1. Introduction
Radar measurements that contain polarization information have shown substantial value in better characterizing ice particle properties in the atmosphere. In particular, these measurements provide particle shape information; based on our best theoretical models for shape evolution during particular ice growth processes, this information provides insights into the relative contributions of the growth processes to the observed precipitation. However, the degree of confidence one can have in these insights critically depends on the uncertainty in how ice particle physical and scattering properties evolve during growth.
One important application of polarimetric radar measurements where these uncertainties are evident is the early aggregation of branched planar crystals. As these particles grow from vapor deposition, aggregation occurs when the particles collide during free fall. How the shapes of the early aggregates evolve after the first few collection events is uncertain. For example, Moisseev et al. (2015) hypothesize that the early aggregates are relatively flat with low aspect ratios. These particles may then produce the observed signature of increases in differential reflectivity (Z_{DR}), specific differential phase (K_{dp}), and reflectivity at horizontal polarization (Z_{H}). However, other researchers hypothesize that aggregation quickly results in more randomly shaped, sparser particles that have nearzero Z_{DR} and produce minimal contributions to K_{dp}, with the observed polarimetric enhancement attributed mostly to the coexisting pristine ice crystals (e.g., Kennedy and Rutledge 2011; Andrić et al. 2013; Schrom et al. 2015).
Additionally, the orientation behavior of aggregates is important in both understanding the polarimetric radar measurements and modeling ice microphysical processes. In the case of the former, the radiative properties of an ice particle population depend on the distribution of particle orientations in the sensing volume. With respect to ice growth processes, orientation behavior can impact both the collision and collection efficiency of the particles. The evolution of particle shape through both aggregation and riming will also depend on the orientation of the ice particles during collection. Owing to observed positive Z_{DR} and K_{dp} in many cases of ice particle growth (e.g., Ryzhkov and Zrnić 1998; Kennedy and Rutledge 2011; Matrosov et al. 2017) and simplified hydrodynamic considerations, ice particle orientation distributions are commonly assumed to have mean values corresponding to horizontal orientation (i.e., the angle of the longest axis of the particle with respect to Earth’s surface is 0°) and varying degrees of dispersion about the mean.
The lack of detailed aggregation simulations coupled to detailed simulations of scattering properties has impeded efforts to understand the polarimetric radar signatures of aggregates. Some progress has been made in this area by using idealized shapes to approximate real ice particles; scattering calculations of these idealized shapes (typically spheres and/or spheroids) are highly efficient using the Tmatrix method (Waterman 1969) or Rayleigh theory (Rayleigh 1897). However, the resulting scattering properties may poorly approximate the true scattering properties of natural ice particles (e.g., Botta et al. 2013; Schrom and Kumjian 2018), despite these simplified shapes correctly capturing the bulk physical properties of the particles (e.g., maximum dimension, aspect ratio, effective density, and mass). Prior studies have carried out quasiphysical Monte Carlo simulations of aggregation (e.g., Westbrook et al. 2006; Schmitt and Heymsfield 2010; Botta et al. 2010; Kuo et al. 2016; Eriksson et al. 2018), and the interaction of microwave radiation with ice particles is well understood from Maxwell’s equations. However, simulating the polarimetric radar signatures of detailedshaped aggregates is computationally demanding because of the high spatial resolution needed to capture the particle shapes and the need to perform independent calculations for many particle orientations.^{1}
Some limited studies examining the polarimetric scattering properties of aggregates have been performed. Tyynelä et al. (2011) conduct scattering calculations for aggregates assumed to have fixed horizontal orientations, and their resulting linear depolarization ratios found to be larger than that of comparable spheroids. Subsequently, Tyynelä and Chandrasekar (2014) perform scattering calculations for aggregates with assumed random orientations, and find that both circular depolarization ratio and Z_{DR} are reduced relative to that for pristine particles. Munchak et al. (2022) shows that detailed scattering calculations for a single aggregate of plates produce physically reasonable retrievals of bulk ice properties, with some potential to estimate the degree of horizontal alignment of the aggregate. However, more rigorous analyses of the impacts of particle flutter on these signatures, as well as the natural variability of scattering properties for similar aggregates have yet to be performed.
The main goal of this study is therefore to explore the variability in polarimetric scattering properties of aggregates given certain bulk particle properties and evaluate the potential to efficiently predict their polarimetric scattering properties from these bulk properties. We approach this problem by first simulating aggregation with varying assumptions about the monomer properties, attachment behavior, and their assumed orientation distributions. We use ensembles of these randomly generated aggregates to compute scattering properties for a variety of orientation angles using the Amsterdam discrete dipole approximation (ADDA; Yurkin and Hoekstra 2011) code. The resulting sets of scattering properties are then linked to the physical properties by conducting a spectral analysis of the scattering properties by transforming the scattering calculations from orientation angle space to Legendrecoefficient space. We then find the dominant coefficients for each scattering property and relate them to the physical properties of the particles.
2. Aggregation simulations
a. Description of aggregation procedure
Several prior studies have generated aggregates for the purposes of understanding how physical properties of these particles evolve and for populating scattering databases of ice particles (e.g., Westbrook et al. 2006; Schmitt and Heymsfield 2010; Xie et al. 2011; Botta et al. 2013; Leinonen et al. 2013; Nowell et al. 2013; Kuo et al. 2016; Brath et al. 2020). The common strategy for generating these particles is to select monomers (i.e., single pristine ice crystals) and randomly attach them to a growing aggregate or an initial monomer at the onset of aggregation. The attachment points of the monomer with the aggregate are then calculated as the closest nonintersecting point between an aggregate and a monomer along some direction. Generally, the relative angles between the monomer and the aggregate are randomly selected, as is the direction that one particle moves toward the other.
Assuming purely random orientations of monomers and aggregates when generating these synthetic aggregates will bias the physical properties and the corresponding scattering properties, especially given the evidence of preferred horizontal orientation from polarimetric radar measurements of ice precipitation. Additionally, we hypothesize that monomer motions that occur after impact such as pivoting and slipping will alter the aggregate structure. Owing to the complexity of these motions, they have not been considered in prior simulations of aggregation, and we will focus only on the effect of monomer pivoting during aggregation. To simulate pivoting (illustrated in Fig. 1), we assume that the particle pivots about its initial attachment point p_{1} and that p_{1} is the closest distance between the monomer and the aggregate along the specified attachment direction (assumed to be along the negative z axis; Fig. 1a). Once p_{1} is found, the monomer will then rotate about this point with a rotation axis r_{1} defined by the cross product of the vector from p_{1} to the monomer center of mass and the negative z axis (Fig. 1b). This rotation continues until a second attachment point p_{2} is made between the monomer and aggregate. A second pivoting then occurs about the rotation axis r_{2} = p_{2} − p_{1} (Fig. 1c). As with the first pivot, this rotation continues until a third attachment point between the monomer and aggregate is found, and the monomer is then incorporated into the aggregate.
We use the Euler angles α, β, and γ to specify particle orientation herein; an illustration of how these angles define the particle orientation is shown in Fig. 2. The orientation of the aggregate during collection will depend on the coupling between the surrounding air motion and the particle, requiring intensive fluid dynamic calculations to accurately model this system. Additionally, observations of aggregate orientation in natural clouds are limited. Owing to this lack of knowledge, we use simplified assumptions about the aggregate orientation behavior to better understand how this element of aggregation impacts the evolution of the bulk physical properties. We first assume that the two largest principal axes of an aggregate’s inertia momentum tensor are oriented horizontally (i.e., along Earth’s surface). We then rotate the aggregate randomly prior to each monomer collection event under two different assumptions:

uniform random distributions of α, cosβ, and γ [referred to herein as fullEuler random (ER)] and

uniform random distributions of α, with β = γ = 0 [referred to herein as azimuthally random (AR)].
A depiction of the Euler angles that define particle orientation relative to the base reference frame of the x, y, and z axes shown in black. The first rotation angle α is applied about the z axis, resulting in axes x′ and y′ shown with the red dashed lines. The second rotation angle β is applied about the y′ axis and results in a z′ axis shown with the purple dashed lines. The third rotation angle γ is then applied about the z′ axis.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
A depiction of the Euler angles that define particle orientation relative to the base reference frame of the x, y, and z axes shown in black. The first rotation angle α is applied about the z axis, resulting in axes x′ and y′ shown with the red dashed lines. The second rotation angle β is applied about the y′ axis and results in a z′ axis shown with the purple dashed lines. The third rotation angle γ is then applied about the z′ axis.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
A depiction of the Euler angles that define particle orientation relative to the base reference frame of the x, y, and z axes shown in black. The first rotation angle α is applied about the z axis, resulting in axes x′ and y′ shown with the red dashed lines. The second rotation angle β is applied about the y′ axis and results in a z′ axis shown with the purple dashed lines. The third rotation angle γ is then applied about the z′ axis.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
The AR assumption has been observed in laboratory studies of aggregates in free fall (e.g., Westbrook and Sephton 2017; McCorquodale and Westbrook 2021); the ER assumption provides a useful baseline to evaluate the impact of preferred orientations on the aggregate properties. For each of these aggregate orientation assumptions, we perform two sets of aggregation simulations: one set with monomer pivoting and one set without monomer pivoting. In the case of no pivoting, the monomers are simply incorporated into the structure upon their first contact with the aggregate. Additionally, we perform separate aggregation simulations for branched planar crystal monomers and for solid hexagonal column monomers, since aggregates of both types of these pristine habits exist in nature. We also aim to test how their physical and scattering properties differ. For the branched planar crystal aggregates, we focus on the early formation stage of these aggregates and generate particles sequentially from two to six monomers. The complex nature of the branched planar crystal shapes makes the aggregation simulations relatively inefficient. The relative simplicity of columns allows us to aggregate a much greater number of them efficiently, and therefore, we construct each column aggregate by collecting a random number of column monomers from 10 to 100.
Description of the assumptions and properties used to simulate the different sets of aggregates. The abbreviations used in the aggregate experiment names are defined as follows: ER corresponds to fullEulerrandom orientations, AR corresponds to azimuthally random orientations, BPA corresponds to branched planar crystal aggregates, CA corresponds to column aggregates, NP corresponds to no pivoting, and P corresponds to pivoting. The aggregation monomer increment refers to whether an aggregate is saved sequentially after each monomer collection event (as in the case of branched planar crystals) or an aggregate is saved after a random number of monomers are collected.
b. Simulated physical properties
Randomly selected examples of the simulated aggregates are plotted in Fig. 3. For each synthetic aggregation experiment, there is a fair amount of variability in the structure of the resulting ice particles. Generally, the ER nonpivoting branched planar aggregates have more chaotically oriented monomers than the ER pivoting, AR pivoting, and AR nonpivoting branched planar aggregates. Both sets of column aggregates have more sparse and more spherical shapes than the branched planar aggregates.
Plots of five randomly selected synthetic aggregates (i.e., each column of the figure is a different realization) from each aggregation experiment as described in Table 1. Each branched planar crystal aggregate has six monomers in these plots; the number of monomers for the column aggregates are random. Note that some of the monomers within individual aggregates are hidden because of the plot perspective (e.g., the ERBPAP aggregate farthest to the right).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Plots of five randomly selected synthetic aggregates (i.e., each column of the figure is a different realization) from each aggregation experiment as described in Table 1. Each branched planar crystal aggregate has six monomers in these plots; the number of monomers for the column aggregates are random. Note that some of the monomers within individual aggregates are hidden because of the plot perspective (e.g., the ERBPAP aggregate farthest to the right).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Plots of five randomly selected synthetic aggregates (i.e., each column of the figure is a different realization) from each aggregation experiment as described in Table 1. Each branched planar crystal aggregate has six monomers in these plots; the number of monomers for the column aggregates are random. Note that some of the monomers within individual aggregates are hidden because of the plot perspective (e.g., the ERBPAP aggregate farthest to the right).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Aside from mass, many particle physical properties have arbitrary definitions. As such, we define the physical properties in ways that are most relevant to the distribution of mass of the particle, since the mass distribution is most closely tied to the nearfield interactions of dipoles within a particle that generate the observed polarimetric signals (e.g., Lu et al. 2014). We use the convex hull of points associated with an aggregate to calculate the particle volume and maximum dimension; the latter is the maximum distance between two convex hull points. We then determine the effective density of the particle as the ratio of the particle mass to the convex hull volume. This method for calculating effective density will produce higher values compared to using circumscribing idealized shapes such as ellipsoids or spheres, since the convex hull more tightly encompasses the particle. However, this estimate of density is a more direct measure of the relative sparseness of the aggregate internal structure.
Side and top view perspectives of an aggregate and a corresponding ellipsoid. The dimensions of the ellipsoid correspond to the characteristic dimensions of the particle (a, b, and c).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Side and top view perspectives of an aggregate and a corresponding ellipsoid. The dimensions of the ellipsoid correspond to the characteristic dimensions of the particle (a, b, and c).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Side and top view perspectives of an aggregate and a corresponding ellipsoid. The dimensions of the ellipsoid correspond to the characteristic dimensions of the particle (a, b, and c).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
The simulated particles have mass and maximum dimension values that generally fall within empirical relations derived from in situ measurements (e.g., Locatelli and Hobbs 1974; Mitchell et al. 1990) shown in Fig. 5a, with a range in mass up to 0.5 mg at a given maximum dimension. The measurements of maximum dimension from these studies have uncertainties associated with the impact of the particle with the collection surface, measurements taken at a single viewing angle, and the small sample sizes, limiting our ability to rigorously compare them to the aggregates generated herein. The twocolumn aggregate sets reach larger sizes and masses compared to the branched planar crystal aggregates. The nonpivoting ER branched planar crystals generally have the lowest mass relative to maximum dimension.
Scatterplots of (a) mass vs maximum dimension, (b) effective density vs mass, and (c) the two fitted ellipsoid aspect ratios for the synthetic aggregates described herein. The contours in (a) encompass 75% of each aggregate set; in (b) and (c), the dashed and solid contours encompass 75% and 25% of the particles, respectively. The empirical relations in (a) labeled SP Agg. and Col. Agg. correspond to aggregates of side planes and aggregates of columns, respectively.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Scatterplots of (a) mass vs maximum dimension, (b) effective density vs mass, and (c) the two fitted ellipsoid aspect ratios for the synthetic aggregates described herein. The contours in (a) encompass 75% of each aggregate set; in (b) and (c), the dashed and solid contours encompass 75% and 25% of the particles, respectively. The empirical relations in (a) labeled SP Agg. and Col. Agg. correspond to aggregates of side planes and aggregates of columns, respectively.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Scatterplots of (a) mass vs maximum dimension, (b) effective density vs mass, and (c) the two fitted ellipsoid aspect ratios for the synthetic aggregates described herein. The contours in (a) encompass 75% of each aggregate set; in (b) and (c), the dashed and solid contours encompass 75% and 25% of the particles, respectively. The empirical relations in (a) labeled SP Agg. and Col. Agg. correspond to aggregates of side planes and aggregates of columns, respectively.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
There are more substantial differences between the aggregation sets with respect to effective density. Both column aggregate sets have low densities, generally <50 kg m^{−3} (Fig. 5b). The branched planar crystal aggregates have larger effective densities relative to the column aggregates, and show a general decrease in effective density with respect to mass. On average, the AR branched planar aggregates with pivoting have the highest densities with ∼25% of the values between 125 and 275 kg m^{−3}. Of the branched planar crystal aggregates, the ER branched planar aggregate without pivoting set has the lowest densities on average with values mostly <100 kg m^{−3} (Fig. 5b).
There are also noticeable differences in the relations between the two aspect ratios ϕ_{1} and ϕ_{2} among the aggregate sets (Fig. 5c). The column aggregate sets have ϕ_{1} and ϕ_{2} values that are generally closer to 1 than the branched planar aggregates, indicating that these particles are more spherical. The ER nonpivoting branched planar aggregates have ϕ_{1} values relatively close to ϕ_{2} values, indicating that these particles are closer to prolate shapes than oblate shapes. These particles occupy a similar location in dualaspectratio space as the retrieved data from Jiang et al. (2019). The aggregate set with the lowest aspect ratios is the AR pivoting branched planar aggregates, with ϕ_{2} values mostly between 0.1 and 0.4. Values of ϕ_{1} for these particles are >0.5, indicating shapes that are more similar to oblate spheroids than prolate spheroids.
3. Scattering calculations
a. Simplified assumptions for the particle orientation distribution
Particle fall behavior is typically parameterized in terms of orientation angle distributions (e.g., Ryzhkov et al. 2011), and we use the Euler angles herein to describe such distributions. Observational evidence as well as laboratory studies with idealized analog particles (e.g., Westbrook and Sephton 2017) suggest that aggregates have a tendency to rotate about a vertical axis (i.e., rotations of α). However, there is still a great deal of uncertainty regarding the conditions where this mode of rotation dominates the orientation behavior during free fall. Unsteady oscillations for particles with large Reynolds numbers may lead to more tumbling over chaotic rotation axes (McCorquodale and Westbrook 2021). Additionally, turbulent air motion in the atmosphere is also likely to perturb the rotation axis. We make the simplifying assumption that α and γ are uniformly distributed. This assumption allows us to characterize the particle orientation distribution as a function of β, the angle of tilt of the particle from its z axis in the base or laboratory reference frame (Fig. 2).
Beta distributions for
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Beta distributions for
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Beta distributions for
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
b. Spectral representation of the radar moment orientation variability
Legendre polynomials up to order l = 6.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Legendre polynomials up to order l = 6.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Legendre polynomials up to order l = 6.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Orientation distribution weights for the Legendre series coefficients as functions of b. The weights are normalized by a factor of
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Orientation distribution weights for the Legendre series coefficients as functions of b. The weights are normalized by a factor of
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Orientation distribution weights for the Legendre series coefficients as functions of b. The weights are normalized by a factor of
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Despite the simplified forms of the radar moments for uniformly distributed α and γ, calculations for completely asymmetric particle shapes must still be conducted on a set of unique values for all three Euler angles. The chosen set of Euler angle triplets determines the maximum number of spectral coefficients that can be resolved, as shown in detail by McEwen et al. (2015).
c. ADDA simulation configuration
We use a general configuration for our scattering simulations that is the same for each synthetic aggregate set. This configuration includes the particle orientation angles, the frequencies of the radiation, the scattering angles, and the direction of incident radiation (set fixed along the x axis of the base reference frame). The particle orientation angles are set according to Gauss–Legendre quadrature nodes for α and β (as in Wieczorek and Meschede 2018), and seven equally spaced nodes between 0° and 360° are used for γ. This angular grid allows for relatively efficient projection onto spherical harmonics using the Python SHTOOLS wrapper (Wieczorek and Meschede 2018), though more efficient quadrature methods are known for the spherical nodes (e.g., Lebedev quadrature). Our choice of orientation angles corresponds to representations of spherical harmonics to order 6. We use the Python SHTOOLs wrapper to calculate the C^{l} values from the calculations performed on this set of orientation angles for each particle and radar moment. The onedimensional orientation angle grids for α, β, and γ that form the three dimension grid of orientation nodes (resulting in 588 independent calculations) are listed in Table 2. We perform the calculations at Ku and Ka band; the precise wavelengths are given in Table 2.
Specifications for the ADDA scattering calculations used herein.
The additional necessary input to ADDA are dimensionless indices for the x, y, and z locations of the dipoles defining the aggregate shape. The length of the dipole is then needed to incorporate the correct physical size into the calculations. We use 15.2 μm for the dipole size to be consistent with the spacing of points used in generating the aggregation simulations and to satisfy the conditions where DDA is valid.
4. Results
a. Analysis of the scattering moment coefficients
The mean spectra for each scattering property over each particle set are shown in Fig. 9. In general, the spectra have peaks in the even coefficients and troughs in the odd coefficients indicating that the symmetric Legendre polynomials relative to β = 90° dominate the scattering behavior with respect to orientation (Fig. 7). As such, we only discuss the even coefficients herein. This symmetry is a consequence of averaging the scattering properties over α and γ, effectively making the particles appear rotationally symmetric. For the horizontal and vertical power coefficients, the zerothorder coefficients are dominant, followed by the secondorder and fourthorder coefficients (Fig. 9). For the K_{dp} coefficients, the secondorder coefficient is dominant, with the zerothorder and fourthorder coefficients substantially (>70 dB) lower in magnitude. The
Power spectra for (a) C_{hh} at Ku band, (b) C_{hh} at Ka band, (c) C_{vv} at Ku band, (d) C_{vv} at Ka band, (e) C_{dp} at Ku band, (f) C_{dp} at Ka band, (g) C_{re} at Ku band, (h) C_{re} at Ka band, (i) C_{im} at Ku band, and (j) C_{im} at Ka band. The colors indicate the aggregation experiment set and the lines are the mean spectra for each aggregation set. The dashed black lines are provided as a visual aid for the relative magnitude of the spectral coefficients.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Power spectra for (a) C_{hh} at Ku band, (b) C_{hh} at Ka band, (c) C_{vv} at Ku band, (d) C_{vv} at Ka band, (e) C_{dp} at Ku band, (f) C_{dp} at Ka band, (g) C_{re} at Ku band, (h) C_{re} at Ka band, (i) C_{im} at Ku band, and (j) C_{im} at Ka band. The colors indicate the aggregation experiment set and the lines are the mean spectra for each aggregation set. The dashed black lines are provided as a visual aid for the relative magnitude of the spectral coefficients.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Power spectra for (a) C_{hh} at Ku band, (b) C_{hh} at Ka band, (c) C_{vv} at Ku band, (d) C_{vv} at Ka band, (e) C_{dp} at Ku band, (f) C_{dp} at Ka band, (g) C_{re} at Ku band, (h) C_{re} at Ka band, (i) C_{im} at Ku band, and (j) C_{im} at Ka band. The colors indicate the aggregation experiment set and the lines are the mean spectra for each aggregation set. The dashed black lines are provided as a visual aid for the relative magnitude of the spectral coefficients.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Generally, the behavior of these coefficient spectra are similar between Ku and Ka bands, with the fourthorder and sixthorder coefficients decreasing slower with order at Ka band compared to Ku band. The increasing contribution of the higherorder coefficients at Ka band results from increased nonRayleigh scattering behavior. For the
b. Relations between scattering properties and physical properties
The dependence of scattering properties on the exact shape of ice particles leads to a fundamental uncertainty in the relations between scattering properties and a set of physical properties (e.g., Schrom and Kumjian 2018), and a corresponding uncertainty in retrieving physical properties from remote sensing measurements. However, the scattering calculations we present show that the scattering property coefficients have clear relations to the physical properties, especially mass and effective density. Since
Figure 10 shows the zerothorder and secondorder coefficients for the horizontal and vertical powers at Ku and Ka band.
Scatterplots of (a)
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
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Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
At Ka band, there is more variability in the scattered power at horizontal and vertical polarization. The
At both Ku and Ka band, the K_{dp} spectral coefficients depend on both mass (Fig. 11) and density, with the latter implied by the density variation between aggregate sets (Fig. 5b). As shown in Fig. 9, the contribution of
Scatterplots of (a)
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Scatterplots of (a)
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The two components of the covariance scaled by
Scatterplots of (a)
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
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Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
The scattering properties of the Monte Carlo aggregates generated herein show minimal dependence on the particle aspect ratio, similar to the results of Ekelund and Eriksson (2019). One reason aspect ratio is less important is that it may be erroneously influenced by elements of the ice structure that have limited impact on the scattering properties. In particular, aggregates composed of only a few monomers have highly nonuniform distributions of mass within the bulk shape (e.g., convex hull, ellipsoidal fits) implied by the aspect ratios, and the aspect ratio provides negligible information about the monomer orientations within the aggregate or the monomer aspect ratios. For aggregates with a larger number of monomers (e.g., the column aggregates generated herein), the mass distribution becomes more homogeneous. However, these particles generally have relatively small polarimetric enhancements (cf. Figs. 10–12), making it difficult to assess whether the bulk aspect ratios are important for the scattering properties of aggregates with larger numbers of monomers.
The simulated polarimetric radar observables averaged over each aggregation set have varying dependencies on the assumed orientation dispersion parameter (b). We show these relations in Figs. 13 and 14, where we first calculate the orientationaveraged polarimetric radar variables over a range of individual b values for each aggregate in the set, and then average the polarimetric radar variables over each set. At Ku band, the average Z_{DR} and K_{dp} per mass have nearly identical relative increases with increasing b, with the highest Z_{DR} values of 2.5–3.5 dB at b = 100 found for both sets of branched planar aggregates with pivoting and the set of AR branched planar aggregates without pivoting (Fig. 13). The ER and nonpivoting branched planar aggregates have smaller Z_{DR} values at b = 100 of ∼1 dB that are similar to the Z_{DR} of the AR column aggregates. The ER column aggregates have the smallest maximum Z_{DR} values with Z_{DR} at b = 100 of <0.5 dB (Fig. 13). Values of the mean orientationaveraged ρ_{hv} minimize near b = 4 with values between 1 and 0.97, increases to >0.995 at b = 100. As expected, the aggregate sets with the smallest ρ_{hv} minima tend to have the largest Z_{DR} maxima. In terms of the standard deviation of the polarimetric radar observables for each aggregation set, the values increase with b, with the ER column aggregates having values less than half that of the other aggregate sets (Fig. 13d).
Orientationaveraged Kuband (a) mean of Z_{DR}, (b) mean of ρ_{hv}, (c) mean of K_{dp}, (d) standard deviation of Z_{DR}, (e) standard deviation of ρ_{hv}, and (f) standard deviation of K_{dp}. The line color indicates the aggregation experiment as labeled in (f).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Orientationaveraged Kuband (a) mean of Z_{DR}, (b) mean of ρ_{hv}, (c) mean of K_{dp}, (d) standard deviation of Z_{DR}, (e) standard deviation of ρ_{hv}, and (f) standard deviation of K_{dp}. The line color indicates the aggregation experiment as labeled in (f).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
Orientationaveraged Kuband (a) mean of Z_{DR}, (b) mean of ρ_{hv}, (c) mean of K_{dp}, (d) standard deviation of Z_{DR}, (e) standard deviation of ρ_{hv}, and (f) standard deviation of K_{dp}. The line color indicates the aggregation experiment as labeled in (f).
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
As in Fig. 13, but for Ka band.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
As in Fig. 13, but for Ka band.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
As in Fig. 13, but for Ka band.
Citation: Journal of the Atmospheric Sciences 80, 4; 10.1175/JASD220149.1
The relations between these orientationaveraged scattering properties and b are similar at Ka band to those shown at Ku band. The maximum values of Z_{DR} (found for the pivoting branched planar aggregates and nonpivoting AR branched planar aggregates) are ∼0.5 dB higher at Ka band (Fig. 14a). The shapes of the mean orientationaveraged ρ_{hv} curves have somewhat more variability at Ka band than those at Ku band, with the ρ_{hv} minima occurring at a wider range of b values (Fig. 14b). Values of the mean K_{dp} per mass are proportionally higher at Ka band compared to Ku band, owing to the shorter wavelength (Fig. 14c). The values of ρ_{hv} at b = 100 are also lower at Ka band than those at Ku, suggesting that some small flutter can still produce measurable ρ_{hv} reductions at this wavelength, especially for the ER branched planar aggregates. The variability in the orientationaveraged ρ_{hv} across the branched planar aggregate sets is much larger at high b values compared to the variability at Ku band at these b values. Both sets of ER branched planar aggregates have their highest standard deviations of ρ_{hv} at b = 100, suggesting certain particles may have much lower ρ_{hv} when assumed to flutter lightly (Fig. 14e).
5. Discussion and conclusions
The shapes, orientation behavior, and resulting scattering properties of aggregates are complex. The simplified, Monte Carlo aggregation experiments presented herein show a wide range of bulk physical properties are possible from initial assumptions about the monomer properties, their attachment behavior, and the fall behavior of the aggregate. The synthetic aggregates assumed to fall with uniform azimuthal rotations about their vertically aligned axes tend to have flatter shapes with denser structures than the synthetic aggregates assumed to fall with completely random orientations. Additionally, the inclusion of monomer pivoting upon attachment produces a similar effect of flatter and denser structures. The nonpivoting, completely random branched planar aggregates have the sparsest structures. The column aggregates show comparatively lowerdensity structures with more isotropically distributed mass (i.e., less flat) than the branched planar aggregates. The degree of horizontal orientation of the monomers is related to density because consistently more aligned particles pack together more efficiently in space compared to more randomly aligned monomers.
We find that the polarimetric scattering properties of the aggregates depend most strongly on mass and effective density (defined based on the convex hull volume of the particles). Our spectral analysis of the scattering coefficients shows that for the zerothorder coefficient that corresponds to random uniform particle orientation, backscatter is highly correlated to mass and increases monotonically with mass at both Ku band and Ka band. Increases in effective density between 50 and 150 kg m^{−3} are associated with rapid increases in the secondorder backscatter coefficient at horizontal polarization and rapid decreases in the secondorder backscatter coefficient at vertical polarization. Similarly, the components of the copolar covariance at these densities decrease rapidly with density. The changes in these coefficients with density are likely due to the increasing inhomogeneity of the aggregate internal structure of partially aligned monomers.
For orientation distributions with increasing horizontal alignment, the secondorder coefficients enhance the orientationaveraged backscatter at horizontal polarization, and reduce the orientationaveraged backscatter at vertical polarization, resulting in increases in Z_{DR}. These relations with respect to effective density are likely a reflection of how the degree of monomer horizontal alignment and packing within an aggregate tend to increase with effective density, especially when considering the effective density in terms of the convex hull of the aggregate. With increasing horizontal alignment of the monomers, the nearfield interactions of the dipoles enhance the internal electric fields at horizontal polarization and reduce the internal electric fields at vertical orientation. Z_{DR} and K_{dp} increases monotonically with the orientation b parameter, while ρ_{hv} has minima between b values of 2–5. These unique behaviors suggest the potential for these variables to provide complementary information about aggregate shapes and orientation distributions.
Our results suggest that conventional measures of aspect ratio have indirect relations to the polarimetric scattering properties. This lack of importance of aspect ratio is likely due to the polarization effects being influenced by the monomer sizes, orientations relative to the polarization state of the incident radiation, and proximity to each other, and this information is not captured by aspect ratio. Individual ice crystals and rimed particles are more likely to show strong scattering property dependencies on aspect ratios, since the mass within these particles is more uniformly distributed and the particle symmetries reduce the ambiguity in defining the aspect ratio.
Our numerical experiments shown herein are relatively simplified, and therefore, the true hydrodynamics of falling aggregates may vary substantially to the assumptions we make herein. For example, the fall behavior of the aggregates is likely to change as increasing numbers of monomers become incorporated into the structure, changing the mass and projected area (e.g., Heymsfield and Westbrook 2010). These more complex natural orientation distributions may lead to polarimetric scattering properties that cannot be reproduced with our more simplified assumptions. Additionally, monomers may slip as well as pivot during their attachment. This slipping effect may further increase the density of the aggregate structures as the monomers become more closely packed.
Finally, natural aggregates are typically composed of a variety of ice crystal habits, partially rimed particles and other aggregates; our synthetic aggregates have only single habits and only include pristineaggregate collection. However, it is possible that aggregates composed of mixed habits may have physical and scattering properties that are similar to those simulated herein, with the possible exception to aggregates composed of irregular ice particles. For these aggregates of irregular ice particles, it is possible that the mass is more homogeneously distributed compared to aggregates of branched planar crystals and aggregates of columns, and bulk properties such as aspect ratio may be more important than characterizing the monomer properties.
To best evaluate model simulations of ice clouds, it is important to accurately simulate the scattering properties associated with the simulations. We have shown that the effective density of the simulated aggregates depend substantially on the monomer habits, attachment assumptions of the monomers, and orientation of the aggregates as they fall. However, the relations between the scattering properties and mass and density suggest the potential for the scattering properties to be predicted without explicit knowledge of the detailed structure of the monomers within an aggregate. Such predictions could be used in a forward model such as in Schrom and Kumjian (2019) to efficiently predict probabilities of scattering properties given a set of physical properties such as mass and effective density. These probabilities may then be used in a forward simulation to characterize the mean and uncertainty of modelsimulated radar measurements to more quantitatively use radar measurements to inform microphysical model parameterizations.
Some solvers that allow for the calculation of scattering properties at different orientations efficiently after an intensive initial computation do exist (e.g., Mackowski 2002).
Acknowledgments.
This work is supported by the NASA GPM Ground Validation program and the NASA Postdoctoral Program, administered by Oak Ridge Associated Universities. We thank the reviewers for their thorough comments that substantially improved this manuscript.
Data availability statement.
The ADDA simulations for the aggregate experiments and the corresponding geometric property files are stored in an online repository (available at https://github.com/rskschrom/aggregate_scattering).
APPENDIX A
Representation of Scattering Properties for Azimuthally Uniform, Fluttering Particles
a. General representation
b. Representation with uniform γ
c. Representation with uniform α and γ
APPENDIX B
Analytical Orientation Averaging of Scattering Properties Expressed with Beta Distributions
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