Thermals from finite sources in stable and unstable environments

Abstract

The rise of thermals in the atmosphere has attracted a lot of attention since the early work of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23), who proposed that entrainment into a thermal was proportional to the surface area of the thermal and to the mean vertical velocity of the thermal. This paper presents new analytical solutions for the heights of rise of buoyant thermals in both stably and unstably stratified environments, for both negatively and positively buoyant sources, and where the sources have different size and strength (momentum) characteristics. The limiting cases of these analytical solutions are consistent with previous work. These analytical solutions do not appear elsewhere, and provide a compact set of equations that are easy to apply to a wide range of circumstances. The solutions are dependent upon the entrainment hypothesis, which is of course only an approximation, but the simplicity of the analytical solutions allows easy calculation and additional insights. These include the fact that while heights of rise are strongly dependent on both source strength and size for flows in stable environments, the dilution at the top of rise is independent of the source momentum. Further, in a stable environment, there is a conserved quantity that has dimensions proportional to vertical momentum. For negatively buoyant flows in an unstably stratified environment, thermals having low initial momentum will reach a maximum height, while thermals with high initial momentum will entrain sufficient buoyant environmental fluid that they will eventually become positively buoyant and continue to rise indefinitely.