 Solution

The strong plume's outgoing flux equals the total lower-layer supply, $Q_2(h_2)=Q$. Apply the additional closure in this part and the lower-layer balance from part (b):Dividing by $C{B}_2^{1/3}{h}_1^{5/3}$ givessoFor $0<\psi <1$ this puts the second matching level above the first. It tends to one as $\psi \to 0$ and to $2^{2/5}$ as $\psi \to 1$. At the latter endpoint the two warm layers already have identical density, so $h_2$ is only a formal matching label, not a distinct observable interface. The expression follows from the stipulated closure; modeling the actual plume in the changed ambient requires the matched continuation discussed in part (d).