Abstract Ice formation in metastable, super-cooled droplets, which are frequently found in the atmosphere, influences the appearance and characteristics of atmospheric clouds significantly, for example regarding precipitation. Its numerical investigation can provide deep insight into the underlying physical mechanisms and supports the deduction of models that describe these processes on the microscale; those models are required for a description of the macrophysical system. However, even the processes on the microscale span about four orders of magnitude. A semi-analytical sub-scale model based on similarity solutions is thus deduced in order to narrow the gap between the different scales describing the initially spherical ice growth in a super-cooled droplet, which can be reduced to a radially symmetric, but highly non-linear Stefan-type problem. All relevant physical effects, e.g. the reduction of the melting temperature, the expansion of the water phase due to the decrease of density upon solidification and high degrees of supercooling, are taken into account in contrast to classical approaches. The maximum relative error in terms of the freezing time, which is given explicitly as well as the temperature fields, is less than 10% at a degree of supercooling of 35 K and decreases rapidly as the ambient temperature increases.

中文翻译：

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关于过冷液滴中初始冰生长的分析模型

摘要 大气中常见的亚稳态、过冷液滴中的冰形成显着影响大气云的外观和特征，例如降水。其数值研究可以深入了解潜在的物理机制，并支持推论在微观尺度上描述这些过程的模型；描述宏观物理系统需要这些模型。然而，即使是微尺度的过程也跨越了大约四个数量级。因此推导出基于相似解的半解析子尺度模型，以缩小描述过冷液滴中初始球形冰生长的不同尺度之间的差距，该模型可以简化为径向对称，但高度非-线性斯蒂芬型问题。与经典方法相比，所有相关的物理效应，例如熔化温度的降低、由于凝固时密度降低和高度过冷导致的水相膨胀都被考虑在内。明确给出的冻结时间和温度场方面的最大相对误差在 35 K 的过冷度下小于 10%，并且随着环境温度的升高而迅速减小。