Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 43 / 3 / b / Solution

In a stably stratified ambient, entrained fluid is progressively lighter with increasing height. Relative to that ambient, the plume's buoyancy flux satisfies $B^{\prime }=-N^{2}Q<0$. It therefore becomes neutrally buoyant before its upward momentum vanishes. It overshoots, becomes negatively buoyant, and eventually turns and spreads laterally near its neutral-buoyancy level. The top-hat ascent equations cease to describe the overturning cap and lateral intrusion.

For constant $N$, the balances give a useful exact relation:for an ideal zero-momentum source. The rising branch begins at $B=B_0$, attains its largest momentum at $B=0$, and reaches its terminal zero-momentum limit at $B=-B_0$. Entrainment and decreasing buoyancy make the associated rise interval finite.

An estimate follows already from the unstratified pure plume flux: $N^2{\int }_0^{H}Qdz\sim B_0$, so $N^2{B}_0^{1/3}{H}^{8/3}\sim B_0$. HenceThe coefficient depends on the entrainment coefficient and the distinction between neutral height, maximum rise and final spreading level. The scaling can also be obtained from the dimensions $[B_0]=L^4{T}^{-3}$ and $[N]=T^{-1}$. The sketch distinguishes the neutral level from the maximum rise.