1. Introduction

Recent general circulation model (GCM) studies have demonstrated the impact assumed fall velocities have on predictions of cloudiness and radiative forcing. Sensitivity studies using the European Centre for Medium-Range Weather Forecasts (ECMWF) model has been reported by Heymsfield and Iaquinta (2000) and Jakob (2002). These studies examined the relationship of global, radiative flux divergence to ensemble-mean ice crystal sedimentation rate, V_m. Relatively small changes in fall speed led to significant changes in the net global flux divergence. For V_m, using the expected range of 25 to 75 cm s⁻¹, the net flux varied over 8 W m⁻². Iacobellis et al. (2003) found in a 1D-column model that, as particle fall speed increases, the overall cloud fraction, cloud height, and grid-averaged ice water path decreased. The outgoing longwave radiation differed by up to 4 W m⁻² over the range of fall speeds they examined. Given that 1 W m⁻² converts roughly to 0.67 K in the earth’s mean surface temperature, for climate models, reliable estimates of V_m are essential. In Part II of this study, in situ microphysical observations from Heymsfield et al. (2007, hereafter Part I) have been used with Doppler radar measurements to better constrain estimates of ice particle ensemble mass-dimensional relationships and mean, mass-weighted terminal velocities. In the past, these relationships have been difficult to constrain and evaluate.

Most GCMs now use some form of cloud microphysical parameterization to predict the mixing ratio of the condensate at a given time step and grid point, from a water mass continuity equation. For GCM models there are two ways to convert ice condensate to ice precipitation. One method uses a Kessler-type autoconversion scheme producing ice precipitation. The application of this scheme produces and removes condensate that can be converted to vapor or fall to the ground. Other GCMs allow precipitated ice to fall between cloud levels and to be retained from one time step to the next. Using the characteristics of the particle size distributions or relationships reported in the literature; that is, Heymsfield and Donner (1990), the rate of fall is determined by the IWC.

In GCM and weather forecast models, assumptions involved in the treatment of ice and snow fall speeds may lead to errors. As an example, the threshold for precipitation developed through autoconversion may be too large. As all of the particles in such a situation have zero fall velocity, they may only be removed through sublimation and, at low temperatures, sublimation rates are usually very low as well. Because of this effect, when compared with climatological data, these models may produce too-extensive areas of radiatively active cirrus cloud. Another source of error is applying mean sedimentation velocities to full size distributions. This too-quickly precipitates out smaller particles with lower fall speeds, resulting in too-small ice clouds. Another major source of uncertainty is the crystal concentration, or related particle size distribution (PSD) parameters such as the intercept concentration parameter N₀ (see Lin et al. 1983). Varying N₀ changes the mean particle size or slope of the PSD, and V_m. If accurate, mass-weighted fall speeds were available, this problem could be reduced by a double-moment approach in which particle numbers are moved by the number-weighted fall speeds and the masses by the mass-weighted ones.

Figure 1 shows the very wide range of V_m for a given IWC using both earlier and more recent parameterizations. Much of the spread is due to incomplete knowledge of ice cloud particle properties. For most parameterizations, a size distribution function is specified or assumed. The V_m in most schemes is found analytically, from the size-dependent concentration, mass, and fall speed integrated from zero to infinity, divided by the ice water content. In these, an explicit fall velocity relationship, V_t = AD^B, and an adjustment for atmospheric pressure,¹ is used. Some representation of ice particle mass, m, in terms of ice particle maximum dimension, D, or a population-mean density, is required. Pressure or air density dependence is not specified explicitly in V_m parameterizations specific to cirrus, with relatively narrow ranges in pressure making such adjustments less necessary (Heymsfield and Donner 1990; Matrosov and Heymsfield 2000). The V_m–IWC relationships in these schemes are derived empirically, either from measurements of the PSD and estimates of particle masses and fall speeds, or from Doppler radar remote sensing of cirrus clouds. The lowest V_m values are in the Goddard Institute for Space Science (GISS) model and are intentionally so specified to correct for an upper troposphere model dry bias (A. DelGenio 2005, personal communication). The GISS parameterization also does not depend upon pressure. The Lin et al. (1983) one-moment microphysical scheme produces the largest V_m values, higher than Rutledge and Hobbs (1983) even though both methods use the same snow density (0.1 g cm⁻³). This occurs because the two methods use different fall velocity relationships, adjusted slightly differently to account for the ambient atmospheric pressure.

As noted in Part I, an assumed distribution of particle mass with size might yield the correct IWC but not the correct V_m because they are differently weighted by the size distribution. This is illustrated by the curves in Fig. 2, which show the results of calculations of the reflectivity-weighted mean terminal velocity, V_Z. Here V_Z is approximately weighted by the square of the mass (section 2) instead of linearly by mass used in mean, mass-weighted ensemble fall speeds.

In Fig. 2, V_Z is plotted as a function of temperature for different values of the exponent b in the mass-dimensional relationship. The associated coefficient a, with PSD, produced IWCs yielding the measured value. Non-Rayleigh scattering effects, assuming spherical particles, are considered in these calculations (section 2), but are small at 8.6-mm radar wavelength. A method for refining the estimate of b is suggested by Fig. 2 because the assumed b produced a large range in V_Z.

Part I reports temperature-dependent coefficients for the mass-dimensional relationship. Here, in situ and Doppler radar remote sensing observations are used together to derive m(D) relationships from which improved estimates of the lower and higher moments, including V_m, are derived. The methodology is briefly summarized in section 2 and the results of individual case studies are presented in section 3. Section 4 summarizes the principal findings of section 3. Section 5 evaluates representations of V_m used in earlier microphysical parameterizations in light of the new results. Section 6 develops parameterizations representing the results. The primary findings are summarized and conclusions drawn in section 7.

2. Methodology

This study uses in situ observations in middle- and low-latitude ice cloud layers sampled between −60° and 0°C. Five cloud layers referred to as synoptically generated were sampled at midlatitudes, four by the University of North Dakota (UND) Citation over the Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) site in the field program of the spring of 2000. The fifth case, in a deep-layer cloud with embedded convection, was sampled by the National Center for Atmospheric Research (NCAR) C130 during the Alliance Icing Research Study (AIRS-2) near Mirabel, Canada, on 19 November 2003 (Isaac et al. 2005). Ten flights, primarily in convectively generated, low-latitude ice cloud layers, were made by the UND Citation during the Cirrus Regional Study of Tropical Anvil and Cirrus Layers, of the Florida Area Cirrus Experiment (CRYSTAL FACE, hereinafter CF).

Size-spectra measurements and information on ice particle cross-sectional areas, for sizes from about 50 to 1000 μm, were obtained from Particle Measuring Systems (PMS) 2D-C probes, having resolutions of about 33 μm. For the ARM flights, a PMS 2D-P probe, sizing from 200 μm to more than 6 mm, was used. The CF and AIRS-2 programs used a Stratton Park Engineering Company (SPEC) high-volume precipitation spectrometer with a resolution of 200 μm to collect data from 200 μm to >1 cm. Particles objectively determined to have been shattered on the tips of the 2D probes were removed from the dataset (Field et al. 2006).

Relatively few particles are measured in the largest sizes of the imaging probes, thereby leading to statistical uncertainties in the measured PSD and calculated moments, especially the radar reflectivity, which is weighted by the largest but fewest particles. To evaluate the potential effects on Z_e, we specified statistical errors in all size bins containing particles by assuming that the sample in each bin was Poisson distributed. Given that this statistical error is 1/N, where N is the number of particles per size bin per 5-s sample, we introduced an error of ±1 standard deviation about N, altering the concentration in each bin appropriately. We used the mass-dimensional relationship of Brown and Francis (1995), with b = 1.9, for this evaluation. Applying this method to the 9 March (synoptic) and 26 July (CF, high IWC/reflectivity) cases resulted in mean changes in Z_e of +3/−5 dB and +2/−3 dB, respectively. Statistical uncertainties in the PSD from the imaging probes, therefore, will produce insignificant bias about the mean values.

A forward scattering spectrometer probe (FSSP) provided size distributions for particles from 2 to about 50 μm although, in those cases with large crystals, breakup in the probe’s inlet may have led to overcounting (12 March, 19 November, and the CF clouds). Heymsfield et al. (2004, their Fig. A1 and associated text) show that in the IWC derived from the FSSP PSD for the CF clouds crystal breakup could produce an overestimate of the total IWC of about 5%. The effect on V_m would be even less.

FSSP and 2D particles were placed in one of the 48 size bins. Their average size distributions were measured over 5-s intervals, or about 0.75 km of horizontal flight. The cloud particle imager (CPI) probe collected high-resolution particle images. In the three field programs, a counterflow virtual impactor (CVI) directly measured the IWC.

Terminal velocity calculations require estimates of particle diameter, mass, cross-sectional area, drag coefficient, temperature (T), and pressure (P). Particle mass is expressed as a single, power-law relationship in the form

View Expanded

where D is the maximum imaged ice particle diameter. To span its likely range, in Part I, exponent b varied from 1.5 to 2.3. Using assumed values for b across this range, values for the coefficient, a, were determined by summing Eq. (1) across the entire binned size distribution and matching the measured IWC. This approach resulted in obvious errors. Below a given particle diameter, D_t, the mass of the particles exceeded that of solid ice spheres, thus implying that the particles were, in fact, solid spheres or drizzle. Using b = 1.5, however, 18% of the PSD from CF gave D_t > 150 μm, the percentage decreasing to 13% at b = 1.7 and to 12% at b = 2.3. The discussion and figures in the appendix in Part I suggest that many of these instances are due to overestimates of the measured IWC. When the IWC decreases rapidly, residual water is left in the CVI plumbing. Obviously anomalous points have been omitted from the dataset but some, difficult to remove objectively, still remain (Fig. A2; Part I). They have been implicitly removed by placing an upper limit on D_t.

Particle cross-sectional areas, A_c, were derived from 2D images. Each particle was assigned an area ratio, A_r, in which the particle area, A_c, was divided by the area of a sphere of the same maximum diameter. An average ratio, A_r, was found for the particles in each size bin. For particles in the FSSP size range, A_r was set to 1.0. Then, given A_c, D, T, and P, V_t(D) was calculated using Mitchell and Heymsfield (2005).

In the preparation of Fig. 2 and later evaluations, radar reflectivity Z_e (mm⁶ m⁻³) was calculated using Mie scattering code (Bohren and Huffman 1983), assuming the particles to be spheres with the same D but with a density reduced to give the appropriate mass. This procedure is known to be of limited accuracy (Matrosov et al. 2005), because large ice particles are usually not spherical. For 95 GHz, Donovan et al. (2004) show that solid equivalent mass or equivalent area spheres compare well with discrete–dipole approximation (DDA) calculations for a range of idealized crystal shapes for D < 1000 μm. However, for larger values of D the accuracy of equivalent sphere formulations is very uncertain. Accordingly, in this work we emphasize results where our non-Rayleigh calculations indicate that non-Rayleigh effects are small. This applies in general to cold cirrus cases. For the warmer synoptic cases and the CF clouds non-Rayleigh scattering effects can be significant, sensitivity calculations indicating that there may be an uncertainty of up to 3 dB. Reflectivity-weighted fall speed, V_Z, was calculated as was V_m by summing the products of Z_e(D_i) and V_t(D_i) per particle size bin i, integrated over all bins, divided by the (total) Z_e.

For all five synoptic cases, Z_e and the Doppler fall speed, V_D, were derived from upwardly pointing Doppler radars. Here V_D is the sum of V_Z + w, where w is the air velocity. The millimeter-wavelength cloud radar (MMCR; 35-GHz frequency) is located at the SGP site in Oklahoma (36.61°N, 97.50°W). Dual frequency data is from the 33 and 95 GHz, University of Massachusetts’ (UMass) collocated Doppler radars (Li et al. 2001). During the spring program in 2000, the radars, separated by a horizontal distance of 20 km, were located at the airport in Blackwell, Oklahoma (36.75°N, 97.35°W). In AIRS-2, in the fall of 2003, the McGill, X-band (9.4 GHz) Doppler radar operated at Mirabel, Canada (45.69°N, 74.03°W).

3. Results

This section reports the combination of a and b coefficients that may be used to accurately derive moments of the PSD: the second, IWC (approximate); the third divided by the second, V_m (approximate); the fourth, Z_e (approximate); and the fifth, divided by the fourth V_Z (approximate). Figure 2 shows that V_Z and Z_e are strong functions of the exponent b. The approach here is to compare the relationships of V_Z and Z_e from the Doppler radar data around the time of the in situ measurements, to those from the calculations using varying b. Rather than a 1:1 comparison of values obtained when the aircraft was within or close to the radar beam, this comparative approach has several advantages. It increases the sample size. It reduces the errors resulting from the large differences in sample volumes from aircraft probes and radar. It removes the uncertainties about the proximity of the aircraft to the radar and reduces possible errors in situ data noted in Part I.

4. Interpretation of radar–aircraft intercomparison

From section 3, the magnitudes of Z_e (=10^dBZ/10) and V_Z, and their relationship, are a strong function of the exponent b. The aircraft–radar data from 9 March (Figs. 4 and 5a) suggest that b is close to 1.5, but a range of 1.5 to 1.7 is more likely. This is because net updrafts may have produced the cirrus that would have reduced the measured V_Z. We also conclude from the synthesized Doppler radar observations from −30° to −70°C (Fig. 5b) that below −30°C, b is between 1.5 and 1.7. The dWR–dV_Z relationship measured on 12 March and calculated using b = 1.7 and 1.9 (Figs. 9a,c,d) suggests that between −30° and 0°C, b is about 1.75 or 1.8. As shown in Fig. 9f, the calculations using a b that increases linearly with temperature fit the observations quite closely. The MMCR radar observations from −30° to 0°C (Figs. 6 and 10), and for the AIRS-2 case (Fig. 11) are consistent with b = 1.7 to 1.9. In CF clouds between −25° and −52°C, Heymsfield et al. (2005) have shown that b of 1.6 fits coincident radar/aircraft observations.

The results cited above are consistent with observation and theory in fractal mass aggregates in several systems (e.g., colloidal particles in the presence of electrolytes). Although the rosettes dominating the colder temperature observations are not strictly aggregates, rosette aggregates have been observed. In mass fractal aggregates, if b is less than 2, the b exponent is preserved in the projection to cross-sectional area (σ), where σ is the exponent in the area dimensional relationship, a condition known as geometric transparency (Bushell et al. 2002). Because b is evidently less than 2, if the particles are aggregates, b and σ should have the same value. Figure 12 indicates strong dependence on temperature in σ derived from power-law fitted to the 2D particle image areas. The particles are from 5-s PSD given in Part I and the dependence is consistent with inferences drawn from section 3 herein. At warmer temperatures where aggregation dominates, σ tends to 2.0, consistent with the results reported by Westbrook et al. (2004). There is little difference between tendencies in synoptic and CF datasets. If a and b have the same values, an average of 1.75 for b, or one that increases with temperature, would fit the observations well. Table 1 gives curves representing the temperature dependence of a for b of 1.75 and with b varying with temperature. The a coefficients have been derived using the CVI IWC, as in Part I. The corresponding terminal velocity coefficients A and B in the relationship V_t = AD^B, normalized for a pressure level of 1000 hPa, are shown in Table 2. The a and b coefficients are valid for the full range of particle sizes and the A and B are valid for particles 100 μm and above, as in Part I.

5. Evaluation of earlier microphysical model parameterizations

The coincident, in situ, and radar observations given in section 3, present an unique opportunity to evaluate microphysical parameterizations of ensemble mean particle velocities for snow used in the earlier studies, specifically aspects of Lin et al. (1983, hereafter L83) and Rutledge and Hobbs (1983, hereafter RH83). Particularly interesting for evaluation in the two studies are the fall velocity relationships in Locatelli and Hobbs (1974), from which the fall velocities in the parameterizations are based on. We may also compare our results with those reported for cirrus by Heymsfield and Donner (1990, hereafter HD90). Errors in microphysical assumptions are amplified more by the Z_e and V_Z terms than they are by IWC or V_m. Now there are radar and in situ datasets that one can use to test the model calculations. We look first at how these earlier parameterizations predict the actual radar measurements. With a better understanding, it is then possible to evaluate errors in the V_m parameters.

The basic variables in L83 and RH83 are the slope of the exponential PSD, λ, the PSD intercept N₀, the assumed ice density ρ_i, the fall speed relationship V_t = AD^B, and the adjustments for air density. L83 takes N₀ = 0.03 cm⁻⁴ and RH83 as 0.2 cm⁻⁴. The λ is derived analytically from the IWC, ρ_i, and N₀. By integrating these variables to represent Z_e and V_Z, analytical relationships are developed between them. [The derivation of V_Z and Z_e for an exponential PSD is given by Heymsfield et al. (2002) and others. Non-Rayleigh scattering is not included.] The V_Z for the L83 and RH83 schemes are a function of A, B, λ, and indirectly ρ_i through the effect on λ whereas Z_e depends on ρ_i, N₀, and λ.

The values of the PSD parameters in L83 and RH83 were not derived for cirrus clouds but it is interesting to evaluate their applicability in such conditions, especially where non-Rayleigh effects are negligible. The comparisons are given in Fig. 13 where the measured and calculated (for temperature-dependent b from Fig. 12) distributions of V_Z to Z_e in the 9 March ARM case are compared with earlier calculations. The differences in the results from L83 are particularly noticeable.

In Fig. 13b λ and N₀ values from L83 have been used to produce PSD by numerical integration over the size range measured by our particle probes instead of integrating the function from 0 to infinity, which improved the results. In Fig. 13c, V_Z and Z_e are estimated from the PSD from the particle probes and the ρ_i, A, and B from L83, which improved the agreement with radar data. In Fig. 13d, the RH83 coefficients with a higher N₀ and lower terminal velocities than L83 have been used. Here V_Z is within several tens of cm s⁻¹ of the observations but Z_e is close to those of L83 and are as much as 15 dB higher than observed. The V_Z relationship is, therefore, appropriate but the ice densities too high.

Figure 14 shows the same plots as in Fig. 13, except for the warmer 12 March case. Similar results are noted: an ice density of 0.1 g m⁻³ is too large for the larger particles because the Z_e are overpredicted; the V_Z from L83 overpredict the observations and those from RH83 are flat and likely underpredict the observations; and, improvements are gained by using the measured rather than parameterized PSD.

Our goal is to derive accurate V_m over a wide range of IWC for synoptically and convectively generated ice cloud layers and to ascertain factors that lead to errors in earlier derivations. In Fig. 15, our V_m increase IWC, as expected from the results shown in Fig. 1. The mass parameterizations provide results that are comparable to those that use the IWC to constrain the mass estimates (cf. Fig. 15b to Fig. 15a and Fig. 15d to Fig. 15c) and have the advantage that they extend to lower IWC. The higher masses associated with the CF cases yield higher V_m than for the synoptic cases.

The relationship between V_m and IWC from Figs. 15b,d is replotted in Fig. 16a for synoptic cases and in Fig. 17a for CF cases. They are used as references because they extend to IWCs below those measured by the CVI. Figures 16b and 17b show V_m as a function of IWC using 1) the L83 approach, 2) L83 ice densities and fall velocities with the measured PSD, and 3) the measured PSD for exponent b of 1.7 with temperature-dependent particle masses as in Figs. 16a and 17a. With 3), the calculations represent a significant improvement but the mismatch with the results in Figs. 16a and 17a indicate that for small particles or low IWCs V_t from Locatelli and Hobbs (1974) is overpredicted. In Figs. 16c and 17c, V_m from RH83 is unrealistic, showing only slight dependence on IWC. This is probably explained by the use of an unrealistic V_t–D relationship. Other variables in RH83, particularly ice density, are similar to L83. From Figs. 16d and 17d, for the range for which it was developed, the HD90 expression produces good results for IWC < 0.1 g m⁻³. In Fig. 16d, note the scatter in V_m (dashed curve). The HD90 relationship should produce a monotonic curve. The V_m was calculated using b = 1.5, but the IWC, derived from temperature-dependent mass coefficients for all values of b, varies only slightly. In this figure, therefore, the spread represents the spread in IWC and it is remarkable that it is so small. This is due in part to the V_m dependence on IWC, developed in HD90.

6. Parameterizations

Based on the results given in section 5, the temperature and IWC dependencies of V_m are parameterized from the temperature. Two sets of calculations from the results presented in section 4 are examined: one using b = 1.75 and a obtained directly from the IWC (mass specific), and the second using temperature-dependent b and the associated, temperature-averaged a. To generalize the results, the pressure and temperature of the observations have been adjusted to surface (air density) conditions P₀ of 1000 hPa and 0°C according to [ρ₀/ρ(T, P)]^0.54 from Part I. Figure 18 shows a strong dependence of V_m on temperature for the synoptic (Figs. 18a,b) and CF (Figs. 18c,d) datasets. In the figure, the primary difference between the mass-specific and the temperature-averaged coefficients is that the latter, because of the averaging, have a much reduced scatter. At temperatures close to 0°C, the synoptic and the CF datasets are about the same but, at lower temperatures, the CF values are much larger. Given that convective mixing produces more homogeneous particle properties throughout the cloud column, this is not surprising.

In each Fig. 18 panel, and also as noted in Fig. 15, much of the scatter about the means may be attributed to variability in IWC. The trend is further investigated in Fig. 19, specifically for variable b and temperature-averaged a. The contours in this figure are probability distributions enclosing [IWC, V_m] points where the features noted in Fig. 15 exist. In Figs. 19a,b, the nearly horizontal lines show the median values for V_m as a function of IWC, partitioned according to temperature. With a given IWC, V_m increases with temperature, reflecting the broadening of the PSD to larger sizes as the temperature increases.

The median value for V_m as a function of IWC for all of the data combined (dark, solid line in Fig. 19) clearly does not account for the V_m dependence on temperature, and cannot be a curve universally applicable to ice clouds with wide temperature ranges. The IWC–V_m dependence, accounting for temperature has been fitted to the data in Fig. 19 as

View Expanded

where for the synoptic (CF) cases a₀ = 120.4, a₁ = 2.21, b₀ = 0.0487, b₁ = 4.57 × l0⁻⁴ (a₀ = 135.3, a₁ = 1.93, b₀ = 0.0058, b₁ = −0.002 453 9), and the median ratio of the predicted to derived V_m is 0.99 (1.09), with 75% (65%) of the points falling within 20% of the mean.

The combination of spaceborne and ground-based radars presents an opportunity to derive vertical profiles of the fall velocity of cloud ice particle ensembles on a global basis. Figure 20 shows the temperature dependence of the relationship of V_m (temperature-varying b) to Z_e, at the frequency of the CloudSat radar, 95 GHz. The results in this figure are fitted by the relationship

View Expanded

where for the synoptic (CF) cases c₀ = 109.3, c₁ = 1.87, d₀ = 0.782, d₁ = 0.00944 (c₀ = 82.4, c₁ = 0.92, c₀ = 0.8117, c₁ = 0.004 087), and the median ratio of the predicted to derived V_m is 0.99 (0.92). The approximate temperature may be derived from the CloudSat data stream. Synoptic and convective ice clouds may be differentiated on the basis of height- (or temperature-) dependence of Z_e. In Fig. 20, we estimate the uncertainty for Z_e < 20 dBZ to be 3 dBZ or less, if we consider aggregates to be oblate with their long axes horizontal and use the T-matrix scattering calculations of Matrosov et al. (2005). In Fig. 20, scatter in V_m is much less than in Fig. 19, probably from the variability in the particle cross-sectional area in the V_t calculations.

A ground-based Doppler radar operating in a vertically pointing mode can provide more accurate estimates for V_m. Orr and Kropfli (1999) removed vertical motions by averaging over 10-min time scales and quite accurate Vz were then obtained. Figure 21 shows the relationship of V_m/V_z to Z_e at a frequency of 35 GHz. These calculations are derived from the PSD for a temperature-varying b but the ratio is relatively independent of b. The following expresses the temperature dependence of the ratio

View Expanded

where the ratio should be limited to a maximum of 0.95. For the synoptic (CF) cases w₀ = 0.94, w₁ = 0.0044, x₀ = 0.00935, x₁ = 0.00019 (w₀ = 0.78, w₁ = 0.00210, x₀ = −0.00157, x₁ = −5.082 × l0⁻⁵, and the median ratio of the predicted to derived V_m is 1.02 (0.96).

These results suggest that long term estimates of V_m from ARM Doppler radars at several locations and non-ARM sites, like the three CloudNET European sites, could be used to evaluate spaceborne radar data.

7. Summary and conclusions

Part II of this study includes synthesized particle size distributions, direct measurements of ice water content, and vertically pointing Doppler radar measurements. The results produce coefficients in mass-dimensional relationships yielding accurate estimates of lower and higher moments from particle size distributions including ice water content, mean, mass- and reflectivity-related fall speeds, and radar reflectivity. We have also used the results to evaluate earlier bulk-parameterization methods relating ice water content to the mean, mass-weighted fall speed. The methods have been improved by incorporating better estimates of size and pressure-dependent fall speeds. Parameterizations of mass-weighted ensemble fall speeds as a function of the ice water content have also been improved by incorporating temperature dependence. Parameterizations have been developed to derive V_m from a 95-GHz spaceborne radar, and to evaluate these products using ground-based, vertically pointing Doppler radars.

It has been found that earlier, one-moment, bulk parameterizations, developed for cloud modeling studies, rely upon fall speed relationships that produce significant error. Such parameterizations may be reformulated to include temperature-dependent mass coefficients, a and b, (in m = aD^b) and A and B (in V_t = AD^B), from Tables 1 and 2. To account for varying atmospheric pressure (air density), adjustments in V_t can also be improved (Part I). Inclusion of a temperature-dependent, particle size distribution slope parameter, into a one-moment parameterization could reduce the need to assume a constant value for the intercept parameter of the PSD, N₀. These improvements could also be included in two-moment parameterization schemes and for bin-resolved models as well. Fall velocity relationships can be incorporated into two-moment microphysical schemes such as that advanced by Field et al. (2005) and others.

The testing methods advanced in this report should be part of the evaluation of bulk microphysical schemes. In so far as a consistent set of the variables, IWC, V_m, V_Z, and Z_e is obtained at multiple wavelengths, modeling case studies may be compared with coincident, or long-term, vertically pointing Doppler radar data to evaluate the model’s shortcomings. Comparisons at a number of geographically diverse sites such as the SGP and the tropical western Pacific (TWP) ARM sites, and at three CloudNet European sites, may be used to isolate differences between convectively and synoptically generated ice cloud microphysical properties. Opportunity also exists to derive global distributions of V_m from spaceborne radar reflectivity measurements. With them, the ratio of the mass to reflectivity-weighted ice particle fall speeds is found to be about 0.8. Opportunity now exists to evaluate them from geographically diverse, ground-based, vertically pointing Doppler radar measurements at multiple radar wavelengths.