 Solution

Above $h_1$, plume 2 entrains middle-layer fluid rather than the original ambient. That fluid is already warm and light, so the plume's buoyancy relative to its surroundings is reduced. Its pre-interface density contrast and zero-source pure plume formula cannot simply be continued unchanged through the interface.

A consistent model treats the middle layer as a uniform new ambient, assumes a sharp interface and weak overall density differences, and carries the incident volume and momentum fluxes continuously across it. The relative buoyancy flux immediately above the interface isContinue an entraining plume model from these nonzero incident fluxes, using buoyancy measured relative to the middle layer. In that uniform layer the relative buoyancy flux is conserved, while entrainment raises the absolute buoyancy carried relative to the outside environment. This is generally a finite-flux or forced plume continuation, not a newly reset point-source similarity solution. A virtual-origin approximation can be used only after matching its fluxes appropriately.

At its matching level $h_2$, the plume density reaches the upper-layer density; it spreads and detrains, with possible finite-momentum overshoot. The model neglects mutual plume entrainment, wall heat transfer, interface thickness and detailed intrusion dynamics, and assumes each ambient layer is well mixed. In the equal-source limit $B_2^{+}=0$, the continuation is a neutrally buoyant momentum-driven flow, emphasizing why the upper interface loses its independent density meaning.