particle shapes based on environmental supersaturations with a multitude of complexities. Instead, identical monomer properties are used to create aggregates as if using the characteristic properties within a grid box in a bulk ice-microphysical model. While each database aggregate is composed of identical monomers, sufficient variability is captured across all 9 744 000 aggregates (both orientations) from a collection methodology that applies randomization. Deviations attributable to the collection of varied monomers are not presented herein but have been explored. While remarkable consistencies are evident between aggregate aspect ratio distribution modes from multiple simulations up to 150 monomers (not shown), inherent variability in IPAS, even with identical initializations, cannot be disentangled from fluctuations that would arise from varied monomer collection, especially as ϕm increases toward needlelike monomers (see Przybylo et al. 2019, their Fig. 5). Thus, the applicability and validation of database aggregates with varied monomer types is not quantitatively discussed.

By design, every monomer is created with an equivalent volume, which initializes columns to have a longer major axis length than plates, and subsequently, influences aggregate database properties such as the composition of extreme ϕca bins, to be majority prolate. Initializing all monomers with equivalent major axis lengths instead of volumes, regardless of aspect ratio, eliminates the axis length bias but, instead, issues a new bias: the equivalent volume radius of plates would be 4.6 times larger than columns between ϕm = 0.01 and 100.0. In this case, particle mass would differ across all monomer aspect ratios, refuting the commonplace methodology in bulk models that compute aggregation within a grid box according to the bulk ice characteristics, namely, mass and number quantities, which remain unchanged within a single microphysical time step. IPAS aggregate properties are coupled with a bulk ice-microphysical model via lookup tables for evolution from ice to snow with respect to changes in particle dimensions and densities (see Sulia et al. 2021). For compliance and ease of use with the coupled model, meaning constant indices for all monomer properties within an aggregate (ϕm and rm), each monomer is intentionally defined with an equivalent volume, yielding to the inherent implication on the collection of longer columns than plates at extreme ϕm. A sensitivity test for the collection of monomers with a fixed maximum dimension to 150 monomers (not shown) proved analogous trends in ϕca and ϕba to Fig. 6 even during the earliest stages of aggregation, contrary to Dunnavan et al. (2019).

Particle falling orientation is another region with great ambiguity dependent not only on environmental properties such as background flow direction and magnitude (Jiang et al. 2019) but also on aggregate morphology itself (Pruppacher and Klett 1997). In addition, there is significant orientation uncertainty across large volumes of particles due to instrument complications from self-induced turbulence (shear and deceleration of air around probes), particle size in relation to background motion (Brownian motion), and prevailing wind direction (Garrett et al. 2015; Westbrook and Septhon 2017; Jiang et al. 2017). Garrett et al. (2015) found that the distribution of orientation angles for aggregates at the surface with a maximum dimension ≤ 13 mm had a mode of 13°, suggesting a horizontal preference; however, as aggregate aspect ratios increase, a wider range of possible orientations was apparent. Moreover, Karrer et al. (2020) found that tumbling can significantly affect the properties of aggregates with nm < 10, but has a surprisingly small effect on aspect ratio distributions for aggregates with higher nm. To encompass the assortment of observed orientations and lack of consistency between turbulence in aircraft imaging probes and surface-based imagers with aggregates in free-fall, sensitivity tests on multiple orientations are imperative. In IPAS, there are clear distinctions on aggregate properties between orientations (e.g., aspect ratio distribution shapes), especially for low nm, and these effects persist as nm increases but are not more pronounced as collection continues.

The above description and analysis of IPAS indicates its utility as a statistical tool, generating millions of aggregation events. At this point, IPAS is not meant to mimic a dynamical setting with fluid flow, trace trajectories, or temperature dependent sticking efficiencies. Thus, the attachment of each monomer in IPAS is ensured such that there is 100% collection efficiency. In other words, all initialized particles collect (no “misses”), which is different than other studies that use a collection kernel (e.g., Westbrook et al. 2004; Maruyama and Fujiyoshi 2005; Schmitt and Heymsfield 2010; Brdar and Seifert 2018; Karrer et al. 2020). While a collection efficiency of 100% is not realistic, the surface of aggregates are rougher than pristine crystals owing to multiple monomers, which enhance branch interlocking and surface roughness (Phillips et al. 2015). The purpose of IPAS in this work is not to identify collection probabilities, but rather to perform analysis of aggregate geometrical characteristics following formation. The addition of streamline or turbulent flow around particles adds an unnecessary number of complications to trend analysis with already long run times due to weighty simulation quantities and methodologies; however, current simplicities could influence aggregate property evolution. Many different collection methods were considered for particle attachment in the xy plane, though the flow modeling in Schmitt and Heymsfield (2010) did not show a significant regional preference regarding the location at which the collected crystals would impact the aggregate, even with airflow trajectories acting radially away from the aggregate center. Since particle contacts were relatively uniform over the surface even with the consideration of streamlined airflow, particles in IPAS assume no preference in the xy plane. Given the expandability of IPAS, dynamic and thermodynamic implications on collection efficiency could be considered in a geometrical sense; however, this would require extensive redevelopment of the simulator and beyond the purpose of this work.

When considering attachment sites across particles, many other studies rely on discrete dipole approximation (DDA), which divides a particle into small homogeneous spherical volume elements (which behave as dipoles) (Leinonen et al. 2017; Dunnavan et al. 2019; Jiang et al. 2019). DDA requires appropriate resolution of dipoles across ϕm, which adds another variable for consideration in past studies. IPAS uses a consistent resolution guided by the double-precision floating point polygonal vertices, orients each particle using Euler angles for rotation, and leverages the Python Shapely suite of functions (e.g., intersect, closest point) for systematic attachment of polygonal representations of ice crystals.

6. Conclusions

A database of multimonomer aggregates is created from the collection of a variety of primary monomer shapes and sizes. IPAS provides precise dimensional characteristics from single ice particles to multimonomer aggregates through an established and reproducible collection methodology. Parameters such as aggregate aspect ratios and complexities are calculated for the formation of every aggregate to identify and understand aggregate evolution as a function of monomer orientation and number per aggregate. Sensitivity tests on aggregate formation from varied monomer properties could not be accomplished on this level of detail solely using observations.

Since bulk ice-microphysical models do not track the number of monomers within an aggregate, the methodologies presented herein have been designed to account for generalized statistics on the encompassing properties of the aggregate, such as size and shape evolution, which is important information for dynamic processes such as fall speed calculations, vapor diffusional growth, and riming (Harrington et al. 2013; Jensen et al. 2018). The growth of snow through collection and vapor deposition is important to accurately represent in ice-microphysical models as the inherent size and variety of aggregates with diverse fall speeds largely controls the development of precipitation.

In comparison to other theoretical models, IPAS is consistent in not only predicting aggregate particle shape (i.e., oblate versus prolate spheroids) but also in quantitatively producing similar distributions of aspect ratios, proving that results are reputable. The most frequent aspect ratios from the collection of 100 randomly oriented identical monomers is ≈0.5 (with a slight dependence on monomer habit), calculated from the minor and major axes of an encompassing ellipsoid. Despite the amount of random variability inherent in IPAS collection methodologies, definite trends emerge for aggregate characteristics: a majority of the aspect ratio variability occurs during the early stages of aggregation between monomer aspect ratios, randomly oriented monomers generally produce more extreme aggregate aspect ratios by ≈0.25 for ϕca after 150 monomers have collected (relative to the quasi-horizontal orientation), and aggregate aspect ratio distributions are nearly identical by nm ≈50 for both orientations.

IPAS geometrically calculates idealized aggregate axis lengths based on initial particle aspect ratios and radii, which avoids previous requirements of separate habit-dependent mass–dimensional (power-law) relationships (Heymsfield et al. 2002; Woods et al. 2007), reduced-density spheres, or computationally expensive bin models that prognostically evolve mass from habit specific mD relationships (Hashino and Tripoli 2011a,b). Applying nonlinear vapor growth to statistically established aggregate aspect ratios derived from IPAS ensures better representation of ice mass and size, liquid mass, and possibly cloud lifetime by better constraining aggregate characteristics for ice-microphysical models that predict particle habit. Use of IPAS and bulk ice-microphysical models should lead to new insights in ice microphysics, including better quantitative precipitation forecasts both at cloud and ground level through carefully derived snow characteristics, which will serve as a benchmark to develop and refine bulk parameterizations.

1

Euler angles describe the orientation of a rigid body with respect to a fixed coordinate system.

2

The minimum volume enclosing ellipsoid follows the open-source code by Nima Moshtagh, version 1.2.0.0, at https://www.mathworks.com/matlabcentral/fileexchange/9542-minimum-volume-enclosing-ellipsoid.

3

Primary habits are defined by aspect ratio ϕ = c/a (e.g., oblate or prolate), and secondary habits are defined by density.

Acknowledgments.

All authors 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.

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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  • Phillips, V. T. J., M. Formenton, A. Bansemer, I. Kudzotsa, and B. Lienert, 2015: A parameterization of sticking efficiency for collisions of snow and graupel with ice crystals: Theory and comparison with observations. J. Atmos. Sci., 72, 48854902, https://doi.org/10.1175/JAS-D-14-0096.1.

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  • Przybylo, V. M., K. J. Sulia, C. G. Schmitt, and Z. J. Lebo, 2022: The Ice Particle and Aggregate Simulator (IPAS). Part III: Verification and analysis of ice–aggregate and aggregate–aggregate collection for microphysical parameterization. J. Atmos. Sci., 79, 16511667, https://doi.org/10.1175/JAS-D-21-0180.1.

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  • Schmitt, C. G., K. Sulia, Z. J. Lebo, A. J. Heymsfield, V. Przybylo, and P. Connolly, 2019: The fall speed variability of similarly sized ice particle aggregates. J. Appl. Meteor. Climatol., 58, 17511761, https://doi.org/10.1175/JAMC-D-18-0291.1.

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  • Jiang, Z., J. Verlinde, E. E. Clothiaux, K. Aydin, and C. Schmitt, 2019: Shapes and fall orientations of ice particle aggregates. J. Atmos. Sci., 76, 19031916, https://doi.org/10.1175/JAS-D-18-0251.1.

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  • Karrer, M., A. Seifert, C. Siewert, D. Ori, A. von Lerber, and S. Kneifel, 2020: Ice particle properties inferred from aggregation modelling. J. Adv. Model. Earth Syst., 12, e2020MS002066, https://doi.org/10.1029/2020MS002066.

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  • Korolev, A., and G. Isaac, 2003: Phase transformation of mixed-phase clouds. Quart. J. Roy. Meteor. Soc., 129, 1938, https://doi.org/10.1256/qj.01.203.

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  • Leinonen, J., and W. Szyrmer, 2015: Radar signatures of snowflake riming: A modeling study. Earth Space Sci., 2, 346358, https://doi.org/10.1002/2015EA000102.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Leinonen, J., S. Kneifel, and R. J. Hogan, 2017: Evaluation of the Rayleigh-Gans approximation for microwave scattering by rimed snowflakes. Quart. J. Roy. Meteor. Soc., 144, 7788, https://doi.org/10.1002/qj.3093.

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  • Locatelli, J., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particle. J. Geol. Res., 79, 21852197, https://doi.org/10.1029/JC079i015p02185.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Maruyama, K., and Y. Fujiyoshi, 2005: Monte Carlo simulation of the formation of snowflakes. J. Atmos. Sci., 62, 15291544, https://doi.org/10.1175/JAS3416.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Matrosov, S. Y., 2007: Modeling backscatter properties of snowfall at millimeter wavelengths. J. Atmos. Sci., 64, 17271736, https://doi.org/10.1175/JAS3904.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Matrosov, S. Y., R. F. Reinking, and I. V. Djalalova, 2005: Inferring fall attitudes of pristine dendritic crystals from polarimetric radar data. J. Atmos. Sci., 62, 241250, https://doi.org/10.1175/JAS-3356.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meyers, M. P., R. L. Walko, J. Y. Harrington, and W. R. Cotton, 1997: New RAMS cloud microphysics parameterization. Part II: The two-moment scheme. Atmos. Res., 45, 339, https://doi.org/10.1016/S0169-8095(97)00018-5.

    • Crossref
    • Search Google Scholar
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  • Mitchell, D. L., 1996: Use of mass- and area-dimensional power laws for determining precipitation particle terminal velocities. J. Atmos. Sci., 53, 17101723, https://doi.org/10.1175/1520-0469(1996)053<1710:UOMAAD>2.0.CO;2.

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  • Mitchell, D. L., R. Zhang, and R. L. Pitter, 1990: Mass-dimensional relationships for ice particles and the influence of riming on snowfall rates. J. Atmos. Sci., 29, 153163, https://doi.org/10.1175/1520-0450(1990)029<0153:MDRFIP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Morrison, H., and J. A. Milbrandt, 2015: Parameterization of cloud microphysics based on the prediction of bulk ice particle properties. Part I: Scheme description and idealized tests. J. Atmos. Sci., 72, 287311, https://doi.org/10.1175/JAS-D-14-0065.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., J. A. Milbrandt, G. H. Bryan, S. A. Tessendorf, G. Thompson, and K. Ikeda, 2015: Parameterization of cloud microphysics based on the prediction of bulk ice particle properties. Part II: Case study comparisons with observations and other schemes. J. Atmos. Sci., 72, 312339, https://doi.org/10.1175/JAS-D-14-0066.1.

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  • Noel, V., and K. Sassen, 1983: Study of planar ice crystal orientations in ice clouds from scanning polarization lidar observations. J. Appl. Meteor., 22, 653664, https://doi.org/10.1175/JAM2223.1.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., M. Formenton, A. Bansemer, I. Kudzotsa, and B. Lienert, 2015: A parameterization of sticking efficiency for collisions of snow and graupel with ice crystals: Theory and comparison with observations. J. Atmos. Sci., 72, 48854902, https://doi.org/10.1175/JAS-D-14-0096.1.

    • Crossref
    • Search Google Scholar
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  • Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds and Precipitation. Kluwer Academic, 954 pp.

  • Przybylo, V. M., K. J. Sulia, C. G. Schmitt, Z. J. Lebo, and W. C. May, 2019: The Ice Particle and Aggregate Simulator (IPAS). Part I: Extracting dimensional properties of ice–ice aggregates for microphysical parameterization. J. Atmos. Sci., 76, 16611676, https://doi.org/10.1175/JAS-D-18-0187.1.

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  • Przybylo, V. M., K. J. Sulia, C. G. Schmitt, Z. J. Lebo, and W. C. May, 2021: IPAS repository. GitHub, https://github.com/vprzybylo/IPAS.

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  • Przybylo, V. M., K. J. Sulia, C. G. Schmitt, and Z. J. Lebo, 2022: The Ice Particle and Aggregate Simulator (IPAS). Part III: Verification and analysis of ice–aggregate and aggregate–aggregate collection for microphysical parameterization. J. Atmos. Sci., 79, 16511667, https://doi.org/10.1175/JAS-D-21-0180.1.

    • Search Google Scholar
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  • Schmitt, C. G., and A. J. Heymsfield, 2010: The dimensional characteristics of ice crystal aggregates from fractal geometry. J. Atmos. Sci., 67, 16051616, https://doi.org/10.1175/2009JAS3187.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schmitt, C. G., and A. J. Heymsfield, 2014: Observational quantification of the separation of simple and complex atmospheric ice particles. Geophys. Res. Lett., 41, 13011307, https://doi.org/10.1002/2013GL058781.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schmitt, C. G., A. J. Heymsfield, P. Connolly, E. Jarvinen, and M. Schnaiter, 2016: A global view of atmospheric ice particle complexity. J. Geol. Res., 43, 11 91311 920, https://doi.org/10.1002/2016GL071267.

    • Search Google Scholar
    • Export Citation
  • Schmitt, C. G., K. Sulia, Z. J. Lebo, A. J. Heymsfield, V. Przybylo, and P. Connolly, 2019: The fall speed variability of similarly sized ice particle aggregates. J. Appl. Meteor. Climatol., 58, 17511761, https://doi.org/10.1175/JAMC-D-18-0291.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Seifert, A., J. Leinonen, C. Siewert, and S. Kneifel, 2019: The geometry of rimed aggregate snowflakes: A modeling study. J. Adv. Model. Earth Syst., 11, 712731, https://doi.org/10.1029/2018MS001519.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Straka, J. M., and E. M. Mansell, 2005: A bulk microphysics parameterization with multiple ice precipitation categories. J. Atmos. Sci., 44, 445466, https://doi.org/10.1175/JAM2211.1.

    • Search Google Scholar
    • Export Citation
  • Sulia, K. J., and J. Y. Harrington, 2011: Ice aspect ratio influences on mixed-phase clouds: Impacts on phase partitioning in parcel models. J. Geophys. Res., 116, D21309, https://doi.org/10.1029/2011JD016298.

    • Search Google Scholar
    • Export Citation
  • Sulia, K. J., J. Y. Harrington, and H. Morrison, 2013: A method for adaptive habit prediction in bulk microphysical models. Part III: Applications and studies within a two-dimensional kinematic model. J. Atmos. Sci., 70, 33023320, https://doi.org/10.1175/JAS-D-12-0316.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sulia, K. J., Z. J. Lebo, V. Przybylo, and C. G. Schmitt, 2021: A new method for ice–ice aggregation in the adaptive habit model. J. Atmos. Sci., 78, 133154, https://doi.org/10.1175/JAS-D-20-0020.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, G., P. Field, R. Rasmussen, and W. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. J. Atmos. Sci., 136, 50955115, https://doi.org/10.1175/2008MWR2387.1.

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  • Fig. 1.

    Theoretical aggregates from the collection of identical monomers with both the monomers and resulting aggregates quasi-horizontally oriented. Corresponding ϕm values for the aggregate are listed above the aggregates for (left) ϕm < 1.0 and (right) ϕm ≥ 1.0 from (top) most extreme to (bottom) least extreme.

  • Fig. 2.

    Database characteristics truncated at 5 mm for (left) randomly oriented monomers and aggregates and (right) quasi-horizontally oriented monomers and aggregates. Aggregate aspect ratio ranges (ϕca) are shown on the x axis and aggregate radii (ra) are shown on the y axis for (a),(b) the average monomer radius, (c),(d) the number of monomers within the aggregate (nm), (e),(f) the majority shape of the monomers within the aggregates, and (g),(h) aggregate complexity.

  • Fig. 3.

    A sampling of IPAS-generated theoretical aggregates in the database with corresponding complexity values for the (top) random orientation and (bottom) quasi-horizontal orientation. Higher values are more complex. View is in the xy plane.

  • Fig. 4.

    Schematic of (right) oblate and (left) prolate ellipsoids used to encompass IPAS aggregates. Aggregates are prolate when ϕba = ϕca (diagonal line) and oblate when ϕba = 1.0 (vertical line, right). The region in between is split into aggregates with aspect ratios that would be either closer to prolate (horizontal hatching) or oblate (vertical hatching). This schematic aids the interpretation of Fig. 5.

  • Fig. 5.

    Evolution of ϕba (x axis) and ϕca (y axis) for (top to bottom) different monomer aspect ratios (ϕm) and (left to right) number of monomers (nm). Each aggregate is formed 300 times every time a new monomer is added for the random orientation (red) and quasi-horizontal orientation (blue) to create a multivariate kernel density estimate (contoured). Distribution modes are shown in red and white dots for each orientation. rm = 10 μm for all monomers that make up the aggregates. Percentage oblate values (out of 300) are marked in the upper-left corner of each subplot for randomly oriented monomers (red) and quasi-horizontally oriented monomers (blue). If a majority of the aggregates are defined as oblate (prolate) spheroids (percentage values > 50%) the subplot is shaded on the right (left) based on the orientation. Purple shading represents a majority prolate aggregates, where both orientations are shaded on top of each other.

  • Fig. 6.

    Aggregate aspect ratio evolution as a function of the number of monomers that comprise the aggregate (x axis) for (top) ϕca and (bottom) ϕba. Best-fit lines from the modes of each kernel density estimate (e.g., dots in Fig. 5) are plotted for the random orientation (dashed lines) and the quasi-horizontal orientation (solid lines). Monomer aspect ratios (ϕm) are distinguished by different colors. Aspect ratios closer to ϕca(ba) = 0.0 represent more extreme aggregates, and values closer to ϕca(ba) = 1.0 represent more spherical aggregates.

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