Droplets in clouds form on atmospheric aerosols known as cloud condensational nuclei. More than 90% of cloud droplets nucleate within a few tens of meters above the base of clouds, where the air is supersaturated. The microphysical processes of drop formation within this thin layer above the cloud base determine the structure of the entire cloud, which can be 10–14 kilometers deep. Cloud structure in turn determines the cloud's effects on local and global weather and climate, so the accurate representation of droplet nucleation processes is important for modeling atmospheric processes on scales from single clouds to global scales. However, even with advanced supercomputers, cloud models cannot simulate the small-scale processes in the thin layer above the cloud base, so analytical solutions are needed.