Diffusion ofwater vapor is the dominant growth mechanism for smallwater droplets and ice crystals in clouds. In current cloud models, Maxwell’s theory is used for describing growth of cloud particles. In this approach the local interaction between particles is neglected; the particles can only grow due to changes in environmental conditions, which are assumed as boundary conditions at infinity. This assumption is meaningful if the particles are well separated and far away from each other. However, turbulent motions might change the distances between cloud particles and thus these particles are no longer well separated leading to direct local interactions. In this study we develop a reference model for investigating the direct interaction of cloud particles in mixed-phase clouds as driven by diffusion processes. The model is numerically integrated using finite elements. Additionally, we develop a numerical method based on generalized finite elements for including moving particles and their direct interactions with respect to diffusional growth and evaporation. Several idealized simulations are carried out for investigating direct interactions of liquid droplets and ice particles in a mixed-phase cloud. The results show that local interaction between cloud particles might enhance life times of droplets and ice particles and thus lead to changes in mixed-phase cloud life time and properties.
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This paper is concerned with the generalization of the finite element method via the use of non-polynomial enrichment functions. Several methods employ this general approach, e.g. the extended finite element method and the generalized finite element method. We review these approaches and interpret them in the more general framework of the partition of unity method. Here we focus on fundamental construction principles, approximation properties and stability of the respective numerical method. To this end, we consider meshbased and meshfree generalizations of the finite element method and the use of smooth, discontinuous, singular and numerical enrichment functions.
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