A test-tube model for rainfall

Department of Mathematics and Statistics, The Open University - Walton Hall, Milton Keynes, MK7 6AA, England, UK

Received: 19 January 2014
Accepted: 25 April 2014

Abstract

If the temperature of a cell containing two partially miscible liquids is changed very slowly, so that the miscibility is decreased, microscopic droplets nucleate, grow and migrate to the interface due to their buoyancy. The system may show an approximately periodic variation of the turbidity of the mixture, as the mean droplet size fluctuates. These precipitation events are analogous to rainfall. This paper considers a theoretical model for these experiments. After nucleation the initial growth is by Ostwald ripening, followed by a finite-time runaway growth of droplet sizes due to larger droplets sweeping up smaller ones. The model predicts that the period and the temperature sweep rate ξ are related by , and is in good agreement with experiments. The coefficient C has a power-law divergence approaching the critical point of the miscibility transition: , and the critical exponent η is determined. It is argued that while the mechanism does not provide a quantitative description of terrestrial rainfall, it may be a faithful model for precipitation on other planets.