For each bullet rosette measured with the ICR software, Fig. a shows mean branch length vs. measured Dmax. 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 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 Dmax is reasonably fit by a line passing through the origin.

For the same crystals, Fig. b and c show mean W, and the ratio L/W=αe, 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 minimum and maximum branch projected widths to the maximum (equivalent to the 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 6 (out of 4–10 measured), consistent with recent analyses from tropical and Arctic field campaigns . Rosettes with more branches are seen to have systematically smaller W and larger αe, consistent with competition for vapor during growth. However, a simple least squares fit of W to Dmax gives W>Dmax when extrapolated to small crystal size, which is not physical; unfortunately, 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 measured properties smoothly to sizes smaller than measured. A similar approach was taken by to improve calculated cirrus crystal fall speeds over those obtained from independently derived mass–dimensional and area–dimensional relations. We first assume branch width for rosettes consistent with the six-rosette model considered in , 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. b, and resolves at least gross discrepancy of W>Dmax.

Figure c shows that adopting this bullet model results in a smooth increase in branch aspect ratio αe=L/W from smallest to largest sizes, suitable as a basis for calculating optical properties. L/W is constant at the smallest sizes, where 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.. As shown, the relationship of branch aspect ratio to Dmax also agrees with that used by in derivation of the mass–dimensional relation for Dmax>100 µm listed in Table  and discussed further below.

The model assumes that all bullets are at 90∘ angles to one another, giving true maximum dimension of 2L, which is ∼ 40 % greater than Dmax shown in Fig. a. Measured Dmax being a randomly oriented value can account for less than 30 % discrepancy. Another source of difference is the commonly seen deviations of arm locations from 90∘ separations, which can only decrease Dmax from 2L. Since a more quantitative explanation is beyond the scope of this initial study, we adopt the randomly oriented Dmax as our only defined maximum dimension, an assumption that has also been made in past studies using two-dimensional images e.g.,.

The bullet model described above now allows calculation of crystal surface area (As) and m (see Appendix A1 for details). To calculate m from the geometrical dimensions, we assume ice bulk density (ρi) 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 Ap of a six-branch rosette are seen to reasonably represent the scatter of individual crystal properties (solid lines in Fig. e and f). The ratio of measured Ap to calculated As is found to be about 0.11 (Fig. d), smaller for these concave particles than the Ap/As of 0.25 for convex shapes, consistent with theoretical results and reasonably independent of Dmax across measured sizes.

In Fig. e derived m(Dmax) is compared with power-law relations from previous literature that have been used in similar bin microphysical schemes (Table ). To our knowledge, only one unpublished data set has provided direct measurements of bullet rosette mass, consisting of 45 crystals with a range of 2–5 arms as reported by , but that data set is not the basis of commonly used relations. As used in , for instance, the relation is based on calculation of effective density from a combination of ice water content and particle size distribution measurements; coefficients in Table  cf. are calculated from their Eq. (22), based on crystals with Dmax of 200—2000 µm fit in their Fig. 15. As also used in , combined crystal volume expressions with size-dependent bulk densities of ∼ 0.78 g cm-3 to obtain an m-Dmax relation for crystals with Dmax of 200–1000 µm (Table  values are taken from their Eq. 32); for crystals smaller than 100 µm, proposed a mass–dimensional relation using ad hoc estimates of crystal mass (Table  values are taken from their Table 3).

The difference between our calculated m(Dmax) and that from is roughly a factor of 4 at measured crystal sizes, which results in a similar discrepancy in fall speeds and effective diameters, as shown below. We can attribute lower m in to four factors: (i) L is substantially shorter based on the approximation Dmax=2L cf.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 extension Dmax/2 (see Fig. b), (iii) five branches are assigned instead of six found here, and (iv) ρi is ∼ 0.78 g cm-3 instead of 0.917 assumed here. All else being equal, increasing their branch number and ρi would together increase m by only about 40 %, but m scales roughly linearly with L and geometrically with W. The trilateral pyramid ends taken from 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 , available for Dmax>100 µm (Fig. c), suggests that differences in m are primarily attributable to differing approaches to defining Dmax. 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 natural crystals.

Our calculated m(Dmax) is also nearly a factor of 2 greater than that from for ice particle ensembles (all habits, dominated by bullet rosettes) measured over the same Oklahoma location. In , Dmax is taken from 2-D Cloud and Precipitation 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 (ρe, the density of a sphere with diameter Dmax) to Dmax. Whereas our approach is subject to uncertainty in ICR measurements and assumed ρi, the 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 , but the bin-wise uncertainty in particle size distribution measurements are generally unquantified; we consider it beyond the scope of this study to undertake the detailed analysis required to resolve such differences. Although it is not used in the m-Dmax relationship adopted by and listed in Table 1, also derive a typical ρi of 0.82 ± 0.06 g cm-3 for bullet rosettes based on independent 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(Dmax) is substantially greater than that previously used in studies with size-resolved microphysics, our measured Ap(Dmax) is similar or smaller. The relationship of Ap and Dmax derived by their Table 1 independently from m(Dmax) for five-branched bullet rosettes in a manner similar to that here, is nearly identical to ours (cf. Fig. e). The less widely available Ap–Dmax 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 , which are cited from but not apparent in .

Figure  allows a closer examination of the extrapolation from manually measured rosette properties (Dmax>200µm) 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 shape obtained from a 2DS probe in mid-latitude cirrus, derived a mean ρe 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 ρe derived here happens to exhibit a similar behavior (Fig. a), where the discontinuity using our bullet model represents the transition to truncated branch caps. derived polynomial m-Dmax relations for synoptic cirrus clouds warmer than -40 ∘C from single-particle measurements of m, Dmax and Ap obtained during the 1985–1987 Sierra Cooperative Pilot Project (SCPP) , or by applying a habit-independent m-Ap relation derived from the SCPP data set , shown in Fig. e, to 2DS measurements obtained during 13 SPARTICUS flights (at colder temperatures). Although the SCPP data set does not contain bullet rosettes or spatial crystals , report that ice water content derived by applying that habit-independent m-Ap relation to a combination of tropical anvil and synoptic cirrus measurements agreed with CVI measurements to within 20 %. At Dmax<100 µm, m values were calculated from CPI measurements of Ap and α assuming hexagonal column geometry cf.their Appendix B, and effective densities are similar to those derived here. At larger sizes and especially colder temperatures, effective densities are smaller than derived here, consistent with the m-Ap relation giving lower per-particle m than derived here.

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, Ap is reported for all imaged crystals and can be directly compared with the bullet model. Analogous to ρe but dimensionless, the measured ratio of Ap to that of a sphere with diameter Dmax can also be compared with the bullet model. Figure b shows that literature power law relations can become unphysical for the smallest particle sizes (projected areas greater than for a sphere of diameter Dmax); to correct the greatest deviations for the purposes of parcel calculations below, we adopt a constant ratio of Ap to sphere projected area where Dmax<100µm when using relations. When considering all rosettes automatically identified (not all of which were measurable using the Ice Crystal Ruler), the bullet model Ap(Dmax) agrees quite well with median measurements and with m-Ap relations for bullet rosettes and budding bullet rosettes from (Fig. c). However, when considering all crystals (Fig. d), there is a wider range of Ap(Dmax) and the bullet model underestimates median Ap(Dmax), as addressed further below; the polynomial m-Ap fit from the SCPP data set (their warmest-temperature fit) and the m-Ap power law agree best with the full data set where Dmax>100µm.

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 observed in laboratory grown ice and ice-analog crystals , 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 diameter 10 µm and then assume that arms emerge with initial width 4 µm. If we assume that L falls linearly to zero at Dmax equal to the core dimension (Fig. a) and W correspondingly falls linearly to its minimum initial width (Fig. b), then branch αe is relatively constant near the mean observed (Fig. c). As and m can now be calculated using this Bucky ball model, except that Ap of the smallest crystals must be interpolated to bridge the geometry of a sphere (Dmax< core diameter) and that of a rosette; to do this, we calculate a mass-weighted sum of Ap obtained from the linear relation in Fig. d and that of a sphere (see Appendix A2). Thus, as m converges to that of a sphere, so does Ap. Results are similar to those of the bullet model at larger particle sizes (Fig. d-f), with m(Dmax) still larger than previous estimates and Ap(Dmax) still similar or smaller.

However, using this simplified Bucky ball model, a developing six-arm rosette has a systematically smaller ρe and Ap than it did with the bullet model (Fig. ). Although this particular version of a Bucky ball model, with only six arms even at small sizes, gives substantially smaller Ap(Dmax) than measured for automatically classified rosettes at small Dmax (Fig. c), it does serve to provide quite a close match to the minimum area relative to that of a sphere over the full particle data set (Fig. d), and is therefore included in parcel calculations below. In reality it seems likely that not all budding arms grow evenly. For instance, 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 substantial 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 complexity here, we have simply assumed six arms throughout.

We return now to the distribution of habits during the 1–2 April flights, and consider the properties of crystals in the observed cirrus deck that are not identified as bullet rosettes. The rosettes are most common in the upper cloud regions at temperatures colder than -40 ∘C (Fig. ), 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, 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. ; see Appendix A3), where the mean and median branch numbers are found to be 12 per aggregate, consistent 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 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 a plate growth regime (Fig. ), a phenomenon well documented in previous cirrus field observations and laboratory measurements . In the lower cloud regions at temperatures warmer than -40 ∘C, modified bullets have been described as mostly “platelike polycrystals, mixed-habit rosettes, and rosettes with side planes” , where side plane growth on columns may be attributable to facet instability on prism faces .

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 cobwebs that lead to blocky ice particle shapes (e.g., Fig. ). In such a process, particle m could increase without rapid expansion of particle Dmax. 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 from the rosette center, consistent with an important role for defect and dislocation sites . Toward cloud base, sublimation then increasingly rounds crystal edges (Fig. ). Rosettes that did not enter a side plane growth stage appear now with rounded arms that can still 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 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 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 than rosette crystals (Fig. ). 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 follows.

We first calculate the additional projected area that can be attributed to side plane growth. Considering all unclassified crystals, a fit of measured Ap to calculated bullet surface area (based on measured maximum dimension and assuming a bullet model rosette with six arms) yields a slope of 0.15 (Fig. a), 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 projected 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. c, solid line). For this crude representation of plate-like growth on the bullet model, the calculated crystal properties agree reasonably well with effective density in the limit of small Dmax (Fig. e) and with the area ratio as a function of Dmax over all unclassified crystals (Fig. f) 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 Lc (see Appendix A4 for details). Where Dmax>100 µm, the resulting polycrystal model Ap agrees closely with the fit for warmest-temperature synoptic cirrus, but resulting m is now also correspondingly greater than and further from than in the bullet model (cf. Fig. a).

The foregoing results for this polycrystal model are dependent upon the underlying bullet model assumed, the assumed ratio of As to Ap, 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 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 further 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.

Figure  shows a point calculation of fall speeds (vf) at 350 mb and 233 K for comparison with their Fig. A1. Our bullet model gives vf(Dmax) values that are more than a factor of 1.5–2 greater than derived from Mitchell and Heymsfield ice properties and used by in a microphysics model similar to ours. Increasing crystal m calculated from the literature by a factor of 0.917/0.78 can account for relatively little of the difference (not shown), indicating that the main differences are attributable to crystal geometries. In the case of Mitchell properties, as discussed above, the main difference may be traceable to differing Dmax definition used to calculate m, whereas Ap(Dmax) is very similar. In the case of Heymsfield properties, m(Dmax) is closer to ours but Ap(Dmax) is also larger, a factor that should be relatively more easily resolved in future studies since both Ap and Dmax can be directly measured. As shown in Fig. c, for instance, an m-Ap relation from earlier mid-latitude cirrus measurements agrees well with SPARTICUS rosette measurements and with our bullet model. At the warmest temperatures considered by , Ap(Dmax) is similar to or even greater than the bullet or polycrystal models but substantially lower m(Dmax) leads to substantially lower vf(Dmax); the trend toward greater decrease in m than Ap with decreasing temperature 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 results in vf values that appear similar to those of and are also within roughly 10 % of those calculated using the method described in (not shown). However, our aggregate model gives fall speeds roughly one-third reduced from similar-sized bullet model ice, which is a substantially larger difference than that using Heymsfield ice properties for aggregates and their rosettes shown in . We can trace this greater difference in part to substantially larger m(Dmax) derived here, as shown above. Overall, we conclude from comparison of our results with those of that the precise method of calculating vf as a function of m and Ap appears to be responsible for relatively little spread, but differences in ice properties themselves (m and Ap) introduce vf(Dmax) differences that are substantially larger than expected, as discussed further below.

Owing to the dependence of parameterized capacitance on bullet arm aspect ratio alone (see Sect. ), capacitance differences are nearly negligible for crystals larger than ∼ 400 µm across all bullet models derived here, in sharp contrast to factor of 2 differences in fall speed at such sizes. Because assumed or derived bullet arm aspect ratios vary most where Dmax 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 ice properties are similar to ours where Dmax>100 µm (see Fig. ), 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 , taken here as representative of polycrystals with unspecified aspect ratios. A similar assumption would be required for ice properties; since parcel simulations are also not configured for changes in ice crystal properties during a single simulation, we omit ice properties from the remaining calculations.

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 Sect. , 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 of in-cloud w from the beginning of the first flight examined here to the end of the second flight (Fig. ). Additionally, 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 a similar framework has been used to test cirrus models , 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  shows the ice particle size distribution (PSD) for each simulation at -55 ∘C, which is close to the average of in-cloud temperatures observed during the 1–2 April flights. Figures – show the ice crystal number concentration (Ni), number-weighted mean diameter (Di), total projected area (Ai), and relative dispersion (ν), respectively, as a function of parcel ascent distance. In the absence of sedimentation, ice mass is essentially distributed across differing crystal sizes depending upon w and ρe. The magnitude of w primarily determines Ni: when w is strongest, vapor growth competes least with nucleation, resulting in the greatest Ni (Fig. a). Nucleated 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.,. The Heymsfield ice properties give roughly a doubling of Ni relative to other ice properties, owing to the densest small ice 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 Ni, leads to the correspondingly smallest Di (Fig. a) and the greatest Ai (Fig. a). Where Ni is insensitive to ice properties, the sensitivity of Di and Ai 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 2 range of Di at lowest w and nearly a factor of 4 range of Di at highest w. Whereas Di is variable with ice properties, Ai at all w falls into two groups: Mitchell and Heymsfield properties, with relatively large Ai, and all other ice properties including spheres and our models derived here, with Ai systematically smaller by roughly a factor of 3 at all w. Dispersion (ν) exhibits up to factors of 2–3 difference (Fig. e). The ice properties associated with the lowest effective density (Mitchell) have the greatest Di and ν. However, at the lowest w the Bucky ball model exhibits substantially greater Di but similar ν as spheres, which can be attributed to a weak dependence of ρe on Dmax at Dmax>100 µm that is more similar to spheres than other ice models (Fig. ).

In summary, in the simple case of a non-sedimenting parcel, differing ice property assumptions lead to a factor of 2 difference in Ni and factor of 3 in Ai. Up to a factor of 2 increase in ν is also induced by ice properties that exhibit a trend in ρe across the relevant size distribution relative to ice properties with constant ρe. Differences in ρe across ice properties considered here (regardless of trend) also lead to factors of 2–3 difference in Di.

When sedimentation is included with an assumed parcel depth of 100 m following , results are largely unchanged at the strongest w since vf≪w (cf. Fig. ); the only notable change is roughly a factor of 2 reduction in Ni by -55 ∘C, seen primarily as a uniform downward shift of the PSDs between Fig. a and Fig. b. However, at w≪1 m s-1, the parcel behavior changes rather dramatically because sedimentation reduces 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. d) and at least 10 times more frequently when w=0.01 m s-1 (Fig. f). At intermediate w, nucleation occurs roughly 50 % less frequently for the slowest 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, and the spread in Ni 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.  shows that some size distributions happen to be in a period with small crystals present whereas others do not.

Although sedimentation only reduces parcel m and Ai, maximum parcel Ni may be increased over their values without sedimentation by more than a factor of 2 owing at least in part to faster aerosol freezing at colder temperatures. Nonetheless, in parcels subject to repeated nucleation events, sedimentation reduces time-averaged Ni by nearly an order of magnitude and time-averaged Ai by even more. Di and νi experience briefer discontinuities associated with nucleation events, Di dropping and νi increasing each time new crystals appear. The bullet and polycrystal models derived here exhibit a lagged transition in νi after each nucleation event compared with the other ice properties, which can be attributed to evolution between ρe varying not all with Dmax<100 µm (giving minimum νi) to ρe decreasing with Dmax>100 µm (increasing νi only when new crystals grow past 100 µm).

At the greatest w, sedimentation results in Di and ν similar in magnitude to that without sedimentation (e.g., Fig. f vs. Fig. c), but the addition of sensitivity to fall speed increases the spread across Ni and Ai. 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 ρe, leading to roughly factor of 2–3 differences in Di and νi in this parcel framework. We note that these parcel simulations with and without sedimentation generate results that span the range of Ni, Di, Ai, and νi observed in situ during SPARTICUS, but we do not attempt any direct comparisons owing to the lack of realism of this simulation framework.

Extinction cross sections, scattering asymmetry parameters, 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.,. However, particle shape does affect the relevant shortwave optical properties substantially. Detailed, accurate calculations of optical properties of non-spherical ice particles are generally computationally expensive. Existing databases and calculations of optical properties e.g., assume crystal geometries based on sparse measurements and ad hoc assumptions that generally do not match the geometries derived here. As an alternative, approaches such as those of and 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 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 volume (V=m/ρi), Ap, and αe and roughness of crystal components. Ice refractive indices are taken from .

The parameterization is based on geometric optics calculations. Accordingly, it assumes the extinction efficiency (Qe) 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 Qe at wavelength λ for particles with effective size parameter P=2πV(mr-1)/(λAp) less than π/2, where mr is the real part of the ice refractive index . We also apply the edge effect adjustment given by . Both adjustments depend on V and Ap.

The single-scattering albedo (ws) is parameterized as a function of V, Ap, and αe of the crystal components. All models use αe 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 model, we use only bullet arm αe, neglecting the slight increase of ws owing to the thinness of the plates between arms. For the Bucky ball model, the αe of the arms as given in Fig. c is limited to values of unity or greater to roughly account for the influence of the compact core where budding arms remain shorter than they are wide.

The asymmetry parameter (g) depends on particle V, Ap, and αe values, as well as the crystal surface roughness, which may substantially lower g e.g.,. In the parameterization, the level of surface distortion is specified by a roughness parameter δ as defined by . The ray-tracing code perturbs the normal of the crystal surface from its nominal orientation by an angle that, for each interaction with 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 or Gaussian statistics rather than uniform distributions. However, and demonstrated that the same roughness parameter value defined through a Weibull, Gaussian or uniform distribution represents very similar crystal microscale surfaces and yields largely equivalent scattering properties. Unfortunately, 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 . Since and 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 Ap cf.. 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.  appear relatively transparent, we assume smooth surfaces for the plates (i.e., δ=0).

Calculated Qe, g, and we are shown in Fig.  as a function of crystal Dmax. Also shown are the optical properties of the six-branch bullet rosette model calculated by . The geometry of the bullet rosettes assumed by is taken from and is similar to that of shown in Fig. . 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 cancel out when integrated over size distributions . The calculated g values generally increase with size because of increasing αe with size (cf. Fig. c). At visible wavelengths, g of the bullet, Bucky ball, and aggregate models, as well as the 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. Additionally, we note that assuming plates with δ=0.5 reduces 0.5 µm g by only about 0.01 in the limit of large Dmax (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.,. The 2.1 µm g from is generally lower than our results because ws is generally greater. At a given λ, ws of an ice crystal is mostly determined by the particle effective diameter Deff=3V/(2Ap) . Figure  shows Deff as a function of Dmax. For the bullet rosettes, Deff(Dmax) is generally substantially smaller than for our models, which is consistent with our generally greater V and smaller Ap compared to the bullet rosettes (see Fig. ).

Figure  shows the shortwave optical properties integrated over the model size distributions at -55 ∘C shown in Fig. . Extinction efficiencies generally increase slightly with λ as P decreases and are therefore generally greater for the cases with sedimentation owing to the smaller crystal sizes. At λ∼2.8 µm, a Christiansen band is present where a combination of strong absorption and refractive indices near or less than unity leads to a decrease in Qe cf.. The ws is generally greater for cases with sedimentation since these simulations lead to small Deff∼20µm, whereas Deff produced by simulations without sedimentation range from about 50 to 500 µm, primarily depending on w (cf. Fig. ). For the same reason, g for cases with sedimentation is generally lower.