Size distribution of convective thermals in the unstable stratified layer

An ensemble of convective thermals is considered in the layer of penetrative turbulent convection over a homogeneous heated horizontal surface. A model of an unsteady spontaneous jet is proposed to describe the dynamics of an isolated convective element. The model has an exact self-similar solution, which is used to derive a dynamic invariant that relates the velocity, temperature, and radius of the jet. The size distribution of spontaneous convective jets is derived, which agrees with the description of the fine structure of the atmospheric surface layer. The statistical size distribution of thermals is substantiated using the Fokker-Planck kinetic equation