 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-76/6/b/solution

Let

g_{i}^{'} = g Δ ρ_{i} / ρ

denote the density deficit of each warm layer relative to outside. The integrated hydrostatic driving head is

H = g_{1}^{'} (h_{2} - h_{1}) + g_{2}^{'} (H - h_{2}) .

The same pair of orifice laws therefore gives

Q_{b} = Q_{t} = Q = A^{*} H .

Both 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 gives

Q = q_{1} + q_{21} = C (B_{1}^{1/3} + B_{2}^{1/3}) h_{1}^{5/3} .

In 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}^{'} Q = B_{1} + B_{2}

. These balances assume the plumes remain spatially separate and wall heat loss is negligible.

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