The penetration of interfaces by cylindrical thermals

A theory is presented which connects the proportion of a cylindrical thermal which indefinitely penetrates any specified interfacial discontinuity of density with the distribution of density excess or deficit within similar thermals moving through unstratified surroundings. The relation between the proportion of the thermal which so penetrates and the strength of the inversion has been determined experimentally; the results are expressed by the empirical formula Y = (0·93 S + 1) exp (‐ 0·93 S ), where Y is the fraction of the mass of the substance released which penetrates, and S = Δ p 1 A 1 / M , where Δ p 1 is the change of density across the interface in the direction of motion of the thermal, A 1 is the measured cross‐sectional area of the thermal when its widest part is at the initial level of the interface, and M is the initial mass excess of the thermal per unit length. The theory then predicts that, in cylindrical thermals moving through unstratified surroundings, T /A = exp (‐ Δ p A/M ), where A is the average cross‐sectional area of the thermal, M is its mass excess or deficit per unit length, and T is the total average cross‐sectional area of all regions within which the density excess or deficit, respectively, is greater than or equal to Δ p . Mean cross‐sectional distributions of a passive pollutant have been determined experimentally in three thermals moving through unstratified surroundings. The results were roughly consistent with the expression α/ A = exp (‐ $C/$C ), where A has the previous meaning, $C is the average concentration of the pollutant over the whole thermal, and α is the mean total cross‐sectional area of all regions within which the concentration of pollutant, averaged over the length of the thermal, is everywhere greater than or equal to ΔC . These and other observations seem to show that cylindrical thermals are usually appreciably non‐uniform along their lengths.