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

A focus is on how a 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 a 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 a. The height zc also corresponds to the thermal top height when buoyant fluid along the thermal's vertical axis is fully replaced by entrained nonbuoyant environmental fluid rising from below the thermal. The time scale for this process is controlled by the vertical velocity of parcels rising upward through the thermal's core.