Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 76 / 6 / b / Solution

Let $g_i^{\prime }=g\Delta {\rho }_i/\rho$ denote the density deficit of each warm layer relative to outside. The integrated hydrostatic driving head isThe same pair of orifice laws therefore givesBoth plumes remove fluid from the lower layer. If $q_1=C{B}_1^{1/3}{h}_1^{5/3}$ and $q_{21}=C{B}_2^{1/3}{h}_1^{5/3}$ are their fluxes at its upper interface, lower-layer conservation also givesIn steady state the weak plume supplies the middle layer, while the strong plume entrains that supply above $h_1$ and carries the total volume into the upper layer. Consequently $Q_2(h_2)=Q$ and $g_2^{\prime }Q=B_1+B_2$. These balances assume the plumes remain spatially separate and wall heat loss is negligible.