A note on non-Boussinesq plumes in an incompressible stratified environment

Abstract

The recent work of Rooney & Linden (1996) is generalized to describe the motion of non-Boussinesq plumes in both uniform and stratified environments. Using an integral model in which the horizontal entrainment velocity is assumed to take the form u_ε=α(ρ¯/ρ_e) ^1/2_w where α is the entrainment coefficient, ρ¯ is the plume density, w the plume velocity and ρ_e the ambient density, it is shown that the vertical scale over which non-Boussinesq effects are significant is given by z_B=5/3 (B²_o/ (20α⁴g³))^1/5 where B_o is the buoyancy flux at the source. In a uniform environment, the system admits similarity solutions such that the location of the source of a real plume lies a distance z_B[mid ]ρ_o/Δρ[mid ] ^−5/3 beyond the point source of the similarity solution. The above entrainment law implies a fundamental difference between the motion of upward and downward propagating non-Boussinesq plumes, with the radius of upward propagating plumes being greater than that of the equivalent Boussinesq plume, while the radius of downward propagating plumes is smaller. In a stratified but incompressible environment the model predicts that non-Boussinesq effects are confined close to the source and that at each height, the plume velocity and the fluxes of mass, momentum and buoyancy coincide exactly with those of the equivalent Boussinesq plume. Furthermore, at the neutral buoyancy height, the plume radius equals that of the equivalent Boussinesq plume.