Derivation of physical and optical properties of mid-latitude cirrus ice crystals for a size-resolved cloud microphysics model

Single-crystal images collected in mid-latitude cirrus are analyzed to provide internally consistent ice physical and optical properties for a size-resolved cloud microphysics model, including single-particle mass, projected area, fall speed, capacitance, single-scattering albedo, and asymmetry parameter. Using measurements gathered during two flights through a widespread synoptic cirrus shield, bullet rosettes are found to be the dominant identifiable habit among ice crystals with maximum dimension (Dmax) greater than 100μm. Properties are therefore first derived for bullet rosettes based on measurements of arm lengths and widths, then for aggregates of bullet rosettes and for unclassified (irregular) crystals. Derived bullet rosette masses are substantially greater than reported in existing literature, whereas measured projected areas are similar or lesser, resulting in factors of 1.5–2 greater fall speeds, and, in the limit of large Dmax, near-infrared single-scattering albedo and asymmetry parameter (g) greater by ∼ 0.2 and 0.05, respectively. A model that includes commonly imaged side plane growth on bullet rosettes exhibits relatively little difference in microphysical and optical properties aside from∼ 0.05 increase in mid-visible g primarily attributable to plate aspect ratio. In parcel simulations, ice size distribution, and g are sensitive to assumed ice properties.


Introduction
It is well known that cirrus clouds substantially impact radiative fluxes and climate in a manner that depends upon their microphysical and macrophysical properties (e.g., Stephens et al., 1990). With respect to microphysical properties, obser-30 vations of cirrus cloud particle size distributions and underlying ice crystal morphology still remain subject to large uncertainties, in part owing to lack of instrumentation adequate to provide artifact-free and well-calibrated measurements of size-distributed ice particle number and mass concentrations 35 (e.g., Baumgardner et al., 2011;Lawson, 2011;Cotton et al., 2012). With respect to single-crystal properties, the Cloud Particle Imager (CPI) instrument provides high-resolution images of crystals at 2.3 µm per pixel (Lawson et al., 2001), but to our knowledge no airborne instrumentation to date pro-40 vides a direct measurement of the most fundamental quantity: single-particle mass. How important is advancement of such microphysics observations? On one hand, for instance, simulated climate sensitivity has been reported sensitive to cirrus ice fall speeds (e.g., Sanderson et al., 2008). On the 45 other hand, statistical properties of cirrus simulated at the cloud-scale have been reported relatively insensitive to ice crystal habit assumptions (e.g., Sölch and Kärcher, 2011). Such an insensitivity to ice habit presents a contrast to mixedphase cloud simulations, which are found sensitive to even 50 relatively minor changes in the specification of ice microphysical properties such as habit, fall speed, and size distribution shape (Avramov and Harrington, 2010;Avramov et al., 2011;Fridlind et al., 2012;Ovchinnikov et al., 2014;Simmel et al., 2015). 55 It is also well known that ice crystals in the atmosphere exhibit a profound degree of diversity in morphology that impacts microphysical process rates and radiative properties (e.g., Pruppacher and Klett, 1997). Within parcel, cloud-resolving and climate model microphysics schemes, ice properties are simplified in a variety of ways, generally based on some degree of observational guidance. Early observational studies using single-crystal measurement approaches commonly reported power-law relations between particle mass and a relevant particle dimension, such as column length or aggregate maximum dimension, generally valid over a relatively short range of dimensions measured for any particular crystal habit class (e.g., Locatelli and Hobbs, 1974). Later work identified the importance of projected area to fall speed, reported observation-based area-dimensional Avramov et al., 2011). The fact that the majority of ice crystals in natural clouds are not generally pristine owing at least in part to the commonality of polycrystalline growth and the curving sides and edges caused by sublimation (e.g., Korolev et al., 1999) has been increasingly recognized in lit-100 erature that addresses the consequences of morphological diversity for factors such as single-scattering properties (e.g., McFarquhar et al., 1999). Later laboratory and measurement analyses have specifically aimed to provide more generalized guidance on complex morphologies, offering revisions 105 to earlier diagrams of habit as a function of temperature and supersaturation (e.g., Bailey and Hallett, 2002;Korolev and Isaac, 2003;Bailey and Hallett, 2009).
Currently, based on CPI imagery, automated identification of ice habit is relatively commonly reported (e.g. , Lawson 110 et al., 2006b;McFarquhar et al., 2007). However, analysis of quantitative single-crystal data on within-habit diversity to inform the representation of microphysical and radiative properties of ice for modeling studies of observed case studies (or, by extension, cloud system classes such as cir-115 rus) remains nearly absent. The widespread occurrence of polycrystals and aggregates further complicates ice properties substantially. In relatively thick mixed-phase clouds, for instance, cycles of riming and vapor growth may result in a wide variety of plate-like fin structures grown on highly 120 rimed substrates (e.g., Magono and Lee, 1966, R3c habit) as seen during the M-PACE campaign (Fridlind et al., 2007), creating crystal properties so diverse that it is essentially impossible to find quantitative, measurement-based guidance from analyses available in the literature to date. 125 Perhaps not yet as widely considered in models are the difficulties of consistently assigning ice crystal component aspect ratio, roundness, and microscale surface roughness for accurate calculation of radiative properties (e.g., van Diedenhoven et al., 2014a). When using mass-and area-dimensional 130 relations as a foundation for ice properties in a model, as most commonly done, it is possible to assign a surface roughness and aspect ratio, and to calculate optical properties based on columns and plates that match ice volume, projected area and aspect ratio for any given ice class and size 135 (e.g., Fu, 1996Fu, , 2007. Guidance can be obtained from past studies of cirrus that quantify the variability of bullet arm aspect ratio, for instance (e.g., Iaquinta et al., 1995;Heymsfield and Iaquinta, 2000;Um and McFarquhar, 2007). However, the aspect ratio of whole 140 crytals and their crystalline elements are relatively scarcely reported and analyzed for natural ice crystals (e.g., Korolev and Isaac, 2003;Garrett et al., 2012;Um et al., 2015), making necessary some relatively poor approximations for specific natural conditions that may be encountered in the field 145 (e.g., Fridlind et al., 2012). Finally, for a size-resolved microphysics scheme, obtaining continuity of ice particle properties over the full size range required to represent relevant cloud microphysics generally requires awkward concatenation of aspect ratio-, mass-and area-dimensional relations 150 relevant for limited size ranges (e.g., Sölch and Kärcher, 2010), which can easily lead to unphysical discontinuities in derived quantities such as fall speed or capacitance. Erfani and Mitchell (2016) recently provided polynomial mass-and area-dimensional relations 155 that surmount lack of continuity and simplify to analytically integrable power laws that closely approximate the full solution over a local size range.
Here we analyze single-crystal ice crystal field data with the primary objective of deriving physically continuous ice 160 microphysical and optical properties over the size range required (1-3000 µm). As a well-defined starting place, and a foundation for large-eddy simulations, we focus narrowly on the morphological properties of a well-developed midlatitude synoptic cirrus case study, taking advantage of an ex- 165 isting extended analysis of single-crystal images (Um et al., 2015). Because the most accurate representation of cirrus optical properties requires consideration of polycrystal element aspect ratios (e.g., van Diedenhoven et al., 2014a), which are commonly a function of particle size in obser-170 vations, the polycrystal elements are adopted as the foundation for treating mass and projected area rather than vice versa (as required if area-and mass-dimensional relationships are instead adopted as the foundation, as most commonly done); a similar approach was taken by Heymsfield and Iaquinta (2000) for the purpose of deriving physically based expressions for cirrus crystal terminal velocities, such as bullet rosettes with varying numbers of arms. Parcel simulations are used to compare the ice properties derived in this work with ice properties available in existing literature that 180 have been used in large-eddy simulations of cirrus with sizeresolved microphysics (Sölch and Kärcher, 2010). Because the derivations here are based on crystal component geometries and do not yield continuous analytic relationships, equations are provided in Appendix A and derived ice properties 185 are provided for download as the Supplement.

Observations
In situ observations are analyzed from a well-sampled cirrus system observed during 1 April (flight B) and 2 April (flight A) during the 2010 Small Particles in Cirrus  TICUS) field campaign (Mace et al., 2009). Based on an extensive analysis of atmospheric states during SPARTICUS, Muhlbauer et al. (2014) classified the 1-2 April conditions as ridge-crest cirrus (Fig. 1). Relative to the other nonconvective cirrus states identified during SPARTICUS, ridge-195 crest cirrus were characterized by formation within the coldest environments at cloud top, within considerable ice supersaturation, and were statistically associated with the highest ice crystal number concentrations and lowest ice water contents.

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Previous studies using SPARTICUS data can be considered in at least five general categories: characterization of the environmental properties observed (e.g., Muhlbauer et al., 2014), characterization of the ice crystal morphology or size distribution characteristics observed (e.g., Mishra et al.,205 2014; Um et al., 2015;Jackson et al., 2015), cirrus cloud process modeling studies (e.g., Jensen et al., 2013;Muhlbauer et al., 2015), evaluation of satellite retrievals (e.g., Deng et al., 2013), and evaluation of climate model cirrus properties (e.g, Wang et al., 2014). The work here is in the second 210 category, and is based primarily on single-crystal ice crystal properties using data obtained from a CPI probe on the Stratton Park Engineering Company (SPEC) Inc. Learjet 25 aircraft. The ice crystals imaged by the CPI are first classified by habit using the scheme described by Um and McFarquhar 215 (2009). Images classified as bullet rosettes and aggregates of bullet rosettes are then further analyzed using output from the recently developed Ice Crystal Ruler (ICR) software (Um et al., 2015) to obtain the imaged width and length of each branch.

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To provide context for parcel simulations, we also use Learjet ice particle size distributions derived from a 2D Stereo Probe (2DS) equipped with tips that reduce effects of shattering (Lawson, 2011) and analyzed as reported by Jackson et al. (2015), together with in-cloud vertical wind speed 225 retrievals from profiling Doppler radar measurements (Kalesse and Kollias, 2013).

Model description
The overall objective of this study is to use analyzed CPI image data to derive consistent representations of ice 230 physical and optical properties for a size-resolved ice microphysics scheme, and to compare results with existing literature. The target microphysics scheme is based on the Community Aerosol-Radiation-Microphysics Application (CARMA) code (Jensen et al., 1998;Ackerman et al., 235 1995). The CARMA model allows selection of an arbitrary number of mass bins to represent the size distributions of an arbitrary number of aerosol and ice classes. Within each ice class, the mass in each bin is a fixed multiple of the mass in the preceding bin.

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In this work the ice crystal properties in each ice mass bin are represented using the approach developed by Böhm (1989Böhm ( , 1992bBöhm ( , c, a, 1994Böhm ( , 2004, as previously applied to represent the ice crystals in mixed-phase stratus in Avramov et al. (2011, dendrites and their aggregates) and 245 Fridlind et al. (2012, radiating plates). The Böhm scheme provides an integrated treatment of terminal fall speeds and collision efficiencies for non-spherical ice that is based not on specification of a particular habit but rather on four properties that are quantitatively defined for both pristine and non-250 pristine shapes: particle mass m, a characteristic maximum dimension D and projected area A, and aspect ratio α. The foundational physical quantity of this parameterization is fall speed, so the characteristic quantities A, D and α are best defined by fall orientation, which can perhaps most simply 255 be considered as the maximum projected area (which determines the fall orientation), the maximum dimension of a circumscribed circle around that projected area, and the aspect ratio of thickness normal to the fall orientation to that maximum dimension (cf. Böhm, 1989). Bodily aspect ratio α is 260 defined as 1 for ice crystals without a preferential fall orientation (e.g., bullet rosettes), less than 1 for oblate bodies (e.g., plates), and greater than 1 for prolate bodies (e.g., columns). Throughout this work, α is fixed at 1 based on the geometries discussed below, and D and A by extension assumed 265 equal to randomly oriented maximum dimension (D max ) and randomly oriented projected area (A p ).
In each ice mass bin, quantities that are not considered in the Böhm scheme but that should ideally be specified in an integrated manner are capacitance and radiative scattering 270 and absorption coefficients. For a given crystal, the capacitance can be either specified from the literature in the case of a pristine habit or else estimated from prolate or oblate spheroids (Pruppacher and Klett, 1997, their Eqns. 13-78 and 13-79). Here we take the former approach for bullet rosettes 275 and their aggregates and polycrystals analyzed below: given bullet arm length L and arm width W (twice the hexagon side length), we specify D max -normalized capacitance (C) as 0.4(L/W ) 0.25 based on the fit to calculations for six-arm rosettes by Westbrook et al. (2008).
Scattering and absorption properties assuming randomly oriented ice crystals in each mass bin are computed following van Diedenhoven et al. (2014a), which, in addition to m and A p , also requires specification of elemental aspect ratio (α e ) and a microscale surface roughness or crystal distortion 285 (δ), as defined by Macke et al. (1996). Here α e required for the optical properties is identical to the bodily aspect ratio α in the case of a single-component crystal (e.g., plate or column), but for a polycrystal such as a bullet rosette α e is the aspect ratio of constituent arms or other component crystals 290 (cf. Fu, 2007). In this work α e values are derived from ICR measurements where possible. Additional details are given in Section 5.4.
Parcel simulations are used to test ice properties in a simplified framework, following Ackerman et al. (2015), prior 295 to use in computationally expensive 3D large-eddy simulations in future work. All simulations include adiabatic expansion, aerosol homogeneous freezing, diffusional growth of ice crystals, and latent heating. Heterogeneous freezing is neglected. Parcels are initialized at 340 mb, 233 K, and 300 80% relative humidity. Saturation vapor pressures are related to water vapor mixing ratio following Murphy and Koop (2005). Each simulation is assigned a fixed updraft speed (w) of 0.01-1 m s −1 . Parcel expansion is treated by assuming dry adiabatic ascent and iterating three times on parcel 305 air pressure, temperature, and density assuming hydrostatic conditions and using the ideal gas law. Latent heat is computed in accord with diffusional growth of the ice. A default time step (∆t) of 1 s is variably reduced to a minimum value of 0.1 s, which is reached when fast processes 310 such as aerosol freezing are active, and parcel height is incremented by w∆t each time step. Ice sedimentation, when included, assumes a vertical length scale of 100 m as in Kay and Wood (2008). Gravitational collection is neglected. We use the Koop et al. (2000) parameterization for aerosol freez-315 ing, including the Kelvin effect on surface vapor pressure, and assume that aerosol are at equilibrium with atmospheric water vapor. Aerosol are initialized with a concentration of 200 cm −3 lognormally distributed with geometric mean diameter 0.04 µm and geometric standard deviation 2.3 as in 320 Lin et al. (2002), except that composition is assumed to be ammonium bisulfate. We assume a fixed ice accommodation coefficient of 1, which is within the range of recent laboratory measurements (Skrotzki et al., 2013), and account for Knudsen-number-dependent gas kinetic effects (cf. Zhang 325 and Harrington, 2015). Growth across mass bins is treated with the piecewise parabolic method of Colella and Wood-ward (1984). Simulations use 50 bins with a mass ratio of 1.65 from one bin to the next, starting with D max of 2 µm, suitable for use in 3D large-eddy simulations.

4 Derivation of ice single-crystal properties
Considering all CPI images collected during the 1-2 April flights, automated analysis places roughly half of all ice crystals in the small quasi-sphere category, and remaining crystals are primarily unclassified (Fig. 2). However, consider-335 ing only ice crystals with D max greater than 100 µm, bullet rosettes emerge as the most common classified habit. Subjective examination of images suggests that bullet rosettes are the dominant habit in the coldest crystal growth regions with significant ice water content (Fig. 3), as discussed fur-340 ther below. We therefore begin with an analysis of ICR measurements of bullet rosette arm lengths and widths, which are suitable to describe the physical and optical properties for a cloud composed entirely of growing rosettes.

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For each bullet rosette measured with the ICR software, Fig. 4a shows mean branch length versus measured D max . Since branches that are not aligned with the viewing plane are foreshortened, we take L as the average of all measured branch lengths minus half of randomly oriented projected 350 end plate diameter, multiplied by a factor of 4/π to account for random orientation to first order (see Appendix A1). The relationship of L to D max is reasonably fit by a line passing through the origin.
For the same crystals, Fig. 4b and Fig. 4c show mean W , 355 and the ratio L/W = α e , respectively. To account for random orientation, W is taken as the average of all measured branch widths divided by a factor of (1+ √ 3/2)/2 = 0.933, which is the ratio of the arithmetic mean of mimimum and maximum branch projected widths to the maximum (equivalent to the 360 ratio that would be found if measurements of projected width were made for a sufficiently large number of orientations of a bullet arm of known W ). Both mean and median number of branches is six (out of four to ten measured), consistent with recent analyses from tropical and Arctic field campaigns 365 (Um et al., 2015). Rosettes with more branches are seen to have systematically smaller W and larger α e , consistent with competition for vapor during growth. However, a simple least squares fit of W to D max gives W > D max when extrapolated to small crystal size, which is not physical; unfortunately, 370 measurements are not available to provide guidance at such sizes.
Because we seek a continuous description of ice properties across all sizes, here we take the approach of adopting a physical model of crystal geometry to extrapolate mea-375 sured properties smoothly to sizes smaller than measured. A similar approach was taken by Heymsfield and Iaquinta Fridlind et al.: Cirrus ice properties for a size-resolved microphysics model 5 (2000) to improve calculated cirrus crystal fall speeds over those obtained from independently derived mass-and areadimensional relations. We first assume branch width for rosettes consistent with the six-rosette model considered in Westbrook et al. (2008), but using a fixed angle of 44 • between opposing edges of the hexagonal pyramids that cap each branch (sensitivity of results to choice of cap angle is discussed at the end of Appendix A1). Selecting a fixed angle and using the linearly fit branch width at all sizes allows determination of the cap contribution to L; L is found to consist entirely of a truncated cap at the smallest sizes and corresponding W is taken as the truncated cap base width. This model results in the line slope discontinuity seen in Fig. 4b, 390 and resolves at least gross discrepancy of W > D max . Fig. 4c shows that adopting this bullet model results in a smooth increase in branch aspect ratio α e = L/W from smallest to largest sizes, suitable as a basis for calculating optical properties. L/W is constant at the smallest sizes, where 395 only the cap contributes and both W and L are varying at the same relative rate. The range of aspect ratios measured (2-6) and the fitted trend from near-unity at the smallest sizes to roughly 5 at the largest sizes is consistent with several past studies (cf. Heymsfield and Iaquinta, 2000). As shown, 400 the relationship of branch aspect ratio to D max also agrees with that used by Mitchell (1994) in derivation of the massdimensional relation for D max > 100 µm listed in Table 1 and discussed further below.
The Westbrook et al. (2008) model assumes that all bul-405 lets are at 90 • angles to one another, giving true maximum dimension of 2L, which is ∼40% greater than D max shown in Fig. 4a. Measured D max being a randomly oriented value can account for less than 30% discrepancy. Another source of difference is the commonly seen deviations of arm loca-410 tions from 90 • separations, which can only decrease D max from 2L. Since a more quantitative explanation is beyond the scope of this initial study, we adopt the randomly oriented D max as our only defined maximum dimension, an assumption that has also been made in past studies using two-415 dimensional images (e.g., Heymsfield et al., 2002;Baker and Lawson, 2006a). The bullet model described above now allows calculation of crystal surface area (A s ) and m (see Appendix A1 for details). To calculate m from the geometrical dimensions, we 420 assume ice bulk density (ρ i ) of 0.917 g cm −3 ; any bullet arm hollows are neglected here owing to lack of quantitative guidance, as discussed further below. Calculated m and A p of a six-branch rosette are seen to reasonably represent the scatter of individual crystal properties (solid lines in Fig. 4e and 425 4f). The ratio of measured A p to calculated A s is found to be about 0.11 (Fig. 4d), smaller for these concave particles than the A p /A s of 0.25 for convex shapes, consistent with theoretical results (Vouk, 1948) and reasonably independent of D max across measured sizes.

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In Fig. 4e derived m(D max ) is compared with power-law relations from previous literature that have been used in sim-ilar bin microphysical schemes (Table 1). To our knowledge, only one unpublished data set has provided direct measurements of bullet rosette mass, consisting of 45 crystals with a 435 range of 2-5 arms as reported by Heymsfield et al. (2002), but that data set is not the basis of commonly used relations. As used in Sölch and Kärcher (2010), for instance, the Heymsfield et al. (2002) relation is based on calculation of effective density from a combination of ice water content and  Table 1 values are taken from their equation 32); for crystals smaller than 100 µm, Mitchell et al. (1996) proposed a mass-dimensional relation using ad hoc estimates of crystal 450 mass (Table 1 values are taken from their Table 3).
The difference between our calculated m(D max ) and that from Mitchell (1994) is roughly a factor of four at measured crystal sizes, which results in a similar discrepancy in fall speeds and effective diameters, as shown below. We can at-455 tribute lower m in Mitchell (1994) to four factors: (i) L is substantially shorter based on the approximation D max = 2L (cf. Iaquinta et al., 1995, including assumed trilateral pyramidal end following their Fig. 1), (ii) W is substantially thinner based on earlier cited literature that relates W to L and by ex-460 tension D max /2 (see Fig. 4b), (iii) five branches are assigned instead of six found here, and (iv) ρ i is ∼0.78 g cm −3 instead of 0.917 assumed here. All else being equal, increasing their branch number and ρ i would together increase Mitchell (1994) m by only about 40%, but m scales roughly linearly 465 with L and geometrically with W . The trilateral pyramid ends taken from Iaquinta et al. (1995) would result in slightly greater m than ours, all else being equal. The close agreement between our arm aspect ratio L/W and that following Mitchell (1994), available for D max > 100 µm (Fig. 4c), sug-470 gests that differences in m are primarily attributable to differing approaches to defining D max . However, we are unable to quantitatively confirm that because randomly oriented maximum dimension cannot be calculated analytically for either the idealized geometries derived here or for CPI images of 475 natural crystals.
Our calculated m(D max ) is also nearly a factor of two greater than that from Heymsfield et al. (2002) for ice particle ensembles (all habits, dominated by bullet rosettes) measured over the same Oklahoma location. In Heymsfield et al. 480 (2002), D max is taken from 2DC and 2DP probe measurements and m is derived from coincident ice water content measurements from a Counterflow Virtual Impactor (CVI) via a linear fit of effective particle density (ρ e , the density of a sphere with diameter D max ) to D max . Whereas our ap-485 proach is subject to uncertainty in ICR measurements and assumed ρ i , the Heymsfield et al. (2002) approach is subject to uncertainty in the measurement of ice particle size distribution, uncertainty in the measurement of ice water content, and the importance of any deviations of the particle ensemble from bullet rosettes. Uncertainty in CVI probe measurements are reported to be 10% for ice water contents larger than 0.2 g m −3 (Twohy et al., 1997), but the bin-wise uncertainty in particle size distribution measurements are generally unquantified; we consider it beyond the scope of this 495 study to undertake the detailed analysis required to resolve such differences. Although it is not used in the m − D max relationship adopted by Sölch and Kärcher (2010) and listed in Table 1, Heymsfield et al. (2002) also derive a typical ρ i of 0.82±0.06 g cm −3 for bullet rosettes based on independent 500 photographic evidence for hollow bullet rosette arm ends; we make no such reduction here, as discussed above, and doing so is not a dominant cause of the differences in m.
Whereas our calculated m(D max ) is substantially greater than that previously used in studies with size-resolved micro-505 physics, our measured A p (D max ) is similar or smaller. The relationship of A p and D max derived by Mitchell et al. (1996, their Table 1) independently from m(D max ) for five-branched bullet rosettes in a manner similar to that here, is nearly identical to ours (cf. Fig. 4e). The less widely available A p -D max 510 relations are surprisingly more difficult to trace, considering that they can be more directly derived from CPI images, and we are unable to identify the observational sources of the relations reported in Sölch and Kärcher (2010), which are cited from but not apparent in Heymsfield et al. (2002). 515 Figure 5 allows a closer examination of the extrapolation from manually measured rosette properties (D max > 200 µm) to smaller sizes using our bullet model, and shows comparisons to additional published fits. From in situ measurements of total ice water content and ice crystal size distribution and 520 shape obtained from a 2DS probe in mid-latitude cirrus, Cotton et al. (2012) derived a mean ρ e of 0.7 g cm −3 below a threshold size of 70 µm and a power law decrease of density to 0.5 g cm −3 at roughly 100 µm and 0.05 g cm −3 at roughly 1000 µm. The mean ρ e derived here happens to 525 exhibit a similar behavior (Fig. 5a), where the discontinuity using our bullet model represents the transition to truncated branch caps. Erfani and Mitchell (2016) derived polynomial m − D max relations for synoptic cirrus clouds warmer than −40 • C from single-particle measurements of m, D max and 530 A p obtained during the 1985-1987 Sierra Cooperative Pilot Project (SCPP) (Mitchell et al., 1990), or by applying a habit-independent m − A p relation derived from the SCPP data set , shown in Figure 4e, to 2DS measurements obtained during 13 SPARTICUS flights 535 (at colder temperatures). Although the SCPP data set does not contain bullet rosettes or spatial crystals , Lawson et al. (2010) report that ice water content derived by applying that habit-independent m − A p relation to a combination of tropical anvil and synoptic cir-540 rus measurements agreed with CVI measurements to within 20%. At D max < 100 µm, Erfani and Mitchell (2016) m val-ues were calculated from CPI measurements of A p and α assuming hexagonal column geometry (cf. Erfani and Mitchell, 2016, their Appendix B), and effective densities are similar 545 to those derived here. At larger sizes and especially colder temperatures, Erfani and Mitchell (2016) effective densities are smaller than derived here, consistent with the  m − A p relation giving lower per-particle m than derived here.

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Although m cannot be calculated in this study for bullet rosettes that are not measurable with the ICR software or for crystals with unclassified habit, A p is reported for all imaged crystals and can be directly compared with the bullet model. Analogous to ρ e but dimensionless, the measured ratio of A p 555 to that of a sphere with diameter D max can also be compared with the bullet model. Figure 5b shows that literature power law relations can become unphysical for the smallest particle sizes (projected areas greater than for a sphere of diameter D max ); to correct the greatest deviations for the purposes of 560 parcel calculations below, we adopt a constant ratio of A p to sphere projected area where D max < 100 µm when using Mitchell (1994) relations. When considering all rosettes automatically identified (not all of which were measurable using the Ice Crystal Ruler), the bullet model A p (D max ) agrees 565 quite well with median measurements and with m − A p relations for bullet rosettes and budding bullet rosettes from Lawson et al. (2006a) (Fig. 5c). However, when considering all crystals (Fig. 5d), there is a wider range of A p (D max ) and the bullet model underestimates median A p (D max ), as ad-

Bucky ball model
To consider uncertainty in the geometry of the smallest crystals, we next consider an alternative proposed model for early bullet rosette shape: budding Bucky balls . The so-called budding rosette shape has been 580 observed in laboratory grown ice and ice-analog crystals (Ulanowski et al., 2006;Bailey and Hallett, 2009), and the CPI does not have the resolution necessary to distinguish such a shape from the bullet model geometry assumed above. Here we approximate the Bucky ball core as a sphere of di-585 ameter 10 µm and then assume that arms emerge with initial width 4 µm. If we assume that L falls linearly to zero at D max equal to the core dimension ( Fig. 6a) and W correspondingly falls linearly to its minimum initial width (Fig. 6b), then branch α e is relatively constant near the mean observed 590 (Fig. 6c). A s and m can now be calculated using this Bucky ball model, except that A p of the smallest crystals must be interpolated to bridge the geometry of a sphere (D max < core diameter) and that of a rosette; to do this, we calculate a mass-weighted sum of A p obtained from the linear relation in Fig. 4d and that of a sphere (see Appendix A2). Thus, as m converges to that of a sphere, so does A p . Results are similar to those of the bullet model at larger particle sizes ( Fig. 6d-f), with m(D max ) still larger than previous estimates and A p (D max ) still similar or smaller.
However, using this simplified Bucky ball model, a developing six-arm rosette has a systematically smaller ρ e and A p than it did with the bullet model (Fig. 7). Although this particular version of a Bucky ball model, with only six arms even at small sizes, gives substantially smaller A p (D max ) 605 than measured for automatically classified rosettes at small D max (Fig. 7c), it does serve to provide a quite close match to the minimum area relative to that of a sphere over the full particle data set (Fig. 7d), and is therefore included in parcel calculations below. In reality it seems likely that not all bud-610 ding arms grow evenly. For instance, Um and McFarquhar (2011) propose a Bucky ball model with 32 regular and irregular hexagonal arms, one growing from each of the ball's 20 hexagonal and 12 pentagonal planes. From this study, it is apparent that only up to about 12 arms commonly reach sub-615 stantial lengths, and most commonly only six such arms are seen. Faced with the problem of how to introduce geometry that smoothly transitions from an unknown larger number of sub-100-µm arms to roughly six arms at larger sizes with no quantitative basis for how to introduce such added complex-620 ity here, we have simply assumed six arms throughout.

Aggregate model
We return now to the distribution of habits during the April 1-2 flights, and consider the properties of crystals in the observed cirrus deck that are not identified as bullet 625 rosettes. The rosettes are most common in the upper cloud regions at temperatures colder than −40 • C (Fig. 8), consistent with previous findings that rosette shapes in the temperature range −40 to −55 • C are mostly pristine . In this case, at slightly warmer temperatures, 630 aggregates of bullet rosettes become most common. Using ICR measurements for aggregates of bullet rosettes, it is straightforward to extend the bullet model to rosette aggregates ( Fig. 9, see Appendix A3), where the mean and median branch numbers are found to be 12 per aggregate, consistent 635 with aggregation of two typical bullet rosettes. Compared to single rosettes, aggregate properties are generally similar to those of single rosettes except shifted in size to a larger maximum dimension. We do not dwell here on the properties at the smallest sizes since aggregates are born from fully 640 formed bullet rosettes and this study is focused on crystal growth (neglecting sublimation).
However, aggregates of pristine rosettes also represent a small fraction of ice crystals observed in this case, at least on a number basis. CPI images show that some rosettes reach 645 a plate growth regime (Fig. 10), a phenomenon well documented in previous cirrus field observations and laboratory measurements (Bailey and Hallett, 2009). In the lower cloud regions at temperatures warmer than −40 • C, modified bullets have been described as mostly "platelike polycrystals, 650 mixed-habit rosettes, and rosettes with side planes" , where side plane growth on columns may be attributable to facet instability on prism faces (Bacon et al., 2003).

655
For the purposes of considering how plate-like growth impacts rosette single-crystal properties, it is notable from the SPARTICUS images in this case that radiating side plane elements appear to increasingly fill the space between the arms of rosettes and rosette aggregates, giving the impression of 660 cobwebs that lead to blocky ice particle shapes (e.g., Fig. 10). In such a process, particle m could increase without rapid expansion of particle D max . Such a tendency for crystals to become less florid may be related to the finding of side plane growth on rosettes in the laboratory exclusively originating 665 from the rosette center, consistent with an important role for defect and dislocation sites . Toward cloud base, sublimation then increasingly rounds crystal edges (Fig. 11). Rosettes that did not enter a side plane growth stage appear now with rounded arms that can still 670 be counted, whereas rosettes that did experience substantial side plane growth emerge from sublimation zones as relatively large quasi-spheres, which appear as a non-negligible percentage of large particle habit; the existence of such large quasi-spheres would be otherwise difficult to explain. The 675 smallest sublimated crystals appear occasionally as sintered chains.
We next consider an approximate model for the physical and optical properties of these more common, irregular crystals. In the data set examined here, we are unable to find 680 a consistent increase in projected area ratio with increasing temperature that would be expected if rosettes are modified by side plane growth during sedimentation from colder to warmer temperatures, but we do find that unclassified crystals at all temperatures exhibit consistently larger area ratios 685 than rosette crystals (Fig. 12). To account for rosette shape evolution in a manner amenable to calculation of radiative and microphysical properties at least for growing crystals, we attempt to coarsely estimate the side plane mass added to pristine rosettes and its associated elemental aspect ratio as 690 follows.
We first calculate the additional projected area that can be attributed to side plane growth. Considering all unclassified crystals, a fit of measured A p to calculated bullet surface area (based on measured maximum dimension and assum-695 ing a bullet model rosette with six arms) yields a slope of 0.15 (Fig. 13a), which is larger than the slope of 0.11 found using the bullet model for measured rosettes, consistent with greater area ratios for unclassified crystals. If we make the ad hoc assumption that the relationship of surface area to pro-700 jected area is close to that for bullet rosettes, we can attribute the surface area beyond that of the bullet model to plates. If we make the ad hoc assumption that a plate-like side plane grows on each of six arms and neglect plate thickness, the plate or side plane surface area can be considered as the sum of hexagonal faces of the six plates, and the plate diameter can be calculated. If we further relate plate thickness to plate diameter as described in Appendix A4, then mass can now be calculated as the sum of bullet and plate contributions for a typical particle (e.g., Fig. 13c, solid line). For this crude 710 representation of plate-like growth on the bullet model, the calculated crystal properties agree reasonably well with Cotton et al. (2012) effective density in the limit of small D max (Fig. 13e) and with the area ratio as a function of D max over all unclassified crystals (Fig. 13f) if the following choices are made: the cap angle β is increased to 25 • , plates are assumed present only where branches extend beyond truncated caps, and the plate surface area is assumed to increase inverse exponentially to its terminal value with a length scale equal to the diameter at L greater than L c (see Appendix A4 for de-720 tails). Where D max > 100 µm, the resulting polycrystal model A p agrees closely with the Erfani and Mitchell (2016) fit for warmest-temperature synoptic cirrus, but resulting m is now also correspondingly greater than and further from Erfani and Mitchell (2016) than in the bullet model (cf. Fig. 5a).

725
The foregoing results for this polycrystal model are dependent upon the underlying bullet model assumed, the assumed ratio of A s to A p , and the assumed plate or side plane geometry, for which no quantitative guidance exists in the current data set. This polycrystal model is intended only as 730 a relatively simple example of ice properties that is guided by available observations and allows calculation of internally consistent physical and radiative properties in a continuous fashion over all crystal sizes that need to be represented in our microphysics model. In order to evaluate the need for fur-735 ther consideration of ice properties in greater detail, we next consider parcel simulations to evaluate the influence of ice models on predicted size distributions and optical properties. properties, as discussed above, the main difference may be traceable to differing D max definition used to calculate m, whereas A p (D max ) is very similar. In the case of Heymsfield properties, m(D max ) is closer to ours but A p (D max ) is also larger, a factor that should be relatively more easily resolved 755 in future studies since both A p and D max can be directly measured. As shown in Fig. 5c, for instance, an m − A p relation from earlier midlatitude cirrus measurements  agrees well with SPARTICUS rosette measurements and with our bullet model. At the warmest tempera-760 tures considered by Erfani and Mitchell (2016), A p (D max ) is similar to or even greater than the bullet or polycrystal models but substantially lower m(D max ) leads to substantially lower v f (D max ); the Erfani and Mitchell (2016) trend toward greater decrease in m than A p with decreasing temperature 765 leads to increasing divergence between the models derived here and their results.
Given literature ice properties, using our model to calculate crystal fall speed as detailed in Avramov et al. (2011) results in v f values that appear similar to those of Sölch and 770 Kärcher (2010) and are also within roughly 10% of those calculated using the method described in Heymsfield and Westbrook (2010) (not shown). However, our aggregate model gives fall speeds roughly one-third reduced from similarsized bullet model ice, which is a substantially larger dif-775 ference than that using Heymsfield ice properties for aggregates and their rosettes shown in Sölch and Kärcher (2010). We can trace this greater difference in part to substantially larger m(D max ) derived here, as shown above. Overall, we conclude from comparison of our results with those of Sölch 780 and Kärcher (2010) that the precise method of calculating v f as a function of m and A p appears to be responsible for relatively little spread, but differences in ice properties themselves (m and A p ) introduce v f (D max ) differences that are substantially larger than expected, as discussed further be-785 low.
Owing to the dependence of parameterized capacitance on bullet arm aspect ratio alone (see Section 3), capacitance differences are nearly negligible for crystals larger than ∼400 µm across all bullet models derived here, in sharp con-790 trast to factor of two differences in fall speed at such sizes. Because assumed or derived bullet arm aspect ratios vary most where D max is less than 300 µm, capacitance differences up to roughly 25% are most pronounced at those sizes. Although aspect ratios used in derivation of the Mitchell 795 ice properties are similar to ours where D max > 100 µm (see Fig. 6), no such aspect ratios are provided for smaller Mitchell crystals or for Heymsfield ice properties. For parcel calculations, we therefore adopt a C value of 0.25 derived for aggregates (Westbrook et al., 2008), taken here as rep-800 resentative of polycrystals with unspecified aspect ratios. A similar assumption would be required for Erfani and Mitchell (2016) ice properties; since parcel simulations are also not configured for changes in ice crystal properties during a single simulation, we omit Erfani and Mitchell (2016)   To grossly evaluate the potential effect of different model ice properties on ice crystal nucleation and growth, we first consider parcel simulations without the complication of sedimentation. Since aggregation is neglected, aggregate ice properties are not considered. As described in Section 3, parcels begin at 233 K (−40 • C), 340 mb, and 80% relative humidity. Vertical wind speed (w) is fixed at 0.01, 0.1 or 1 m s −1 , within the range of millimeter cloud radar retrievals 815 of in-cloud w from the beginning of the first flight examined here to the end of the second flight (Fig. 16). Aside we note that a parcel simulation is not a realistic rendition of natural cirrus cloud evolution, which is characterized by extensive growth and sublimation during particle sedimentation. But 820 a similar framework has been used to test cirrus models (Lin et al., 2002), and in this case it allows a simple comparison of particle growth to sizes that span the range observed during SPARTICUS, as discussed further below. Figure 17 shows the ice particle size distribution (PSD) for of parcel ascent distance. In the absence of sedimentation, ice mass is essentially distributed across differing crystal sizes depending upon w and ρ e . The magnitude of w primarily determines N i : when w is strongest, vapor growth competes least with nucleation, resulting in greatest N i (Fig. 18a). Nu-835 cleated number concentrations range from several per liter when w is 0.01 m s −1 to several per cubic centimeter when w is 1 m s −1 , consistent with past studies (e.g., Lin et al., 2002). The Heymsfield ice properties give roughly a doubling of N i relative to other ice properties, owing to the densest small ice 840 accompanied by a fixed capacitance for non-spheres (in the absence of obvious means of transitioning capacitance from spheres to non-spheres). The strongest w and associated fastest aerosol freezing, which leads to largest N i , leads to the correspondingly small-845 est D i (Fig. 19a) and the greatest A i (Fig. 20a). Where N i is insensitive to ice properties, the sensitivity of D i and A i to ice properties at a given w can be seen as simply scaling inversely with effective density and area per unit mass, respectively. Ice properties assumptions lead to roughly a factor of 850 2 range of D i at lowest w and nearly a factor of 4 range of D i at highest w. Whereas D i is variable with ice properties, A i at all w falls into two groups: Mitchell and Heymsfield properties, with relatively large A i , and all other ice properties including spheres and our models derived here, with A i 855 systematically smaller by roughly a factor of 3 at all w. Dispersion (ν) exhibits up to factors of 2-3 difference (Fig. 21e). The ice properties associated with the lowest effective density (Mitchell) have greatest D i and ν. However, at lowest w the Bucky ball model exhibits substantially greater D i but 860 similar ν as spheres, which can be attributed to a weak dependence of ρ e on D max at D max > 100 µm that is more similar to spheres than other ice models (Fig. 7).
In summary, in the simple case of a non-sedimenting parcel, differing ice property assumptions lead to a factor of 2 865 difference in N i and factor of 3 in A i . Up to a factor of 2 increase in ν is also induced by ice properties that exhibit a trend in ρ e across the relevant size distribution relative to ice properties with constant ρ e . Differences in ρ e across ice properties considered here (regardless of trend) also lead to 870 factors of 2-3 difference in D i .

Parcel simulations with sedimentation
When sedimentation is included with an assumed parcel depth of 100 m following Kay and Wood (2008), results are largely unchanged at the strongest w since v f w (cf. 875 Fig. 14); the only notable change is roughly a factor of two reduction in N i by −55 • C, seen primarily as a uniform downward shift of the PSDs between Fig. 17a and Fig. 17b. However, at w 1 m s −1 , the parcel behavior changes rather dramatically because sedimentation reduces 880 surface area sufficiently to allow aerosol freezing events repeatedly as the parcel ascends, every 250-500 m when w = 0.1 m s −1 (Fig. 18d) and at least ten times more frequently when w = 0.01 m s −1 (Fig. 18f). At intermediate w, nucleation occurs roughly 50% less frequently for the slow-885 est falling ice (Mitchell, Heymsfield) than for other ice properties. At the greatest w, nucleation does occur eventually if parcel ascent is continued for several kilometers (not shown). Thus, the frequency of nucleation events is impacted by the differing assumptions about ice properties and capacitance, 890 and the spread in N i seen for w = 1 m s −1 can be viewed as a frequency difference with a very long period. With sedimentation at −55 • C, Fig. 17 shows that some size distributions happen to be in a period with small crystals present whereas others do not.

895
Although sedimentation only reduces parcel m and A i , maximum parcel N i may be increased over their values without sedimentation by more than a factor of two owing at least in part to faster aerosol freezing at colder temperatures. Nonetheless, in parcels subject to repeated nucleation 900 events, sedimentation reduces time-averaged N i by nearly an order of magnitude and time-averaged A i by even more. D i and ν i experience briefer discontinuities associated with nucleation events, D i dropping and ν i increasing each time new crystals appear. The bullet and polycrystal models de-905 rived here exhibit a lagged transition in ν i after each nucleation event compared with the other ice properties, which can be attributed to evolution between ρ e varying not all with D max < 100 µm (giving minimum ν i ) to ρ e decreasing with D max > 100 µm (increasing ν i only when new crystals grow 910 past 100 µm).
At the greatest w, sedimentation results in D i and ν similar in magnitude to that without sedimentation (e.g., Fig. 19f versus Fig. 19c), but the addition of sensitivity to fall speed increases the spread across N i and A i . Thus, we conclude that with or without sedimentation, a chief effect of varying ice properties is on the size distribution of ice owing to differing ρ e , leading to roughly factor of 2-3 differences in D i and ν i in this parcel framework. We note that these parcel simulations with and without sedimentation generate results that span the range of N i , D i , A i , and ν i observed in situ during SPARTICUS, but we do not attempt any direct comparisons owing to the lack of realism of this simulation framework.

Optical properties
Extinction cross sections, scattering asymmetry parameters, 925 and single-scattering albedos that are consistent with the derived crystal geometries are needed for interactive radiative calculations in cloud-resolving simulations and for calculation of diagnostic fluxes and radiances to be compared with measurements (e.g., . Infrared radiative transfer is dominated by emission, which is affected by particle size, but its sensitivity to crystal shape is minimal (e.g., Holz et al., 2016). However, particle shape does affect the relevant shortwave optical properties substantially. Detailed, accurate calculations of optical properties of non-935 spherical ice particles are generally computationally expensive. Existing databases and calculations of optical properties (e.g., McFarquhar, 2007, 2011;Yang et al., 2013) assume crystal geometries based on sparse measurements and ad hoc assumptions that generally do not match the ge-940 ometries derived here. As an alternative, approaches such as those of Fu (1996Fu ( , 2007 and van Diedenhoven et al. (2014a) can be used to approximate the optical properties of complex crystals based on those of hexagonal prisms that serve as radiative proxies. Here we adopt the van Diedenhoven 945 et al. (2014a) parameterization to approximate the optical properties of our derived crystal geometries. This parameterization provides the extinction cross section, asymmetry parameter, and single-scattering albedo at any shortwave wavelength for ice particles with any combination of crystal vol-950 ume (V = m/ρ i ), A p , and α e and roughness of crystal components. Ice refractive indices are taken from Warren and Brandt (2008).
The van Diedenhoven et al. (2014a) parameterization is based on geometric optics calculations. Accordingly, it as-955 sumes the extinction efficiency (Q e ) to be 2 for all particles and wavelengths. To partly correct this simplification for small particle sizes, here we apply anomalous diffraction theory to adjust Q e at wavelength λ for particles with effective size parameter P = 2πV (m r − 1)/(λA p ) less than π/2, where m r is the real part of the ice refractive index (Bryant and Latimer, 1969). We also apply the edge effect adjustment given by Nussenzveig and Wiscombe (1980). Both adjustments depend on V and A p .
The single-scattering albedo (w s ) is parameterized as a 965 function of V , A p , and α e of the crystal components. All models use α e of bullet arms for this calculation. In the case of the bullet and aggregate models, the arm length is taken to include the cap, and the width is taken as the cap base width where arms comprise only caps. In case of the polycrystal 970 model, we use only bullet arm α e , neglecting the slight increase of w s owing to the thinness of the plates between arms. For the Bucky ball model, the α e of the arms as given in Fig. 6c is limited to values of unity or greater to roughly account for the influence of the compact core where budding 975 arms remain shorter than they are wide. The asymmetry parameter (g) depends on particle V , A p , and α e values, as well as the crystal surface roughness, which may substantially lower g (e.g., Macke et al., 1996;van Diedenhoven et al., 2014a). In the van Diedenhoven 980 et al. (2014a) parameterization, the level of surface distortion is specified by a roughness parameter δ as defined by Macke et al. (1996). The Macke et al. (1996) ray-tracing code perturbs the normal of the crystal surface from its nominal orientation by an angle that, for each interaction with 985 a ray, is varied randomly with uniform distribution between 0 and δ times 90 • . Similar commonly used parameterizations of particle roughness perturb the crystal surfaces using Weibull (Shcherbakov et al., 2006) or Gaussian (Baum et al., 2014) statistics rather than uniform distributions. However, fortunately, the roughness parameter cannot be constrained by the CPI data used here. Laboratory studies demonstrate that the microscopic structure of ice crystals is dependent on the environmental conditions in which they grow (Neshyba et al., 2013;Magee et al., 2014;Schnaiter et al., 2016). Since  2014b) show that a roughness parameter of 0.5 best fit observations, that is the default value we adopt here. For the Bucky ball model, we average core and arm g values, weighted by their relative contributions to total A p (cf. Fu, 2007;van Diedenhoven 1005van Diedenhoven et al., 2015. For the polycrystal model, the arm and plate g values are averaged in the same way. Since the plate-like structures on the polycrystals shown in Fig. 10 appear relatively transparent, we assume smooth surfaces for the plates (i.e., δ = 0).

1010
Calculated Q e , g, and w e are shown in Fig. 22 as a function of crystal D max . Also shown are the optical properties of the six-branch bullet rosette model calculated by Yang et al. (2013). The geometry of the bullet rosettes assumed by Yang et al. (2013) is taken from Mitchell and Arnott (1994) and is 1015 similar to that of Mitchell et al. (1996) shown in Fig. 4. Yang et al. (2013) calculate the optical properties using a combination of improved geometric optics and other methods, which reveals resonances in the extinction efficiencies that are not seen in our results. However, such resonances largely can-1020 cel out when integrated over size distributions (Baum et al., Fridlind et al.: Cirrus ice properties for a size-resolved microphysics model 11 2014). The calculated g values generally increase with size because of increasing α e with size (cf . Fig 4c). At visible wavelengths, g of the bullet, Bucky ball, and aggregate models, as well as the Yang et al. (2013) bullet rosettes, converge at about 0.81 at large sizes. Because of the addition of thin smooth plates to the polycrystal model, its g is generally greater. Aside we note that assuming plates with δ = 0.5 reduces 0.5-µm g by only about 0.01 in the limit of large D max (not shown), indicating that plate aspect ratio is the main cause of g increase. At 2.1 µm, g values increase owing to ice absorption (e.g., van Diedenhoven et al., 2014a). The 2.1-µm g from Yang et al. (2013) is generally lower than our results because w s is generally greater. At a given λ, w s of an ice crystal is mostly determined by the particle effective diam-  (1996) bullet rosettes (see Fig. 4). Figure 24 shows the shortwave optical properties integrated over the model size distributions at −55 • C shown in Fig. 17. Extinction efficiencies generally increase slightly with λ as P decreases and are therefore generally greater 1045 for the cases with sedimentation owing to the smaller crystal sizes. At λ ∼ 2.8 µm, a Christiansen band (Arnott et al., 1995) is present where a combination of strong absorption and refractive indices near or less than unity leads to a decrease in Q e (cf. Baum et al., 2014). The w s is generally greater for cases with sedimentation since these simulations lead to small D eff ∼ 20 µm, whereas D eff produced by simulations without sedimentation range from about 50 to 500 µm, primarily depending on w (cf. Fig. 17). For the same reason, g for cases with sedimentation is generally lower.

6 Discussion and conclusions
In preparation for large-eddy simulations with size-resolved microphysics for a case study of mid-latitude synoptic cirrus observed on 1-2 April 2010 during the SPARTICUS campaign (Muhlbauer et al., 2015), here we use CPI image anal-1060 ysis to develop ice crystal geometries that are physically continuous over the required crystal size range and suitable to calculate internally consistent physical and optical properties. The model to be used employs the Böhm ( , 2004 approach to calculate fall speeds and pairwise collision rates (based on crystal mass m, maximum projected area A, corresponding maximum dimension D, and bodily aspect ratio α) and the van Diedenhoven et al. (2014a) approach to calculate radiative properties (based on crystal mass m, maximum projected area A, and crystal or polycrystal element aspect ratios α e ). Assuming bullet rosettes as a typical geometry, we approximate α as unity (no preferred fall orientation), consistent with adoption of measured (randomly oriented) max-imum dimension D max and projected area A p for physical and optical properties. We then take an approach to estimat-1075 ing mass from CPI image data that begins with derivation of geometric crystal components suitable for calculation of optical properties, based on available ICR measurements. We also use derived α e values in calculation of capacitance for vapor growth. This approach to ice crystal properties offers 1080 an advance over our past, ad hoc approach of using piecewise mass-and area-dimensional relations as a foundation, and then separately assigning aspect ratios based on sparse literature sources (e.g., Avramov et al., 2011;Fridlind et al., 2012;.

1085
Our results using a typical bullet model of rosettes give m(D max ) systematically larger than literature values used in similar past size-resolved microphysics simulations (Sölch and Kärcher, 2010), and A p (D max ) systematically smaller or similar. Taken together, these differences lead to v f greater 1090 by a factor of 1.5-2, and w s and g respectively greater by about 0.2 and 0.05 in the limit of large D max at near-infrared λ. A polycrystal model that estimates side plane growth on bullet rosettes increases v f by only about 15%, indicating that the effect of increased m outweighs that of increased 1095 A p given the relatively ad hoc assumptions made here. In the polycrystal model, side plane growth also increases g by about 0.05, primarily owing to plate aspect ratio.
In parcel simulations with and without sedimentation, differing ice properties lead to factors of 2-4 difference in crys-1100 tal number concentration N i , number-weighted mean diameter D i , total projected area A i , and size distribution relative dispersion ν i . When crystal effective density ρ e is smaller, D i is larger; when ρ e varies with size, ν i is larger. When v f w, faster falling crystals are associated with more fre-1105 quent nucleation events, by roughly 50% at w = 0.1 m s −1 .
Overall, it appears that the main differences between our models and past literature arise from differences in bullet rosette geometry (i.e., single-particle mass) or its representation (i.e., definition of D max ). Where available, A p (D max ) 1110 and arm α e (D max ) appear more similar, by contrast. Based on ad hoc assumptions made here, the chief potential impact of side plane growth could be an increase in g by ∼0.05 in the mid-visible. More detailed observational analysis would be needed to confirm side plane properties assumed here. How-1115 ever, differences between our polycrystal and bullet properties are surprisingly substantially less than the differences between our bullet properties and those in past literature, which may prioritize better establishing the baseline bullet rosette model over working out details of irregular crystal proper-1120 ties.
Evolution of newly nucleated ice crystals may proceed from amorphous shapes to more defined habits (e.g., Baker and Lawson, 2006b;Schnaiter et al., 2016) in a manner that may depend in part on nucleation mode (e.g., Bacon 1125Schnaiter et al., 2016), but observations considered here are inadequate to derive a robust geometric model for D max smaller than roughly 100 µm, as in other recent work (e.g., Erfani and Mitchell, 2016). However, we find that growth from a budding Bucky ball shape versus an idealized 1130 bullet rosette shape could lead to non-negligible differences in normalized capacitance of nearly 0.1 (cf. Fig. 15). If such geometry is important to predicted PSD evolution, deriving a statistically decreasing number of arms with increasing size could be needed to simultaneously represent the evolution of 1135 crystal m, A p , and α e . Or more accurate geometries could be established (e.g., Nousiainen et al., 2011;Schnaiter et al., 2016) and relevant physical and optical properties made appropriately consistent and continuous for modeling purposes. Evident diversity of both small and large crystal properties at 1140 a given D max , even when most rigorously defined, could also be relevant.
It may be the case that uncertainties in ice crystal m and its relationship to morphological properties, which together determine factors such as v f and radiative properties, are not 1145 sufficiently considered in current literature. Single-crystal mass measurements that were made laboriously in studies decades ago (e.g., Kajikawa, 1972;Mitchell et al., 1990) have not been replaced by improved measurements or substantially augmented since that time. In the case of bullet 1150 rosettes, for instance, we are aware of only one unpublished data set comprising 45 crystals, we are aware of no such measurements made at cirrus elevations, and it appears that those ground-level measurements may be biased to fewer branches, as discussed above. From analysis of the SPARTICUS data 1155 here, we can see that such a bias in branch number could likely be correlated with a bias in α e and m. The degree to which a single habit-independent m − A p power law applied to 2DS PSDs leads to accurate calculation of m(D max ) for both anvil and synoptic cirrus crystal conditions may also 1160 warrant additional investigation (cf. Fig. 4e). As discussed by Baker and Lawson (2006a), for instance, particles are not entirely randomly oriented in the petri dish measurement approach used in the SCPP data set; to the extent that nonrandom orientation favors a higher ratio of A p /m on a petri 1165 dish, the derived m(A p ) could be biased correspondingly low when applied to randomly oriented crystal images.
With respect to classification of morphological properties, it also appears to be the case that classification algorithms may give substantially differing results. For instance, 1170 whereas here roughly 80% of crystals with D max greater than 100 µm are unclassified (irregular), the algorithm reported by Lindqvist et al. (2012) classifies more than 50% of crystals as rosettes in a similar mid-latitude cloud. The fact that their study places fewer than 20% of crystals in an irregular class 1175 across tropical, Arctic, and mid-latitude conditions suggests that it is fundamentally different from the algorithm applied here. The fact that unclassified crystals here differ relatively little in derived properties from bullet rosettes (with the possible exception of g, given some relatively ad hoc assump-1180 tions) suggests that algorithms may currently differ in the allowable degree of deviation from a pristine state. It may be useful to establish comparable statistics from differing algo-rithms to allow comparison of circumference or other nonhabit-dependent measures.

1185
Overall, the results obtained here motivate the use of our derived ice properties in comparison with more widely used values in 3D simulations of the April 1-2 SPARTICUS conditions, which can in turn be compared with in situ ice size distribution observations. 1190 Appendix A: Ice crystal models A fundamental geometric element of all ice models considered below is the regular hexagonal column with length L and width W , defined here as twice hexagon side length. In all cases the true mean branch width W is taken as the 1195 mean of the measured widths of all branches divided by a factor of (1 + √ 3/2)/2 to correct for random orientation. Thus, true mean branch width W is about 7% wider than measured mean branch width.
Since branch length measurements extend from crystal 1200 center to the outermost edge of projected randomly oriented branches, the true mean total branch length (including cap or core contributions, depending on the model) is taken as the mean of the measured lengths less one-half of the mean of the measured widths times π/4 (the contribution of randomly 1205 oriented projected base to measured length), all multiplied by a factor of 4/π to account for branch foreshortening by random orientation. Thus, true mean branch length is about 30% longer than measured mean branch length corrected for the contribution of column base projection.

1210
All non-aggregate models (bullet, Bucky ball, and polycrystal) assume six branches, consistent with mean and median number found over all bullet rosettes measurable with the ICR software.
Derived ice properties are supplied as the Supplement.

A1 Bullet model
The bullet model assumes that each hexagonal column has a single cap, and that the six caps meet at a point in the center of the crystal. If the cap is a hexagonal pyramid with a fixed angle β between pyramid edges and the line defining 1220 pyramid height, then cap length L c scales with column width according to Thus, wider branches have longer caps. Here we assume fixed β, and assign a value of 22 • ; generally wider angles 1225 have been assumed in previous work, as discussed further at the end of this section. For the bullet model, total true branch length includes both hexagonal column length L and hexagonal pyramid cap length (L c ). A least squares linear fit of total mean branch 1230 length L+L c to measured maximum dimension (uncorrected for random orientation throughout, see Section 3) gives a line nearly through the origin. Adopting only the slope (cf. Fig. 4a) gives (A2)

1235
A least squares fit of mean branch width W to D max gives (cf. Fig. 4b, µm units) In the limit of zero D max , Equation A3 would give a nonzero branch width. As a simple physical solution, we assume 1240 that branch width is equal to cap base width wherever predicted cap length per Equations A1 and A3 is greater than total branch length per Equation A2, designated as D max,L0 (where branch length L is zero). Thus, for the bullet model, crystals with mean branch width less than D max,L0 ∼50 µm comprise hexagonal pyramids without developed hexagonal columns extending from them; we note that no ICR measurements were possible at such small sizes. With crystal geometry now defined using the bullet model, it is straightforward to calculate the aspect ratio W/(L+L c ), 1250 surface area A s , and mass m for each measured crystal (symbols in Fig. 4c-e). The crystal model derived and used in simulations is that for a corresponding typical crystal with number of branches fixed to six, equal to both mean and median of measured branch numbers (solid lines in Fig. 4c-e), 1255 with mass therefore defined as where ρ i is the bulk density of ice, taken here as 0.917 g cm 3 , and surface area defined as However, the randomly oriented projected area A p is not analytically defined. A least squares fit of measured crystal projected areas to bullet model crystal surface areas results in a line nearly through the origin (cf. Fig. 4d), and this is used with Equation A5 to define model projected area where λ b = 0.107. The linear relationship and slope < 0.25 are consistent with theory for convex particles (Vouk, 1948), as discussed above. If the cap angle β is increased, effective density and pro-1270 jected area ratio increase for ice crystals with D max smaller than about 90 µm. In the limit of small D max , a β of 22 • is selected to give effective density and projected area ratio no larger than that calculated for any measured rosettes (cf. Fig. 5a, b). Calculated fall speeds are not strongly sensi-1275 tive to changes in β because effective density and projected area increase or decrease together. Regarding choice of β, Iaquinta et al. (1995) have noted that a bullet rosette with a six-faced pyramidal end and a 56-• angle between opposing faces has been assumed in past work but cannot fit to form 1280 a multibranched bullet rosette, leading to their adoption of a trilateral pyramidal end as "only an idealized form of the sharp end of natural ice crystals"; we select β values here in the same spirit.

1285
The Bucky ball model assumes that each hexagonal column grows initially from a Bucky ball face. The core is approximated as a sphere with diameter D c of 10 µm, and budding columns are assigned an initial width W min of 4 µm. In order to insure a branch length of zero when D max is equal to 1290 that of a sphere with core diameter D c , a slope is fit to L as a function of D max − D c (Fig. 6a), giving Similarly, in order to insure a branch width of W min when the maximum dimension is equal to the core diameter D c , a 1295 slope is fit to W −W min as a function of D max −D c (Fig. 6b), giving With crystal geometry now defined using the Bucky ball model, it is straightforward to calculate the branch aspect 1300 ratio (L/W ) and mass for each measured crystal. For the canonical crystal with six arms, where D max < D c , then values are those of a sphere with diameter D max and density ρ i . Otherwise, using W and L from Equations A7 and A8, Rigorous calculation of A s for the model crystal with typical six branches is less straightforward. Here we take the ad hoc approach of first estimating the surface area of measured crystals as the total of branches with one end each, neglecting the inner end faces (Equation A5 without the third term that 1310 represents cap surface area). A fit of A p measured to A s,est so estimated gives a slope 0.0921 (Fig. 6d). The model crystal with six arms is then assigned A s (D max ) as a weighted average of estimated A s and that of a sphere with diameter D max , where m r is the ratio of m to that of a sphere with diameter D max , m r,max is the value in the limit of large D max (roughly 0.24), Relative to the bullet model, the Bucky ball model exhibits a stronger increase of W with increasing D max (cf. Fig. 4b  and 6b) and a nearly constant aspect ratio at D max greater than 100 µm (cf. Fig. 4c and 6c). Smooth variation of radiative properties and capacitance in the limit of small D max is achieved with and (Fig. 15)

A3 Aggregate model
The 'aggregate model' is an extension of the bullet model. A least squares fit of mean branch length L+L c to D max gives a line again nearly through the origin. Adopting only the slope as in the bullet model (cf. Fig. 9a) gives The mean and median measured number of arms is twelve, consistent with aggregates primarily of two typical single rosettes. The roughly 30% reduction in slope compared with 1340 single rosettes can be attributed to the overlap of aggregate arms, compounded by random orientation when two crystals create a linearly aligned pair that will be rarely normal to the viewing angle. A least squares fit of mean branch width W to D max gives (cf. Fig. 9b, µm units) 1345 W = 0.0886D max + 44.9.
To handle unphysical branch widths in the limit of zero D max , we again assume that branch width is equal to cap base width wherever predicted cap length per Equations A1 and A16 would be greater than total branch length per Equation A15.

1350
Using this model for aggregates, mass and projected area are simply twice that of bullet rosettes, where ρ i is the bulk density of ice, taken here as 0.917 g cm 3 , and surface area defined as Using Equation A18 with A1, A15 and A16 to calculate A s , measured A p is found to be 10% of calculated A s (cf. Fig. 9d), roughly 1% lower than found for single rosettes using the bullet model, consistent with branch entanglement 1360 that reduces A p but not A s relative to a pair of single rosettes.

A4 Polycrystal model
The polycrystal model is derived for unclassified crystals using plate growth on the bullet model as a basis. When the unclassified crystals are initially assumed to follow the bul-1365 let model, and measured A p is regressed against calculated bullet surface area (A s,b ) following Equation A5, assuming six arms per crystals, the slope is greater than found for the bullet model, consistent with systematically greater projected area than rosettes demonstrated in Fig. 12. We adopt the 1370 ad hoc assumption that additional projected area can be attributed to side plane growth, represented here for simplicity as growth of hexagonal plates. Continuing with the six-arm bullet model as a basis, we further make the ad hoc assumption that a single plate is grown on each arm with sufficient 1375 total plate surface area (A s,p ) to restore A p /(A s,b + A s,p ) to a value near λ b in the limit of large D max . Based on trial and error, taking the foregoing assumptions as a recipe, the following prescription was found to match effective density from Cotton et al. (2012) in the limit of 1380 small D max (Fig. 13e) and median measured A p (D max ) for unclassified crystals at all sizes (Fig. 13f) to the extent possible without exceeding the effective diameter of equivalentsized spheres. First, to increase effective density relative to the bullet model where crystals are entirely truncated caps 1385 (D max < D max,L0 ), β is increased to 25 • . If A s,b is then calculated following Equation A5 for measured crystals, a slope λ i = 0.147 is found (Fig. 13a), larger than λ b = 0.107 found for rosettes using the bullet model for measured rosettes (Equation A6). Next A p /A s is matched to allow zero plate 1390 contribution to surface area where D max < D max,L0 (noting that increased β slightly reduces D max,L0 relative to that for the bullet model) and maximum contribution to surface area within an ad hoc scale length of 2D max,L0 using 1395 and, taking λ p = 0.1 as the ratio of A p to A s for polycrystals (reduced by an ad hoc amount from that for bullets on the basis that A s has increased relatively more than A p , but not so much that effective density exceeds that of equivalent-sized spheres), For the model crystal with six arms, the plate contribution to surface area is then If plate surface area is approximated as twice the face ar-1405 eas (neglecting edge contributions), then per-plate diameter (D p ), defined for D max > D max,L0 , can be calculated from Plate thickness L p , which is neglected in the addition of plate surface area to A s but is included in calculation of plate contribution to crystal mass, is taken as (Pruppacher and Klett, 1997, their In the limit of zero plate size, L p is not permitted to exceed 0.1D p . Thus, for radiative calculations, the maximum 1415 plate aspect ratio α e,p = L p /D p = 0.1 and the bullet arm aspect ratio remains as α e,b = (L + L c )/W . Normalized capacitance is calculated as for a bullet rosette with L(β) and W (β), neglecting the presence of plates, for lack of another obvious strategy.
Plate contribution to polycrystal mass can be calculated as Total mass is then m p plus bullet mass following Equation A4 with β = 25 • , and total projected area A p = 1425 λ(A s,p + A s,b ).
effective density of small ice particles obtained from in situaircraft observations of mid-latitude cirrus, Quart. J. R. Meteorol. Soc., 139, 1923-1934, doi:10.1002/qj.2058 Evaluation of several A-Train ice cloud retrieval products with in situ measure-1525 ments collected during the SPARTICUS campaign, J. Appl. Meteorol. Clim., 52, 1014-1030, doi:10.1175/JAMC-D-12-054.1, 2013 L.: Developing and bounding ice particle mass-and area-dimension expressions for use in atmospheric models and remote sensing, Atmos. Chem. Phys., 16, 4379-4400, doi:10.5194/acp-16-4379-2016supplement, http://www.atmos-chem-phys.net/16/4379/2016/ acp-16-4379-2016-supplement.pdf, 2016 project. Phase 1: The critical components to simulate cirrus initiation explicitly, J. Atmos. Sci., 59, 2305-2329. particle tracking, Quart. J. R. Meteorol. Soc., 136, 2074-2093, doi:10.1002/qj.689, 2010: Process-oriented large-eddy simulations of a midlatitude cirrus cloud system based on observations, 1990. Twohy, C. H., Schanot, A. J., and Cooper, W. A.: Mea-surement of condensed water content in liquid and ice clouds using an airborne counterflow virtual impactor, J. Atmos. Oceanic Technol.,14,[197][198][199][200][201][202]doi:10.1175/1520-      . Bullet model: measured and calculated properties of imaged bullet rosettes with six arms (black symbols), fewer than six arms (blue symbols), and more than six arms (red symbols). Line types indicate derived ice properties as follows (see legend in panel e): a sphere, a six-arm bullet rosette per the bullet model (see Section 4.1 and Appendix A1), five-arm rosettes from Mitchell et al. (1996), and cirrus crystals from Heymsfield et al. (2002). Also shown is the habit-independent m − Ap relation derived by Baker and Lawson (2006a).  5. Bullet model: measured and calculated properties of imaged ice crystals (red symbols) emphasizing the transition to the smallest sizes. Effective density and projected area for bullet rosettes with ICR measurements (a, b), and projected area for all bullet rosettes identified (c, including those not measurable with the ICR software), and for all crystals imaged during the April 1-2 flights (d). Within Dmax doubling bins, the median of measurements is shown where a bin contains more than 100 measurements (thick solid line segments, c and d only). Other line types indicate derived ice properties as follows (see legend in panel b): a sphere, a six-arm bullet rosette per the bullet model (see Section 4.1 and Appendix A1), five-arm rosettes from Mitchell et al. (1996), and cirrus crystals from Heymsfield et al. (2002) and Cotton et al. (2012). Also shown (see legends in c and d): fits to measured areas of bullet rosettes and budding bullet rosettes from Lawson et al. (2006a), and polynomial fits from Erfani and Mitchell (2016) for synoptic cirrus crystals at −55 to −65 • C (coldest range fitted) and −20 to −40 • C (warmest, see text).      Erfani coldest Erfani warmest Figure 14. Ice crystal fall speeds at 350 mb and 233 K for derived ice properties as follows (see legend): a sphere, six-arm rosettes following the bullet and Bucky ball models, twelve-arm aggregates following the bullet model, the polycrystal model, five-arm rosettes from Mitchell et al. (1996), and cirrus crystals from Heymsfield et al. (2002) and from Erfani and Mitchell (2016) assuming ice crystal properties at −55 to −65 • C (coldest range fitted) and −20 to −40 • C (warmest, see text).  Fig. 14. In the absence of specified αe for some or all crystal sizes, a constant value is taken for Mitchell et al. (1996) and Heymsfield et al. (2002) ice properties (see text).   Figure 18. Simulated ice crystal number concentration as a function of parcel distance from initiation at −40 • C, with ice properties as in Fig. 14 (see legend) and updraft speeds of 1, 0.1, and 0.01 m s −1 without sedimentation (top row) and with sedimentation (bottom row). Parcel level corresponding to −55 • C corresponding to size distributions in Fig. 17 is shown as dotted yellow line.