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.
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.
Type 0.5–1 cm 1–1.5 cm 1.5–2 cm 2–2.5 cm >2.5 cm >0.5 cm I 45 − − − − 45 II 91 − − − − 91 III 506 87 16 1 − 609 IV 985 229 80 12 − 1304 V 2656 463 57 21 13 3204 Total 4283 779 153 34 13 5262
Table 1. Number of hailstone trajectories for various hail diameters (larger than 0.5 cm)
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.
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.
|Type||0.5–1 cm||1–1.5 cm||1.5–2 cm||2–2.5 cm||>2.5 cm||>0.5 cm|