Turbulent gravitational convection from a point source in a non-uniformly stratified environment

We examine the turbulent gravitational convection which develops above a point source of buoyant fluid in a stably stratified environment in which the buoyancy frequency varies with height according to N2=N2s (z/zs)β. This generalizes the classical model of turbulent buoyant plumes rising through uniform and uniformly stratified environments originally developed by Morton et al. (1956). By analogy, the height of rise of a plume with initial buoyancy flux Fs has the form Hp= Apεp−1/2Fs1/4Ns−3/4hp (λ, β) where εp is the entrainment constant for plume motion, Ap is an O(1) constant, and the non-dimensional plume height, hp is a function of &λ=Apεp−1/2Fs1/4Ns−3/4/zs and β. In the case β>0, the stratification becomes progressively stronger with height, and so plumes are always confined within a finite distance above the origin. Furthermore, the non-dimensional height of rise h decreases with λ. In contrast, in the case β<0, the stratification becomes progressively weaker with height, and so the non-dimensional plume height increases monotonically with λ. For slowly decaying stratification, β>−8/3, the motion is confined within a finite distance above the source. However, for each value of β with β<−8/3, there is a critical value λc(β) such that for λ<λc a plume is confined to a region near the source while for λ[ges ]λc the motion is unbounded. In the unbounded case, the motion asymptotes to the solution for a buoyant plume rising through a uniform environment, with asymptotic buoyancy flux F∞(λ)