Motion of a rising thermal

The simple model used by Wang [C. P. Wang, Phys. Fluids 16, 744 (1973)] to describe the motion of a turbulent thermal in a uniform or a stably stratified atmosphere is modified and extended. A simple expression is derived which enables one to determine the entrainment coefficient from data on the rise and the size of a thermal. Analytical solutions are obtained for the maximum rise and the corresponding size of a thermal in a stable atmosphere. The model is applicable to thermals of all sizes, but comparisons with large thermals formed by strong explosions have been stressed. The results show good agreement between theory and observation and indicate that the entrainment coefficient for these thermals is of the same order of magnitude as those for other thermals (α ≃ 0.25) reported in the literature. The numerical result reported by Wang is shown to be a special case of the more general analytical solution derived in the present work.