雨滴尺寸分布的随机理论
Stochastic Theory of the Size Distribution of Raindrops
AI总结：
本文提出一种融合热力学、流体动力学与随机过程的理论，无需经验拟合即可预测雨滴尺寸分布，与全球0.4-40毫米/小时降雨强度数据吻合，并给出约10毫米的最大直径上限。
摘要：
一个多世纪以来，雨滴尺寸分布一直是广泛科学研究的主题，通常通过包括Marshall-Palmer指数方程、伽马分布、威布尔分布、对数正态分布及其他数学函数在内的模型来描述。在本工作中，我们提出了一种将热力学和流体动力学中的确定性原理与随机元素相结合的理论，以预测预期的雨滴直径并描述地面水平的雨滴尺寸分布。重要的是，我们的方法避免了假设特定的雨滴尺寸离散过程或依赖多变量经验数据拟合。我们推导了关键雨滴参数（如中值直径、预期最小和最大直径）以及作为降雨强度函数的雨滴尺寸分布的解析方程。我们的理论预测与大量已发表的实验数据吻合良好，这些数据覆盖了全球不同地点、降雨强度从0.4到40毫米/小时的范围。此外，与超大水滴的观测结果一致，我们的理论表明地面水平雨滴最大直径限制约为10毫米。我们建立了气象参数（如Marshall-Palmer常数和总雨滴浓度）的解析表达式，并针对经验数据进行了验证。这里提出的理论方法可广泛应用于气候建模、喷雾和气溶胶动力学、气泡和颗粒、地外降雨以及古气候学等领域。
英文摘要：
For over a century, raindrop size distributions have been a subject of extensive scientific study, typically described by models including the Marshall-Palmer exponential equation, gamma, Weibull, lognormal, and other mathematical functions. In this work, we present a theory that integrates deterministic principles from thermodynamics and fluid dynamics with stochastic elements to predict expected raindrop diameters and describe ground-level drop-size distributions. Importantly, our approach avoids assuming specific drop-size dispersion processes or relying on multi-variable empirical data fitting. We derive analytical equations for key raindrop parameters (e.g., median diameter, expected minimum and maximum diameters) and drop-size distributions as functions of rainfall intensity. Our theoretical predictions align well with extensive published experimental data, covering rainfall intensities from 0.4 to 40 mm/h across diverse global locations. Additionally, consistent with observations of super-large raindrops, our theory suggests a maximum ground-level raindrop diameter limit of around 10 mm. We establish analytical expressions for meteorological parameters like the Marshall-Palmer constant and total raindrop concentration, validated against empirical data. The theoretical approach presented here can find broad applications in climate modeling, sprays and aerosol dynamics, bubbles and particles, extraterrestrial rainfall, and paleoclimatology.
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