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  • View in gallery

    Normalized histograms (total equal to unity) of the zeroth moment (concentration for D > 100 μm) generated from 10-s snow PSDs from midlatitude (gray) and tropical (solid) datasets. The histograms cover six temperature ranges (5° to −55°C in 10°C intervals). The vertical bars represent the Poisson counting error in each bin. The inset vertical axis is the normalized frequency. Here N is the number of points (one point for each PSD) in the histogram, μgeo is the geometric mean, and σgeo is the geometric standard deviation.

  • View in gallery

    As in Fig. 1 except for the second moment (IWC for D > 100 μm if mass ∝ D2) of the 10-s PSDs: Midlatitude (gray) and tropical (solid) datasets.

  • View in gallery

    As in Fig. 1 except for the fourth moment (radar reflectivity Z for D > 100 μm if mass ∝ D2 and pure Rayleigh scattering) of the 10-s PSDs: Midlatitude (gray) and tropical (solid).

  • View in gallery

    As in Fig. 1 except for the characteristic size (mean mass weighted size for D > 100 μm if mass ∝ D2) of the 10-s PSDs: Midlatitude (gray) and tropical (solid).

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    Variation of histogram parameters for the second moment as a function of averaging time over which the snow PSD was compiled: (a) variation of mean of the second moment and (b) variation of fractional standard deviation (σ/μ).

  • View in gallery

    Rescaled PSDs for tropical and midlatitude clouds and fits (solid lines, Table 2). Each point represents a 10-s average rescaled concentration from each size bin of the PSDs. (a) Log–log plot of data from the tropical “a” dataset; (b) log–linear plot of data from tropical “a” dataset; (c) log–log plot of data from the tropical “b” dataset with fit from tropical “a”; (d) log–log plot of data from the midlatitude “a” dataset; (e) log–linear plot of data from midlatitude “a” dataset; and (f) log–log plot of data from the midlatitude “b” dataset with fit from midlatitude “a.” Also shown are the fits from F05 and a rescaled exponential size distribution, Exp.

  • View in gallery

    Scatterplots of the measured moments of the PSDs vs the predicted moments using the parameterization given in Table 3. Each point represents the value obtained from a 10-s PSD. The grayscale is intended to provide a qualitative indication of temperature (lightest = −60°C, darkest = 0°C); (a) fourth, (b) third, (c) first, and (d) zeroth moments. The 1:1 lines are also shown.

  • View in gallery

    Quadratic fits to the parameters in the moment estimation parameterization (Table 3). The filled circles are the values obtained using 10-s PSDs. The solid line is the least squares fit quadratic to the 10-s data. Results obtained when midlatitude, tropical, and 120-s PSDs only are used to estimate the moments are also shown.

  • View in gallery

    Number of 10-s data points used in 10°C temperature bins for the comparison from the tropical “b” (dashed) and midlatitude “b” (solid) datasets.

  • View in gallery

    Test of the moment estimation scheme assuming known second moment and temperature. (top pair) Tropical “b” comparison of (upper row) geometric mean μgeo of the ratio of estimated to measured moment as a function of temperature in 10°C temperature bins and of (lower row) geometric standard deviation σgeo of the ratio of estimated to measured moment as a function of temperature. (bottom pair) As for “Tropical b” but for the midlatitude “b” dataset. Columns from left to right show the results for moments 0, 1, 3, and 4, respectively. Results from this study, F05, and H79 are shown. Legend is in the bottom left panel.

  • View in gallery

    As in Fig. 10 but assuming the second and third moments of the PSD are known. Columns from left to right show results for moments 0, 1, 2, 3, and 4, respectively.

  • View in gallery

    Estimated ratio of total moments to measured truncated moments if the PSDs are extrapolated assuming that the rescaled size distribution given in Table 2 is valid for all sizes (section 5) and the measured second and third moments are unaffected by truncation errors. (a) Tropical zeroth moment; (b) midlatitude zeroth moment; (c) tropical first moment; (d) midlatitude first moment; (e) tropical fourth moment; and (f) midlatitude fourth moment.

  • View in gallery

    Normalized histograms (total area equal to unity) of the extrapolated total concentration generated from 10-s snow PSDs assuming that the rescaled size distribution given in Table 2 is valid for all sizes (section 5) and the measured second and third moments are unaffected by truncation errors: Midlatitude (gray) and tropical (solid) datasets. The histograms cover six temperature ranges (5° to −55°C in 10°C intervals). Vertical bars represent the Poisson counting error in each bin, and the inset vertical axis is the normalized frequency: N is the number of points (one point for each PSD) in the histogram, μgeo is the geometric mean, and σgeo the geometric standard deviation.

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Snow Size Distribution Parameterization for Midlatitude and Tropical Ice Clouds

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  • 1 National Center for Atmospheric Research, Boulder, Colorado
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Abstract

Many microphysical process rates involving snow are proportional to moments of the snow particle size distribution (PSD), and in this study a moment estimation parameterization applicable to both midlatitude and tropical ice clouds is proposed. To this end aircraft snow PSD data were analyzed from tropical anvils [Tropical Rainfall Measuring Mission/Kwajelein Experiment (TRMM/KWAJEX), Cirrus Regional Study of Tropical Anvils and Cirrus Layers-Florida Area Cirrus Experiment (CRYSTAL-FACE)] and midlatitude stratiform cloud [First International Satellite Cloud Climatology Project Research Experiment (FIRE), Atmospheric Radiation Measurement Program (ARM)]. For half of the dataset, moments of the PSDs are computed and a parameterization is generated for estimating other PSD moments when the second moment (proportional to the ice water content when particle mass is proportional to size squared) and temperature are known. Subsequently the parameterization was tested with the other half of the dataset to facilitate an independent comparison. The parameterization for estimating moments can be applied to midlatitude or tropical clouds without requiring prior knowledge of the regime of interest. Rescaling of the tropical and midlatitude size distributions is presented along with fits to allow the user to recreate realistic PSDs given estimates of ice water content and temperature. The effects of using different time averaging were investigated and were found not to be adverse. Finally, the merits of a single-moment snow microphysics versus multimoment representations are discussed, and speculation on the physical differences between the rescaled size distributions from the Tropics and midlatitudes is presented.

Corresponding author address: Dr. Paul R. Field, NCAR, 3450 Mitchell Lane, Boulder, CO 80301. Email: prfield@ucar.edu

Abstract

Many microphysical process rates involving snow are proportional to moments of the snow particle size distribution (PSD), and in this study a moment estimation parameterization applicable to both midlatitude and tropical ice clouds is proposed. To this end aircraft snow PSD data were analyzed from tropical anvils [Tropical Rainfall Measuring Mission/Kwajelein Experiment (TRMM/KWAJEX), Cirrus Regional Study of Tropical Anvils and Cirrus Layers-Florida Area Cirrus Experiment (CRYSTAL-FACE)] and midlatitude stratiform cloud [First International Satellite Cloud Climatology Project Research Experiment (FIRE), Atmospheric Radiation Measurement Program (ARM)]. For half of the dataset, moments of the PSDs are computed and a parameterization is generated for estimating other PSD moments when the second moment (proportional to the ice water content when particle mass is proportional to size squared) and temperature are known. Subsequently the parameterization was tested with the other half of the dataset to facilitate an independent comparison. The parameterization for estimating moments can be applied to midlatitude or tropical clouds without requiring prior knowledge of the regime of interest. Rescaling of the tropical and midlatitude size distributions is presented along with fits to allow the user to recreate realistic PSDs given estimates of ice water content and temperature. The effects of using different time averaging were investigated and were found not to be adverse. Finally, the merits of a single-moment snow microphysics versus multimoment representations are discussed, and speculation on the physical differences between the rescaled size distributions from the Tropics and midlatitudes is presented.

Corresponding author address: Dr. Paul R. Field, NCAR, 3450 Mitchell Lane, Boulder, CO 80301. Email: prfield@ucar.edu

1. Introduction

Accurate prediction of cloud evolution is important for the quantitative prediction of precipitation, cloud extent, occurrence of supercooled water, and feedback to the larger scale dynamics (e.g., Forbes and Clarke 2003). Many of the microphysical rate equations in numerical weather models depend upon the representation of snow through assumptions about mass–dimension, fall speed–dimension relations, capacitance, and the snow particle size distribution (PSD). Current microphysical schemes that prognose vapor, liquid, and ice (e.g., Wilson and Ballard 1999) make use of an exponential size distribution that has an “intercept parameter” controlled by temperature. Typically, these microphysical schemes have the exponential PSD assumption hard-wired into the rate equations and given ice water content (IWC) and temperature; microphysical process rates involving snow can be estimated. However, these schemes can be generalized so that many of the process rates can be written in terms of the moments of an arbitrary PSD (e.g., Zawadzki et al. 2000; Thompson et al. 2007, manuscript submitted to Mon. Wea. Rev.) provided that the physical attributes (e.g., mass, fall speed) of ice crystal aggregates (snow) are written as power laws of the particle dimension, as is commonly done. Two-moment schemes for snow are now being introduced (e.g., Ferrier 1994; Morrison et al. 2005; Milbrandt and Yau 2005; Seifert and Beheng 2006) that predict both IWC and number concentration. It can be shown empirically that estimating moments of a PSD derived from two given moments is superior to using only a single moment (e.g., Lee et al. 2004). However, the strength of these two-moment schemes relies on the accuracy of the representation of microphysical processes such as ice nucleation and aggregation, which we will return to in the discussion.

It would be advantageous if a parameterization could be produced that could be used globally without consideration of the regime being simulated. To that end we have used aircraft-measured snow PSDs obtained during campaigns in both the anvils of tropical convective storms and midlatitude stratiform cloud. Previously, Field et al. (2005) used just midlatitude stratiform cloud from around the British Isles to develop a moment estimation parameterization. The intention of this paper is to produce a single-moment bulk snow parameterization based on the second moment of the PSD and temperature that will estimate any other moment of the PSD required for predicting microphysical process rates involving snow applicable to both midlatitude and tropical regimes. The reason for choosing the second moment of the PSD as the reference moment is because many empirical studies (e.g., Brandes et al. 2007; Heymsfield et al. 2004; Mitchell et al. 1990; Locatelli and Hobbs 1974) and theoretical studies (Westbrook et al. 2004a) find that the exponent in the mass–dimension power law for aggregates of ice crystals, that is, snow, is two. Thus, a common prognostic of numerical weather models, IWC, is directly proportional to the second moment of the snow size distribution.

While a moment estimation parameterization is sufficient to determine the microphysical process rates involving snow it may be desirable for some applications to produce a representative PSD. In part of this study we have rescaled the snow PSDs to investigate the underlying form of the distribution. Many others have previously rescaled both raindrop distributions (e.g., Sekhon and Srivastava 1971; Willis 1984; Sempere-Torres et al. 1998; Illingworth and Blackman 1999; Testud et al. 2001; Lee et al. 2004) and ice size distributions (e.g., Field and Heymsfield 2003; Westbrook et al. 2004a, b; Delanoë et al. 2005; Field et al. 2005) using both single- and double-moment rescaling methods. For a comprehensive review of the rescaling technique we recommend Lee et al. (2004) or more briefly Westbrook et al. (2004b). The motivation for rescaling is to attempt to tease out the underlying form of the PSD. Originally, Gunn and Marshall (1958) found that an underlying exponential form described their data and various adaptations to this form have been suggested since. With rescaling it is possible to explore the underlying shape of the PSD without imposing any a priori expectation for any specific analytic form. Once the rescaling has been carried out analytic fits to the rescaled size distribution can be found. These fits have invariably been based on an exponential form (e.g., gamma, generalized gamma distributions). In Field et al. (2005) an exponential term was used to represent the small end of the size distribution and a gamma distribution was used to represent the large end of the PSD.

The structure of the paper is as follows. Section 2 deals with the aircraft cloud probe data used in this study. Section 3 compares histograms of moments obtained from tropical and midlatitude ice cloud. Section 4 looks at the effect of the averaging time on the moments of the PSD. Section 5 considers the rescaling of the PSDs. Section 6 presents the moment estimation parameterization. Section 7 suggests how the parameterization can be implemented, and it is then tested in section 8. The discussion and conclusions form sections 9 and 10.

2. Data

The data used in this study were obtained from four campaigns that we have divided into tropical [Tropical Rainfall Measuring Mission/Kwajelein Experiment (TRMM/KWAJEX) and Cirrus Regional Study of Tropical Anvils and Cirrus Layers-Florida Area Cirrus Experiment (CRYSTAL-FACE)] and midlatitude [First International Satellite Cloud Climatology Project Research Experiment (FIRE) and Atmospheric Radiation Measurement Program (ARM)]. Table 1 lists the geographical locations, dates, amount sampled, and the optical array cloud probes used. The tropical data are largely composed of sampled ice cloud within the anvils of both continental- (CRYSTAL-FACE) and maritime- (TRMM/KWAJEX) dominated convection. The midlatitude clouds consist of stratiform ice clouds associated with midlatitude cyclonic systems. So, when we refer to tropical and midlatitude regimes, we are considering anvil and stratiform ice clouds, respectively.

The snow particle size distributions using the particle maximum dimension were derived from combining data from two optical array probes that covered the smaller [100–1000 μm, two-dimensional cloud (2D-C)] and the larger [>1000 μm, two-dimensional precipitation (2D-P) or High Volume Precipitation Spectrometer (HVPS); see Table 1] ends of the distribution. The Particle Measuring Systems (PMS) 2D-C nominally sizes particles in the size range 25–800 μm with a pixel size of 25 μm. The PMS 2D-P nominally sizes particles in the range 200–6400 μm and has a pixel size of 200 μm. The Stratton Park Engineering Company HVPS nominally measures particles in the size range 200–51 200 μm and has a pixel size of 400 μm along the flight direction (for an airspeed of 100 m s−1) and 200 μm perpendicular to the flight direction. For particles that have sizes similar to that of the 2D-C array width it is possible to reconstruct partially imaged particles. This was done for the 2D-C following the method described in Heymsfield and Parrish (1978). Only complete images were used in the analysis of the HVPS and 2D-P data. Particles are rejected for the following reasons. For the HVPS or 2D-P, image records with area ratios (measured pixel area/area of best-fit circle) <0.1 are rejected. This eliminates “streakers” and images where multiple particles are present in a single image record. Due to the width of the HVPS imaging array it is possible for two particles to appear side by side. However, because there is no automatic way to decide if these particles are separate, naturally occurring particles or shatter products we have chosen to be conservative and eliminate them. For the 2D-C, image records with area ratios <0.1 or area ratios <0.2 and lengths >600 μm are rejected. This eliminates streakers and multiple particle images. In addition, rejection of particles associated with interarrival times of less than approximately 2 × 10−4 s was also carried out (this threshold is dynamically determined for every 5 s of data and varies slightly) to remove artifacts associated with shattering of ice crystals on the probe housing (Field et al. 2006b). The 2D-C (for D > 100 μm) and HVPS or 2D-P (see Table 1) PSDs were joined at 1000 μm over 5-s periods. The overlap region between 1000 and 1400 μm showed good agreement. Particles smaller than 100 μm were not considered because of problems with detection by the 2D-C (Strapp et al. 2001) and uncertainty caused by shattering on the housing of the forward-scattering spectrometer probe when large particles are present (Field et al. 2003).

The moments of the distribution were computed from 100 μm to 20 mm using the measured PSD:
i1520-0469-64-12-4346-e1
where n is the moment order, D is the particle maximum dimension, N(D) is the bin width normalized concentration in size bin D, and ΔD is the width of the size bin. Any size bins containing fewer than five counted particles were ignored. Because we cannot observe the entire PSD, the moments that we will deal with up to section 9 are necessarily truncated. In the discussion we will consider the concept of complete moments for the hypothetical situation where the complete PSD can be observed. We have also compiled PSDs that have concentrations greater than 1 L−1 and their respective moments over 5-, 10-, 30-, 60-, and 120-s intervals, corresponding to approximately 0.7, 1.4, 4.2, 8.4, and 16.8 km of along-track distance for a true airspeed of 140 m s−1.

In the following sections the datasets were divided into tropical “a” and “b” and midlatitude “a” and “b” sets. We note that, although the two datasets comprise different flight data, they are still derived from the same probes. Therefore, if any systematic errors are introduced by the probe analysis, they will be present in both datasets. For the development of the parameterization and subsequent testing the two new datasets were formed by combining the tropical and midlatitude “a” and the tropical and midlatitude “b” sets to form two new independent sets: “a” and “b.” The parameterization that will be presented later has been derived from set “a” and then tested using set “b.”

3. Tropics versus midlatitude

It is instructive to compare and contrast histograms of PSD moments from the tropical and midlatitude regimes using all of the 10-s PSDs. Figures 1 –3 show histograms of the zeroth (concentration for D > 100 μm), second (proportional to IWC when particle mass is proportional to D2), and fourth moments (proportional to radar Rayleigh scattering reflectivity since Z ∝ mass2), respectively, while Fig. 4 shows the ratio of the third and second moments: characteristic or mass-weighted mean size. All of the histograms were constructed using 10-s mean values. The histogram geometric mean and the geometric standard deviation are given. Differences to these values introduced by altering this averaging time are explored in the next section.

The distribution of the moments and characteristic size are approximately lognormal with broad spreads (0.5 to 1 order of magnitude) within each temperature range (Figs. 1 –4) for both the tropical and midlatitude datasets. Unfortunately, there is very little data in the midlatitude dataset for temperatures warmer than −5°C. The mean concentration (Fig. 1) for midlatitudes is between 5 and 10 L−1 in agreement with previous studies of midlatitude ice cloud (e.g., Korolev et al. 2000; Gultepe et al. 2001; Field et al. 2005; Boudala and Isaac 2006), while for the tropical data the mean is 10–50 L1, again similar to previous independent measurements (e.g., McFarquhar and Heymsfield 1996). In addition, we note that the concentration of particles from the tropical data with D > 100 μm can reach extreme values of 500 L−1 (Fig. 1).

For the midlatitude data the second moment (Fig. 2) extends up to 0.03 m−1 [∼2 g m−3 assuming mass = 0.069D2, SI units, Wilson and Ballard (1999)], whereas the tropical data extend to nearly double that value: 0.05 m−1 (∼3.5 g m−3). The midlatitude histograms of the second moment tend to broaden toward higher values with increasing temperature that increases the mean. In contrast the histograms of the second moment obtained from the tropical data show a narrowing with increased temperature. The loss of the smaller values leads to an increasing mean with temperature. At 0°C, the second moment for both tropical and midlatitude exhibits a decrease when compared to the values at −10°C.

Figure 3 shows the histograms for the fourth moment. For pure Rayleigh scattering, radar reflectivity will be proportional to M4. Therefore, an order of magnitude difference in the fourth moment will translate to 10 dBZ. While the spread of radar reflectivity values for the midlatitude and tropical data are more comparable at warmer temperatures (e.g., −10°C), there are large differences at colder temperatures. At −50°C the difference in maximum reflectivities is of the order of 20 dBZ, with the tropical data extending to the largest values. This difference is related to both the higher values of IWC (second moment) and the particle sizes at these temperatures.

The characteristic size (Fig. 4) displays a monotonic increase with temperature, but the histograms derived from the tropical data exhibit more spread than the midlatitude data and even indicate some bimodality (e.g., −10°C). At −50°C the histogram of characteristic size determined from the tropical data extends to 2 to 3 mm, which is 3 to 4 times larger than the largest characteristic sizes exhibited by the midlatitude data at this temperature. At −10°C, the histograms of characteristic size are more comparable, although the mean of the tropical data is still 30% greater than the midlatitude mean.

4. Time averaging

Numerical weather prediction models use a variety of grid spacings controlled by operational practices. Often parameterizations are used at resolutions that differ from the scale at which the data or relationships were obtained. The problem of reconciling model and data gathering scales is not simple, but here we will briefly assess what are the effects of changing the interval over which the cloud probe data is averaged on the histograms discussed in the previous section. Figure 5 shows the geometric means and standard deviations of the histograms of the second moment (other moments exhibit similar behavior) for averaging periods ranging from 5 to 120 s that have cloud fractions (defined by the fraction of 5-s periods that contain cloud particles detected by the 2D-C probe) greater than 0.9. It can be seen that the histogram mean of the second moment remains approximately constant for a 24 times increase in averaging period. In contrast, the width of the distributions shows a general decrease with increasing accumulation time. We expect a decrease in the spread of the histogram due to increasing the sampling period, but the observed decrease is slower than would be expected for sampling of a random lognormal distribution and is most likely the result of the larger-scale structure of the clouds. The decrease in the width of the histograms is about 30% in the midlatitude cloud for an increase in averaging by a factor of 24 and less for the tropical cloud. For the rest of the analysis we must choose an averaging period with which to work. Mesoscale and global models are approaching grid spacings of order 1 and 10 km, respectively. Given the small changes seen in the variation of the mean moment with averaging time (for ratios of moments the differences are even smaller) we have elected to adopt an averaging period of 10 s for the rest of the analysis.

5. Two-moment rescaling

Snow PSDs can be rescaled using two moments (e.g., Testud et al. 2001; Lee et al. 2004; Delanoë et al. 2005; Field et al. 2005) to reveal an underlying distribution Φ that is determined by the physics that control the snow PSD evolution:
i1520-0469-64-12-4346-e2

Here we have made use of the second and third moments to carry out the rescaling (Fig. 6), but any other pairs of moments can be used: Φ23(x) is given in terms of the dimensionless size x, which is simply the measured maximum size divided by the characteristic size of the PSD (L23 = M3/M2). Overall we find the same result as Delanoë et al. (2005), namely, that the midlatitude rescaled distribution has lower rescaled concentrations at both small and large values of dimensionless size x than the tropical rescaled distribution and exhibits a conspicuous mode that is also absent in the tropical case. Differences in rescaled distributions reveal the action of different physical effects. Delanoë et al. (2005) have previously suggested that this difference is due to the nature of the mechanisms that transport water vertically through the atmosphere: convective and large-scale uplift.

For some applications the generation of a PSD from the predicted moments is desired so that radiative transfer calculations can be carried out (e.g., Doherty et al. 2007; Baran 2007) or non-Rayleigh radar scattering can be computed. Therefore we have provided a fit to the midlatitude and tropical rescaled distributions. There are three properties that the fits to the rescaled distributions should possess. (i) All the moments (n ≥ 0) of the rescaled distribution should be finite (this avoids encountering infinite concentrations, etc.). (ii) The tail of the rescaled distribution should tend toward exponential (the expected large x behavior; Westbrook et al. 2004b). (iii) The second and third moments of the rescaled distribution should equal unity for self consistency (since we have rescaled using the second and third moments of the PSD). These constraints are satisfied by the form used in Field et al. (2005), where an exponential and a gamma distribution were used to represent the small and large ends of the rescaled distribution, respectively. Figure 6 shows the fits (given in Table 2) to the tropical and midlatitude rescaled PSDs. The fits were found by using a constrained minimum Chi-squared random search for the three free parameters using dataset “a” (see Field et al. 2005 for details). Also shown in Fig. 6 is a comparison of the fits with the rescaled distributions generated from dataset “b” that shows good agreement. The rescaled distribution (using the second and third moments of the PSD) given in Field et al. (2005) shows fairly good agreement with the fit presented in this study for the midlatitude data, apart from at very small dimensionless sizes. Comparison with the tropical PSDs shows that the Field et al. curve is similar to the fit presented here for tropical PSDs when x < 2.5, but the exponential tail is too steep at larger x. A rescaled exponential distribution [Φ23,exp = 13.5 exp(−3x)] is very similar to the Field et al. curve for x > 1 but has much lower values for x < 1. Potential users should be aware that when reconstructing PSDs using Φ23 this should strictly be done for D > 100 μm only. Reconstruction to smaller sizes represents an extrapolation of the data used in this study.

6. Moment parameterization

The motivation for developing an empirical moment prediction scheme for snow PSDs is based on the fact that, if parameters associated with ice crystal aggregates such as mass and fall speed can be written as power laws applicable to all sizes, then many microphysical process rates can be written in terms of moments of the snow PSD. For diffusional growth of snow and hence vapor sink to snow, riming, and sedimentation the moments of interest depend upon the choice of power law exponents used for the mass–dimension and fall speed–dimension relations. For example, ignoring ventilation effects, diffusional growth = ∫ kd DN(D) dD = kdM1, where the constant, kd, represents various thermodynamic variables and the dimensionless capacitance of the ice crystal aggregate. Similarly, the sedimentation rate, when mass∝ D2 and fall speed ∝ D0.42 (e.g., Ferrier 1994) is ∫ ksD2D0.42N(D) dD = ksM2.42, where ks is the product of the prefactors from the mass–dimension and fall speed–dimension power laws appropriate for ice crystal aggregates. To encompass a wide range of moments we have constructed empirical relationships using the zeroth to the fifth moment of the PSD. As mentioned in the introduction, we have based our relations on a fit to the second moment (proportional to IWC when mass ∝ D2) and temperature. Unlike the rescaled distributions where a choice needs to be made for which regime the PSD is to be generated, the moment parameterization presented here is applicable to both midlatitude and tropical ice clouds. The apparent inconsistency between the development of a single-moment estimation parameterization and the difference in rescaled distributions highlighted in the previous section will be returned to in the discussion.

For the parameterization we have adopted the following relational form (in SI units):
i1520-0469-64-12-4346-e3
where Tc is the temperature in °C, M2 is the second moment, Mn is the moment of order n to be predicted, and A(n), B(n), and C(n) are constants that are quadratic functions of n only. This is essentially the same form that was used by Hogan et al. (2006) to describe the relation between reflectivity and IWC. The inclusion of an exponential function in Eq. (3) derives from a theoretical argument based on the result (see Westbrook et al. 2007) that aggregation will lead to an increase in the characteristic size of a PSD as an exponential function of fall time. If we combine a constant lapse rate with the assumption that the mean fall speed of ice crystal aggregates is approximately constant, then we can see that we may expect that the characteristic size will be proportional to an exponential function of temperature.

Figures 7a–d show scatterplots, derived from dataset “a,” of the measured moments plotted against the moments computed using the relation in Eq. (3) with n = 4, 3, 1, and 0, respectively. As expected, the scatterplots lie along a 1:1 line and the data generally converge toward low values as the PSD narrows [see Field et al. (2005) for more discussion concerning this convergence effect]. The grayscale provides a qualitative indication of temperature (cold = light, warm = dark) and shows that there should be little temperature-dependent bias in the parameterization. Table 3 gives the quadratic equations describing A(n), B(n), and C(n) that were found by least squares fitting of Eq. (3) for n = 0, 1, 2, 3, 4, and 5. Subsequently, a least squares quadratic fit was carried out for A(n), B(n), and C(n) as a function of n. The fits and relationships were determined using the 10-s dataset “a” described in section 2 and are shown in Fig. 8. Also shown in Fig. 8 are the results obtained for the parameters using just the tropical “a” set, the midlatitude “a” set, and the results for the complete “a” dataset when the PSDs are averaged over 120 s. It can be seen that the points agree quite well with the results obtained for the 10-s PSDs, although the midlatitude values for C are slightly lower. We note that the value for C(4) = 1.56 is in agreement with that suggested by Hogan et al. (2006) for deriving IWC from radar reflectivity.

7. Implementation

To estimate moments of the snow PSD no knowledge of the regime is required. For a grid box with a given temperature and power law relating mass m to size (m = αDβ) the IWC in the grid box is related to the snow PSD moments through
i1520-0469-64-12-4346-e4
where Mβ is the βth moment of the snow PSD. If β = 2, as we have suggested, then M2 = IWC/α. The grid box values of M2 and temperature can then be inserted into Eq. (3) and any moment, n, of the snow PSD can be estimated using the relations given in Table 3. If β ≠ 2, that is, the mass–dimension relation used in the microphysical scheme is not proportional to D2, then Eq. (3) can be inverted first to find M2 given Mβ.

The moment estimation parameterization can be used in conjunction with the fits to the rescaled distributions given in the previous section to generate a realistic PSD. Given an estimate of the second moment and temperature, the third moment of the PSD can be estimated regardless of regime. Then N(D) can be found by inverting Eq. (2) [N(D) = Φ23(x)M42/M33], inserting the values for the second and third moments, and choosing the required tropical or midlatitude rescaled distribution from Table 2. Similarly, the dimensionless size given in Eq. (2) can also be inverted to generate actual particle maximum dimension.

8. Testing the parameterization

We have used dataset “b” to test the moment prediction equation given in section 6 and split the comparison between midlatitude and tropical data for moments of order 0, 1, 3, and 4. In each 10°C size bin centered on −60° to 0°C the number of 10-s data points used in the comparison is shown in Fig. 9. It can be seen that, while the temperature bins are well populated throughout the temperature range for the tropical data, the midlatitude data is sparse for temperatures warmer than −15°C and colder than −55°C. To assess the bias and spread inherent in the parameterization, we have plotted the geometric means and standard deviations of the ratio of predicted moment over the measured moment as a function of temperature for the midlatitude and tropical data (Fig. 10). Also overplotted are results from the Field et al. (2005, hereafter F05) moment estimation parameterization and computed moments assuming an exponential PSD and using the relations obtained in Houze et al. (1979, hereafter H79).1

For the tropical data (Fig. 10, top row), the moment estimation scheme developed in this study straddles the unity line and provides good estimates of the moments. For moments 0, 1, and 3, the mean ratio generated from the moment estimation scheme developed here is between 0.8 and 1.2 at all temperatures, but the parameterization underestimates the fourth moment (geometric mean of 0.5 to 0.8). The Field et al. (2005) relation shows good agreement at temperatures warmer than −20°C for all of the moments but is biased high for moments less than two and low for moments greater than two at temperatures colder than −20°C. The exponential distribution does well at −30°C, but for moments less than two it is biased high for temperatures colder than −30°C and biased low for warmer temperatures. For moments greater than two the opposite trend in the bias of the moments estimated from the exponential PSD is observed. For all of the schemes the geometric standard deviation increases with temperature for the tropical data (Fig. 10, second row) with large spreads (factors of 3 or 4) evident at around −10°C for the zeroth and fourth moments.

Examining the midlatitude results (Fig. 10, third row) it can be seen that the mean ratios generated by the moment estimation scheme developed in this study generally lie within 0.8 and 1.2 at all temperatures and for all the moments. The Field et al. (2005) relation appears slightly biased high for moments less than two and biased low for moments greater than two. The exponential distribution is biased high for moments less than two and temperatures colder than −20°C. For moments greater than two the mean ratio for the exponential distribution increases from below one at −60°C to greater than one for temperatures warmer than −30°C. The geometric standard deviations (Fig. 10, bottom row) for the midlatitude data are smaller than for the tropical data and are approximately constant with temperature. We note that all three schemes exhibit a similar spread and conclude that additional parameters will be required to reduce this spread still further, such as distance from cloud top (e.g., van Zadelhoff et al. 2004) or knowledge of additional moments.

Figure 11 shows the geometric means and standard deviations of the ratio of predicted moment over the measured moment as a function of temperature for the midlatitude and tropical data if both second and third moments are known. These moments (obtained from the observed PSDs) were combined with the rescaled size distribution function found in section 5, the fit for the second and third moments rescaled distribution given in Field et al. (2005), and one appropriate for an exponential after rescaling using the second and third moments, to produce synthetic size distributions as outlined in section 7. By integrating these synthetic size distributions for sizes greater than 100 μm we are able to assess any biases in the moments introduced by the functional fit to the rescaled PSDs. Because of the truncation effect we have limited the comparison to PSDs with characteristic sizes exceeding 300 μm (the smallest mean size seen in the midlatitude histogram for −50°C). Figure 11 is similar to Fig. 10, but we have also included the results for the second moment. Because the second and third moments are known we might expect zero bias for those two moments. While the third moment has mean ratio close to 1.0, the second moment becomes progressively underestimated with decreasing temperature. This effect is due to the truncation of the PSD at 100 μm. If, instead, we limited the comparison to PSDs with characteristic sizes greater than 500 μm, the effect of the truncation would be diminished. Overall, the biases and spread of the data are much improved over the results depicted in Fig. 10. For the fits provided by this study the geometric mean of the ratio of the moments is generally between 0.8 and 1.2 for temperatures warmer than −50°C for the tropical data and between 0.9 and 1.1 for the midlatitude data. The biggest differences between the fits presented here and those from the Field et al. (2005) study and an exponential distribution are for moments less than two in the tropical regime. For the midlatitude data, there is little variation between the different fits when two moments of the distribution are known.

9. Discussion

a. Power laws

The successful employment of the moment estimation parameterization in a microphysical scheme depends on the assumption that the power laws used to represent mass and fall speed as a function of maximum size are applicable to particles of all sizes. While this is expected to be true for aggregates of ice particles that display self similar properties, it is not necessarily correct for small particles (e.g., D < 300 μm) that will tend to exhibit the properties of their monomer habits. A possible solution to this problem is to use the predicted characteristic size to dictate the power laws used: for example, for large characteristic sizes assume aggregate relations and for small characteristic sizes assume pristine ice relations. To avoid discontinuities it would be advantageous to smoothly vary the power law from pristine to aggregate as a function of characteristic size.

b. Ice nucleation and two-moment microphysics schemes

Bulk microphysical schemes either represent snow and ice with a single category or divide the ice into cloud ice and snow categories with some nominal threshold size of around 100 μm. Each category can be represented by an ice water content only or number concentration and ice water content together. We acknowledge that two-moment microphysics schemes that predict both number and mass are inherently more accurate than a single-moment scheme that predicts mass alone, because they represent more of the essential physics involved, but point out that the successful implementation of a more complex two-moment, multispecies ice scheme relies upon the accurate representation of various microphysical rates and processes. For example, once the number concentration of cloud ice due to nucleation has been determined, the microphysical scheme must compute the decrease in number concentration due to aggregation, sublimation, and the transfer rate from cloud ice to the snow category. This last process is analogous to the autoconversion process in liquid clouds. Number concentration in the snow category then depends upon aggregation, sublimation, and autoconversion. We suggest that many of the rates and processes are not well constrained, at present. Currently, state-of-the-art parcel models predict homogeneously nucleated ice particle concentrations that have a factor of 25 spread mainly due to differences in microphysical constants assumed (Lin et al. 2002). Aggregation rate estimates vary widely (e.g., Field and Heymsfield 2003), and any effect of temperature or habit on this value is not well characterized.

We anticipate that using the proposed single-moment bulk microphysics scheme will on average perform better at predicting microphysical process rates associated with ice crystal aggregates (snow) than a two-moment scheme until the uncertainties related to the microphysical processes outlined above can be reduced. While the use of a single-moment scheme may be adequate for numerical weather prediction, it does not easily allow for investigations into the effects of different aerosol loading on the climate system through interaction with cloud microphysics. One possible avenue of research would be to try various representations of ice nucleation, aggregation, and autoconversion within a two- (or more) moment microphysical scheme. Then, truncated moments could be generated using Eq. (1) and the simulated moment relationships could be compared with those presented here until a good match was found. It may be the case that feedbacks driven by aggregation and diffusional growth will lead to well-constrained concentrations of particles larger than 100 μm, which in itself would be a result worth further investigation.

c. Aggregation and diffusional growth

Numerous studies have documented the exponential variation of the slope or characteristic size of the PSD with height, although these studies fold in changes in IWC as well. It has been suggested that this observed relationship is due to aggregation increasing the mean size of the distribution. Mitchell (1988) showed this for the slope of an exponential distribution and Westbrook et al. (2007) argue that a similar relation for the characteristic size of the PSD can be found. The parameterization given in section 6 generates a relation for the characteristic size L23 = M3/M2 = A(3) exp[B(3)Tc]M0.242, which is exponentially dependent upon temperature but also has some weak dependence on the second moment. For a PSD undergoing only aggregation the second moment will be nearly constant and the characteristic size will then vary exponentially. Observations made during a Lagrangian spiral descent in which a PSD evolved through aggregation alone (Field et al. 2006a) show a factor of 1.5 increase in characteristic size over a 6°C temperature change. This observed increase is larger than given by this parameterization (factor of 1.15) when the second moment is held constant.

The relationship between moments at a given temperature is proportional to MC(n)2 where we have found C(n) empirically. For small amounts of diffusional growth or loss we would expect that, if we could observe the entire PSD, then the actual concentration including particles smaller than 100 μm would remain constant. We can use the relationship [e.g., Field et al. (2005), where a generalized equation for any pair of moments can be found]
i1520-0469-64-12-4346-e5
where the starred moments represent the complete moments of the PSD computed from zero to infinity, m02,n is the nth moment of the distribution rescaled using the second and zeroth moments, ∫0 xnΦ02(x) dx (not presented here), and is therefore constant. If M*0 is considered constant, we can see that for diffusional growth or loss M*nM*n/22. Here C(n) is fairly well approximated by n/2 for n > 2, but this simple relation fails for n < 2. The main reason for the relationship’s lack of agreement with C(n) is due to the truncation of the PSD at 100 μm. The effect of the truncation is that particles can appear to be created and destroyed by crossing this size threshold through the action of diffusional growth/loss. Even if we could sample the entire PSD, real evaporation and particle production during diffusional growth/loss would still lead to lower (higher) values of C for n > 2 (n < 2).

Therefore, we can view the components of the parameterization as approximately representing the physics relating to aggregation (exponential part) and diffusional growth or loss and possibly including the effects of new particle production (power law part). The differences in aggregation and diffusional growth identified above highlight the fact that this parameterization is simply a “climatological” representation rather than physical. These differences underline the need for reducing the uncertainties in the microphysical processes mentioned above so that two-moment snow schemes that explicitly reproduce the action of aggregation can be implemented.

d. Difference in rescaled distributions

Delanoë et al. (2005) attribute the difference in the midlatitude and tropical rescaled distributions to the fact that the midlatitude clouds are generated through large-scale uplift, whereas the tropical cases result from convection. We would like to speculate more on the difference by noting that the variations in the underlying rescaled size distribution are related to the physics governing the aggregation of the ice particles. In midlatitude stratiform cloud electric field strengths can range from 1 to 10 kV m−1, but in continental convection the field strengths within the anvils can be 10 times larger (MacGorman and Rust 1998). There is less lightning associated with maritime convection, but it is likely that field strengths will still exceed those found in midlatitude stratiform cloud. Where electric fields are significant they may enhance aggregation and modify the collection kernel when compared to the collection kernel appropriate for midlatitude stratiform cloud. Tan et al. (2000) have shown that the morphology (specifically the mass–dimensional relationship) of aggregates can change as a function of applied electric field and produce increasingly elongated aggregates with increasing electric field strength reminiscent of the laboratory studies of Saunders and Wahab (1975). Aggregation modeling using collection kernels that have the influence of electrical effects included needs to be carried out to critically assess this speculation.

Other potential reasons for the difference between the tropical and midlatitude rescaled distributions could be related to differences in the monomer population. It is possible that the convective activity can grow pristine single crystals to quite large sizes (∼500 μm) before they are detrained into the anvil, as compared to their counterparts in midlatitude cloud. Potential differences in the initial populations of monomer crystals could lead to differences in rescaled distributions until sufficient aggregation had occurred. Again, it should be possible to test this speculation with aggregation modeling.

The rescaled distributions shown in Fig. 6 are obviously different, yet the estimation of moment parameterization is still applicable for both regimes (section 7). At first glance, this appears contradictory, but, if we write an analogous expression to Eq. (4) for the second and third complete moments (Field et al. 2005), we obtain M*n = M*(n−2)3M*(3−n)2m23,n, where m23,n is the nth moment of the rescaled distribution. If we assume that A(n), B(n), and C(n) are the same for both midlatitude and tropical datasets (as we have found), then by eliminating M3 using Eq. (3) we will only get the same estimates for Mn if the moments of the midlatitude and tropical rescaled distributions are the same. Computing m23,n(n = 0, 1, 2, 3, 4, 5) for the tropical and midlatitude rescaled distributions we obtain [25.1, 2.35, 1.0, 1.0, 1.61, 3.5] and [12.2, 1.61, 1.0, 1.0, 1.25, 1.83], respectively. It can be seen that the first and fourth moments of the rescaled distributions are ∼35% different, while the zeroth and fifth moments are about a factor of 2 different. So, although the forms of the rescaled distributions appear different, their moments are perhaps similar enough to allow the use a single-moment estimation parameterization for both tropical and midlatitude clouds.

e. Possible effects of truncation

All of the analysis has been carried out for particles larger than 100 μm. Therefore, there are likely to be systematic differences between the moments as defined by Eq. (1) and those that would be obtained if we could have observed the whole PSD and integrated from zero rather than 100 μm. One way to assess the possible effects of the truncation is to assume that the rescaled size distributions found in section 5 (Table 2) are applicable to all sizes and that the measured second and third moments are accurate. McFarquhar and Heymsfield (1997) show comparisons of IWC estimates from the 2D-C and the Video Ice Particle sampler in the 10–100-μm size range that indicate that, when the total IWC is 0.01 g m−3 (M2 ∼ 10−4), then the IWC contribution from particles smaller than 100 μm is equal to the IWC contained in the larger particles. These values are at the lower end of the observations presented here, so the majority of the second-moment estimates are likely to be unaffected by truncation. Figure 12 shows the ratio of the estimated “complete” moment to the measured “truncated” moment, using the above-stated assumptions, as a function of characteristic size. The results are shown for the zeroth, first, and fourth moments for both tropical and midlatitude clouds (using dataset “b”). It can be seen that, potentially, the total concentration could be over 10 times greater than that observed for D > 100 μm when the PSD is narrow. The effects of the truncation are less dramatic for the other moments that span the range important for predicting the microphysical process rates involving snow. We note that, although the systematic errors may be larger for PSDs with smaller characteristic sizes, the absolute magnitude of the microphysical process rates will also be small. As the magnitude of the microphysical rates involving snow becomes larger so does the characteristic size and the accuracy of the parameterization. Finally, using the assumptions stated above, we can look at histograms similar to Fig. 1 for the total concentration after extrapolating below 100 μm. Interestingly, Fig. 13 shows that the “total extrapolated concentration” is now more comparable for both midlatitude and tropical clouds with means generally around 100 L−1.

The effects of the truncation are still uncertain because we are not yet able to satisfactorily observe the PSD for D < 100 μm within thick ice cloud. The above analysis is just one possibility for an exponential extrapolation of the PSD to smaller sizes, and reality could well be different.

10. Conclusions

We present a parameterization valid for both midlatitude and tropical clouds to estimate moments of the snow PSD given temperature and an estimate of the second moment (proportional to IWC if mass ∝ D2). These moments can then be used in numerical weather models to predict the microphysical process rates involving snow that control the evolution of the cloud. Additionally, we have presented rescaled PSDs from the Tropics and midlatitudes that can be used in conjunction with the moment prediction parameterization to generate PSDs if required. (The parameterization is available as FORTRAN or IDL routines from the authors.)

Acknowledgments

We gratefully acknowledge the hard work and dedication provided by the University of North Dakota’s Citation aircraft personnel and the NCAR Research Aviation Facility in obtaining the data used in this study. We also thank three anonymous referees for their comments that helped improve the manuscript.

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Fig. 1.
Fig. 1.

Normalized histograms (total equal to unity) of the zeroth moment (concentration for D > 100 μm) generated from 10-s snow PSDs from midlatitude (gray) and tropical (solid) datasets. The histograms cover six temperature ranges (5° to −55°C in 10°C intervals). The vertical bars represent the Poisson counting error in each bin. The inset vertical axis is the normalized frequency. Here N is the number of points (one point for each PSD) in the histogram, μgeo is the geometric mean, and σgeo is the geometric standard deviation.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 2.
Fig. 2.

As in Fig. 1 except for the second moment (IWC for D > 100 μm if mass ∝ D2) of the 10-s PSDs: Midlatitude (gray) and tropical (solid) datasets.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 3.
Fig. 3.

As in Fig. 1 except for the fourth moment (radar reflectivity Z for D > 100 μm if mass ∝ D2 and pure Rayleigh scattering) of the 10-s PSDs: Midlatitude (gray) and tropical (solid).

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 4.
Fig. 4.

As in Fig. 1 except for the characteristic size (mean mass weighted size for D > 100 μm if mass ∝ D2) of the 10-s PSDs: Midlatitude (gray) and tropical (solid).

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 5.
Fig. 5.

Variation of histogram parameters for the second moment as a function of averaging time over which the snow PSD was compiled: (a) variation of mean of the second moment and (b) variation of fractional standard deviation (σ/μ).

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 6.
Fig. 6.

Rescaled PSDs for tropical and midlatitude clouds and fits (solid lines, Table 2). Each point represents a 10-s average rescaled concentration from each size bin of the PSDs. (a) Log–log plot of data from the tropical “a” dataset; (b) log–linear plot of data from tropical “a” dataset; (c) log–log plot of data from the tropical “b” dataset with fit from tropical “a”; (d) log–log plot of data from the midlatitude “a” dataset; (e) log–linear plot of data from midlatitude “a” dataset; and (f) log–log plot of data from the midlatitude “b” dataset with fit from midlatitude “a.” Also shown are the fits from F05 and a rescaled exponential size distribution, Exp.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 7.
Fig. 7.

Scatterplots of the measured moments of the PSDs vs the predicted moments using the parameterization given in Table 3. Each point represents the value obtained from a 10-s PSD. The grayscale is intended to provide a qualitative indication of temperature (lightest = −60°C, darkest = 0°C); (a) fourth, (b) third, (c) first, and (d) zeroth moments. The 1:1 lines are also shown.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 8.
Fig. 8.

Quadratic fits to the parameters in the moment estimation parameterization (Table 3). The filled circles are the values obtained using 10-s PSDs. The solid line is the least squares fit quadratic to the 10-s data. Results obtained when midlatitude, tropical, and 120-s PSDs only are used to estimate the moments are also shown.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 9.
Fig. 9.

Number of 10-s data points used in 10°C temperature bins for the comparison from the tropical “b” (dashed) and midlatitude “b” (solid) datasets.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 10.
Fig. 10.

Test of the moment estimation scheme assuming known second moment and temperature. (top pair) Tropical “b” comparison of (upper row) geometric mean μgeo of the ratio of estimated to measured moment as a function of temperature in 10°C temperature bins and of (lower row) geometric standard deviation σgeo of the ratio of estimated to measured moment as a function of temperature. (bottom pair) As for “Tropical b” but for the midlatitude “b” dataset. Columns from left to right show the results for moments 0, 1, 3, and 4, respectively. Results from this study, F05, and H79 are shown. Legend is in the bottom left panel.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 11.
Fig. 11.

As in Fig. 10 but assuming the second and third moments of the PSD are known. Columns from left to right show results for moments 0, 1, 2, 3, and 4, respectively.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 12.
Fig. 12.

Estimated ratio of total moments to measured truncated moments if the PSDs are extrapolated assuming that the rescaled size distribution given in Table 2 is valid for all sizes (section 5) and the measured second and third moments are unaffected by truncation errors. (a) Tropical zeroth moment; (b) midlatitude zeroth moment; (c) tropical first moment; (d) midlatitude first moment; (e) tropical fourth moment; and (f) midlatitude fourth moment.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Fig. 13.
Fig. 13.

Normalized histograms (total area equal to unity) of the extrapolated total concentration generated from 10-s snow PSDs assuming that the rescaled size distribution given in Table 2 is valid for all sizes (section 5) and the measured second and third moments are unaffected by truncation errors: Midlatitude (gray) and tropical (solid) datasets. The histograms cover six temperature ranges (5° to −55°C in 10°C intervals). Vertical bars represent the Poisson counting error in each bin, and the inset vertical axis is the normalized frequency: N is the number of points (one point for each PSD) in the histogram, μgeo is the geometric mean, and σgeo the geometric standard deviation.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2344.1

Table 1.

Details of the data used.

Table 1.
Table 2.

Rescaled PSD fits for tropical and midlatitude cloud: Φ is the rescaled dimensionless concentration and x is the dimensionless size (D/L23).

Table 2.
Table 3.

Moment estimation parameterization (SI units): T (°C) is the in-cloud temperature, Mn is the moment of order n to be predicted, and M2 is the second moment of the PSD [see Eq. (1)].

Table 3.

1

Repeating this exercise using dataset “a” produces similar results to those presented for “b.” This indicates that both datasets appear to sample the population equally well.

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