On the analytical modelling of the initial ice growth in a supercooled liquid droplet

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.