Integral bubble and jet models with pressure forces

Abstract

Modifications of integral bubble and jet models including the pressure force are proposed. Exact solutions are found for the modified model of a stationary convective jet from a point source of buoyancy and momentum. The exact solutions are compared against analytical solutions of the integral models for a stationary jet that are based on the approximation of the vertical boundary layer. It is found that the modified integral models of convective jets retain the power-law dependences on the altitude for the vertical velocity and buoyancy obtained in classical models. For a buoyant jet in a neutrally stratified atmosphere, the inclusion of the pressure force increases the amplitude of buoyancy and decreases the amplitude of vertical velocity. The total amplitude change is about 10%. It is shown that in this model there is a dynamic invariant expressing the law of a uniform distribution of the potential and kinetic energy along the jet axis. For a spontaneous jet rising in an unstably stratified atmosphere, the inclusion of the pressure force retains the amplitude of buoyancy and increases the amplitude of vertical velocity by about 15%. It is shown that in the model of a spontaneous jet there is a dynamic invariant expressing the law of a uniform distribution of the available potential and kinetic energy along the jet axis. The results are of interest for the problems of anthropogenic pollution diffusion in the air and water environments and the formulation of models for statistical and stochastic ensembles of thermals in a mass-flux parameterization of turbulent moments.