## 1. Introduction

Tropospheric ice clouds have a major role in the radiative balance of the earth–atmosphere system. On one hand, they reflect solar shortwave (SW) radiation back to space, weakening the SW radiative heating of the surface. On the other hand, they absorb longwave (LW) radiation emitted by the surface and the lower atmosphere, decreasing the amount of LW radiative cooling. Due to the low temperature of these clouds, their own LW emission to space is typically weak. Overall, their SW cooling and LW warming impacts are almost balanced; the global net cloud forcing by high, optically thin clouds is estimated to be only about 2 W m^{−2} (Hartmann et al. 1992).

Compared to warm clouds composed of liquid water droplets, ice clouds are more challenging to model because of the more complex ice-phase microphysics and the nonspherical shapes of the ice crystals. The latter means that the ice content is not unambiguously connected to the optical cross section, and the optical properties are challenging to compute. Additional complications arise from the possible preferred orientations of ice crystals, the fact that crystal shapes are more complicated than the idealized models used to represent them, and the temporal evolution of shapes due to deposition, sublimation, and aggregation.

Knowledge of ice crystal shapes is thus a key issue in modeling ice–cloud radiative interactions. There are many ways to obtain such information, none of which is sufficient on its own. Remote sensing provides reasonable spatial and temporal resolution, but retrieving crystal shape distributions is a severely underconstrained inversion problem. In situ observations of crystal habits, such as the ones used in the present study, provide detailed information at a specific location and time, but are severely lacking in coverage.

The smallest ice crystals are particularly difficult to study since their shapes cannot be accurately resolved from currently used cloud-particle probes. Yet, these small ice crystals may have a major contribution to the radiative impact of ice clouds, and thus potentially on climate (Mitchell et al. 2008). Some new microphysical probes, such as the Small Ice Detector (SID) probes (Kaye et al. 2008), could alleviate this problem by measuring scattering patterns rather than imaging crystals. Scattering patterns are sensitive to shape even for wavelength-scale crystals, whereas reliable imaging requires the particle sizes to be considerably larger than the wavelength. Deriving the three-dimensional shape from scattering patterns is, however, likely to be even more difficult than deriving it from two-dimensional silhouettes. Further, since the measured scattering patterns do not cover the whole scattering-angle range, the measurements do not provide the optical properties of crystals directly. However, they should provide usable references for model simulations.

This study focuses on determining the shapes and single-scattering properties of small quasi-spherical ice crystal in tropical cirrus, where “small” hereafter applies to ice crystals with maximum dimension *D*_{max} < 100 μm. These crystals appear roundish and generally nonhexagonal and can be characterized as irregular in the sense that their shapes cannot be described by a simple regular function. Prior observations have consistently shown that small crystals exhibit quasi-circular shape regardless of the formation mechanisms or geographical location (e.g., McFarquhar and Heymsfield 1996; Korolev et al. 1999; Nousiainen and McFarquhar 2004). However, there has yet to be a study to examine whether the characteristics of these quasi-spherical particles vary in different geographical locations. This paper uses the techniques introduced by Nousiainen and McFarquhar (2004) to characterize the shapes and single-scattering properties of small ice crystals in tropical cirrus, and thereafter compares the properties against those derived by Nousiainen and McFarquhar (2004) for midlatitude cirrus. The relevance for climate considerations is estimated based on the criteria given by Vogelman and Ackerman (1995): for asymmetry parameters around 0.8, about a 2% or 5% change in *g* at cloud optical depths of 12 and 2, respectively, is deemed necessary for a significant impact in radiative fluxes.

The in situ observations of the ice crystals are described in section 2. Section 3 summarizes the light scattering theory, statistical shape model, and the statistical shape analysis. The results are given in section 4, followed by the discussion and conclusions in section 5.

## 2. Microphysical data

A comprehensive dataset describing tropical cloud systems and their environmental setting was collected during Tropical Warm Pool International Cloud Experiment (TWP-ICE) campaign over Darwin, Northern Australia, in January and February of 2006 (May et al. 2008). The Scaled Composite’s *Proteus*, a high-altitude aircraft, made measurements of cloud microphysics during TWP-ICE. In this study, images obtained by the Stratton Park Engineering Company’s (SPEC Inc.) Cloud Particle Imager (CPI) on board the *Proteus* on three flights conducted on 27 January, 29 January, and 2 February 2006 were used to characterize the shape of the small quasi-spherical ice crystals. These flights were selected for analysis because all of the cloud microphysical probes were operational during these flights (McFarquhar et al. 2007).

McFarquhar et al. suggested that the shattering or bouncing of ice crystals off the inlet or protruding shroud of one of the microphysical probes used during TWP-ICE, the Cloud and Aerosol Spectrometer, led to artificial amplifications in the small ice crystal concentrations. Because the CPI, whose images form the basis of the analysis presented here, also has protruding components on which large ice crystals may shatter, the potential impact of shattering on the CPI images was investigated and is discussed in the appendix.

Table 1 summarizes the flight intervals that were analyzed for this study. The horizontal legs on 27 and 29 January were flown in a very different environment than the spiral ascents and descents executed on 2 February. On 2 February, an anvil produced from a convective tower that formed off the Tiwi Islands was sampled a couple of hours after it reached its maximum intensity of about 10–15 dB*Z* at a height of 12 km. The *Proteus* flew a pair of spiral descents and an ascent (t2, t3, and t4) in a trailing anvil that dissipated rapidly due to low humidity (McFarquhar et al. 2007). Thereafter the *Proteus* moved southeastward and conducted two more ascents and a descent behind a different parent cell (t5, t6, and t7). The observations made on 27 January were conducted in a moister ambient environment. These observations were made in aged cirrus bands associated with a low pressure system south of Darwin (McFarquhar et al. 2007). Three legs were analyzed that were flown in very thin cirrus at 15 km (t8_{a}), in thicker cirrus between 13 and 14 km in the presence of a generating cell (t8_{b}), and during a descent through the cirrus (t8_{c}). The horizontal legs analyzed from 29 January (t1_{a}–t1_{h}) were flown through cirrus bands of varying ages associated with a deep quasi-stationary low over the central Northern Territory (Um and McFarquhar 2009) and were flown at altitudes ranging from 11 to almost 15 km.

Summary of the TWP-ICE flights from which the microphysical data used in the study originates.

In this study two-dimensional high-resolution (2.3 μm) CPI images of ice crystal silhouettes were analyzed. The CPI, which can image and size particles with maximum dimensions larger than about 20 μm well, was run in a trigger mode to maximize the number of images acquired. Although many large particles had pristine shapes, almost all crystals with *D*_{max} < 100 μm were quasi-circular. Only a small fraction of the small quasi-spherical ice crystals were considered with about 1600 images selected for shape analysis, about 400 of which fell outside the time intervals eventually chosen for analysis. Example images that passed the selection criteria described below are shown in Fig. 1.

The image selection strategy was designed to provide an unbiased sample of manageable size. Every attempt was made to keep the selection criteria as close as possible to that of Nousiainen and McFarquhar (2004). The selection process included an objective and a subjective step. In the objective step, the following criteria were imposed: 1) the crystals had to have *D*_{max} between 20 and 100 μm; 2) the focus of the instrument had to be greater than 25%; and 3) a maximum 2% of the particle area could reside out of the boundary of the region of interest (ROI).

Images from the intervals of interest that satisfied these conditions were then manually inspected to choose images for analysis. At this subjective phase, the following criteria were used: 1) the ratio of the maximum dimension and the orthogonal width (aspect ratio) had to be below ~2; 2) the crystal was required to be compact and irregularly shaped (not hexagonal, nor a fragment or aggregate: one or two quasi-planar facets were allowed if the particle was otherwise rounded); 3) the image quality had to be sufficient for the perimeter extraction (this typically meant that there were no diffraction-induced features distorting the perimeter); 4) the shapes had to be starlike [i.e., the particle perimeter must be a single-valued function of polar angle from an origin inside the particle (here the origin was at the center of mass and the number of nonstarlike quasi-spherical ice crystals encountered was very small)]; and 5) the shapes had to be asymmetric. The last criterion was due to the difficulty of reliably distinguishing some regular ice crystals habits from quasi-spheres when they were very small owing to blurring of the crystal edges.

From the crystals that passed both the objective and subjective criteria, the first five images from each starting minute were chosen for the shape analysis. The main difference in the selection criteria from that of Nousiainen and McFarquhar (2004) is that an objective step is used to replace some of their subjective conditions. In addition, it is possible that the subjective impressions evolve over time and therefore images now considered too pristine might have been accepted in Nousiainen and McFarquhar, or vice versa. However, it is our belief that any subjective impacts on the shape statistics derived are small.

## 3. Theoretical aspects

### a. Light scattering theory

*I*,

*Q*,

*U*, and

*V*(Bohren and Huffman 1983). The Stokes parameters for the incident and scattered radiation are related bywhere subscripts

*i*and

*s*stand for the incident and scattered light, respectively;

*C*

_{sca}is the scattering cross section,

*d*is the distance from the scatterer, and

*P*are the elements of the so-called phase matrix

_{ij}*C*

_{sca}and

The scattering cross section describes the total scattered power in terms of a normal surface area upon which an equal amount of radiative power is incident. Similarly, the absorption cross section *C*_{abs} specifies the total absorbed power. The extinction cross section, *C*_{ext} = *C*_{sca} + *C*_{abs}, describes the total power removed from the incident radiation. The relative strength of scattering and absorption is customarily specified by the single-scattering albedo, *ϖ* = *C*_{sca}/*C*_{ext}.

*P*

_{11}element, also called the scattering phase function, specifies the normalized angular distribution of scattered intensity for unpolarized incident radiation. This important quantity can be conveniently characterized by the asymmetry parameter

*g*, given bywhere

*θ*is the scattering angle, that is, the angle between the propagation directions of the incident and scattered light, and Ω is a solid angle.

_{s}*x*of interest, defined here asFor the ice crystals with

*D*

_{max}> 20 μm considered here the size parameters are about 100 or more at visible wavelengths. This allows use of the ray-optics approximation with good accuracy (Mishchenko and Macke 1999) for determining the single-scattering properties. The ray-optics solution consists of the sum of two parts: the geometric optics and diffraction (van de Hulst 1981). Accordingly, the single-scattering quantities also consist of two parts; for example,andwhere the superscripts

*D*and

*G*refer to the diffraction and geometric optics, respectively (Muinonen et al. 1996). Further,

*A*〉 is the ensemble-averaged cross-sectional surface area.

### b. Gaussian random sphere geometry

The shapes of quasi-spherical ice crystals are modeled here using the Gaussian random sphere geometry (e.g., Muinonen et al. 1996; Muinonen 2000). The use of this statistical shape model has several advantages. First, three-dimensional model shapes can be generated from the shape statistics of the two-dimensional ice crystal images. This results from the assumption of statistical isotropy of the shapes (shape statistics are independent of the viewing direction), which also renders the generated model shapes randomly oriented. No preferred orientation has been noted for the small quasi-spherical ice crystals in past studies or in the analysis here. Second, large ensembles of model shapes can be generated randomly and obey the specified shape statistics. Third, the shape statistics can often be parameterized to a good accuracy by using only a few free parameters, so the shapes of ice crystals can be specified with a very small number of parameters. For example, a two-parameter approach is used here.

*r*,

*ϑ*,

*φ*) aswhere

*a*is the mean radius,

*σ*the relative standard deviation of radius,

*s*the so-called logradius,

*ϑ*,

*φ*), and

*Y*the orthonormal spherical harmonics with complex weights

_{lm}*s*, degree

_{lm}*l*, and order

*m*.

*s*based on statistics defined by a covariance function of radius

_{lm}*a*

^{2}Σ

_{r}, whereThe

*P*are the Legendre polynomials with weights

_{l}*c*and

_{l}*γ*the angular distance between two directions (

*ϑ*

_{1},

*φ*

_{1}) and (

*ϑ*

_{2},

*φ*

_{2}). In practical considerations, the covariance function expansion is truncated at a specified

*l*

_{max}. A detailed description of how to generate realizations of a Gaussian random sphere can be found in Muinonen (2000) and Nousiainen and McFarquhar (2004).

The use of the Legendre polynomial expansion of the covariance function is beneficial because values of the *s _{lm}* coefficients depend only on the weight

*c*of the same degree. Nousiainen and McFarquhar (2004) also found that the

_{l}*c*for small, quasi-spherical ice crystals in midlatitudes closely followed a power-law dependence

_{l}*c*∝

_{l}*l*

^{−ν}, where

*ν*is the power-law index. For a power-law covariance function, the shapes are thus statistically fully described by only two parameters,

*σ*and

*ν*.

### c. Statistical shape analysis

The purpose of the statistical shape analysis is to invert the *c _{l}* coefficients from the two-dimensional images of observed crystals. The analysis was conducted almost exactly as in Nousiainen and McFarquhar (2004) and is only summarized here. The one notable difference is that Nousiainen and McFarquhar smoothed the extracted perimeters using a three-point floating average if the perimeters were nonstarlike owing to, for example, a small dent in a perimeter. As the images considered here were somewhat more noisy, a three-pixel boxcar averaging was used to smooth the images prior to the perimeter extraction. This considerably facilitated the perimeter extraction and largely eliminated the need to use the floating average for the perimeters. According to tests, the results of the perimeter extraction were practically identical to those obtained using the methodology of Nousiainen and McFarquhar (2004).

The perimeters were obtained by finding a specific shade of gray that divides the images into the particle and the background and was unique for each image. Next, the center of mass of each extracted silhouette was calculated. This was considered as the origin for the radius vectors, obtained by discretizing the perimeter curve with 1° angular resolution with respect to the origin. At this point, it was checked whether the shape was starlike, namely whether the perimeter was a single-valued function of the polar angle. This is a necessary requirement for the covariance function of radius to exist, so all nonstarlike shapes were discarded. Only about half a percent of about 1600 analyzed perimeters were found to be nonstarlike.

The covariance function of the radius was then computed from the extracted sets of radius vectors. For each perimeter, an autocovariance between *r _{ϕ}* and

*r*

_{ϕ}_{+γ}was computed, where

*r*is the length of a radius vector at a polar angle

_{ϕ}*ϕ*and

*γ*is the angular distance between the pair. The (auto)covariance was calculated over all radius pairs in a single perimeter and over all the perimeters in the ensemble. The

*γ*value was varied in 5° increments from 0° to 180°. Since the covariance between

*r*and

_{ϕ}*r*

_{ϕ}_{+γ}is identical to that between

*r*

_{ϕ}_{+γ}and

*r*, the covariance for angles between 180° and 360° corresponds to those of 360° −

_{ϕ}*γ*. Naturally, the covariance at

*γ*= 0° equals the variance

*a*

^{2}

*σ*

^{2}.

Once the covariance function was obtained, the Legendre polynomial expansion, as given by Eq. (9), was derived. At this stage, the *c*_{0} and *c*_{1} terms were set to zero to eliminate the impact of size variation or the location of the origin to the retrieved shape statistics.

The fact that silhouettes of particles were analyzed, while Gaussian random sphere statistics apply to intersections, was taken into account. According to Nousiainen and McFarquhar (2004), silhouettes underestimate *σ* by about 10% compared to intersections. Thus, when model crystals were generated, the derived *σ* were always multiplied by a factor of 1.1.

## 4. Results

### a. Shape analysis

Similar to the case of midlatitude cirrus (Nousiainen and McFarquhar 2004), Fig. 2 shows that the logarithm of retrieved *c _{l}* depends linearly on the logarithm of the degree

*l*. Thus, the covariance functions of tropical quasi-spherical ice crystals closely resemble power-law covariance functions. The largest deviations to the power-law dependence are seen with high degree

*c*whose values are below about 0.01. But, as Nousiainen and McFarquhar showed, these values are unreliable due to poor signal-to-noise ratio. To quantify the power-law dependence, a linear least squares fit of log

_{l}*c*on log

_{l}*l*was performed. Following Nousiainen and McFarquhar, fits were obtained both for the

*l*∈ [2, 10] and

*l*∈ [2, 15] intervals. However, since the

*c*values for about

_{l}*l*> 10 fall below the 0.01 threshold and are thus unreliable, the results for the latter range should be treated with caution and are only reported here for the sake of consistency.

The results of the shape inversion are summarized in Table 2. Although the retrieved *σ* vary by about a factor of 2 and *ν* by tens of percent between flights, there are some general conclusions. First, the *σ* values are generally smaller than those obtained for midlatitude cirrus (Nousiainen and McFarquhar 2004, Table 4). Excluding the smallest ice crystals analyzed by Nousiainen and McFarquhar for which *σ* appears to be inflated by the poor image resolution, number-weighted *σ* averaged 0.148. Here, the obtained *σ* only exceeded 0.148 for the t1_{b} flight interval. All other intervals had smaller *σ*, five of which had *σ* < 0.100. The number-weighted *N*_{tot} average *σ* was 0.117. Interestingly, the t8 flight intervals had substantially smaller *σ* than the other intervals. Further, flights t2–t4, which are subsequent spiral descents/ascents in a decaying cirrus, show a monotonous decrease in *σ* with time. This might be connected to a continuing sublimation of ice crystals that should cause more roundish ice crystals (see, e.g., Nelson 1998 and references therein) and hence smaller *σ*.

Overview of the shape inversion results for the selected TWP-ICE flight intervals. See text for the symbol definitions.

The *ν* values show considerable spread between flights, varying between 2.3 and 3.3 when the *l* ∈ [2, 10] retrieval range is used. Interestingly the mean, 2.86, is practically identical with the mean of *ν* = 2.9 suggested for the corresponding crystals in midlatitude cirrus by Nousiainen and McFarquhar (2004). Thus, the correlation functions for small quasi-spherical ice crystals in tropical and midlatitude cirrus appear to be practically identical, with the differences in the shape statistics entirely in *σ*. The one standard deviation uncertainty range for *ν* has been obtained from the least squares fit under the assumption that the power-law covariance function is a good model for the observed covariance functions (Press et al. 1992). As the computed *χ*^{2} values are much smaller than the degree of freedom for each case, the power-law parameterization appears to be appropriate (Press et al. 1992).

The retrieved area ratios (AR: cross-sectional area divided by the area of a maximum-diameter circle), computed using *l*_{max} = 10, vary in range from 0.704 to 0.824, with the number-weighted mean of 0.771. The area ratios thus vary considerably from flight to flight. Nousiainen and McFarquhar (2004) do not provide area ratios for comparison, but they can be computed from the reported mean shape statistics to obtain 0.716. The corresponding value for the tropical case is 0.769. This is slightly different from the number-weighted mean above due to a nonlinear impact of shape parameters on the AR. Thus, small quasi-spherical ice crystals in tropical cirrus have over 7% larger projected area than their midlatitude counterparts with equal maximum dimension, showing that for the samples analyzed the tropical crystals are closer to spherical. The retrieved volume ratios (VR: volume divided by the volume of a maximum-diameter sphere) are also reported. Not surprisingly, they correlate well with the area ratios. The volume ratios obtained for the mean shape statistics are 0.390 for the midlatitude and 0.480 for tropics. Note that the volume ratios have been computed from real three-dimensional maximum diameters of the model particles, which can be larger than the maximum diameters observable from ice crystal images.

The impact of the variability of the retrieved shape parameters on the ice crystal shapes is illustrated in Fig. 3. The left-hand particle was generated using the smallest *σ* and largest *ν* obtained, resulting in the most spherical shape. Likewise, the right-hand particle was produced using the largest *σ* and smallest *ν* retrieved, resulting in the most nonspherical shape. To facilitate the comparison, the same random numbers were used to generate both shapes, so the differences are solely due to different shape statistics.

### b. Light-scattering simulations

Two types of light-scattering simulations were conducted. First, the impact of retrieved shape statistics on scattering was investigated. Second, certain modifications to the modeling approach were tested and the sensitivity of the computed single-scattering properties to these modifications assessed.

#### 1) Impact of retrieved shape statistics

The retrieved shape statistics show substantial variations in both *ν* and *σ*. To assess how these differences affect single-scattering properties, light-scattering simulations were conducted separately for the shape statistics retrieved from each flight interval. Simulations were also carried out using the corresponding best-fit power-law covariance functions to assess how accurately they describe the shapes of these particles in terms of the single-scattering properties. For the simulations, a mean radius of *a* = 20 μm was used for all flight intervals, corresponding to the mean radius of the analyzed crystals. This means that the differences in the obtained single-scattering properties are entirely due to different shapes. The simulations were carried out at wavelength *λ* = 550 nm, and the refractive index of nonabsorbing pure ice, *m* = 1.310 + *i*0.0, was assumed.

Geometric optics and diffraction solutions were computed using the publicly available Siris package of Karri Muinonen and Timo Nousiainen (http://www.atm.helsinki.fi/~tpnousia/siris.html). In Siris, the target shapes are represented by a triangle mesh. Similar to Nousiainen and McFarquhar (2004), 100 rows and a total of 20 000 triangles were used to represent the three-dimensional target shapes. The geometric-optics calculations were conducted using an ensemble of 500 randomly generated target shapes using 500 light rays per target. This was deemed sufficient for dampening the statistical noise associated with the Monte Carlo ray tracing. The diffraction part was solved for equivalent cross-sectional area spheres.

Figure 4 shows how the computed asymmetry parameters varied from flight to flight and between the retrieved covariance functions and the corresponding power-law parameterizations. For the retrieved shape statistics, the asymmetry parameter varies in a range *g* ∈ [0.771, 0.799], with a number-weighted mean of 〈*g*〉 = 0.783. The corresponding values for the fitted power-law covariance functions are *g* ∈ [0.778, 0.801] and 〈*g*〉 = 0.788. Thus, the power-law covariance functions provide slightly more forward-scattering shapes, but the half-percent difference is not significant for climate considerations. Similarly, the flight-to-flight variations are insignificant.

A sensitivity study of the dependence of scattering on *ν* was conducted to supplement that on the impact of *σ* carried out by Nousiainen and McFarquhar (2004). For the simulations, the other free parameters were fixed at *a* = 20 μm, *λ* = 550 nm, *m* = 1.310 + *i*0.0, and *σ* = 0.13. The *σ* value used corresponds to the observed average corrected for the bias of using silhouettes. Since the power-law covariance function expansion is infinite and the expansion coefficients cannot be reliably retrieved for the high-degree terms, the impact of the expansion truncation at different *l*_{max} on scattering was also assessed.

The *ν* values were varied over the observed range from 2.3 to 3.3 and the values of *l*_{max} from 10 to 75. Example phase functions from the simulations are shown in Fig. 5. A decrease in *ν* flattens the phase function at side-scattering angles, with the effect more pronounced for large *l*_{max}. This effect is most likely due to the roughness of the model-particle surface: small-scale details in the shape are caused by high-degree spherical harmonics that attain more weight with smaller *ν*. Further, these higher degree terms exist only for sufficiently large *l*_{max}. Roughening the surface makes scattering more neutral and featureless because the surface reflections and transmitted rays are spread over a wider angular range.

The *g* values, shown in Table 3, vary from 0.744 to 0.796 with the largest values corresponding with the largest *ν* and smallest *l*_{max}, namely the most spherical shapes. The *g* almost systematically decrease as the deviations from the spherical shape increase. This is consistent with earlier studies indicating that spherical bodies have exceptionally large *g* in the ray-optics domain (e.g., Muinonen et al. 1996; Nousiainen and Muinonen 1999). For all *ν* tested, an increase in *l*_{max} results in a decrease in *g* that appears to converge toward some limiting value very close to that obtained with *l*_{max} = 75. Consistent with the findings of Nousiainen and McFarquhar (2004), this implies that, even though the power-law covariance function expansion is in principle infinite, in practice it can usually be truncated. Unfortunately, accurate predictions of *g* require longer covariance function expansions than what is accurately retrieved using the shape analysis. This means that the estimates of *g* may be systematically too large. However, because it is not known whether the extrapolation of the covariance function using the power-law dependence is valid for high-degree terms, the magnitude of the error cannot be established. In the worst case, if the power-law dependence applies ad infinitum, the impact of truncation at *l*_{max} = 10 would lead to an overestimation of *g* by as much as 3.5%.

The dependence of the asymmetry parameter *g* on *l*_{max} for three different *ν*. Ray-optics computations for small quasi-spherical ice crystals with *a* = 20 μm, *λ* = 550 nm, *σ* = 0.13, and *m* = 1.31 + *i*0.0.

Finally, the optical properties of small, quasi-spherical ice crystals in the tropics and midlatitudes are compared. The comparison is limited to the mean shape statistics for the power-law covariance function that are known for both geographical locations: *σ* = 0.13 and *ν* = 2.9 for tropics; *σ* = 0.17 and *ν* = 2.9 for midlatitudes. Interestingly, these values were considered already by Nousiainen and McFarquhar as part of a sensitivity study. Figure 15 of Nousiainen and McFarquhar (2004) shows that the phase functions of these two cases are very similar. The corresponding asymmetry parameters are *g* = 0.776 for the tropics and *g* = 0.764 for the midlatitudes, so *g* differs only by about 1% and, thus, the differences are not significant for climate studies.

#### 2) Impact of inclusions, facets, and sixfold symmetry

The model shapes generated using the retrieved shape statistics resemble many of the quasi-spherical ice crystals seen in the CPI data. There are, however, features in real ice crystals that the model crystals do not have: 1) some of the observed small crystals appear to be partially faceted; 2) real ice crystals may possibly possess small-scale surface roughness, or 3) have internal structure, such as bubbles of trapped air. Here, a modified ray-optics code RODS (Muinonen et al. 2009) was used to investigate how these features affect scattering. Here we used 2000 crystals and two million rays per simulation to reduce the statistical noise from the sensitive polarization quantities.

Some partially faceted ice crystals are included in Fig. 1. For example, the fourth crystal in the second row exhibits two planar faces at about a 120° angle from each other. The abundance of such features in the ice crystals considered has not been estimated because of the lack of a suitable tool to do such an analysis objectively. Partial faceting is also easier to identify from larger crystals, which could bias the results.

To generate partially faceted random crystals, a convex-hull transformation was applied to the original Gaussian random sphere, replacing all concavities with planar faces (Fig. 6b). To enhance the sixfold symmetry in the model crystals, the *c*_{6} coefficient of the covariance function was increased. An example crystal with *c*_{6} multiplied by 50 is shown in Fig. 6c. As an extreme case, a crystal with *c*_{6} = 1 (so that *c _{l}*

_{≠6}= 0) was also considered (Fig. 6d). The shapes in Figs. 6c and 6d resemble those of droxtals that have also been used to represent quasi-spherical ice crystals (Yang et al. 2003; Zhang et al. 2004). The main difference from droxtals is that the model crystals considered here are random. The interfacial angles are not constant because the sixfold symmetry is applied over a deformed sphere instead of over a cylindrical body.

The light-scattering properties for the partially faceted crystals are presented in Fig. 7. The introduction of facets (Fig. 6b) and the sixfold symmetry (Figs. 6c,d) affect scattering markedly. Compared to the original shapes, the modified shapes have less side-scattering and more backscattering. The modified shapes also enhance the angular dependence of *P _{ij}*/

*P*

_{11}elements that are connected to different polarization components. In particular, they produce a more positive linear polarization. The

*g*values obtained for modified shapes are given in the legend. The introduction of a convex hull increases the asymmetry parameter from 0.785 to 0.808, which is further increased to 0.818 when the sixfold symmetry is maximally enhanced. Incidentally, the

*g*values obtained for these modifications are similar to those obtained for Chebyshev ice crystals (McFarquhar et al. 2002). The impacts of both the sixfold symmetry and the convex-hull transformation can be significant for radiative fluxes, especially when considering cirrus with large optical depths. Um and McFarquhar (2011) more thoroughly compare the single-scattering properties of different idealized models used to represent small quasi-spheres.

The image resolution of the CPI (2.3 μm) does not allow identification of small-scale surface roughness. Further, since the light source is right behind the particle, interference between transmitted and diffracted light often causes minor wavelike structures along the perimeter. Nevertheless, the sensitivity tests conducted in section 4b(1) demonstrate the impact surface roughness could have on scattering.

Similarly, internal structures cannot be identified from the CPI small-crystal images. The RODS model allows filling the crystal interior with a cloud of scatterers with specific single-scattering albedo, phase matrix, and the mean free path length *δ*_{in}. Here, the phase matrix was specified using the Rayleigh scattering matrix with two modifications: 1) the −*P*_{12}/*P*_{11} element, connected to the degree of linear polarization, was normalized to a maximum value of 0.4 and 2) the phase function was replaced with a parameterized Henyey–Greenstein phase function with a specific asymmetry parameter *g*_{0}. The choice of the maximum value of 0.4 is rather arbitrary but was intended to take into account that −*P*_{12}/*P*_{11} is below unity for the wavelength scale and larger scatterers considered. According to Muinonen et al. (2009), this parameter has little impact on the results. Mie simulations indicate that the asymmetry parameter of spherical air bubbles in ice varies in a range from 0.2 to 0.9 for *x* ∈ [1, 50] at *λ* = 550 nm. Accordingly, this range was used for *g*_{0}. The single-scattering albedo was fixed to unity. When the *δ*_{in} was varied, a fixed *g*_{0} = 0.6 was used. Likewise, when *g*_{0} was varied, *δ*_{in} was fixed at 10 μm.

The impacts of internal structures on scattering are shown in Figs. 8 and 9. As can be seen from Fig. 8, high values of *g*_{0} enhance scattering near the forward angles. Particles with small *g*_{0}, on the other hand, produce more neutral scattering with weak angular dependence in scattering patterns. Further, small *g*_{0} produce the strongest depolarization; that is, *P*_{22}/*P*_{11} deviates most from unity. At certain scattering angles, *P*_{22}/*P*_{11} could change as much as 0.45 when *g*_{0} was changed from 0.2 to 0.9. These findings can be understood by considering how *g*_{0} affects the ray paths. For large *g*_{0}, most rays interacting with internal scatterers are scattered predominantly in the forward direction so that most scattered internal rays propagate more or less parallel to their original direction and the impact on scattering is weak. When *g*_{0} is decreased, the probability of scattering to nonforward angles increases, making the distribution of internally scattered light more isotropic. This flattens the phase function but also strengthens depolarization, as now light rays with different polarization states (due to different refraction angles) can more easily scatter in the same direction.

Values of *δ*_{in} as large as 100 μm, 5 times the mean radius, have a clear impact on scattering (Fig. 9). Decreasing *δ*_{in} to 1 μm increases backscattering up to threefold and makes scattering more depolarizing by up to 0.65 at certain angles, while reducing the angular dependence of the phase-matrix elements. Mean free path lengths much smaller than the radius make ice crystals scatter very similarly to Lambertian particles (Nousiainen et al. 2003), while the −*P*_{12}/*P*_{11} element resembles the polarization pattern of large, perfectly reflecting spheres.

Not surprisingly, the impact of internal scatterers on *g* follows the asymmetry parameter *g*_{0} of internal scatterers: decreasing *g*_{0} decreases *g*. This is consistent with the findings by Macke et al. (1996) for hexagonal ice crystals. Increasing *g*_{0} from 0.2 to 0.9 increased *g* from 0.533 to 0.713. Such a change would have a dramatic impact on radiative fluxes. It is noted that increasing *g*_{0} cannot be used to make *g* larger than it would be without internal scatterers, and the impact depends on *δ*_{in} and disappears at the limit of infinite *δ*_{in}. The latter can be seen from Fig. 9, where increasing *δ*_{in} from 1.0 to 1000 μm increases *g* from 0.377 to 0.782, to a value almost identical to that obtained without internal scatterers (Fig. 7). These results are also in accordance with the findings by Macke et al. (1996). Unfortunately, we are not aware of any observational data that might facilitate choosing a realistic value for *δ*_{in}. The real impact of internal scatterers thus cannot be assessed. The simulations conducted here, however, show that internal structures can affect the single-scattering properties of small, quasi-spherical ice crystals quite significantly for some choices of *g*_{0} and *δ*_{in}.

## 5. Discussion and conclusions

In this paper, the shapes and optical properties of small, quasi-spherical ice crystals were studied using CPI images of real ice crystals acquired in tropical cirrus during TWP-ICE. Using methodology that Nousiainen and McFarquhar (2004) applied to midlatitude ice crystals, the images were used to invert shape statistics that, together with the Gaussian random sphere geometry, were used to generate three-dimensional model crystals. The ray-optics approximation was then applied to compute the single-scattering properties.

Similar to the midlatitude case, the retrieved covariance functions (of radius) closely resembled power-law covariance functions. The relative standard deviation of radius *σ* and the power-law index *ν* varied in a range of *σ* ∈ [0.081, 0.168] and *ν* ∈ [2.32, 3.29] from flight to flight. The number-weighted mean of 〈*ν*〉 = 2.9 was identical to that suggested for small, quasi-spherical ice crystals in midlatitude cirrus. In contrast, the number-weighted mean 〈*σ*〉 = 0.117 is significantly smaller than *σ* = 0.15 obtained for midlatitudes. Thus, the quasi-spherical ice crystals in tropical cirrus are more spherical than the corresponding crystals in midlatitude cirrus, and their area and volume ratios are higher.

The simulated optical properties showed minor differences between TWP-ICE flights. For example, the asymmetry parameters varied within about 3%, with a mean value of 0.783 (original shape statistics) or 0.788 (power-law fits). The mean values were within about 1% of the corresponding midlatitude asymmetry parameter. Thus, the differences in *g* between midlatitudes and tropics appear not to be significant from the standpoint of climate studies.

The retrieved *ν* varied considerably, with its impact on scattering depending on how many terms of the Legendre expansion of the covariance function are accounted for. With larger *l*_{max}, *ν* has a larger impact on scattering. This behavior is most likely connected to small-scale surface roughness of the crystals, which is pronounced when *ν* is small and *l*_{max} is large. While *g* tends to converge toward some limiting value when *l*_{max} is increased, this limiting value may be up to 4% smaller than the value obtained with *l*_{max} = 10. This means that, if quasi-spherical ice crystals possess small-scale surface roughness similar to that obtained by extrapolating the power-law dependence of *c _{l}* to high degrees, the estimates of

*g*will be significantly in error.

A second sensitivity test investigated the impact of crystal facets on scattering by applying a convex hull transformation to the model crystals and by modifying the covariance function to enhance the sixfold symmetry of the model crystals. Together, these modifications led to an increase in *g* by about 4%.

A third sensitivity test investigated how internal structures, for example of trapped air bubbles, impacted the optical properties. The simulated asymmetry parameters varied in a range 0.377–0.782 when the properties of internal scatterers were varied. The impact, however, depended on the amount and characteristics of internal scatterers, which are not known. Even mean free path lengths 5 times the mean particle radius are sufficient to decrease *g* by up to 4%. This is a strong indication that the internal structure of small ice crystals is potentially a very important factor. Data about the internal structure of ice crystals in cirrus clouds would thus be extremely valuable.

According to Vogelman and Ackerman (1995), for asymmetry parameters around 0.8, about 2% or 5% accuracy in *g* at cloud optical depths of 12 and 2, respectively, is deemed necessary for climate considerations. According to these criteria, the partial facets as well as the surface roughness and internal inhomogeneity can all have a significant impact on *g*—at least at high optical depths. The internal inhomogeneity can have an impact even for thin ice clouds, while the differences in the optical properties between tropical and midlatitude cirrus appear not to be significant.

Data were obtained from the ARM program archive, sponsored by the United States Department of Energy (DOE), Office of Science, Office of Biological and Environmental Research (BER), Environmental Sciences Division. The research was supported by the Academy of Finland (Contracts 212979 and 125180) and by the Office of Science (BER) of U.S. DOE (DE-FG02-02ER63337, DE-FG02-07ER64378, DE-FG02-09ER64770, and DE-SC0001279).

# APPENDIX

## Potential of Large Ice Crystal Shattering on CPI

Two types of analyses were carried out to assess the potential impact of how the shattering of large ice crystals on the protruding components of the CPI affects the images of small quasi-spheres obtained by the CPI. First, Um and McFarquhar (2011) analyzed 180 905 particles imaged by the CPI on the three specified days during TWP-ICE. Of these particles, 178 102 (98.451%) particles were the only particles in their frame, while 2596 (1.435%), 186 (0.102%), 16 (0.009%), and 5 (0.003%) occurred in frames with 2, 3, 4, and 5 particles, respectively. Because the CPI imaging array occupies an area of only 2.3 mm × 2.3 mm and the cluster of particle fragments generated by shattering can constitute a horizontal dimension along the direction of airflow as large as 10 cm (Korolev et al. 2011), there is a possibility that a single particle passing through the CPI field of view could be the remnant of a shattered particle. Assumptions about the potential role of shattering on CPI images should be carefully examined for any future dataset of CPI particles that are used in any image analysis.

In addition, CPI images were sorted into four classes depending on the concentration of large ice crystals with *D*_{max} > 100 μm (*N*_{>100}) measured by the Cloud Imaging Probe (CIP), where the classes were defined as follows: A) *N*_{>100} = 0 L^{−1}; B) 0 < *N*_{>100} < 0.1 L^{−1}; C) 0.1 < *N*_{>100} < 1.0 L^{−1}; and D) *N*_{>100} > 1.0 L^{−1}. The results are summarized in Table A1. The number of crystals in class B was too small so the shape statistics were not derived. Further, the number of samples for categories A and C was quite small. Thus, the bootstrap technique (Press et al. 1992) was used to derive the mean shape parameters and their variances for the classes. Class A with no large crystals present was the reference against which the shape statistics of the other classes were compared.

Shape parameters *σ* and *ν* and the asymmetry parameter *g* as a function of large-ice-crystal concentration; *N* denotes the number of ice crystals in each class.

There is a trend for both *σ* and *μ* to increase with *N*_{>100}. Application of a Student’s *t* test showed that there is over 25% probability that both *σ* and *μ* were the same for classes A and C. On the other hand, the probability of *μ* (*σ*) to be the same between classes A and D was between 1% and 5% (less than 0.1%). Interestingly the *g* were similar for each class because *μ* and *σ* act in different directions when they increase; increasing *μ* tends to make shapes more spherical whereas increasing *σ* makes them more nonspherical.

Thus, whereas the first test suggests that shattering does not seem to affect the timing of the arrival of the small quasi-spheres, the second test does suggest that there may be some differences in the small crystal shapes as a function of large crystal concentration. However, the differences in shape as a function of large crystal concentration cannot necessarily be attributed to shattering. There are likely differences in meteorological conditions, such as humidity, that have had an impact on the large crystal concentrations and thus very possibly on the shapes of the small crystals. This would be consistent with the differences in shape statistics of the small crystals for the 2 February 2006 case in the decaying compared to observations in the 27 and 29 January 2006 cases in more humid conditions. Further, the size of the samples makes it difficult to draw definitive conclusions. Nevertheless, future analysis of CPI images obtained in other geographic locations should carefully apply such tests to further investigate the degree to which the CPI data may be affected by the shattering of large ice crystals.

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