In this study, in situ airborne observations at 1 Hz are provided by instruments aboard the NSF High-Performance Instrumented Airborne Platform for Environmental Research (HIAPER) GV research aircraft. A comprehensive global dataset is compiled based on seven major flight campaigns funded by the NSF, including START08 (Pan et al., 2010), HIPPO deployments 2–5 (Wofsy, 2011), PREDICT (Montgomery et al., 2012), TORERO (Volkamer et al., 2015), DC3 (Barth et al., 2015), CONTRAST (Pan et al., 2017) and ORCAS (Stephens et al., 2018). Table 1 provides a detailed summary of the seven flight campaigns, including location, duration of flights, total flight hours of all temperatures and flight hours for in-cloud and clear-sky conditions at temperatures ≤-40 ∘C only. Maps comparing the flight tracks of in situ observations and the collocated CAM6 nudged simulations (hereafter named “CAM6-nudg” data) are shown in Fig. 1.

For this study, ice particle measurements are provided by the Fast 2-Dimensional Cloud particle imaging probe (Fast-2DC) with a 64-diode laser array for a range of 25–1600 µm. Larger particles can be reconstructed up to 3200 µm. The mass–dimensional relationship of Brown and Francis (1995) is used to calculate IWC for the Fast-2DC probe, which was previously used in other studies of the Fast-2DC probe aboard the NSF GV aircraft (Diao et al., 2014a, b, 2015). Number-weighted mean diameter (Di) is calculated by summing up the size of particles in each bin using the bin center and then dividing it by the total number of particles. In order to mitigate the shattering effect, particles with diameters <62.5 µm (i.e., first two bins) are excluded in the Fast-2DC measurements when calculating IWC, Ni and Di. The Rosemount temperature probe was used for temperature measurements, which has an accuracy and precision of ∼±0.3 and 0.01 K, respectively. All analyses are restricted to temperatures ≤-40 ∘C, in order to exclude the presence of supercooled liquid droplets in this study. Laboratory calibrated and quality-controlled water vapor data were collected using the vertical-cavity surface-emitting laser (VCSEL) hygrometer (Zondlo et al., 2010), with an accuracy of ∼6 % and precision of ≤1 %. Both temperature and water vapor are used at 1 Hz resolution for this analysis. Aerosol measurements were collected from the Ultra-High-Sensitivity Aerosol Spectrometer (UHSAS), which uses 100 logarithmically spaced bins ranging from 0.06–1 µm. RHi is calculated using saturation vapor pressure with respect to ice from Murphy and Koop (2005). The combined RHi uncertainties from the measurements of temperature and water vapor range from 6.9 % at -40 ∘C to 7.8 % at -78 ∘C. Measurements are separated by cloud condition where in-cloud condition is defined by the presence of at least one ice crystal from the Fast 2-DC probe (Ni>0 L-1). The same in-cloud definition has been used by several previous studies (D'Alessandro et al., 2017; Diao et al., 2014a, b, 2015, 2017; Tan et al., 2016), and all other samples are defined as clear sky. For regional variation analysis, data are binned by six latitudinal regions in the two hemispheres, that is, NH polar (60–90∘ N), SH polar (60–90∘ S), NH midlatitudes (30–60∘ N), SH midlatitudes (30–60∘ S), NH tropics (0–30∘ N) and SH tropics (0–30∘ S). The majority of observations in the SH midlatitude and tropical regions are located over the oceans, while the observations of NH midlatitude and polar regions are predominantly over land.

The vertical profiles of observed in-cloud temperature, clear-sky potential temperature (Θ) and their correlations are shown in Fig. 2. The observations sampled temperatures from -78 to -40 ∘C and altitudes from 5–15 km, while a previous study of Krämer et al. (2020) sampled -91 to -30 ∘C and 5–19 km (their Fig. 2). The lowest temperatures are found in the tropical regions and at the highest altitudes, whereas polar regions show more observations at lower altitudes that satisfy temperature ≤-40 ∘C. Distributions of cirrus cloud properties (i.e., IWC, Ni, Di), in-cloud and clear-sky RHi, and clear-sky water vapor mixing ratio for the observation dataset are shown in Fig. 3. Di increases with decreasing altitudes, IWC slightly increases with decreasing altitudes, and Ni is almost independent of altitudes. Clear-sky RHi and water vapor mixing ratio both increase with decreasing altitudes, while in-cloud RHi is centered around 100 % and shows smaller dependency on altitudes. Compared with Fig. 3 in Krämer et al. (2020), 48 % of their ice particle samples have Di<40 µm, which is below the size cut-off used in this study. The higher Di in this study also leads to lower range of Ni (0.01–1000 L-1) and higher range of IWC (10-5–10 gm-3) compared with that previous study (i.e., Ni from 0.1–105 L-1 and IWC from 10-7–1 gm-3), representing the sampling bias towards larger particles in this study. The relationships of IWC with respect to meteorological conditions (i.e., temperature and RHi) and other microphysical properties (i.e., Ni and Di) are shown in Fig. S1 in the Supplement. The distributions of IWC samples are relatively uniform at various temperature and RHi, while more IWC samples are correlated with Di between 100 and 300 µm.

This study uses model simulations based on the NCAR CAM6 model. Compared with its previous version (the CAM5 model), CAM6 implemented a new scheme, the Cloud Layers Unified by Binomials (CLUBB) for representations of boundary layer turbulence, shallow convection and cloud macrophysics (Bogenschutz et al., 2013). CLUBB is a higher-order turbulence closure scheme that calculates prognostic higher moments based on joint probability density functions (PDFs) for vertical velocity, temperature and moisture (Golaz et al., 2002). An improved bulk two-moment cloud microphysics scheme has been implemented (Gettelman and Morrison, 2015) that replaces diagnostic treatment of rain and snow with prognostic treatment of all hydrometeors (i.e., rain and snow). This is coupled with a four-mode aerosol model (MAM4) (Liu et al., 2016) for simulations of aerosols and aerosol–cloud interactions. It allows ice crystals to form via homogeneous freezing of sulfate aerosols and heterogeneous nucleation of dust particles (Liu et al., 2007; Liu and Penner, 2005). The model uses Wang et al. (2014b) for ice nucleation, which implemented and improved Hoose et al. (2010) by considering the probability density function of contact angles for the classical nucleation theory. The model also uses Shi et al. (2015) for modifications of pre-existing ice. Finally, the deep convection scheme (Zhang and McFarlane, 1995) has been tuned to include sensitivity to convection inhibition.

Results from in situ observations are compared with two types of CAM6 simulations: nudged and free-running simulations. Simulations are based on a finite-volume dynamical core (Lin, 2004) with a horizontal resolution of 0.9∘ × 1.25∘ and 32 vertical levels. All simulations are conducted using prescribed sea-surface temperature and present-day aerosol emissions and include a 6-month spin-up time. CAM6 nudged simulations are nudged spatially and temporally with meteorological data (i.e., 2-D horizontal wind and temperature) from the Modern-Era Retrospective Analysis for Research and Applications version 2 (MERRA2) (Gelaro et al., 2017) and collocated with aircraft flight tracks in space and time. A nudged simulation was conducted for each campaign independently and was combined into one dataset to compare with observations. One free-running simulation was conducted for the duration of all flight campaigns from July 2008 to February 2016. To reduce the size of model output when comparing with observations, a total of 24 instantaneous output data from the free-running simulation are combined into one dataset (“CAM6-free” hereafter), which includes 00:00 and 12:00 UTC for the first day of each month in 2010. Additional sensitivity tests on different model output from the free-running simulation show very minor differences in the statistical distributions of cirrus microphysical properties and the correlations with their controlling factors when selecting different years, seasons and days in a month.

In order to examine observations and simulations on more comparable scales, a running average of 430 s was calculated for meteorological parameters (i.e., temperature and RHi) and microphysical properties (i.e., IWC, Ni and Di), which translates to ∼100 km horizontal scales since the mean true air speed below -40 ∘C for all campaigns was 230 ms-1 (Fig. S2). When applying the running average, both in-cloud and clear-sky conditions (i.e., where Ni and IWC values are zero) are included in the averages. Grid-mean quantities from model output are used in comparisons with observations, including “IWC”, “NUMICE”, “QSNOW” and “NSNOW”, which are mass and number concentrations of ice particles and snow, respectively. Another type of comparison between 1 Hz observations and in-cloud quantities from model output is shown in the Supplement. Both methods have been previously used in model evaluation, such as D'Alessandro et al. (2019), who compared 200 s averaged aircraft observations with simulated grid-mean quantities, and Righi et al. (2020), who compared 1 Hz aircraft observations with simulated in-cloud quantities.

A summary of the ranges of meteorological conditions and ice microphysical properties for in situ observations and simulations is shown in Table 2. Simulated RHi is calculated from simulated specific humidity and temperature, and the calculation of saturation vapor pressure with respect ice is based on the equation from Murphy and Koop (2005). Simulated ice and snow are restricted to ≥62.5 µm based on the size cut-off of the Fast-2DC probe by applying methods from Eidhammer et al. (2014). Based on their equations 1 to 5, we followed their assumption that the shape parameter μ equals 0 when calculating the slope parameter λ. Mass and number concentrations of ice and snow are further calculated based on integrals of incomplete gamma functions from 62.5 µm to infinity. The simulated values of IWC, Ni and Di are calculated based on the combined ice and snow population after applying the size restriction. In-cloud conditions in simulations are defined by concurring conditions of IWC >10-7 gm-3 and Ni>10-4 L-1 based on size-restricted grid-mean quantities. These thresholds are the lower limits from observations after calculating the 430 s averages. Note that due to the ice crystal size constraint, some thin cirrus may not be detected. In addition, analysis of simulated cirrus clouds is restricted to similar pressure ranges as those measured in the seven campaigns. An additional constraint on cloud fraction >10-5 was applied to both nudged and free-running simulations to exclude extremely low values.

To visualize the impact of the size truncation on simulated data, we employed methods similar to Gettelman et al. (2020) and reconstructed the simulated particle size distributions for snow and ice in Fig. 4a using gamma functions from Morrison and Gettelman (2008). Compared with the observations, the number density for combined ice and snow is overestimated for smaller particles (<400 µm) and underestimated for larger particles (>1000 µm). After applying the size restriction, the PDFs of simulated Ni and IWC show increasing probability of small Ni and decreasing probability of small IWC due to the removal of small particles (Fig. 4b and c).

Finally, simulated aerosol number concentrations are further categorized by diameters >500 and >100 nm (i.e., Na,500 and Na,100, respectively), by summing the size-restricted concentrations of the Aitken, accumulation and coarse aerosol modes. Previously, field experiments found that Na,500 correlates well with INP number concentrations (DeMott et al., 2010). Even though that correlation was only determined based on observations warmer than -36 ∘C, the separation of Na,500 and Na,100 can help to examine the effects of larger and smaller aerosols in this work.

Three cirrus cloud microphysical properties (IWC, Ni and Di) are examined in relation to temperature at six latitudinal regions (Fig. 5). The standard deviations of the IWC, Ni and Di in each temperature bin are shown in Fig. S3. The 1 Hz observations of IWC and Ni in the NH indicate clear latitudinal differences with the highest values occurring in the midlatitudes, followed by tropics, then polar regions for temperatures between -40 and -60 ∘C, while for colder temperatures the NH tropical region shows the highest IWC. In the SH, the highest IWC and Ni occur in the tropics, followed by the polar regions and midlatitudes. Comparing the two hemispheres, IWC and Ni show significant reductions by ∼1 order of magnitude from NH midlatitudes to SH midlatitudes (Fig. 5b, e). These hemispheric differences in midlatitudes may be due to air-mass differences between NH (more continental) and SH (more oceanic) and/or more anthropogenic emissions in the NH. The IWC, Ni and Di are relatively similar between NH and SH tropical regions, while IWC and Di are higher in the SH polar region than in NH polar regions.

The simulations are further compared with averaged observations at a similar horizontal scale of ∼100 km. After applying 430 s running averages for observations, the average IWC and Ni values decrease by 0.5–1.5 orders of magnitude compared with 1 Hz observations depending on temperature and geographical region. Hemispheric differences are mostly consistent between 1 and 430 s averaged observations except for polar regions. CAM6-nudg data show a similar trend of average IWC, Ni and Di with respect to temperature as seen in observations; that is, the average IWC increases with increasing temperature, consistent with previous observational studies (Krämer et al., 2016; Luebke et al., 2013; Schiller et al., 2008), average Ni shows no clear trend with temperature, and average Di increases with increasing temperature. Differing from observations, CAM6 produces the highest IWC and Ni in the tropical regions, followed by midlatitudes, then polar regions for both hemispheres. The simulated IWC, Ni and Di also show smaller differences between hemispheres and latitudes. The CAM6-nudg data underestimate and overestimate IWC in the NH and SH by 0.5–1 orders of magnitude, respectively, with the largest discrepancies in the midlatitudes. The simulations overestimate Ni in the tropics and polar regions in both hemispheres by 0.5–1 orders of magnitude and overestimate Ni in the SH midlatitudes by 1–2 orders of magnitude. The simulated Di is about half of the observed values in most regions except polar regions. This result indicates “too many” and “too small” simulated ice in most regions. The low bias of simulated Di indicates possible misrepresentation of ice particle growth and sedimentation in the model parameterization.

A sensitivity test is conducted by comparing 1 Hz observations with in-cloud quantities from model output (Fig. S4). Larger differences are seen between simulated and observed IWC and Ni in Fig. S4 compared with Fig. 5. The directions (i.e., positive or negative) of model biases of IWC, Ni and Di are generally consistent in both comparisons.

A previous study by Righi et al. (2020) evaluated the ice microphysical properties in the EMAC-MADE3 aerosol–climate model (i.e., ECHAM/MESSy Atmospheric Chemistry – Modal Aerosol Dynamics model for Europe adapted for global applications, third generation) by comparing in-cloud quantities from model output with 1 Hz in situ observations of multiple aircraft field campaigns from 75∘ N to 25∘ S (Krämer et al., 2009, 2016, 2020). Although that study included a greater number of smaller ice particles (3–1280 µm) compared with this study, they still showed low biases of simulated Di at 190–243 K and low biases of simulated IWC at 205–235 K, as well as high biases of simulated Ni above 225 K, which are generally in the same direction as the biases we found in CAM6 model. Note that Righi et al. (2020) implemented different cloud microphysics parameterizations compared with the CAM6 model, including a two-moment cloud microphysics scheme of Kuebbeler et al. (2014) and the ice nucleation parameterization for cirrus clouds (T<238.15 K) from Kärcher et al. (2006), which account for both homogeneous and heterogeneous nucleation and the competition between the two mechanisms. Additional future intercomparison studies of these models are warranted to examine the reasons behind the similar biases.

Regional distributions of RHi for clear-sky and in-cloud conditions are shown for 1 Hz observations (Fig. 6), 430 s averaged observations (Fig. 7) and simulations (Fig. 8). The 1 Hz observations show RHi magnitudes ranging from <5 % up to ∼180 % in both clear-sky and in-cloud conditions, and are mostly located below the homogeneous freezing line except for the NH tropical region. A few samples exceed the liquid saturation line but are within the measurement uncertainties of RHi. This result agrees with the RHi distributions based on previous midlatitudinal observations (Cziczo et al., 2013). Differing from 1 Hz observations, 430 s averaged observations show much lower RHi magnitudes for both clear-sky and in-cloud conditions, ranging from <5 % to 120 %–140 %. For clear-sky conditions, the majority of the observed and simulated RHi values are below 100 %, while the CAM6-nudg data show fewer RHi exceeding ice saturation. For in-cloud conditions, both 1 Hz observations and simulations show that RHi frequently occur within ∼20 % of ice saturation, consistent with previous observation and modeling studies (Diao et al., 2014a, 2017; D'Alessandro et al., 2017, 2019; Krämer et al., 2009), while almost no simulated RHi data exceed the homogeneous freezing threshold. The higher RHi observed in the NH tropical region was also observed by Krämer et al. (2009). Such features can be explained by the competition between higher updrafts seen in the tropics and the depletion of water vapor from newly nucleated ice particles as discussed in Kärcher and Lohmann (2002). For the polar regions, in-cloud RHi is skewed towards ISS in both observations and simulations, indicating less effective water vapor depletion likely due to lower Ni values (Fig. 5e). Note that the simulation samples in the tropical regions show peak frequencies at certain temperatures due to larger bin sizes of pressure levels in the lower latitudes.

Regional distributions of the variance of w (σw) for 1 Hz observations at 40 and 430 s scales and CAM6 nudged simulations are shown in Figs. 9, 10 and 11, respectively. σw in the observations is calculated as the variance of w within each 40 and 430 s of data, which correspond to a horizontal scale of ∼10 and 100 km, respectively. The σw in simulations is based on the “wsub” variable, which is calculated from the square root of turbulent kinetic energy (TKE) (Gettelman et al., 2010). Observed σw shows the highest values in the tropical and midlatitude regions, reaching up to ∼3 ms-1, while the polar regions show updrafts up to ∼1 ms-1. A similar decreasing trend of maximum σw is seen in the simulations from the lower to higher latitudes. The observations show similar σw maximum values between clear-sky and in-cloud conditions, while the simulations show much higher maximum σw for in-cloud conditions in the tropics (1 ms-1), midlatitude (1 ms-1) and polar regions (0.5 ms-1), compared with those values in clear sky (i.e., 0.5, 0.25 and 0.1 ms-1, respectively). This result suggests that the model has a stronger dependence on higher σw for cirrus cloud formation compared with observations. We further examine the potential impact of convection in simulations and observations. Figure S5 shows the locations where w>1 ms-1 is seen in the observations as well as where wsub>0.5 ms-1 is seen in the CAM6-nudg data for in-cloud conditions. Since wsub in CAM6 is based on the turbulent scheme, higher wsub values indicate that the convection scheme may be active and produce detrained ice in convective outflows. The majority of observed and simulated in-cloud samples do not appear to have high w or wsub, indicating that detrained ice from the convection is unlikely a significant contribution. More future investigation is needed to track cirrus cloud origins and quantify impacts from convection.

PDFs of temperature, RHi and σw are shown in Fig. 12. The PDFs are normalized by the total number of samples of both clear-sky and in-cloud conditions. The observations are located mostly around -68 to -40 ∘C, and the simulations show similar temperature distributions. For the PDFs of RHi, the observations and simulations all show a peak position around 100 % for in-cloud condition. However, a secondary peak is shown in simulations at 80 % RHi, which is likely due to the parameter of RHimin for ice cloud fraction calculation being set at 80 % for representing variance of humidity in a grid box (more details on RHimin are described in Gettelman et al., 2010). In addition, the maximum RHi values for in-cloud conditions are 170 %, 154 %, 160 % and 257 % for 1 Hz observations, 430 s averaged observations, CAM6-nudg and CAM6-free, respectively. The maximum RHi values for clear-sky conditions are 175 %, 135 %, 108 % and 181 %, respectively. The CAM6-free data show higher maximum RHi values than CAM6-nudg data, likely due to additional data from tropical regions at temperatures below -70 ∘C (Fig. 12j). When using a lower size cut-off (1 µm) of ice particles for the simulation data, the number of in-cloud samples increases by 4 % (Fig. S6). However, negligible differences are seen in the PDFs of temperature, RHi and σw for the two simulations between Figs. 12 and S6.

PDFs of σw show consistent results with Figs. 9–11, with simulations showing much higher maximum σw for in-cloud conditions than clear-sky conditions, while observations show similar maximum σw in both conditions. The lower maximum values of σw in simulations are most likely a result of the model missing representations of gravity waves from topography, fronts and convection, and only including σw from turbulence.

The relationships between ice microphysical properties and RHi are examined in Fig. 13. For the 1 Hz observations, the maximum IWC and Ni occur slightly above ice saturation at 110 % RHi, while the maximum Di occur at 130 % RHi. The average IWC and Ni increase 1.5 orders of magnitude from 40 % to 110 % RHi and decrease 0.5 orders of magnitude (i.e., a factor of 3) from 110 % to 130 % RHi. The maximum IWC and Ni do not occur at the highest RHi most likely due to the consumption of water vapor by ice deposition. High Di values at lower RHi (∼30 %) are likely a result of sedimenting large ice crystals, which has been previously observed by Diao et al. (2013) when investigating the evolutionary phases of cirrus clouds. For 430 s averaged observations, the peak IWC, Ni and Di occur at 100 %, 100 % and 115 % RHi. The maximum IWC and Ni values are nearly the same between 1 and 430 s averaged observations (i.e., 0.04 gm-3 and 10 L-1, respectively) near saturation, while the 430 s averaged observations show lower minimum IWC and Ni at very low RHi (<10 %), which are 1.5 orders of magnitude lower than 1 Hz observations. This feature is due to in-cloud segments being longer around saturation compared with subsaturated conditions as shown in Diao et al. (2013), which means that fewer clear-sky conditions are being included in the 430 s averages around saturation and therefore show little reduction of the IWC and Ni due to spatial averaging.

In contrast to observations, both CAM6-nudg and CAM6-free simulations show bimodal distributions of IWC and Ni with the primary peak at 100 % RHi and the secondary peak at 80 % RHi. The secondary peak at RHi 80 % is likely produced by the RHimin parameter reflecting subgrid-scale RHi variance as mentioned above (Gettelman et al., 2010), which was set at the default value (80 % RHi) for both simulations. The primary peak at 100 % RHi is likely a result of the minimum threshold for heterogeneous ice nucleation being set at 120 % as well as a subgrid variability scaling factor of 1.2 being considered (Wang et al., 2014a). Similar to 430 s averaged observations, IWC and Ni show steep increase (i.e., 3–4 orders of magnitude) from 40 % to 100 %. Increases of average IWC and Ni are seen in the simulations as RHi increases from 120 % to 160 %, differing from the decreasing trend seen in the observations. These higher values of IWC and Ni near 160 % are possibly due to RHi reaching the homogeneous nucleation thresholds, where ice nucleation becomes more dependent upon temperature and updraft speed (Liu and Penner, 2005). Note that at this same point as IWC and Ni increase, there is a decrease in Di, which also suggests homogeneous nucleation in the model. For Di–RHi correlations, both simulations show similar results to the observations, with the maximum Di around 130 % RHi and some large ice particles in the subsaturated conditions. The large variability of observed ice microphysical properties is also significantly underestimated in the model for ISS conditions. Standard deviations are 0.5–1 orders of magnitude lower for IWC and Ni and a factor of 2 lower for Di compared with observations.

Comparing the correlations with σw (Fig. 14), the simulations show increasing IWC and Ni with higher σw, which agree with observations, although the increases of IWC and Ni are smaller in the simulations than the 430 s observations. The simulated Di is relatively constant with increasing σw, which differs from the slight positive correlation between Di and σw in the observations. This slight positive Di–σw correlation is likely due to the growth of ice particles as cirrus clouds evolve with continuous updrafts that supply excess water vapor above ice saturation, which was previously discussed in a cirrus cloud evolution analysis (Diao et al., 2013). The simulations may overlook this positive correlation for several reasons, such as the lack of temporal resolution to resolve cirrus evolution in the growth phase, the lack of vertical velocity subgrid variabilities (as discussed in Zhou et al., 2016) and a dry bias (i.e., lower RHi) in the model (as discussed in Wu et al., 2017).

Comparing the performance of two types of simulations, both CAM6-nudg and CAM6-free show bimodal distributions for IWC–RHi and Ni–RHi correlations, and they both show positive correlations for IWC–σw and Ni–σw. This result indicates that the general trends in these correlations are statistically robust and less affected by sampling sizes and geographical locations. For correlations with RHi, the maximum IWC value in CAM6-nudg and CAM6-free is lower than the 430 s averaged observations by a factors of 25 and 100, respectively. The maximum Ni value in CAM6-nudg is similar to the 430 s averaged observations, while that value in CAM6-free is lower by a factor of 3. For correlations with σw, there are no significant differences for the maximum IWC between the two simulation types. The maximum Ni value in CAM6-nudg and CAM6-free is higher than the 430 s averaged observations by factors of 3 and 10, respectively. These results show that CAM6-nudg data, which are collocated with flight tracks, produce IWC and Ni values closer to the 430 s averaged observations than CAM6-free, possibly due to the variabilities of IWC and Ni in different geographical locations as shown in Fig. 5.

The effects of larger and smaller aerosols (i.e., Na,500 and Na,100) on ice microphysical properties are further examined for observations and CAM6-nudg data (Figs. 15 and 16). Cloud fraction is calculated in each temperature–Na bin by normalizing the number of in-cloud samples with the total number of samples in that bin for both observations and simulations. For three cirrus microphysical properties (i.e., IWC, Ni and Di), positive correlations are seen in 1 Hz observations with respect to Na,500 and Na,100. In addition, higher Na,500 (>10 cm-3) and Na,100 (>100 cm-3 values are associated with significant increases in cloud fraction. At -70 to -60 ∘C, higher IWC, Ni and cloud fraction are seen when Na,500 is observed, with positive correlations of IWC and Ni with respect to Na,500. This finding indicates that larger aerosols provide an effective pathway of ice particle formation for colder conditions. The higher IWC and Ni are only shown in much higher Na,100 (>100 cm-3) between -70 and -60 ∘C, demonstrating that larger aerosols facilitate ice formation more effectively than smaller aerosols at this temperature range, possibly due to the activation of larger aerosols as INPs for heterogeneous nucleation. Compared with 1 Hz observations, 430 s averaged observations show weaker correlations of IWC, Ni and Di with respect to Na. However, they do still show higher IWC and Ni between -70 and -40 ∘C associated with higher Na (i.e., Na,500>1 cm-3 and Na,100>30 cm-3).

The CAM6-nudg simulation shows increasing average IWC, average Ni and cloud fraction with increasing Na,500, consistent with the observations. But at temperatures below -60 ∘C, simulated IWC and Ni do not show a sudden increase with higher Na,500 as shown in the observations. The simulated Di slightly decreases with increasing Na,500, differing from the increasing trend seen in observations. For aerosol indirect effect analysis based on Na,100, the comparison results are similar to Na,500; that is, the CAM6-nudg simulation is able to represent positive correlations of IWC, Ni and cloud fraction with respect to Na,100. However, the CAM6-nudg simulation shows smaller (larger) increase of IWC (Ni) at very high Na (i.e., Na,500>1.6 cm-3 and Na,100>30 cm-3) compared with the 430 s averaged observations. The model also misses positive correlations between Di and Na seen in both 1 Hz and 430 s averaged observations.