伽瑪雨滴粒徑分布中之斜率參數Λ和形狀參數μ之間的關係藉由長期地面雨滴譜儀觀測的資料也已完成分析與研究。其中發現降水終端速度與斜率參數Λ或形狀參數μ之間存在一指數型態的經驗關係，而根據此經驗關係式，進一步可得到μ和Λ的數學解析關係式，在降雨率大於5的情況下，μ和Λ的經驗式與解析式之間有良好的一致性。此外發現ln(No)、μ和Λ的機率密度函數可分別用對數常態分布、伽瑪分布與對數常態分布描述之。而根據其統計特性，伽瑪雨滴粒徑分布將可被模擬出來並獲得相對應的μ和Λ間之關係式，利用其相關物理限制條件小心地濾除資料後，可發現利用此一方式獲得的μ和Λ的關係式與地面雨滴譜得到的μ和Λ的經驗關係式相當地吻合，此一結果可說明伽瑪雨滴粒徑分布參數的統計特性在決定μ和Λ的經驗關係中扮演了相當關鍵的角色。此外從豐富的地面雨滴譜儀數據來看，μ-Λ經驗關係式的特徵確實代表了雨滴下落過程中碰撞破碎與聚合之間的平衡。;With a theoretical derivation and long-term ground-based disdrometer measurements, we investigate vertical wind effect on the relation between slope (Λ) and shape (μ) parameters of the Gamma rain drop size distribution (DSD) estimated from the precipitation echoes of the Chung-Li VHF coherent scatter radar. We derive approximate equations to estimate μ and Λ from radar-measured precipitation terminal velocity and Doppler spectral width. The result shows that there is a tendency for the estimated μ and Λ values to increase with the increase of the upward wind velocity. In addition, the terminal velocity and the spread of the DSD estimated from the precipitation echoes bear a strong relation to the vertical air velocity. With increasing upward vertical air velocity, the terminal velocity tends to be large and the spread of the DSD has a tendency to be small. The dependence of the estimated μ and Λ values on the vertical wind velocity is very likely caused by the updraft that can support and carry away the smaller rain drops in the original drop size distribution in the radar volume, leading to a truncated DSD that is characterized by specific μ and Λ values. Numerical calculations of the intentionally truncated disdrometer-measured DSDs are in good agreement with the radar observations. These results can account for the difference in the patterns of μ-Λ scatter distributions between radar estimation in the air and disdrometer measurement on the ground.
On the basis of disdrometer-measured rain drop size distribution (DSD) for the period from 2000 to 2008, the relation between shape (μ) and slope (Λ) parameters of the Gamma DSD are analyzed and investigated. We find that the empirical relation between the shape (or slope) parameter and the precipitation terminal velocity (VT) can be well described by an exponential function. With the help of the empirical μ-VT or Λ-VT relation, we derive an analytical μ–Λ relation and find that it almost perfectly matches the empirical 2nd order polynomial of the disdrometer-measured DSDs with rainfall rates greater than 5 mm hr-1. It is found that the probability density functions of ln(No), μ and Λ can be described by lognormal, Gamma and lognormal functions, respectively. With these statistical properties, the Gamma DSD is simulated and the empirical 2nd order polynomial of the corresponding μ–Λ relation can thus be obtained. After carefully sifting realistic data from the randomly generated DSD data based on physical constraints, the μ–Λ relation of the simulated Gamma DSDs is in good agreement with that of the disdrometer measurement. These results suggest that the statistical property of the Gamma DSD parameters plays crucial roles in determining the empirical μ–Λ relation. In addition, judging from the rich ground-based DSD data, the characteristics of the empirical relationship of μ–Λ relation represent the balance between the breakup and coalescence of raindrops and the aggregation process during their falling process.
