Introduction

In nature, 40–60% of clouds between 0 °C and −30 °C are mixed-phase clouds; they have various impacts on the atmospheric radiation budget and hydrologic cycle, which subsequently complicates modeled scenarios of future climate change1,2,3. However, accurate representation of mixed-phase cloud microphysics is still challenging in numerical weather prediction models (NWPMs) and general circulation models (GCMs)4, in part due to the complicated liquid-ice interactions and the unresolved sub-grid spatial variations of hydrometeors’ spatial distributions5,6. Quantifying the heterogeneous distribution of liquid and ice and parameterizing its impacts on microphysical processes are important to improve the capability of models in simulating mixed-phase clouds, and deepen our understanding of how sub-grid microphysics variations may influence the cloud radiative feedback and precipitation formation7,8.

It has been demonstrated by observations that the liquid and ice particles are heterogeneously distributed in mixed-phase clouds on scales down to 100 m or even smaller9,10,11, and the complicated phase partitioning in clouds is affected by various factors. Using airborne in-situ measurements in mixed-phase stratiform clouds, Korolev and Milbrandt10 showed that liquid and ice are distributed in clusters, and the frequency distribution of cluster scales follows the −5/3 power law, which demonstrates that turbulence plays an important role in controlling the spatial distribution of particles. Yang et al.11 further showed that the frequency distribution of particle clusters varies from cloud top to base, and depends on the generation and growth of ice particles. For convective clouds, it is found that the characteristics of phase partitioning are different from those in stratiform clouds9,12. The particle distribution in convective clouds changes significantly throughout the cloud life cycle12. There is a higher occurrence of liquid and mixed-phase clusters in the developing stage, and pure ice regions are more frequently observed in the dissipating stage.

Modelling studies have shown that the sub-grid heterogeneous particle distribution would affect the microphysical processes in clouds8,13. Tan and Storelvmo14 suggested that the clustering distribution of liquid and ice pockets would shrink the contact volume between liquid and ice, declining the ice growth rate through the Wegener-Bergeron-Findeisen (WBF) process. Zhang et al.8 investigated the impact of declined liquid-ice contact volume on ice growth through the WBF process in a GCM. Their results suggest that the modeled cloud phase partitioning can be much improved through adjusting the WBF process by taking into account the clustering of liquid and ice. Yang et al.15 recently developed an observationally constrained parameterization of sub-grid heterogeneous liquid-ice mixing, which improves the capability of modelling the WBF process in GCM. Wang et al.16 showed that sub-grid scale cloud water variance can significantly affect the simulation of warm rain process, and they demonstrated that an improved representation of sub-grid cloud water variance can improve the capability of modelling cloud microphysics and radiative properties. Beyond the efforts that have been made, there are many other microphysical processes, such as riming, that may be strongly affected by the heterogeneous particle distribution in clouds17,18, but are not well understood.

Riming is the growth of ice particles by collecting supercooled liquid droplets during their free fall19. It is a vital process to generate intense precipitation, especially in convective clouds. Riming is often explicitly parameterized in NWPMs, and is implicitly included as part of snow growth in GCMs, and both NWPMs and GCMs usually cannot resolve isolated cumulus clouds. Improving the parameterization of riming is of great importance to increasing the capability of NWPMs and GCMs in predicting precipitation20 and to understanding its impacts on cloud life cycle21,22.

Some studies have investigated the riming processes at typical sub-grid modeling scales, and demonstrated that the riming growth rate is controlled by various factors. For example, airborne measurements23 suggested that riming is stronger within the temperature range from −5 °C to 0 °C, and a stronger updraft24,25 can strengthen the interaction between liquid and ice to enhance riming to different extents. Waitz et al.20 found that riming exhibits spatial variability by comparing observational data of mixed-phase clouds from three major regions: the Arctic, the Southern Ocean, and the U.S. East Coast. Due to the stronger updrafts, the riming fraction in convective clouds is significantly higher than in stratiform clouds. Additionally, differences in the size, habit, and orientation of ice particles affect the collision efficiency, thereby influencing the riming fraction20. Klingebiel et al.26 found that riming is active in organized cloud street regions, while it almost disappears in isotropic cloud regions. This difference occurs because the strong vertical wind shear and high turbulent kinetic energy (TKE) in cloud street regions promote the collision between supercooled droplets and ice particles, whereas the low turbulence in isotropic cloud regions inhibits this process.

Therefore, how liquid and ice are mixed in clouds would certainly influence the quantification of riming growth rate27,28,29,30, and it is necessary to quantify the impact of sub-grid heterogeneous particle distributions on riming in models. Let’s consider two different scenarios. Firstly, if liquid and ice are clustered at a given level (e.g., Fig. 1b, corresponding to Level 1 in Fig. 1a), riming cannot take place as ice particles cannot collect any liquid drops during their free fall. However, for any model assuming that liquid and ice are homogeneously distributed in a grid box, the modelled riming growth rate would be positive. Secondly, if liquid and ice are mixed everywhere but the PSDs are not homogeneously distributed (e.g., Fig. 1c, corresponding to Level 2 in Fig. 1a), ice is able to grow through riming, but the actual riming growth rate is different than that assuming homogeneous particle distribution, as riming is a non-linear process.

Fig. 1: Conceptual diagram of a mixed-phase cumulus cloud and the different liquid-ice distributions.
Fig. 1: Conceptual diagram of a mixed-phase cumulus cloud and the different liquid-ice distributions.
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a Schematic diagrams illustrating the microphysics in mixed-phase clouds. Red and black arrows represent updrafts and downdrafts, respectively. b A conceptual diagram in which liquid and ice are distributed in a clustered way, which corresponds to the microphysical properties of Level 1 in (a), and (c) liquid and ice are mixed everywhere, which corresponds to the microphysical properties of Level 2 in (a). \({{\rm{R}}}_{{\rm{rim}}}\) is the actual riming growth rate, and \({{\rm{R}}}_{{\rm{rim}},{\rm{homo}}}\) indicates the riming growth rate assuming particles are homogeneously distributed.

Until now, the sub-grid scale variability of riming growth rate has not been considered in NWPMs and GCMs. This study aims to quantify the impact of heterogeneous liquid-ice mixing on riming in isolated cumulus clouds using airborne measurements conducted in the interior western United States during High Plain Cumulus (HiCu) project. The results are useful for deepening our understanding of sub-grid scale microphysics and for improving the parameterization of riming in models.

Results

Characteristics of microphysics in isolated cumulus clouds

First, we investigate the characteristics of microphysics and the homogeneity of liquid-ice mixing in isolated cumulus clouds measured in HiCu project. The concentrations and mass of cloud droplets and ice particles are directly related to the riming process in the clouds. According to Fig. 2a–c, the observed LWC has no significant change with temperature, and the mean LWC varied between 0.2 and 0.7 g m−3. The above data are consistent with previous airborne observations in moderate convection, and lower than those in stronger convection31. The vertical variations of IWC and ice particle concentration are similar, both of them increase by approximately two orders of magnitude as the temperature decreases from 0 to −30 °C (Fig. 2b, c). This indicates the IWC is not only related to the growth, but is also strongly controlled by ice generation in HiCu clouds. Both liquid and ice have large spatial variations at a given temperature, as seen from the broad probability densities (Fig. 2a, b). The variation of LWC exceeded 2 orders of magnitude, and that of IWC is 4–5 orders of magnitude. Such a large variation is not only due to the differences among clouds, but also the small-scale variability in individual clouds, as the result is obtained from on 1 Hz data. The riming growth rate is controlled by both liquid and ice. In general, the average Rrim increases with height. Rrim is relatively large between −10 and −15 °C, because the observed ice concentration is higher (Fig. 2c) and there are relatively larger ice particles according to the particle size distributions (not shown) at this temperature.

Fig. 2: Vertical profiles of various parameters in the clouds at different temperature intervals.
Fig. 2: Vertical profiles of various parameters in the clouds at different temperature intervals.
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a liquid water content, (b) ice water content, (c) ice/graupel particle concentration, (d) riming growth rate, (e) liquid-ice mixing homogeneity (χ), and (f) the factor quantifying the impact of heterogeneous liquid-ice mixing on riming (Frim), which is calculated by using the observed mean Rrim in a given cloud divided by the Rrim assuming a homogeneous particle distribution. The blue shaded areas represent probability density, the thin black bars indicate the 10th–90th percentiles, and the thick black bars are the 25th–75th percentiles.

The liquid-ice mixing homogeneity χ (as shown in Fig. 2e, χ = 0 indicates clustering of liquid and ice, and χ = 1 indicates homogeneous liquid-ice mixing, see details in Dataset and Method) is mostly greater than 0.5, suggesting that in most of HiCu clouds, liquid and ice co-exist in the entire or part of the cloud. In nimbostratus, clustering of liquid and ice pockets has been reported in previous observational studies9 due to turbulent eddies. The average χ in HiCu clouds increases from 0.65 to 0.91 as the temperature decreases from 0 to −30 °C; this is because there are more pure liquid regions in clouds at warmer temperatures.

The factor quantifying the impact of heterogeneous liquid-ice mixing on riming (Frim), which is calculated by using the observed mean Rrim in a given cloud divided by the Rrim assuming a homogeneous particle distribution (see detail in Dataset and Method), is shown in Fig. 2f. On average, Frim varied between 0.8 and 1.3, but this is the result of multiple cumulus clouds; the difference in Frim among different clouds is much larger, leading to the broad PDFs (Fig. 2f). Statistically, Frim has the largest variability at −5 to 0 °C, and the PDF narrows as the temperature decreases, suggesting the probability of co-existence of liquid and ice is higher at colder temperatures in HiCu clouds, consistent with the vertical profile of mean χ.

In short, Fig. 2 illustrates that the spatial variation of both liquid and ice-phase microphysics is significant in mixed-phase cumulus clouds, leading to various riming growth rates and non-unity Frim in different clouds. Therefore, it is necessary to consider the impact of heterogeneous liquid-ice mixing on riming growth in models.

Quantifying the impact of inhomogeneous liquid-ice mixing on riming

Figure 3 shows Frim as a function of χ, plotted for all observed ice particles greater than 100 μm (left columns), and for graupel greater than 300 μm (right columns). Each point represents an individual cloud. It is evident from this figure that statistically, Frim has a positive correlation with χ. In theory, Frim tends to be 0 when χ approaches 0. As χ approaches 1, Frim could be greater than 1 as riming is a non-linear process. Note that χ = 1 does not have to mean all the hydrometeors are completely mixed; it also applies when the liquid fractions are the same but the total concentrations vary throughout. Therefore, the riming rate in different sample units differs under such a condition. If riming is linear, we may expect that Frim would be 1 for χ = 1 according to Eq. 6. However, for a non-linear process, averaging Rrim in the numerator of Frim would be different from the mean Rrim after averaging PSD in the denominator. At relatively warmer temperatures (Fig. 3e, f), most of the clouds have a χ ranges between 0.5 and 0.95; while at colder temperatures (Fig. 3c, d), χ ranges mostly between 0.75 and 1, consistent with the analysis shown in Fig. 2e. Since Frim is lower for lower χ, the average Frim is lower at warmer temperatures, which explains the vertical profile shown in Fig. 2f. However, the relationship between Frim and χ has no significant difference among different temperatures. In addition, the magnitudes of Frim, as well as the correlation between Frim and χ for graupel, are similar to those for all ice particles (Fig. 3a, b), suggesting the bulk riming growth rate is controlled mainly by large solid particles. Statistical analysis based on all the data in Fig. 3a, b confirms that there is a positive correlation between Frim and χ for both all ice particles (r = 0.3, p < 0.0001) and graupel particles alone (r = 0.5, p < 0.0001). By comparing these two correlations using a z-test (z = −2.662, p = 0.0078), the results indicate that the positive correlation for graupel only is stronger than that for all ice particles, this statistical analysis supports the above conclusion.

Fig. 3: Scatter plots of the observed χ and Frim colored by ΔRrim,norm.
Fig. 3: Scatter plots of the observed χ and Frim colored by ΔRrim,norm.
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a, b All clouds, (c, d) clouds sampled at temperatures below −15 °C, and (e, f) clouds sampled at temperatures between −15 and 0 °C. The left panels are for all ice/graupel particles larger than 100 μm in diameter, and the right panels are for the graupel larger than 300 μm. The solid lines are fitted curves using Eq. 1 for different ΔRrim,norm.

χ cannot fully explain the variation of Frim. For a given χ, Frim may vary by more than 2. This occurs because χ only represents the homogeneity of liquid-ice mixing, and does not account for particle concentration when calculating χ. In clouds where liquid and ice are well mixed, the particle concentration may still exhibit large variations, leading to substantial variability in Rrim within the clouds. This impact can be intuitively identified by examining the range of normalized Rrim (ΔRrim,norm), which is determined by subtracting the minimum Rrim from the maximum Rrim and then normalizing it by the mean Rrim in each cloud. Statistically, for a given χ, Frim tends to be larger when ΔRrim,norm is greater. This finding is consistent across different temperatures (e.g., Fig. 3c, e). Additionally, this result is similar for clouds with varying widths (not shown), suggesting that the dependency of cloud scale on Frim is minor, at least for the clouds sampled in HiCu.

The relationship between Frim and χ can be fitted using an exponential function (Eq. 1) for different ΔRrim,norm, as shown by the fitted curves in Fig. 3.

$${{\rm{F}}}_{{\rm{rim}}}=\alpha ({e}^{\beta \chi }-1)$$
(1)

where \(\alpha\) and \(\beta\) are fitted coefficients. When taking both χ and ΔRrim,norm into account, the Frim can be still be fitted using Eq. 1, but the coefficient \(\alpha\) is a function of ΔRrim,norm:

$$\alpha ={a\Delta {{\rm{R}}}_{{\rm{rim}},{\rm{norm}}}}^{2}+b\Delta {{\rm{R}}}_{{\rm{rim}},{\rm{norm}}}+c$$
(2)

where, \(a=-0.000233\), \(b=0.031695\), \(c=0.448144\), and \(\beta =1.136207\) when all ice particles are implemented. For graupel only, \(a=-0.000118\), \(b=0.016168\), \(c=0.213294\), and \(\beta =1.765225\). The fitted and observed Frim is compared in Fig. 4. It is seen from the figure that the fitted and observed Frim are generally along the 1:1 line, with a correlation coefficient of 0.76 when all ice particles are used for calculation, and 0.82 for graupel only. Therefore, Frim can be parameterized using χ and ΔRrim,norm, which is potentially helpful to improve the representation of sub-grid scale riming growth in NWPMs and GCMs.

Fig. 4: Comparisons between observed and parameterized Frim.
Fig. 4: Comparisons between observed and parameterized Frim.
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a Ice particles greater than 100 μm and (b) 300 μm in diameter. R, me, and rmse respectively represent the correlation coefficient, mean error, and root mean square error, and the black line is the 1:1 line.

Now, the question becomes what factors control the values of χ and ΔRrim,norm. Since the model cannot prognose χ and ΔRrim,norm, it is necessary to link χ and ΔRrim,norm to predictable variables in models. Deng et al. showed that the liquid-ice mixing homogeneity is higher in stratiform clouds with higher CWC. The physical mechanism responsible for it is that in clouds with higher CWC (or initial LWC), it takes a longer time for glaciation; thus, there is a higher probability of observing mixed-phase clouds rather than pure ice pockets. We did a similar analysis for HiCu clouds here. As shown in Fig. 5a, the CWC generally exhibits a positive correlation with χ, which is observed across all temperature ranges, consistent with the previous findings7. On average, the observed CWC is higher at colder temperatures, leading to a larger χ. However, for a given CWC, the average χ is smaller at colder temperatures, suggesting less homogeneity of liquid-ice mixing.

Fig. 5: χ and ΔRrim,norm as a function of CWC and ice concentration, respectively.
Fig. 5: χ and ΔRrim,norm as a function of CWC and ice concentration, respectively.
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Relationships between (a) χ and mean CWC, and (b) ΔRrim,norm and mean ice particle concentrations calculated for every cloud. The big points represent the average values, and the error bars indicate 25th and 75th percentiles. The dashed lines indicate linear regressions; in the fitted equations in (a), y is log10(CWC), and x is χ; in (b), y is log10(Nice), and x is log10(ΔRrim,norm).

Additionally, it is found that the observed ΔRrim,norm is negatively correlated to the mean ice particle concentration in clouds, as shown in Fig. 5b. The results can be understood from two aspects. Firstly, for a given temperature range, the ΔRrim,norm increases as ice particle concentration decreases. This indicates that in clouds with lower ice concentration, ice particles are more distributed in clusters, resulting in a larger variation of Rrim. Secondly, for a given ice particle concentration, the ΔRrim,norm is larger at colder temperatures. This indicates that although there is a higher probability for ice to be mixed with liquid water at colder temperatures (i.e., larger χ, Fig. 3e), the variation of ice particle concentration is larger at colder temperatures (Fig. 3c), leading to a larger variation of Rrim.

Though the sub-grid scale χ and ΔRrim,norm can be parametrized using the mean CWC and ice particle concentration, we acknowledge that the uncertainties are large as seen from the long error bars, indicating that there are other factors affecting the distributions of liquid and ice. For example, the varying atmospheric instability may result in different vertical air velocities and turbulence strengths in areas, which may influence the phase partitioning in clouds9. Recently, using satellite data, Coopman and Tan32 showed that aerosols could also play a vital role in the heterogeneous distribution of hydrometeors in clouds. The aerosol loading varies significantly over different continents and oceans, which may potentially lead to different liquid-ice mixing properties in different regions. It is worth investigating how these factors can influence the liquid-ice mixing homogeneity and implementing them into parameterizations. With multiple factors considered, machine learning techniques could be useful to explore these non-linear relationships33 in future studies.

The response of Frim to continuous riming

The previous sections show that the heterogeneous distributions of liquid and ice can significantly influence riming growth. Meanwhile, the riming process itself may have feedback on the particle spatial distributions. Therefore, in this section, we investigate how Frim changes with time during riming. Two scenarios are assumed: (1) ice continuously grows through riming but the concentration of cloud droplet remains unchanged, i.e., the cloud can continuously provide supercooled liquid water, which is a condition more likely to be observed in developing clouds; (2) the concentration of cloud droplet decreases by riming, which is more likely to be observed in dissipating clouds lacking updrafts. No matter which scenario is analyzed, it is assumed that riming is taking place in a static condition; the temporal variations of cloud dynamics and thermodynamics are not considered. Additionally, it is assumed that the ice particle size would continuously increase during riming, while it is possible that the particle density increases while size has little change, especially for ice aggregates. Nevertheless, the results can be useful to help us understand how riming itself would affect Frim. For each cloud, riming is modelled for 500 s, and the observed droplet and ice PSDs are used as the initial inputs. The timestep is 0.1 s, and the Frim can be calculated every timestep.

The PDF distributions of Frim, analyzed for all the clouds, are shown in Fig. 6. From the figure, it can be observed that when the droplet concentration remains unchanged (Fig. 6a), Frim slightly broadens over time during riming. Since there is always sufficient liquid water available, in the cloud area where the riming rate is initially higher, its riming growth rate will consistently surpass those in other areas, resulting in stronger clustering of ice and water, increased variability of Rrim and larger Frim over time. Using statistical analysis of the airborne measurements conducted over the US East Coast and the Frim Strait west of Svalbard, Maherndl et al.34 also found riming can increase the probability of ice cluster occurrence; our finding is consistent with their result, though based on different datasets and analysis methods.

Fig. 6: Probability density functions of Frim at different model times for all clouds modelled under two conditions.
Fig. 6: Probability density functions of Frim at different model times for all clouds modelled under two conditions.
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a Cloud droplet concentration remains unchanged, and (b) cloud droplet concentration decreases with time due to riming.

If droplet concentration decreases with time (Fig. 6b), the modeled Frim clearly diminishes during riming. At 500 seconds, Frim approaches 0 in many clouds. This occurs because, in certain cloud regions, liquid water can be completely consumed by riming, while in regions without ice, the liquid water is never depleted, leading to a clustered distribution of pure liquid and ice pockets, which ultimately results in a small χ and Frim. The opposite responses of Frim to riming under different conditions suggest that the riming rate may be underestimated in developing clouds, and overestimated in dissipating clouds in models assuming homogeneous particle distributions.

In short, in mixed-phase cumulus clouds, the heterogeneous spatial distributions of liquid and ice can significantly influence riming growth, and the heterogeneity of particle distribution responds to the riming process. Therefore, it is necessary to implement the impacts of sub-grid scale heterogeneous particle distributions in modelling ice growth through riming.

Discussion

In this study, based on airborne in-situ measurements collected during the HiCu project, we investigate the impact of heterogeneous particle distributions on riming in mixed-phase cumulus clouds. Such an impact is quantified by using an impact factor (Frim), defined as the observed mean riming rate in a given cloud, divided by that assuming a homogeneous particle distribution. The main findings are as follows:

In the mixed-phase cumulus clouds observed in HiCu, the spatial variation of both liquid and ice-phase microphysics is significant, leading to various riming growth rates and non-unity Frim in different clouds. On average, Frim varies between 0.8 and 1.3, and generally increases with height. The PDF of Frim is the broadest just above the freezing level, and narrows as the temperature decreases.

Frim is positively correlated to the homogeneity of liquid-ice mixing (χ), which is quantified using the information entropy theory, and is positively correlated to the range of normalized riming rate (ΔRrim,norm) in a given cloud. Frim can be parametrized using an exponential function of χ and ΔRrim,norm, and the parameterized Frim has a good consistency with the observed Frim, with a correlation coefficient greater than 0.75. χ has a positive correlation with mean CWC, and ΔRrim,norm has a negative correlation with mean ice concentration. These relationships apply across the different temperatures sampled in HiCu. Thus, Frim can be linked to CWC and ice concentration, which are explicitly resolved in models.

Frim responds to continuous riming differently in different conditions. When there is a sufficient water supply, Frim may increase due to riming, while if the liquid water continuously consumed by riming, Frim substantially decreases.

The findings of this study provide a potential way to parametrize the impact of heterogeneous particle distributions on riming growth in models. However, there are several issues that remain unresolved and need more effort in the future. Firstly, the results are obtained based on measurements in isolated cumulus clouds with weak to moderate strength of updrafts. It is necessary to investigate different types of convection under various environments, because the characteristics of phase partitioning vary significantly in clouds sampled in different field campaigns. Additionally, beyond CWC, ice concentration and temperature, there must be other factors that influence χ, and ΔRrim,norm, such as aerosols and atmospheric stability. Much more efforts are needed to investigate the mechanisms that control the phase partitioning in cumulus clouds, and to improve the parametrizations developed in this study. Moreover, the responses of Frim to riming are investigated simply based on assumptions of a static environment using airborne measurements. It may be helpful to improve our understanding of how riming influences the phase partitioning in mixed-phase clouds based on remote sensing techniques and high-resolution simulations.

To sum up, although there are limitations, the present study highlights that the heterogeneous distribution of liquid and ice can influence the riming process in cumulus clouds, and provides a potential way to develop sub-grid parametrization to improve the capability of modelling the riming process in models.

Methods

Airborne measurements

The dataset was obtained from the HiCu project35,36 (Wang et al. 2009; Yang et al. 2016), which was conducted in the interior western United States. During HiCu, there were 35 research flights conducted in July and August of 2002 and 2003. Each research flight lasted for several hours. The research aircraft, University of Wyoming King Air (UWKA) with a complete set of cloud micro-physical probes, penetrated various isolated mixed-phase cumulus clouds at heights of 4–9 km, to measure the cloud microphysics and dynamics between −22.5 °C and 2.5 °C. The updraft strength was weak to moderate in HiCu clouds, with a vertical velocity varying between 0 and 15 m s−1. An FSSP-100 was used to detect cloud droplets and small ice crystals in 0.8 ~ 50 μm. A hot-wire probe was used to detect liquid water content (LWC). 2D-C/P probes were deployed to detect larger particles in 25 ~ 19200 μm. Since the uncertainty in the first 5 bins of 2D-C is large37, only particles greater than 100 μm are used to calculate the riming growth rate. In HiCu clouds, all particles larger than 100 μm were ice as demonstrated by the 2D-C images. In addition, to calculate the concentration of graupel, we use the particles larger than 300 μm and with an aspect ratio greater than 0.6. This method cannot rule out the presence of ice aggregates, but since HiCu clouds were mostly in developing stage, the observed large particles were mostly graupel according to the 2D-C/P particle images. The influence of particle shattering on 2D-C/P was reduced based on the inter-arrival times between particles38. The FSSP measurement was also affected by particle shattering37, but this impact was not significant because the HiCu clouds were dominated by liquid water, with a mean liquid water mass fraction greater than 0.7. Previous studies37 showed that the FSSP had an overestimation (about 20% on average) of LWC compared to the hot-wire probe measurement in HiCu. This average uncertainty will not significantly impact the results because it can be canceled out when comparing the riming rates between the observed clouds and that assuming homogeneous particle distributions (see Eq. 10).

The ice water content (IWC) can be estimated using the mass-size relationship of ice particles39:

$$M=a{D}^{b}$$
(3)
$${IWC}=\mathop{\sum }\limits_{j=1}^{{N}_{2D}}{n}_{j}{M}_{j}$$
(4)

where, \({M}\) (in mg) and \(D\) (in mm) are the mass and diameter of a single ice particle. We use the maximum dimension as the diameter of an ice particle40. Following previous studies of mixed-phase convective clouds, \(a\)=0.029 and b = 2.1 are used41. These coefficients were obtained after calibrating early findings specifically for convective clouds, which better characterize the high-density and irregularly shaped ice particles41, and are consistent with the terminal velocity parameterization adopted42 (shown later). Although the values of coefficients a and b carry certain uncertainties, we believe that the selected values are suitable for the clouds studied here. \({N}_{2D}\) is the number of bins in 2D measurements. \({n}_{j}\) and \({M}_{j}\) are the number concentration and mass of the ice particle in the jth bin. The total condensed water content (CWC) is the summation of LWC and IWC, the liquid mass fraction (LMF) is the ratio between LWC and CWC, and the ice mass fraction (IWC) is the ratio between IWC and CWC.

All the data we use here is stored at 1 Hz. We use a threshold of FSSP concentration greater than 10 cm−3 to identify clouds43,44. Since the present study focuses on riming, pure liquid clouds are not analyzed. In addition, to quantify the liquid-ice mixing homogeneity in each cloud, a single penetration should contain multiple sample units. In this study, we select the penetrations that lasted at least 10 s. Since the UWKA true air speed was about 100 m s−1, this selection implies only clouds broader than 1 km are analyzed, and we are able to investigate the spatial variability of about 100 m. Any clouds with a smaller scale are not considered here, and a total of 616 cumulus clouds that meet the definition of this study were sampled.

Calculation of the riming growth rate

In this study, the riming growth rate (\({{\rm{R}}}_{{\rm{rim}}}\)) is defined as the bulk mass growth rate45 of ice water. For airborne measurements,

$${{\rm{R}}}_{\mathrm{rim}}=\frac{d\mathrm{IWC}}{{dt}}=\mathop{\sum }\limits_{j=1}^{{N}_{2}}\mathop{\sum }\limits_{k=1}^{{N}_{1}}A({D}_{j}){V}_{T}({D}_{j})E\left({D}_{j},{d}_{k}\right)\mathrm{LWC}\left({d}_{k}\right)\cdot n({D}_{j})$$
(5)

where, \({N}_{1}\), \({N}_{2}\) are the maximum bin of FSSP and 2D probes, respectively. \(A({D}_{j})\) is the cross-sectional area that is perpendicular to the direction of air flow of an ice particle with a diameter of \({D}_{j}\). \({V}_{T}\) is the fall speed of ice crystals. \(E(D,{d}_{k})\) is the efficiency of an ice particle with a diameter of \({D}_{j}\) collecting droplets with a diameter of \({{d}}_{k}\). \({LWC}({d}_{k})\) is LWC carried by droplets with a diameter of \({d}_{k}\). \(n({D}_{j})\) is the concentration of ice particles in the jth bin as measured by 2D-C/P probes.

The cross-sectional area and collection efficiency46 during riming can be calculated using:

$$A=\pi {\left(\frac{D}{2}\right)}^{2}$$
(6)
$$E=0.55\cdot \log (2.51K)$$
(7)
$$K=\frac{2\rho {V}_{t}{d}^{2}}{9\eta D}$$
(8)
$$\eta \left(T\right)={\eta }_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{3}{2}}\frac{{T}_{0}+S}{T+S}$$
(9)

where, K is the dimensionless Stokes parameter. ρ is the density of droplets, and η is the dynamic viscosity of the air. D, d are the diameters of ice crystals and droplets. η0 is the dynamic viscosity under reference temperature T0 and S is a constant number of Sutherland. Here, we use η0 = 1.716 × 10−5 Pa∙s when T0 = 273.15 K, and S = 110.4 K for air. The method for cross-sectional area is appropriate for the irregular aggregates and graupel particles in mixed-phase clouds, which are the primary focus of this study. But exhibits limitations when dealing with ice crystal particles dominated by prolate needle shapes.

For \({V}_{T}\), we use the parametrizations in Heymsfield et al.47, which are developed based on airborne measurement in mixed-phase convective clouds. The original parameterizations are developed for 800 hPa, 600 hPa, and 400 hPa, respectively (Table 1). We interpolated it to different heights for HiCu clouds.

Table 1 \({V}_{T}-D\) relationships for particle diameters at 800 hPa,600 hPa and 400 hPa
Full size table

Quantifying the impact of heterogeneous particle distributions on riming

The impact of heterogeneous particle distributions on riming can be investigated by comparing the average riming growth rate in a real cloud to that assuming PSDs are homogenously distributed. For any penetration with N sample units, the impact can be quantified using the formula below:

$${{\rm{F}}}_{\mathrm{rim}}=\frac{\frac{{\sum }_{i=1}^{N}{{\rm{R}}}_{\mathrm{rim},i}}{N}}{{{\rm{R}}}_{\mathrm{rim}}\left(\frac{{\sum }_{{\rm{i}}=1}^{{\rm{N}}}PS{D}_{{\rm{i}}}}{{\rm{N}}}\right)}$$
(10)

where, \({{\rm{R}}}_{{\rm{rim}},i}\) is the riming growth rate in the ith sample unit, and the numerator of \({{\rm{F}}}_{{\rm{rim}}}\) indicates the average riming growth rate. The denominator of \({{\rm{F}}}_{{\rm{rim}}}\) is the riming growth rate assuming homogeneous particle distribution, which is calculated by using the averaged PSDs.

According to the definition above, we expect that \({{\rm{F}}}_{{\rm{rim}}}\) will be close to 0 for the clouds in which liquid and ice are clustered (e.g., Fig. 1b), and \({{\rm{F}}}_{{\rm{rim}}}\) is 1 if particles are perfectly mixed. For clouds in which liquid and ice co-exist in a mixed state but with their size distributions varying throughout (e.g., Fig. 1c), Frim could be either greater or smaller than 1.

Quantifying the liquid-ice mixing homogeneity

\({F}_{{rim}}\) must be related to the heterogeneous liquid-ice mixing in a given cloud. Therefore, to quantify the liquid-ice mixing homogeneity, we introduce the information-entropy theory, which provides a useful framework48 with comprehensive parameter49 to describe the diversity of components within a system and the randomness of their spatial distribution, which has been used in areas such as communication50, ecology51, and aerosols52. For mixed-phase clouds, we use the following formulas:

$${{\rm{LMF}}}_{i}={{\rm{LWC}}}_{i}/{{\rm{CWC}}}_{i}$$
(11)
$${{\rm{IMF}}}_{i}={{\rm{IWC}}}_{i}/{{\rm{CWC}}}_{i}$$
(12)
$${H}_{i}=-{\mathrm{LMF}}_{i}\times ln{\mathrm{LMF}}_{i}-{\mathrm{IMF}}_{i}\times ln{\mathrm{IMF}}_{i}$$
(13)
$$c{(k)}_{i}={{\rm{CWC}}}_{i}/{{\rm{CWC}}}_{{total}}$$
(14)
$${H}_{\alpha }={\sum }_{i=1}^{N}{H}_{i}\times c{(k)}_{i}$$
(15)

N is the number of grids in a cloud. \({{\rm{LMF}}}_{{\rm{i}}}\,,\,{{\rm{IMF}}}_{i}\) is the average mass fraction of liquid and ice in every grid. \({H}_{i}\) is the mixing entropy of every grid, and \({H}_{\alpha }\) is the average mixing entropy for each cloud.

$$\overline{{\rm{LMF}}}={\sum }_{i=1}^{N}{{\rm{LMF}}}_{i}\times c{(k)}_{i}$$
(16)
$$\overline{{\rm{IMF}}}={\sum }_{i=1}^{N}{{\rm{IMF}}}_{i}\times c{(k)}_{i}$$
(17)

where, \(\overline{{\rm{LMF}}}\) and \(\overline{{\rm{IMF}}}\) are the weighted-average LMF and IMF. The population-level mixing entropy \({H}_{\gamma }\) is expressed as:

$${H}_{\gamma }=-\bar{\mathrm{LMF}}\times ln\bar{\mathrm{LMF}}-\bar{\mathrm{IMF}}\times ln\bar{\mathrm{IMF}}$$
(18)

Finally, we get the liquid-ice mixing homogeneity \(\chi\) :

$$\chi =\frac{{D}_{\alpha }-1}{{D}_{\gamma }-1}$$
(19)

where, \({D}_{\alpha }={e}^{{H}_{\alpha }}\,,\,{D}_{\gamma }={e}^{{H}_{\gamma }}\). \({D}_{\alpha }\) and\(\,{D}_{\gamma }\) are per-unit diversity and the population diversity, respectively. To better understand \(\chi\), we use Fig. 1b, c as an example. In Fig. 1b, the liquid and ice are clustered; if dividing it into multiple sample units, in each sample unit, only a single phase was present (liquid or ice), thus, the per-unit diversity is low, but the population diversity is high, leading to \(\chi =0\). In Fig. 1c, liquid and ice are mixed everywhere, so both the per-unit diversity and population diversity are high if the cloud is partitioned into multiple units.