How the three non-dimensionalized variables of interest evolve for the uniform Boussinesq case (dashed, Eqs. (3)–(5)) and the compressible case (solid, Eqs. (9)–(11)). b is the thermal vortex ring diameter, ω = w/(b₀g′₀)^1/2 is its non-dimensionalized mean propagation velocity, is its mean buoyancy perturbation, and τ = t/(b/g′₀)^1/2 is the nondimensional time. The subscript ’ denotes initial values. For this figure we use the following parameters: b₀/H₀ = 1, ω₀ = 0, g′₀/g = 10%, and α = 1/4.