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 N-2 = N-s(2)(z/z(s))(beta). This generalizes the classical model of turbulent buoyant plumes rising through uniform and uniformly stratified environments originally developed by Morton et nl. (1956). By analogy, the height of rise of a plume with initial buoyancy flux F-s has the form H-p = A(p) epsilon(p)(-1/2) F-s(1/4) N-s(-3/4) h(p) (lambda, beta) where epsilon(p) is the entrainment constant for plume motion, A(p) is an O(1) constant, and the non-dimensional plume height, h(p) is a function of lambda = A(p) epsilon(p)(-1/2) F-s(1/4) N-s(-3/4)/z(s) and beta. In the case beta > 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 lambda. In contrast, in the case beta < 0, the stratification becomes progressively weaker with height, and so the non-dimensional plume height increases monotonically with lambda. For slowly decaying stratification, beta > -8/3, the motion is confined within a finite distance above the source. However, for each value of beta with beta < -8/3, there is a critical value lambda(c)(beta) such that for lambda < lambda(c) a plume is confined to a region near the source while for lambda greater than or equal to lambda(c) 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-infinity(lambda) < F-s. We show that in the limiting case lambda = lambda(c), dividing bounded and unbounded motion, as z --> infinity the plume asymptotes to a new similarity solution of the second kind which describes the motion of a plume in a non-uniformly stratified environment. These similarity solutions are unstable in the sense that small perturbations to the initial conditions result in very different behaviour far from the source. Analogous results for an instantaneous release of buoyant fluid from a point source, which forms a thermal, are also presented. The model is applied to describe the motion of plumes and thermals in the upper ocean and in naturally ventilated buildings since in both cases the stratification is typically non-uniform.