What controls the entrainment rate of dry buoyant thermals with varying initial aspect ratio?

Abstract

This study uses theory and numerical simulations to analyze the non-dimensional spreading rate α (change in radius with height) of buoyant thermals as they rise and entrain surrounding environmental fluid. A focus is on how α varies with initial thermal aspect ratio Ar, defined as height divided by width of the initial buoyancy perturbation. An analytic equation for thermal ascent rate wt that depends on α is derived from the thermal volume-averaged momentum budget equation. The thermal top height when wt is maximum, defining a critical height zc, is inversely proportional to α. zc also corresponds to the thermal top height when buoyant fluid along the thermal’s vertical axis is fully replaced by entrained non-buoyant environmental fluid rising from below the thermal. The timescale for this process is controlled by the vertical velocity of parcels rising upward through the thermal’s core. This parcel vertical velocity is approximated from Hill’s analytic spherical vortex, yielding an analytic inverse relation between α and Ar. Physically, this α-Ar relation is connected to changes in circulation as Ar is modified. Numerical simulations of thermals with Ar varied from 0.5 to 2 give α values close to the analytic theoretical relation, with a factor of ~3 decrease in α as Ar is increased from 0.5 to 2. The theory also explains why α of initially spherical thermals from past laboratory and modeling studies is about 0.15. Overall, this study provides a theoretical underpinning for understanding the entrainment behavior of thermals, relevant to buoyantly-driven atmospheric flows.