Ice fall velocity has a strong impact on climate feedback, influencing cirrus cloud coverage and radiative forcing as well as upper troposphere relative humidity. This study aims to provide the atmospheric modeling community with better parameterizations of the ice fall speed in cirrus clouds on the basis of aircraft measurements from recent field campaigns, especially the Tropical Composition, Cloud and Climate Coupling (TC4) campaign in 2007 and the Indirect and Semi-Direct Aerosol Campaign (ISDAC) in 2008. These campaigns provide improved measurements of the ice particle size distribution (PSD) where the concentrations of artifact small ice particles (due to shattering of ice particles on the probe inlet tube) are greatly reduced. In addition to the PSD, the mass-weighted fall velocity (Vm) depends on the ice particle projected area and mass. The calculation of Vm was based on improved direct measurements of the PSD number and area concentration and improved estimates of ice particle mass. The effective diameter (De) was calculated in a similar way. The TC4 analysis has provided a diagnostic relationship that relates Vm to both cloud temperature (T) and ice water content (IWC) with an r2 of 0.78. A similar relationship for De was also obtained with an r2 of 0.82. The Vm relationship and associated Vm-IWC-T measurements were found to agree well with a Vm scheme based on T and cloud radar retrievals of Vm and IWC in tropical cirrus clouds. However, a critical climate-influencing parameter like the ice fall speed needs to be coupled with the cloud microphysics and radiation in climate models. This is made possible through strong correlations between De and Vm regarding TC4 and ISDAC cirrus. Finally, TC4 satellite retrievals of De and Vm are found to be consistent with corresponding observations.
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 The complex interaction between aerosols and cirrus clouds cannot be properly addressed without an accurate representation of ice sedimentation rates that help determine the cirrus cloud lifecycle and coverage. Ice sedimentation rates in cirrus are the product of the mass-weighted fall velocity of the ice particle size distribution (PSD), or Vm, and the corresponding ice water content (IWC). The value of Vm depends upon the PSD mean particle size, the PSD shape such as the degree of bimodality, the ice particle shapes comprising the PSD, and the air temperature and pressure [Mitchell et al., 2010]. Aerosol particles are believed to affect cirrus clouds by changing the rates that ice crystals are nucleated, which changes the number concentration, the mean ice crystal size and possibly the shape of the PSD. In a global climate model (GCM) experiment, the concentration of small ice crystals in bimodal PSDs was shown to have a strong impact on the cirrus ice water path (IWP), the cirrus cloud coverage, and the shortwave and longwave cirrus cloud forcing and heating rates, all resulting from their impact on Vm [Mitchell et al., 2008].
 In a GCM study that relates climate sensitivity to atmospheric GCM model parameters, Sanderson et al.  found that the ice fall speed was the second most important parameter for determining climate sensitivity. A decrease in fall speed was linked to an increase in cirrus cloud coverage, longwave cloud forcing and upper troposphere water vapor. The ice fall speed affects the upper troposphere water vapor amount [e.g., Mitchell and Finnegan, 2009], which appears most critical for determining climate sensitivity [Sanderson, 2011]. An earlier GCM study by Jakob  found the cirrus IWP and longwave cloud forcing was very sensitive to the ice fall speed.
 The parameterization of Vm in GCMs ranges from using a fixed Vm value [Jakob, 2002] to estimating Vm from empirical fall velocity-dimensional relationships for cloud ice and snow [Morrison and Gettelman, 2008] to an ice crystal habit and mass dependence assuming monodisperse ice crystals [Lohmann et al., 2008]. Deng and Mace [2008, hereinafter DM08] used cloud radar Doppler moments to predict Vm in terms of temperature and IWC on the basis of a 7 year cirrus cloud radar data set collected at Atmospheric Radiation and Measurement (ARM) program ground sites. Such Vm treatments are well suited to GCMs having single-moment microphysical schemes that only predict the ice mixing ratio. More recently Vm has been formulated for GCMs in terms of ice particle shape (i.e., temperature) and the degree of riming [Lin et al., 2011; Lin and Colle, 2011].
 In two-moment ice microphysical schemes that predict both the ice mixing ratio and number concentration, the effective diameter De is often predicted on the basis of the IWC and PSD projected area, which require assumptions about the PSD and ice particle shape. Because of a common dependence on the ice particle mass-to-area ratio, Vm is strongly correlated with De as shown by Heymsfield . Predicting Vm from De in two-moment microphysical schemes provides self-consistency between Vm, De, and assumptions about the PSD and ice particle shape, thus making the coupling between the ice microphysics and radiation explicit, provided the ice cloud optical properties are parameterized using De. Since ice cloud coverage depends strongly on Vm, predicting Vm from De also explicitly couples the cloud life cycle to the microphysics. This seems desirable given the important role of Vm in climate feedback.
 In this study Vm and effective diameter De are calculated directly from measurements of cirrus PSD, ice particle projected area and corresponding estimates of ice particle mass. These measurements were made during three recent field campaigns; the Tropical Composition, Cloud and Climate Coupling (TC4) campaign in 2007 (funded by NASA), the NASA African Monsoon Multidisciplinary Analysis (NAMMA) Mission in 2006 and the Indirect and Semi-Direct Aerosol Campaign (ISDAC) in 2008 funded by the Department of Energy's ARM program. Mostly maritime anvil cirrus were sampled during TC4 and NAMMA [e.g., Jensen et al., 2009; Lawson et al., 2010], and Arctic cirrus were sampled during ISDAC [McFarquhar et al., 2011]. This study is based on measurements mostly from TC4 and ISDAC. These measurements are unique because, as explained in section 2, the problem of ice artifact production (contaminating PSD measurements) was much reduced, rendering more realistic estimates of cirrus cloud Vm and effective diameter, De, which were not available before these field campaigns.
 In this paper the field campaigns, measurements, and data analysis are described in section 2, followed by results in section 3. Section 4 compares measured De and Vm with De and Vm derived from satellite radiances, and a summary is given in section 5.
2.1. Field Campaigns
 The TC4 field campaign was conducted near Costa Rica in July and August of 2007, with the NASA DC-8 aircraft sampling anvil cirrus, aged anvil cirrus and in situ cirrus clouds. Anvil cirrus here are relatively young anvils attached to the convective column, whereas aged anvils are not attached to the column. In situ cirrus are cirrus formed by processes unrelated to deep convection. This study uses flight data obtained on 22 July, 24 July, 5 August, and 8 August 2007. On 5 August, the NASA WB57 joined with the DC-8 to vertically profile a deep anvil cirrus deck. To complement the in situ cirrus measured during TC4, in situ cirrus data from the NAMMA campaign were also used in this study for the following flight days: 19 and 26 August and 3 September 2006. Constant temperature transects or legs were flown through the clouds during TC4 and NAMMA, with one measured PSD representing one flight leg.
 The ISDAC field campaign was conducted in April 2009 and focused on mixed phase clouds near Barrow, Alaska. However, ferry flights between Barrow and Fairbanks Alaska flew through cirrus clouds that were sampled during 9 flights on 7 different days. The temperature range of the sampled cirrus ranged from −25°C to −40°C. Sampling of colder clouds was not possible because of the limited altitude capability of the Convair 580 research aircraft. The ISDAC PSD used here are from 1 to 2 min periods of flight time where the cirrus microphysical properties (median mass dimension and extinction coefficient) were not changing rapidly over time. A total of 162 ISDAC PSD were processed from these 9 flights. No evidence for liquid water was found in these clouds on the basis of measurements from the cloud particle imager (CPI), the Rosemont icing rod, the relative humidity sensor, and the forward scattering spectrometer probe (FSSP).
2.2. Field Measurements
 In the noted field campaigns a relatively new probe was used for measuring the PSDs; the 2D-Stereo, or 2D-S, probe [Lawson et al., 2006]. The 2D-S directly measures ice particle length and projected area, and indirectly measures ice particle mass. The ice particle mass in each size bin was determined from 2D-S measurements of ice particle area using the mass-area relationship described by Baker and Lawson . The IWC was calculated by integrating the PSD ice particle masses, and this method was shown to produce IWCs consistent with those measured directly by the counterflow virtual impactor (CVI) during TC4, with differences less than ∼50% and an r2 of 0.88 [Lawson et al., 2010; Mitchell et al., 2010]. The CVI uncertainty in IWC has been estimated to be 13% at water contents of 0.05 to 1.0 g m−3 increasing to 16% at 0.010 g m−3 and to 40% at 0.0025 g m−3 [Heymsfield et al., 2007; Twohy et al., 1997, 2003]. Thus the 2D-S yields direct or indirect measurements of the size distributions for ice particle concentration, area and mass. These three types of size distributions were used to calculate De and Vm, as described below.
 An additional capability of the 2D-S to make improved measurements of ice particle size and projected area is its ability to provide true 10 μm pixel resolution at jet aircraft speeds. In laboratory experiments the 2D-S probe was shown to accurately image an 8 μm fiber rotating at 233 m s−1 [Lawson et al., 2006], which is greater than the true airspeeds that aircraft experienced during these field campaigns. Ice crystals smaller than this appear to have a negligible impact on most anvil cirrus optical properties [Jensen et al., 2009]. This higher resolution improves the measurement of particle projected area.
2.3. Calculation of De and Vm
 Since the ice particle mass is estimated from the 2D-S probe measurements and the 2D-S IWCs compare favorably with the CVI IWCs, it appears that ice particle masses based on the Baker and Lawson  area-mass relationship are realistic regarding the TC4 cirrus anvils. CVI measurements of IWC were not made during ISDAC cirrus flights and it is assumed that ice particle masses derived from the Baker and Lawson  area-mass relationship are valid for ISDAC cirrus. The violation of this assumption should impose a systematic bias on our results, which should not substantially degrade the correlation in the relationships described in section 3.
 With measurements of the concentration of ice particle area, estimated mass and number (to calculate mean bin values of area and mass) for each size bin of the 2D-S probe, these properties can be used directly to calculate the ice particle fall velocity in each size bin, using either the fall speed formulation of Mitchell and Heymsfield  or Heymsfield and Westbrook . The formulation of Heymsfield and Westbrook is said to be more accurate for ice particles having aspect ratios far from unity, such as long columns and needles or “stellar” dendritic ice crystals. This scheme is therefore used in this study, and it is compared with the Mitchell-Heymsfield treatment to evaluate ice particle shape effects.
 The treatment of effective diameter used here is general for liquid, mixed phase and ice clouds [Mitchell, 2002], and is expressed as
where ρ is the bulk density of ice (0.917 g cm−3) in this case and At is the total projected area of the PSD. To apply this definition to the 2D-S measurements, De is calculated as follows:
where IWC(D) is the measurement derived ice mass concentration in a given 2D-S size bin and A(D) is the measured projected area concentration in a given bin. In a similar way, the mass-weighted PSD fall velocity, Vm, is calculated:
where v(D) is the fall speed calculated for a given bin size D on the basis of the mean area Am(D) and the mean mass m(D) for that bin. These quantities are calculated as follows:
where N(D) is the bin number concentration. In this way De and Vm are calculated as directly as possible from measurements. Except when we relate Vm to temperature, Vm calculations reported in this study assume a reference temperature of −20°C and a pressure of 500 hPa. Otherwise pressure is estimated from the observed temperature T as
where temperature is in K, po and To are 500 hPa and 253 K, g is the gravitational constant, R is the gas law constant for dry air, and Γ is the lapse rate (assumed to be 7.5 K km−1).
 In the above calculations, the area ratio AR was calculated for each size bin, where AR is the area of an ice particle divided by the area defined by a circle having diameter equal to the particle's maximum dimension D. All measurements of Am(D) were checked to insure that AR ≤ 1.0. In rare cases when AR > 1.0, then Am(D) was set to the area of a corresponding circle. A similar procedure was adopted for ice particle mass, where mass could not exceed the value of an ice sphere at bulk density (0.917 g cm−3) having the same maximum dimension.
 Since the AR is a function of ice particle shape, ice particle shape differences among PSDs were evaluated using AR. This was done by summing A(D) over the PSD for ice particles greater than 60 μm (smaller ice particles tend to be quasi-spherical [Korolev and Isaac, 2003; Mitchell et al., 2010, 2011]. The same was done using ice spheres (i.e., ice particle maximum dimension defines a sphere from which the area cross section is calculated). Then the area ratio of the PSD was determined as
where As(D) is the area of a sphere having diameter equal to the ice particle maximum dimension and Dmax is the maximum particle size of the PSD. Probability distribution functions (PDFs) of ARPSD were produced for each field campaign to assess how similar ice particle shape attributes were between campaigns.
3.1. Comparing PSDs and Their Area Ratios Between Field Campaigns
 Ice particle size distributions (PSDs) for anvil cirrus and aged anvil cirrus sampled during TC4 are shown in Figure 1, along with in situ cirrus sampled during TC4 and NAMMA and Arctic cirrus sampled during ISDAC. During TC4 a total of 25 anvil cirrus flight legs (corresponding to PSDs) were sampled on 22 and 24 July, 23 aged anvil cirrus flight legs were sampled on 5 and 8 August, and 6 in situ cirrus flight legs were sampled on 22 July and 8 August. During NAMMA on 19 and 26 August 2006 and 3 September 2006, 6 flight legs/PSDs were obtained from in situ cirrus. As noted, 162 cirrus PSDs were selected from the ISDAC ferry flights. These PSDs were binned into temperature intervals of 5°C as shown in Figure 1. If a temperature interval is missing, that means there were no PSDs sampled in that interval.
 There is a tendency for PSDs to evolve from narrower to broader with increasing temperature. A closer analysis reveals that for anvil cirrus, a PSD transition from bimodal to monomodal (with higher concentrations of small ice crystals) occurs around −40°C near the temperature where homogeneous freezing nucleation begins. These cirrus are associated with locally strong updrafts capable of producing supersaturations with respect to ice in excess of ∼40%, a prerequisite for homogeneous freezing nucleation that produces ice crystals at higher rates than heterogeneous processes [Kärcher et al., 2007]. In contrast, aged anvil cirrus are detached from convection and are not likely to have strong updrafts. Nonetheless, there are still narrower PSDs and higher concentrations of small ice crystals for T < −40°C, albeit this is a more subtle transition relative to “fresh” anvil cirrus. For in situ cirrus, there is no PSD transition as temperatures enter the homogeneous freezing domain (T < ∼−38°C). These cirrus are characterized by relatively weak updrafts that may preclude homogeneous freezing from occurring. While other interpretations are possible, it can be said that PSD evolution in these three types of cirrus is consistent with the homogeneous freezing nucleation being active in cirrus associated with relatively high updrafts. For Arctic cirrus, temperatures were never colder than −40°C (because of limitations of the Convair aircraft), and thus never entered the domain of homogeneous freezing nucleation.
Figure 2 shows PDFs of normalized ARPSD for the four types of cirrus clouds, where N indicates the number of PSDs used in the analysis. The greater ARPSD is, the more spherical in appearance the ice particles greater than 60 μm are. For example, compact, high-density ice particles tend to be quasi-spherical. The highest ARPSD values are associated with aged anvil cirrus, while Arctic cirrus are having the lowest values. The dashed line in the Arctic cirrus panel shows the lower bound (0.55) of the other PDFs shown. The substantial contribution of ARPDF less than 0.55 in Arctic cirrus suggests that many ice particles found in Arctic cirrus have unique shapes not found in the other cirrus types sampled. This is consistent with inspection of CPI images from the ISDAC cirrus flights which show that the majority of particles > 50 μm in cirrus clouds are rosette shapes, side planes or combinations thereof having smaller ARs. Conversely, ice particles in fresh anvils rarely contain rosette shapes, and aged anvils contain small percentages of rosette shapes, presumably from subsequent nucleation and growth after the anvil has detached from convection [Lawson et al., 2010].
where m is ice particle mass, g is the gravitational constant, ρa is air density, D is the particle maximum dimension, η is the dynamic viscosity and A is the particle area projected to the flow. The other modification was redefining the limiting pressure drag constant Co = 0.35 and δo = 8.0, which are used to calculate Reynolds number Re from X. These modifications yield significantly more accurate fall speeds for some ice crystals such as thin planar crystals (e.g., stellar dendrites) or long columnar crystals (e.g., long columns and needles), but it differs little from the MH scheme when ice crystals have aspect ratios near unity. Thus a comparison between schemes reveals something about ice particle shape. In Figure 3 (left), Vm is predicted by both schemes for anvil, aged anvil, and in situ cirrus PSDs. The difference between the linear regression line and the 1:1 line is small, indicating that the ice particles were generally compact with aspect ratios near 1. In Figure 3 (right), Vm is predicted by both schemes for Arctic cirrus. The difference between the regression line and the 1:1 line is greater for larger Vm, suggesting that ice particle morphology differences among larger ice particles are significant between tropical anvil/in situ cirrus and Arctic cirrus. The HW scheme is used to calculate Vm in this paper.
3.3. Effective Diameter
 In this section the effective diameter De is related to cirrus cloud temperature and IWC. This may be useful for model validation purposes, including the prediction of cirrus optical properties based on these observed De relationships in contrast to optical properties based on what the model microphysics normally provides. As demonstrated by Ivanova et al. , De depends strongly on the relative concentration of small ice crystals and the ice particle shapes. Two advantages this study has over many previous attempts to characterize De in cirrus clouds are (1) the effects of ice particle shattering have been greatly reduced and (2) it was not necessary to assume any ice particle shape “recipe” when calculating De since De was based on measurements of ice particle projected area and mass estimates (consistent with CVI measurements in the case of TC4 cirrus).
 The effective diameter De was related to cirrus temperature T in Figure 4 regarding anvil, aged anvil, in situ cirrus, and Arctic cirrus. The variance (r2) and number of PSDs sampled (N) are indicated. There is no compelling evidence that indicates the temperature dependence of De depends on cirrus cloud type for the tropical cirrus. For ISDAC Arctic cirrus, the slope of the regression line is similar to tropical cirrus, with the regression lines predicting De ∼ 10 μm smaller for Arctic cirrus than for tropical cirrus. Given the variance, this difference might not be significant. However, this difference does indicate that for the data considered, ice particles in Arctic cirrus on average have more projected area per unit mass than ice particles in the tropical cirrus at the same T on the basis of our definition of De. This would be consistent with the findings in Figures 2 and 3 since ice particles having lower ARs (or relatively high or low aspect ratios relative to unity) tend to have more area per unit mass. While the scatter is clearly greater for Arctic cirrus, the low r2 for Arctic cirrus may be partially due to the narrower range of temperatures employed. Some of the scatter in the Arctic cirrus data may also be due to the fact that Arctic PSDs were sampled over a 1–2 min period whereas TC4 PSDs correspond to flight legs having a mean duration of 15 min (although ranging from 1 to 79 min). Longer sampling times reduce the variance.
Figure 5 relates De to the IWC for all cirrus cloud types. Again, cloud type does not appear relevant for tropical cirrus, where 53% of the variance is explained by the linear regression. For ISDAC Arctic cirrus, there is absolutely no correlation between De and IWC. These results are very different from those reported by Liou et al.  for Arctic cirrus sampled during the MPACE field experiment, where a strong correlation between De and IWC was found over an IWC range of ∼8 to 800 mg m−3 (in this study IWC ranged from ∼1 to 100 mg m−3). Liou et al. also found a good correlation between De and IWC for anvil cirrus clouds with IWCs ranging from 10−3 to 103 mg m−3 (compared to 3 × 10−2 to 103 mg m−3 in this study). In contrast to this study, the Liou et al. study found a poor correlation between De and T for anvil cirrus clouds. However, we would like to point out that in contrast to the 2D-S data analyzed here, all of the results reported by Liou et al.  were based on data collected with scattering probes and the older particle imaging probes that have a slower time response, coarser pixel resolution and have been shown to suffer from artifacts generated by shattered particles [Korolev et al., 2011].
 Since significant correlations were obtained in our study for both T and IWC, both of these variables were related to De. While there are different ways of combining these variables (e.g., DM08), expressions containing more terms and/or higher-order terms did not provide higher correlations than the following simple expression:
The coefficients in equation (9) are given in Table 1. Figure 6 compares De predicted by (9) with measurement derived De, with an r2 of 0.82, indicating a significant improvement relative to predicting De from temperature alone.
Table 1. Coefficients Used in Equations (9) and (10) for Relating De and Vm to IWC and Temperature, Based Mostly on in Situ Measurements From TC4a
Also shown are the coefficients used by Deng and Mace  for their Vm formulation based on cloud radar observations in the tropical western Pacific (TWP). Units for De, Vm, T, and ice water content are μm, cm s−1, °C, and mg m−3, respectively, for equations (9) and (10) and are cm s−1, °C, and g m−3 in the work of Deng and Mace.
3.4. Fall Velocities
 Ice particle fall speeds and De are related mathematically since the fall speed depends on the ice particle mass-to-area ratio while De depends on the ratio IWC/At. Therefore correlations regarding De are bound to be reflected in correlations regarding Vm. This can be seen in Figure 7 where Vm is related to temperature for all cirrus cloud types. The same trends noted above for De also apply to Vm. For example, for a given common temperature, Vm in Arctic cirrus is ∼11–14 cm s−1 lower than Vm in TC4 cirrus on the basis of the regression lines. As noted above, this may be due to ice particles in Arctic cirrus tending to have more area per unit mass than ice particles in TC4 cirrus. PSD shape differences may also be a factor. The dependence of Vm on the cloud IWC is shown in Figure 8, with r2 = 0.496 for tropical cirrus, similar to that obtained for De and IWC. No correlation exists for Arctic cirrus.
 Since Vm is positively related to both T and IWC for TC4 cirrus, a multiple regression using both these variables improves the predictability of Vm, with r2 = 0.78 as shown in Figure 9. While there are many ways to express Vm as a function of IWC and T, the simplest method was best for this data set:
where Vm is in cm s−1, T is in °C, and IWC is in mg m−3. That is, adding additional terms did not improve the correlation. The above coefficients are given in Table 1. The 95% confidence interval is ±15 cm s−1. These results can be used to predict Vm diagnostically in climate models, or can be used to test Vm predicted by a given microphysics module.
3.5. Comparison With Another Study
 In this section we compare Vm from equation 10 with the Vm results of DM08 where Vm was retrieved using cloud radar Doppler moments. DM08 applied a cirrus cloud property retrieval algorithm [Deng and Mace, 2006] to a 7 year record of millimeter cloud radar data collected at two ARM sites: Southern Great Plains (SGP) of the United States and the tropical western Pacific (TWP), which includes the islands of Manus and Nauru. A total of about 30,000 h of cirrus were sampled remotely at these two sites. Since equation (10) is based on mostly anvil cirrus (with some in situ cirrus from TC4 and NAMMA), we compare our results with the TWP retrievals of Vm, IWC and T in DM08, which are from mostly anvil cirrus. DM08 related Vm to T and IWC using the following formulation, and the coefficients of are given in Table 1:
This DM08 equation is contrasted with equation (10) in Figure 10a for three temperatures roughly spanning the range of TC4 measurements. Equations (10) and (11) are similar at −43°C but (10) diverges from (11) near the limits of the TC4 data domain, with (10) being more temperature dependent. Figures 10b, 10c, and 10d compare (10) with (11) at three temperatures where TC4 data density was higher so that Vm derived from in situ measurements, given by the circles, could be compared against the two Vm parameterizations. Measurement derived Vm-IWC-T values correspond to temperature intervals of −29°C to −31°C, −42°C to −45°C, and −55°C to −58°C, respectively. It is seen that both Vm parameterizations coincide with the measurement derived Vm-IWC-T values fairly well. What we find remarkable is the close agreement between DM08 and our parameterization, noting that (11) is based on millimeter cloud radar (MMCR) measurements while (10) is based on in situ measurements. These results strongly support the Deng and Mace  radar retrieval algorithm and suggest that Vm in tropical cirrus can be realistically estimated in terms of T and IWC. The agreement between DM08 and our study also implies that two very different methods for calculating the cloud IWC yielded similar values.
 A comparison (not shown) between the two versions of the Vm scheme of DM08, one based on TWP data and the other based on SGP data, revealed little difference in Vm. Differences were greatest at the highest IWCs and coldest temperatures, up to ∼9 cm s−1 for an IWC of 1 g m−3 and T = −63°C. This suggests that the Vm-IWC-T relationship for tropical anvil cirrus may not differ significantly from that of a mixture of continental anvil and synoptic midlatitude cirrus clouds. This hypothesis will be tested in a future paper comparing the results of this paper with those obtained from the Small Particles In Cirrus (SPARTICUS) field campaign (funded by DOE's ARM program), which was conducted over the SGP and midwestern United States.
3.6. Coupling the Model Microphysics With Radiation
 As discussed under introduction, the ice fall speed can be one of the most important climate-influencing parameters in a climate model. Thus it is critical that Vm be realistically coupled with the treatment of ice cloud microphysics and the ice cloud optical properties. By expressing Vm in terms of De, this is accomplished provided that De is predicted by the microphysics module and is then passed to the radiation component of the model. Treating Vm in terms of T and IWC may render it less sensitive to the microphysics and microphysics-radiation interactions, neutralizing some of the cirrus climate feedback. This was shown to some extent by Mitchell et al. , where Vm, the cirrus PSD and the cirrus cloud optics were all physically coupled in the CAM3 GCM, revealing a strong impact by Vm on model climatology that was otherwise less present.
 The relationship between De and Vm is shown in Figure 11 for all cirrus cloud types sampled during TC4 and for in situ cirrus during NAMMA. As noted, the strong correlation is due to the mathematical similarities between De and Vm. Since the regression line in Figure 11 gives nonphysical (i.e., negative) values of Vm for De < 31 μm, the following equation can be used when De ≤ 51 μm:
where units are cm s−1 and μm. Equation (12), shown by the long-dashed curve in Figure 11, represents the lowest measured Vm values well and roughly corresponds to viscous flow described by Stokes law [Mitchell, 1996]. The Vm values shown in Figure 11 assume a temperature and pressure of −20°C and 500 hPa. To convert to other temperatures and pressures, equation (6) can be used. Although both De and Vm can be estimated in bulk microphysical parameterizations using a common PSD and set of ice particle projected area- and mass-dimensional power law expressions [e.g., Mitchell, 1996], an advantage to this approach is that De and Vm are calculated directly from aircraft measurements, thus avoiding uncertainties associated with the PSD and these m-D and A-D expressions.
Heymsfield  also found a strong correlation between De and Vm. The short-dashed curve in Figure 11 is the best fit equation given for this relationship (Vm = 5.02 × 105 De1.90, units cm s−1 and cm) on the basis of anvil cirrus sampled during the Tropical Rain Measuring Mission (TRMM). When De = 150 μm, this relationship predicts Vm = 172 cm s−1.
 The De-Vm relationship for Arctic cirrus is shown in Figure 12. Again the correlation is very strong due to the ice particle mass-to-area ratio being common to their formulations. For the TC4 cirrus, Vm can be up to ∼10 cm s−1 higher for larger De than for ISDAC cirrus. This could be due to ice particle morphology and PSD shape differences. Otherwise the Vm dependence on De during ISDAC is similar to TC4. This overall similarity may support using a single De-Vm relationship for all cirrus clouds, though more research is needed to determine this.
4. Comparing Observations With Satellite Retrievals of De and Vm During TC4
Mitchell et al.  describes a method for combining satellite measured radiances in the thermal infrared window region (using MODIS) with temperature-dependent PSD schemes (based on T and IWC) to correct small ice crystal (D < 60 μm) concentrations predicted by the PSD schemes, thus providing corrected PSDs. In addition, this methodology provided estimates of De and Vm as a function of cloud temperature, with satellite data based on cirrus cloud fields sampled by aircraft during the TC4 campaign on 22 July (for anvil cirrus) and 5 August 2007 (for aged anvil cirrus). Since these days are analyzed in this study, measurement derived De and Vm from TC4 are plotted against temperature in Figure 13, along with the retrieved temperature dependence of De and Vm, to test the retrieval. The satellite retrieval employed several PSD schemes, two of which are shown here: (1) the CEPEX PSD scheme based on anvil cirrus sampled during the Central Equatorial Pacific Experiment and (2) the Ivanova  PSD scheme based on in situ or synoptic cirrus sampled at midlatitudes.
 Details about the retrieval and its uncertainties are given by Mitchell et al. . While the retrieval is sensitive to the ice particle m-D and A-D expressions assumed, these power laws were derived from the 2D-S measurements obtained during TC4 which should reduce such uncertainties associated with ice particle shape.
 The results for De are shown in Figure 13a, with satellite retrievals shown by the dashed curves. The CEPEX retrievals appear to exhibit the best agreement with observations, although both PSD schemes underestimate De at higher temperatures. This might be expected since a given cirrus cloud was sampled at multiple height levels, and at lower levels the growth process of aggregation acts to produce larger ice particles with relatively low concentrations of smaller ice particles (see Figure 1). The satellite observations correspond to TC4 cirrus having 11 μm channel emissivities less than 0.4, indicating relatively thin cirrus where concentrations of small ice crystals may be higher (i.e., less impact from height-dependent aggregation process), making De lower.
 Comparisons for retrieved and measured Vm are shown in Figure 13b. Again the retrieved Vm using the CEPEX PSD scheme appears to yield better agreement with the observations, although the range of observed Vm at the lower temperatures is quite broad. The Appendix shows that Vm corresponds to the fall speed of an ice particle size slightly larger than the median mass dimension of a PSD, making Vm sensitive to changes in the distribution of larger ice particles. Differences between the Vm retrievals are partially due to differences in the formulation of the large particle mode of the CEPEX (ν = 0) and Ivanova et al.  (ν = 2.64) PSD schemes, as shown by (A1) in the Appendix. Appendix A also shows that De is considerably more sensitive to the concentration of smaller ice particles than Vm. This and the argument above may explain why the agreement between retrieved and observed Vm at higher temperatures is better than for De.
 Overall it appears that the satellite retrieval estimates of De and Vm are in reasonable agreement with observations. If this retrieval is physically valid and accurate, the satellite estimates of De and Vm indicate that the in situ measurements are representative of the MODIS cloud scenes sampled.
 Improved measurements using a 2D-S cloud particle probe to determine the ice particle size distribution (PSD), ice particle projected area and ice particle mass have resulted in better estimates of De and Vm. For the TC4 field campaign these parameters have been related to the cloud temperature and IWC through multiple regression analysis, with cloud temperature accounting for most of the correlation. Strong agreement was found between our TC4 measurement derived values of Vm, IWC and T and those predicted by a Vm parameterization based on 7 years of cloud Doppler radar retrievals of IWC and Vm in the tropical western Pacific [Deng and Mace, 2008], partially validating the this Vm parameterization. For the ISDAC field campaign, De and Vm were unrelated to IWC and exhibited a weak correlation with temperature.
 However, for climate models to predict Vm in terms of T and IWC may obscure some of the coupling between the cloud microphysical and optical properties since Vm also depends on the PSD projected area. This coupling is important since Vm appears to strongly impact climate feedback [e.g., Sanderson et al., 2008; Jakob, 2002]. Mitchell et al.  have shown that Vm is sensitive to the bimodality of the PSD and that this sensitivity can translate to large changes in cirrus cloud radiative forcing and heating rates. Therefore Vm should be coupled to the cloud microphysics and radiative properties, and a simple way to do this is to relate Vm to De, assuming that De is determined by the microphysics and is used to calculate the cloud optical properties. For the TC4 and NAMMA anvil and in situ cirrus data, the De-Vm relationship has an r2 = 0.965, while for ISDAC Arctic cirrus, r2 = 0.911 for the De-Vm relationship. These relationships appear strong enough to represent Vm in climate models for anvil, aged anvil and in situ cirrus (from TC4 and NAMMA) and for Arctic cirrus (from ISDAC). Of course, the use of a De-Vm relationship makes it critical for a model to represent De as realistically as possible.
 Finally, these measurements of De and Vm were used to test a method for retrieving the cirrus PSD using radiances measured by the MODIS instrument aboard satellites in combination with temperature-dependent PSD schemes. The satellite derived temperature dependence of De and Vm was in general agreement with the observations. If the retrieved De and Vm are accurate, then this would indicate that the in situ measurements here are representative of the anvil cirrus and in situ cirrus clouds in the eastern tropical Pacific where TC4 was conducted.
Appendix A:: Moment Positions of De and Vm
 This appendix identifies the “particle size region” or PSD moment that De and Vm relate to, along with other particle size “benchmarks” or moments of the PSD. Here, for purpose of illustration, we consider a PSD composed of ice spheres. First, Vm roughly corresponds to the fall speed of ice particle size Df that divides the PSD mass flux into equal parts [Mitchell et al., 2010], where
where β = 3 for spheres (β is the power in an ice particle mass-dimension power law) and ν and λ are parameters of a gamma function PSD:
The constant b is the power in the ice fall speed expression
where b tends to be ∼1 for ice crystals in cirrus clouds. The median mass dimension corresponding to the third moment is similar to (A1) but is without the fall speed power b. Now compare (A1) to a similar expression for De:
and to the gamma PSD expression for the mean diameter:
Note that is the first moment of the PSD divided by the ice particle number concentration. For an exponential PSD (ν = 0) of ice spheres, De = 3 and Df = 1.56 De.
 Thus, for a given PSD, Vm corresponds to a considerably larger particle size than De.
 This research was primarily sponsored by the Office of Science (BER), U.S. Department of Energy (grant DE-FG02-06ER64201). The acquisition and development of the TC4 and NAMMA data sets were funded through the NASA Radiation Sciences Program and the NASA Atmospheric Dynamics/Precipitation Programs, respectively, corresponding to grants NNX07AK81G and NNX06AC09G, respectively. The acquisition and development of the ISDAC data set was funded through the DOE's Atmospheric System Research Program, grant DE-SC0004024. Min Deng is gratefully acknowledged for her helpful consultation, and we thank the three anonymous reviewers of this manuscript for their thoughtful advice, which improved the quality of this article. The data sets used in this research are freely available to the scientific community; those interested should contact the lead author.