Classification of Hailstone Trajectories in a Hail Cloud over a Semi-Arid Region in China

The distribution of the particle size of ground hailfall plays a decisive role in its catastrophic nature (Picca and Ryzhkov, 2012; Kunz et al., 2018; Lin and Kumjian, 2022). Based on previous studies, storm development characteristics strongly influence hailstone size (Groenemeijer and van Delden, 2007; Kaltenböck et al., 2009; Blair et al., 2017). The hailstone growth trajectory and the accompanying cloud physical processes along the trajectory serve to limit ground hailfall particle size.

The updraft and duration of hailstone growth in the updraft zone are known have a considerable influence on the growth rate of hailstone (Farnell et al., 2022). Some studies have suggested that hail grows during a single passage through an updraft (Miller and Fankhauser, 1983; Nelson, 1983; Foote, 1984); however, hailstone growth during descent can also be significant (Pruppacher and Klett, 1997). The horizontal flow field in hailstorms also plays a vital role in the growth of hailstone (Nelson, 1987; Miller et al., 1988; Tessendorf et al., 2005; Dennis and Kumjian, 2017; Kumjian and Lombardo, 2020). The largest-size hailstones can pass through the entire updraft zone under the action of cyclonic flow fields (Foote, 1984). Besides, there are also lots of observational and numerical simulation studies show that the microphysics in the hailstone growth trajectory, especially the amount of supercooled liquid, has significant influences on hailstone formation and growth (Paluch, 1978; Heymsfield et al., 1980; Conway and Zrnić, 1993; Kennedy and Detwiler, 2003; Johnson and Sugden, 2014; Kumjian and Lombardo, 2020).

Prior studies have found that the internal dynamic structure of the storm caused by low-level convergence, anticyclonic outflow, and storm splitting, together with the response of updraft and supercooled cloud water content to these processes, ultimately lead to changes in the hail formation process (Kumjian et al., 2021). The deepening of the wind shear layer might lead to an increase in the mass of hail produced (Kumjian and Lombardo, 2020). Factors such as greater instability, deeper wind shear, and increased lifting condensation levels might increase the probability of large hailstones (Taszarek et al., 2017). Convective Available Potential Energy (CAPE) values change the updraft intensity, width, horizontal wind field, and liquid water content for hailstone growth. These changes result in subsequent alterations in hailfall diameter and concentration (Kumjian et al., 2021; Lin and Kumjian, 2022).

It can be concluded that the diverse environmental characteristics of hail formation pose significant challenges for models to forecast the probability of hail occurrence and study the influence of climate change on patterns of hail occurrence (Zhou et al., 2021). Recently, many studies were conducted to improve the prediction of hail events, with microphysics parameterization schemes (MPS) for cloud-resolving weather prediction models are often very detailed, multi-moment schemes with several ice-phase categories (Milbrandt and Yau, 2005a, b; Seifert and Beheng, 2006; Morrison et al., 2009; Lim and Hong, 2010; Thompson and Eidhammer, 2014). Significant improvements in the representation of sedimentation, melting, and formation processes of hail particles and storm structures were achieved by simulations using three-moment schemes rather than by those using one- or two-moment schemes (Milbrandt and Yau, 2006; Loftus and Cotton, 2014; Dawson II et al., 2015; Luo et al., 2017, 2018, 2021).

In the Predicted Particle Properties (P3) scheme (Morrison and Milbrandt, 2015; Milbrandt and Morrison, 2016), all ice-phase hydrometeors are represented by a user-specified number of “free” frozen categories, which performs well compared to detailed, state-of-the-art, multicategory MPSs and can realistically simulate a wide range of ice-phase particles in deep convective and stratiform weather systems. Recent developments to P3 include the incorporation of a three-moment rain category and ice particles (Paukert et al., 2019; Milbrandt et al., 2021), the addition of a prognostic variable to predict the liquid fraction of an ice category, thus allowing for mixed-phase particles (Cholette et al., 2019), which can better represent the convective characteristics.

Owing to a lack of observations inside hail clouds, accurate forecasting of hail particle size remains a major challenge in current weather forecasting (Blair et al., 2011, 2017; Gagne II et al., 2017; Ortega, 2018; Adams-Selin et al., 2019; Morrison et al., 2020; Raupach et al., 2021; Lin and Kumjian, 2022). Meanwhile, the study of hail trajectories is shown to be a necessary and effective way to study the influence of storm evolution characteristics on hail formation (Browning and Foote, 1976; Heymsfield, 1983; Miller and Fankhauser, 1983; Nelson, 1983; Ziegler et al., 1983; Foote, 1984; Musil et al., 1986; Rasmussen and Heymsfield, 1987; Miller et al., 1988, 1990; Xu and Duan, 2001; Tessendorf et al., 2005; Chen et al., 2012; Adams-Selin et al., 2019; Kumjian et al., 2021; Lin and Kumjian, 2022).

Numerical simulation of hailstone growth trajectories may compensate for the lack of observation data in hail clouds to a certain extent and improve our scientific understanding of the physical mechanism of hail formation. Notably, hail formation is highly localized, and the physical images of hailstone formation and growth in different regions or background fields may markedly vary. Therefore, an in-depth understanding of the growth trajectory of hailstone in different regions and different types of hail clouds is needed to improve hail forecasting and warning, as well as disaster prevention and mitigation.

Here, we aim to provide background fields in hail clouds based on WRFv3.7.1 model simulations, drive 3D hailstone growth trajectory simulations, and perform cluster analysis of hail trajectories based on K-means cluster analysis. The findings of this study are expected to deepen the current understanding of the characteristics of hailstone formation and growth in the hail event in the semi-arid region of China.

The remainder of this paper is organized as follows. Section 2 briefly introduced the evolution of the hail event and the settings of numerical simulation. Section 3 presents hail cluster analysis. Section 4 discusses the results, and section 5 summarizes the study.

The hail event simulated in this study occurred on 7 June 2017, when hail and short-term heavy precipitation occurred from 0600 to 1200 UTC in Jingyuan, Longde, Xiji, Haiyuan County, Yuanzhou District, Guyuan City, and Zhongwei City (Fig. 1). From 0700 to 0800, strong convective weather was located in Guyuan City, with short-term heavy rainfall and hailfall. The cumulative hailfall depth was 2–3 cm, the maximum ground hailstone diameter was 1.5 cm, and the hailfall duration was 20–40 min.

Figure 1.  Simulation region of the hail event on 7 June 2017, in Ningxia, with the blue box indicating the simulation area of the trajectory model: (A) Guyuan, (B) Jingyuan County, (C) Longde County, (D) Xiji County, and (E) Haiyuan County.

In this study, the ERA-Interim reanalysis data (Dee et al., 2011) from the European Centre for Medium-Range Weather Forecasts were used as the initial conditions for the WRFv3.7.1 model, with a horizontal resolution of 0.75° × 0.75°. Both lateral boundary conditions were updated every six hours. The simulation area adopted a double nesting with both center points at (37.3°N, 106.0°E). The horizontal grid spacings were 3 km and 1 km, respectively, and the vertical resolution was set at 57 layers. The simulation area is shown in Fig. 1.

Case simulations were performed using the following model configurations: the Milbrandt-Yau (MY) double-moment cloud physics scheme (Milbrandt and Yau, 2005a, b), Mellor-Yamada-Janjic (MYJ) boundary layer scheme (Mesinger, 1993; Janjić, 1994), Eta near-surface layer scheme (Janjić, 1994, 1996), Unified Noah land surface process scheme (Tewari et al., 2004), and the Rapid Radiative Transfer Model (RRTM) for the Global Climate Model (GCM) short-wave and long-wave radiation schemes (Iacono et al., 2008). Both the inner and outer nesting did not use a cumulus convection parameterization scheme. As the area where the hail cloud occurred has a relatively clean land background, the cloud condensation nuclei (CCN) concentration in the MY double-moment scheme was set to 200 cm^–3 for the numerical simulation.

The comparison of the numerical simulation results of the WRFv3.7.1 model regarding the development and evolution characteristics of hail clouds, ground precipitation, and hail characteristics observed by radar shows that the model was effective in simulating the ground precipitation distribution and hail location and could satisfactorily reproduce the impact area of hail. Thus, the model and simulation scheme used in WRF v3.7.1 could effectively reproduce the formation, development, and evolution characteristics of hail cloud processes and characteristics as well as ground precipitation in the southern mountainous region of Ningxia. Based on these results, numerical simulation of the hail trajectories can be driven by the simulation results from WRF v3.7.1 (Liu et al., 2021).

The inner nesting layer simulation result of WRF v3.7.1 was used as the initial background field required for the hail trajectory model. The extracted background field information was interpolated to the grid points in the area where the hail trajectory model was located. From the numerical simulation results of the WRF v3.7.1 model, the hail cloud was identified to be in the development stage at 0600 UTC on 7 June, with the formation of graupel in the cloud, which gradually transformed into hail particles. Hence, the hail trajectory model started its integration at 0600 UTC on 7 June 2017, with an integration duration of 125 min and an integration step of 5 s. The horizontal and vertical resolutions of the trajectory model were both 500 m. The number of horizontal grid points was 360 × 360 with 25 vertical layers. The hail embryos were released at the grid points with a mixing ratio of graupel greater than 0.01 g kg^–1 and a number concentration greater than 1 m^–3. One hail embryo was released at each grid point. The initial particle size of the hail embryo was 0.05 cm, and the density was 0.9 g cm^–3.

The three-dimensional hail trajectory model used in the study was developed by Xu and Duan (2001), and more details are available in Wang and Xu (1989). A detailed description of this hail trajectory model is the same as that introduced by Yin et al. (2021).

As the characteristics of hail trajectories differ by location and background, specific identification of subjectivity may exist in the human classification of trajectories. Thus, the K-means clustering method was employed in this study to cluster hail trajectories. The methods are the same as those employed by Jain and Dubes (1988).

Analysis of the sum of the squared error (SSE) against k-values (Fig. 2) showed that SSE displayed a significant decreasing trend that slowed down when k-values were greater than five. Therefore, the k-value was set to five, and the hail trajectory dataset was divided into five classes for clustering.

Figure 2.  Variation of SSE with k-value.

The statistical results from the clustering of hail trajectories revealed (Table 1) that different types of hail trajectories led to different maximum particle sizes of the ground hailstone. A total of 5 262 hail tracks with hailstone diameters greater than 0.5 cm were simulated by the hail trajectory model. Regarding ground hailstone diameter, there were 4 283 trajectories corresponding to small hailstones ranging from 0.5 to 1 cm, accounting for approximately 80%. There were 47 trajectories with hail diameters greater than 2 cm, including 13 tracks with diameters greater than 2.5 cm. From the classification results of hailstone trajectories, both Type I and Type II trajectories correspond to ground hailstone ranging from 0.5–1 cm. Types III and IV trajectories correspond to ground hailstone of a maximum size close to 2.5 cm. Type V trajectories produce a more diverse range of hail size, with the maximum diameter of ground hailstone up to a maximum of 3.17 cm. Figure 3 shows the distribution of hailfall of different trajectory clusters. The different hailstone trajectories might be due to differences in the characteristics of hail clouds in different regions.

Figure 3.  Distribution of hailstone trajectories for different hail diameters.

As depicted in Fig. 4, Type I and II have the smallest number of trajectories, small ground hailstone size, and the shortest hailstone growth duration in the cloud. As the maximum size of hailstone and the number of large-size hailstones on the ground increased, the growth time of hailstone in clouds increased in Type III and Type IV hail trajectories. The growth time of hailtone in Type IV hail trajectories was more prolonged than in Type III trajectories. The number of hailstone trajectories and the maximum hailstone size on the ground were the largest for Type V trajectories. However, the growth time of hailstone for Type V was not longer than that of Type IV. Only hailstone above 2.5 cm had a longer growth time. Based on these results, the growth time of hailtone in the cloud for trajectories with more and larger ground hailstones was generally more prolonged than that for trajectories with weaker ground hailfall. However, the growth time of hailstone in clouds does not always show a systematic increase with the number and maximum size of hailstones on the ground.

Figure 4.  Distribution of (a) growth time, (b) maximum updraft, (c) minimum temperature, and (d) maximum supercooled cloud water content in the growth trajectories of different types of hailstone.

Generally, the maximum updraft velocity during the growth of Types III, IV, and V hail trajectories was markedly more significant than that of the remaining two types. Among them, Type V was greater than Type IV, and Type IV was greater than Type III. A correlation was found between the intensity of updraft velocity in the hailstone growth trajectory and the maximum ground hailstone size corresponding to this type of trajectory.

The growth rate of hailstone in a hail cloud is closely related to the amount of supercooled liquid water content. In the hail event simulated in this study, the primary method of hailtone growth was the riming of supercooled cloud water. As shown in Fig. 4, the increase in hailstone size was accompanied by an increase in the maximum supercooled cloud water content in the environment where it grows. For large-size hailstone, its growth must occur in an environment containing abundant supercooled cloud water. With the strengthening of the updraft, the minimum ambient temperature in the region of hailstone growth generally shows a decreasing trend from Type III to Type IV, and from Type IV to Type V, except for the largest hailstones in each type. Such hailstones can grow longer in cloud areas with more supercooled cloud water and lower maximum growth heights (higher value of the minimum ambient temperature region hailstones can arrive) as they have a larger mass, which causes them to be less susceptible to upward transport by updrafts.

Figure 5 shows that different hailstone trajectories are concentrated in different regions, within which different air-flow characteristics may cause. The cluster clarification was based not only on the final hailfall size on the ground but also on the movement trajectory with time and the growth pattern of the hailstone. The clustering method basically identified and reasonably classified hailstone trajectories.

Figure 5.  Simulated hailstone trajectories (different colors represent different particle sizes during hailstone growth): (a) Type I, (b) Type II, (c) Type III, (d) Type IV, (e) Type V, and (g) sum of all five types; (1) latitude-height projection, (2) longitude-height projection, (3) longitude-latitude projection.

Figure 5.  (Continued)

Figures 6 through 10 show the simulated growth trajectories corresponding to different hailstone sizes on the ground. The growth pattern of hailstonewith similar trajectories show some similarities. As revealed by the hail distribution characteristics, Types I and II trajectories produced ground hailstone with a maximum diameter of 0.65 and 0.76 cm, respectively, and a few corresponding numbers of hailstone trajectories. Despite a minor difference in the diameter of ground hailstone in Types I and II trajectories, the hailstone in Type I trajectories reached a height of approximately 7 km. In comparison, the maximum height of hail stone in Type II trajectories was less than 4 km.

Figure 6.  Simulated 0.5–1 cm hailstone trajectories (different colors represent different sizes during hailstone growth): (a) Type I, (b) Type II, (c) Type III, (d) Type IV, and (e) Type V; (1) latitude-height projection, (2) longitude-height projection, (3) longitude-latitude projection; purple arrows represent the projected wind vectors).

Although both Type I and II trajectories are characterized by an initial fall, subsequent rise, and another fall in the hailstone growth trajectory, the hailstone size grows slowly during movement. In Type I hailstone tracks, due to their small particle size, the hailstone is transported where the updraft is relatively weak. The hail-stone are thus more likely to have a longer horizontal growth path under the action of the horizontal airflow. Type II trajectories have smaller horizontal space and mainly exhibit an “up-down” growth trajectory. In contrast, in Type II trajectories, due to the limited strength of updrafts in the area where the hailstone is located, the hailstone cannot be transported to higher altitudes and grow during the ascent. They reach the maximum height when their terminal velocity closes to the velocity of updrafts, continuing to grow before falling out of the hail cloud.

In comparison, the hailstone growth trajectories of Types III, IV, and V produce larger ground hailstone sizes. The hailstones in these three hail trajectories have a faster growth rate and a more robust updraft strength. A comparison revealed differences in the growth trajectories of hailstone in clouds for hailfall on the ground of the same size range.

As the size of ground hailfall increases, the trajectory of hailstone tends to become more complex. For trajectories that generate ground hailstone ranging from 1 to 1.5 cm, hailstone undergo a specific cyclonic rotation trajectory before falling, during which time the size grows slowly. Generally, as hailstone falls, a rapid growth in its size occurs.

For trajectories that generate ground hailstone ranging from 1.5 to 2 cm, corresponding hailstone growth trajectories show a degree of similarity to those of hailstones ranging from 1 to 1.5 cm. This group of hailstone growth trajectories corresponds to an increase in the intensity of the updraft. As the hailstone growth trajectory corresponds to a stronger updraft region, the corresponding hailstone could deplete the greater abundance of supercooled cloud water in this region to grow rapidly during the growth process; this eventually caused an increase in ground hailfall size.

Figure 9 shows that as size increases when the hailstone is at maximum altitude, the hailstone size on the ground also increases. Hence, the growth rate of the hailstone is closely related to the flow field, updraft, and supercooled liquid water content in the cloud. The size of the hailstone itself also plays a relatively significant role. As the size of hailstone in the cloud increases, the efficiency of its collection of supercooled liquid water increases. Hailstone growth rate and size of ground hailfall also increase accordingly.

Figure 9.  Simulated 2.0–2.5 cm hailstone trajectories (different colors represent different sizes during hailstone growth): (a) Type III, (b) Type IV, and (c) Type V; others as shown in Fig. 6.

Type V hailstone trajectories formed the largest size of ground hailfall, with the strongest updraft velocity at the height of its maximum growth. The size at the largest height reached 2 cm and further increased during descent. Due to the large size at the beginning of the descent, abundant supercooled cloud water during descent prompts hailstone size to become more pronounced, resulting in ground hailfall with larger sizes (Fig. 10).

Figure 10.  Simulated hailstone trajectories with size greater than 2.5 cm in Type V (different colors represent different sizes during hailstone growth); others as shown in Fig. 6.

A comparison with Fig. 9c clearly shows that although the ground hailfall intensity of this group of trajectories increases, the maximum height reached is lower. The growth of hailstone in this group mainly stems from their prolonged growth in regions of stronger updrafts, leading to rapid growth in the specific region. Although the maximum height of hailstone is lower, the updraft is more robust in the region, making it more difficult and requiring a longer time for hailstone to fall from this area. The duration of growth of hailstone in the updraft zone was determined based on the difference between the terminal velocity of hailstone and the updraft, as well as the horizontal width of the updraft zone. A wider updraft zone and a terminal velocity closer to updraft strength allow hailstons to exploit the abundance of supercooled cloud water in the region to grow longer.

In addition to the flow field, the distribution characteristics of the liquid water content in clouds are important factors governing the growth of hailstone. Figures 11 through 15 show the distribution characteristics of hailstone growth trajectories corresponding to cloud water content. These figures show that hailstone growth trajectories corresponding to smaller ground hailfall have lower cloud water content along their growth paths. Areas in trajectories with relatively rich cloud water content may exist in the horizontal motion phase or at the highest altitude reached. The rate of hailstone growthin the hail cloud depends on the distribution of cloud water on the growth trajectory; however, the maximum height reached by the hailstone in the cloud is determined by its mass (particle size) and the updraft velocity in that area.

Figure 11.  Simulated cloud water content variation of 0.5–1 cm hailstone trajectories (different colors represent different cloud water content distribution during hailstone growth): (a) Type I, (b) Type II, (c) Type III, (d) Type IV, and (e) Type V; others as shown in Fig. 6.

In the hailstone growth process, hailstones can either pass through areas with relatively rich liquid water content before slowly growing and falling or can first exploit the air-flow field to reach a region with relatively more abundant supercooled water and stronger updraft before completing the growth process and descent by going through the “up-down” cycle in this region. In Types III, IV, and V trajectories, an abundance of supercooled cloud water existed at the maximum heights reached by the hailstone. These regions were also the regions with the fastest hailstone growth rates. In general, stronger updraft corresponds to increases in supercooled cloud water content. The trajectories may not have formed larger hailstones due to the undulation of updrafts and supercooled cloud water within hailstone growth trajectories. The possibility of an up-down cyclic growth trajectory exists even for hailstone with small sizes.

In Type III trajectories, the growth trajectories of hailstone ranging from 1 to 1.5 cm in the hail cloud exhibited high degrees of undulation accompanied by undulation in updraft velocity along the growth path (Fig. 12). The supercooled cloud water content on their growth trajectory is consistently smaller than that of hailstone in the 1.5 to 2 cm size range (Fig. 13). This result may also be why the trajectory corresponds to smaller ground hailstone. For Types IV and V trajectories that form 1–1.5 cm ground hailstone, the hailstone passes through cloud areas with relatively large supercooled water content during cyclonic rotational ascent. It grows and falls after the hailstone reaches equilibrium between the terminal velocity and updraft. However, this group of trajectories does not have abundant supercooled cloud water during its descent, leading to its limited ground hailstone size.

Figure 12.  Simulated cloud water content variation of 1–1.5 cm hailstone trajectories (different colors represent different cloud water content distribution during hailstone growth): (a) Type III, (b) Type IV, and (c) Type V; others as shown in Fig. 6.

Figure 13.  Simulated cloud water content variation of 1.5–2.0 cm hailstone trajectories (different colors represent different distributions of cloud water content during hailstone growth): (a) Type III, (b) Type IV, and (c) Type V; others as shown in Fig. 6.

As the supercooled cloud water content along the hailstone growth trajectory increased, the size of ground hailfall corresponding to the trajectory increased (Fig. 13). For hailstones ranging from 1.5 to 2 cm in Type III trajectories, the content of supercooled cloud water was generally greater than the other trajectory types for most of the growth process. Type IV hailstone trajectories indicate extended periods of growth with little change in horizontal height; however, fluctuations in updrafts and supercooled water content occur during hailstone growth. This result also indicates that hailstone may grow horizontally in cloud areas with significant differences in supercooled water content. The supercooled cloud water content in Type V hail trajectories showed significant variation in the hail’s horizontal and vertical growth paths. Based on Type V hailstone growth trajectories, hailstone growth may have gone through different updraft zones, hail particles leaving from one updraft zone and undergoing a descent process before entering another stronger updraft zone, exploiting the abundant supercooled cloud water area in the region for more rapid growth.

Strong updrafts and abundant supercooled cloud water were maintained for a long time during the growth of hailstone in the 2–2.5 cm range in Type III trajectories (Fig. 14). Strong updrafts, abundant supercooled cloud water, and sufficient growth times favor the formation of large hailsrone. Types IV and V trajectories formed more hailstones in the 2 to 2.5 cm range than Type III trajectories. Based on a comparison, even if the ground hailstone sizes do not exhibit much difference, their growth trajectories in the cloud may be significantly different. The growth of hailstone in updraft zones with higher supercooled cloud water content may be the main reason for the increased formation of 2 to 2.5 cm hailstones in Type IV and V trajectories.

Figure 14.  Simulated cloud water content variation of 2.0–2.5 cm hailstone trajectories (different colors represent different cloud water content distribution during hailstone growth): (a) Type III, (b) Type IV, and (c) Type V; others as shown in Fig. 6.

Type V hailstone trajectories with ground hailstones of 2.5 cm or larger showed similarities with Type IV trajectories, which have more “up-down” cyclical hailstone growth; however, updraft velocities are relatively larger in Type V trajectories (Fig. 15). The fluctuations in updrafts and supercooled cloud water content on the hailstone growth trajectory as time progresses reflect the cyclical growth of hailstone as they move through repeated up-down cycles.

Figure 15.  Simulated cloud water content variation of hailstone trajectories with sizes larger than 2.5 cm in Type V trajectories (different colors represent different cloud water content distribution during hailstone growth); others as shown in Fig. 6.

For each hailstone size range, the growth of hailstone in cloud areas with strong updrafts and rich supercooled cloud water content corresponds to the period of rapid growth. The longer the time spent in the cloud area with strong updrafts and the richer the supercooled cloud water content, the larger the hailstone growth. Long-term growth in cloud areas with strong updrafts and rich water content in supercooled cloud water is the main reason for hail particle formation above 2.5 cm. In addition, the distribution characteristics of supercooled water along the hailstone growth trajectory show that there may be undulations of supercooled cloud water in both the horizontal and vertical motion trajectories of hailstone. This result implies that the layered structure of hailstone might be partially due to hailstone growth in areas with different cloud water content in the vertical direction. The uneven distribution of cloud water content on its horizontal growth trajectory may also be one of the reasons for the formation of the layered structure for hail particles.

It can be seen from the classification results of hailstone trajectories that different types of trajectories are concentrated in different areas in space. Among the five types of hailstone trajectories, there is no apparent difference in the growth environment of small hailstone (Diameter smaller than 1 cm), which is also the limitation of the hail trajectory classification method employed in this paper. The classification of hail trajectory is not only based on the height and time evolution of the hailstone growth trajectories. However, it is also affected by the horizontal variation of hailstone trajectory to a certain extent.

Although the representative trajectories of hailstones with corresponding size range on the ground are shown in Figs. 6–10, these trajectories are not artificially selected. They are the trajectories corresponding to the largest ground hailstone size of a specific size range corresponding to each type of trajectory. It can be seen in Figs. 7–10 that the hailstone trajectories resolve into certain recognizable patterns for the more complex growth trajectories with larger hailstone sizes on the ground.

Figure 7.  Simulated 1–1.5 cm hailstone trajectories (different colors represent different sizes during hailstone growth): (a) Type III, (b) Type IV, and (c) Type V; others as shown in Fig. 6.

Figure 8.  Simulated 1.5–2.0 cm hailstone trajectories (different colors represent different sizes during hailstone growth): (a) Type III, (b) Type IV, and (c) Type V; others as shown in Fig. 6.

Generally speaking, the hailstone trajectory classification method used in this paper has a certain pattern recognition ability for the trajectories of large-size hailstones, indicating that the classification results are reliable.

Numerical simulation of hailstone growth trajectories is an effective means to study the physical process of hailstone formation and growth, may compensate for the lack of observation data in hail clouds to a certain extent, and improve our scientific understanding of the physical mechanism of hail formation. It is shown that hailstone formation and growth trajectories have different patterns in a hail event. The comparison and cluster analysis of hailstone trajectories shows that the horizontal flow field, updraft, and distribution of supercooled liquid water in hail clouds apparently affect the formation and growth of hailstones. Hence, different catalytic schemes should be adopted in artificial hail prevention operations for different hailstone growth trajectories of hail clouds. Based on this study, the following conclusions can be drawn:

(1) The growth trajectory of hailstone is diversified, and significant differences may exist, even for trajectories with similar ground hailstone sizes. The high similarity of trajectories of the same type based on the cluster method indicates that the classification is appropriate for hailstone growth trajectories, especially those that produce large hailstones. The hailstone growth trajectories show that hailstone grows in clouds mainly in an “up-down” growth mode near areas of strong updrafts to complete growth and descent.

(2) Based on the characteristics of trajectories, not all hailstones are formed through repeated “up-down” growth cycles. Some hailstones complete their primary growth on a horizontal trajectory before rising to reach their maximum height. The airflow field transports some hailstones in the cloud to the cloud area where both updraft and supercooled water content are high before their main growth stage and grow during the falling process. However, other hailstones reach the strong updraft areas, growing while they are transported upward and reaching their maximum height before falling, with the possibility of continued growth during their descent.

(3) There is no direct relationship between the height of hailstone growth and resultinghailstone size. The factors that directly impact hailstone growth include the strength of the updraft along the hailstone growth trajectory, the amount of supercooled cloud water content, and its residence time in the abundant liquid water content area, which is also determined based on the updraft velocity. A longer stay in the area of supercooled water with a strong updraft and rich liquid water content is the direct cause of the generation of hailstone with large sizes.

Some of the results obtained from this work may help us to acquire more detailed knowledge about the formation mechanisms of hailfall in the semi-arid region of China and improve the accuracy of hail prediction and suppression in the local area. However, the present study excluded the effects of differences in hail embryo size in different parts of the hail cloud and the variation of the flow field in the hail cloud with time. In subsequent studies, the growth patterns of hail embryos with different initial sizes and the influence of the variable flow field will be considered.

Acknowledgements. The authors are grateful to two anonymous reviewers and associate editors for providing valuable comments and feedback on this work. This work was supported by the National Natural Science Foundation of China (Grant Nos. 41975176, 42061134009). We acknowledge the High Performance Computing Center of Nanjing University of Information Science and Technology for their support of this work.