On the Temperature Dependence of the Cloud Ice Particle Effective Radius—A Satellite Perspective

Cloud ice particle effective radius in atmospheric models is usually parametrized. A widely‐used parametrization comprises a strong dependence on the temperature. Utilizing available satellite‐based estimates of both cloud ice particle effective radius and cloud‐top temperature we evaluate if a similar temperature‐dependence exists in these observations. We find that for very low cloud‐top temperatures the modeled cloud ice particle effective radius generally agrees on average with satellite observations. For high sub‐zero temperatures however, the modeled cloud ice particle effective radius becomes very large, which is not seen in the satellite observations. We conclude that the investigated parametrization for the cloud ice particle effective radius, and parametrizations with a similar temperature dependence, likely produce systematic biases at the cloud top. Supporting previous studies, our findings suggest that the vertical structure of clouds should be taken into account as factor in potential future updates of the parametrizations for cloud ice particle effective radius.

In addition to the original definition, Field et al. (2007) additionally proposed a latitude-dependent lower limit of , reflecting observations that tropical clouds might typically have larger ice/snow particles than clouds in the Extratropics. Independent of the latter, a pronounced T-dependence continues to be imposed on in this parametrization.
van Zadelhoff et al. (2004) investigated ground-based observations of at Cloudnet and ARM sites for ice clouds with optical depth smaller than 4. They found to be dependent on the vertical position within the cloud layer, with largest values in the middle of the cloud. They also found a pronounced T-dependence of (positively correlated) in the middle sections of the clouds. However, that dependence nearly vanishes at cloud bottom and cloud top, suggesting that the relation of to T has a vertical structure within clouds and, by extension, suggesting that a single parametrization might be inappropriate for some parts of the clouds; for example, a parametrization that features a strong dependence of from T might be inappropriate at cloud top and cloud bottom. As the cloud processes are likely to be different in different parts of the clouds, the results of van Zadelhoff et al. (2004) seem not surprising.
In this study, we extend the observation-driven analysis of the T-dependence of to all ice clouds and to global scales using satellite observations of both and cloud top temperature (CTT). These two cloud top properties are retrieved independently in the three satellite-based cloud datasets used in our study. Acknowledging that the satellite-based cloud data used in this study are retrieval estimates associated with retrieval uncertainties, we will use the term satellite observations throughout the manuscript for convenience.

Modeled
In large-scale atmospheric models, is widely parametrized as a function of T and IWC as formulated by Sun and Rikus (1999) and Sun (2001) When the IWC is given in kg/kg, the conversion to g/m 3 introduces an additional, although relatively weak, sensitivity of to the air pressure (p) via the air density needed in that conversion. Appendix A elaborates Equation 2, which reveals that the dependence of on T is fairly strong, with increasing T leading to increasing . This is visualized in Figure 1, showing as a function of T and IWC calculated using Equation 2 for arbitrary, although realistic, values of T and IWC. The results are first shown for IWC given in g/m 3 (panel a), and second for IWC given in g/kg at three pressure levels representing high-, mid-, and low-level clouds (panels b-d).

Satellite-Observed
This study used three satellite datasets, with the restriction that these products contain both and CTT with global coverage. This is generally only feasible with satellite-borne, passive imaging sensors that measure in the visible through to the infrared spectral range, such as the Moderate Resolution Imaging Spectroradiometer (MODIS) and the Advanced Very High Resolution Radiometer (AVHRR), onboard polar-orbiting satellites. Furthermore, our study required for each potential data set individually that both properties ( and CTT) are independently retrieved. The eventually selected datasets are listed in Table 1 and comprise of Cloud_cci AVHRR-PMv3 (Stengel et al., 2020), CLARA-A2 (Karlsson et al., 2017b) and MODIS Collection 6.1 (C6.1) (Baum et al., 2012;Marchant et al., 2016;Platnick, Meyer, et al., 2017). Cloud_cci and CLARA-A2 provide global composites of the cloud properties of interest on a 0.05° × 0.05° grid as part of their product portfolio from which only NOAA-19 data was used. MODIS Aqua C6.1 swath data (Level-2) products were pre-processed for this study before usage to represent the same global composite. The used data includes all the days of July 2011, with the results being nearly identical when using less or more data and other seasons or years. As solar illumination is required to retrieve , night-time data is not included in our study.
It is important to note that while for all three datasets is defined as given in Equation 1, all three datasets are based on different assumptions regarding the ice crystal habit. Cloud_cci uses the General Habit Mixture (Baum et al., 2011(Baum et al., , 2014, aggregated over a collection of observed particle size distributions. MODIS C6.1 uses aggregates of solid hexagonal columns (Yang et al., 2013), also with a collection of observed particle size distributions. CLARA-A2 uses monodisperse distributions of hexagonal columns (Hess et al., 1998). Similar to CLARA-A2, Sun and Rikus (1999) also assume hexagonal columns with a size distributions following McFarquhar and Heymsfield (1997).  ( ) in μm as function of temperature and cloud water content (given in g m −3 ) following Sun and Rikus (1999). (b) to (d) Modeled ice effective radius (in μm) as function of temperature and cloud water content when given in g kg −1 at 200 hPa (b), 500 hPa (c) and 800 hPa (d). In atmospheric models the cloud water content is often given in g kg −1 , thus a small air pressure dependency is introduced into the calculation of the effective radius as the conversion from g kg −1 to g m −3 requires the air density which is a function of pressure.  (2017) The pixel-based MODIS retrievals of the MYD06 swath product have been collected in time windows of 24 hours to compose daily global composites, basically mimicking the Level-3U/Level-2b products of the datasets above.
Note. Additionally, cloud top pressure (CTP) was extracted to do the stratification by low-, mid-and high-level cloud layers. All data were from July 2011.
Since all three observational datasets are based on passive VIS-IR imaging satellite sensors, they represent the cloud properties near the cloud top only. As this limitation cannot be circumvented, model fields, to which we will compare the satellite observations, have been processed with a satellite simulator to facilitate a fair comparison with more corresponding information given in Section 2.3.
It needs to be mentioned that some uncertainty in quantifying random and systematic errors in the satellite retrievals of ice particle effective radius exists, as there is no ultimate reference data source available. Very few inter-comparison study exist. Kahn et al. (2015), for example, compared MODIS with infrared-based retrievals from the Atmospheric Infrared Sounder (AIRS). While the MODIS derived from the 2.1 micron band was found to be 5-10 μm larger than the AIRS , the MODIS 3.7-micron-derived (as used in this study) was largely unbiased.

Processing Model Fields Through a Satellite Simulator
In this study, the SIMFERA satellite simulator (Stengel et al., 2018) was applied to 6-hourly, un-averaged ERA-Interim data (Dee et al., 2011). The application of the satellite simulator is necessary to identify the cloud top level consistent with the satellite products, for example, to infer cloud-top temperature. The satellite simulator also includes the computation of using the Sun and Rikus (1999) parametrization with the reanalysis cloud fields, that is, IWC and temperature, as input.
Even though the modeled clouds in ERA-Interim might have caveats, we believe that for the comparisons conducted in the study it is not of critical importance to have all individual clouds modeled absolutely correctly, but rather represent the general distribution of cloudiness in space and time, which was shown for ERA-Interim in Stengel et al. (2018). Figure 2 shows the globally collected as a function of CTT, only excluding the polar regions south of 60°S and north of 60°N. For each CTT bin, the mean is given along with the underlying distribution represented by box plots showing 5%, 25%, 50%, 75%, 95% percentiles. Analyzing the model data, the strong dependence of on CTT, as given in Equation 2 and as shown in Figure 1, is preserved in the simulator output in Figure 2, with low (high) values for low (high) temperatures. The observations of agree well with the modeled ones for low temperatures (below −40°C) in terms of mean values per CTT bin, with MODIS C6 presenting slightly higher values than the other datasets, CLARA-A2 and Cloud_cci. The spread, however, is larger in all observational datasets than for the modeled clouds for this temperature range.

Results and Discussion
Considering the temperature range from −40° to 0°C, the deviation between observed and modeled becomes very large. The modeled describes a nearly linear function with increasing more than 1 μm per K. At high sub-zero temperatures, the modeled have a mean near 70 μm. In contrast, the observed remain nearly unchanged for warm sub-zero temperatures. The scatter among the observational datasets is small for the mean values and for the underlying distribution in each temperature bin, and the scatter is much smaller than the deviation to the modeled . The fact that the observational datasets have all different ice crystal habit assumptions, while Sun and Rikus (1999) use the same particles as CLARA-A2 (see Section 2.2), suggests that the assumed habit cannot explain the found deviations between observed and modeled . To emphasize the importance of the findings we analyzed that about 23% of all clouds in the reanalysis data used in our study have an ice cloud top and a CTT in the range from −40°C to 0°C. Thus the findings above concern almost one fourth of all clouds.
For high-level clouds, the observational datasets show a slight tendency to give lower for higher temperatures. This feature is most pronounced for Cloud_cci and for this data set visible for mid-level clouds as well. It is not completely known yet what the cause of this feature could be. It cannot be ruled out that all observational datasets are affected by the same caveat; the possible contaminations by sub-pixel liquid cloud phase in pixels associated with ice phase. Coopman et al. (2019) found that binary phase information has the potential to lead to a low bias in when liquid phase constitutes parts of the satellite pixel. However, this effect is too small to explain the large difference found between observations and modeled clouds for higher sub-zero temperatures. Furthermore, for the Cloud_cci data, which show the most pronounced drop in at higher sub-zero temperatures, we analyzed by comparison with Cloud-Aerosol Lidar with Orthogonal Polarization (Winker et al., 2009) observations that only about 9% of the clouds with CTT between −40°C and 0°C are potentially misclassified as ice.

Summary and Conclusions
In this study, we have used global satellite datasets of CTT and cloud top to evaluate the strong temperature dependence in a commonly used parametrization of . Modeled clouds from ERA-Interim were processed through a satellite simulator to represent (near-) cloud-top information as in the observations and to apply the parametrization. While for cloud temperature below −40°C, the modeled roughly agrees to the observations, the strong temperature dependence that governs the modeled makes them very large for warmer sub-zero temperatures. We do not see any indication in the observations that this is generally the case. The observations suggest nearly a constant cloud top throughout the sub-zero temperature range. Our results extend the findings of an earlier study, that was limited to measurements at a few ground sites, to global scales. Though we do not want to rule out, that for some cloud types such a temperature dependence of the cloud top exists, it is certainly not generally the case on a global scale. Furthermore, as we only investigated the cloud top layers, might be temperature-dependent further down into the clouds. However, our results clearly indicate that cloud top might not be as temperature-dependent as currently implied by a commonly applied parametrization. Thus we suggest revising this parametrization and similar ones to include information on the vertical structure of the clouds and thus potentially to reflect the small temperature dependence of near the cloud top as found in the presented study. As van Zadelhoff et al. (2007) have shown, modifications in the parametrization of the cloud ice particle effective radius can have a modest but significant positive impact on the simulated shortwave radiation budget. However, it might not always be easy to demonstrate the improvements.
With being dependent on ice water content (IWC) in kg/kg, the air density (RHOAIR) in kg/m 3 , the air temperature (PT) in K and RTT which is 273 K. The boundaries applied to ZDESR imply that is between 19.5 and 100.7 μm. Slight deviations of Equation A1 might be implemented in different models.