For plume radius $r$, top-hat speed $U$, and plume reduced gravity $g'_B$, define the <volumetric flow rate>, momentum flux, and <buoyancy flux>
$$
Q_B=\pi r^2U,
\qquad
M_B=\pi r^2U^2,
\qquad
B=\pi r^2Ug'_B.
$$
The integral balances for a steady <axisymmetric pure plume> in a uniform lower layer are
$$
\frac{dQ_B}{dz}=2\pi\alpha rU,
\qquad
\frac{dM_B}{dz}=\pi r^2g'_B=\frac BU,
\qquad
\frac{dB}{dz}=0.
$$
The source is at the plume's virtual origin $z=0$. Solving these equations gives
$$
r(z)=\frac{6\alpha}{5}z,
$$
$$
U(z)=
\left(\frac{25B}{48\pi\alpha^2}\right)^{1/3}z^{-1/3}.
$$
It is useful to define
$$
C_P=\frac{6\alpha}{5}
\left(\frac{9\alpha}{10}\right)^{1/3}\pi^{2/3}.
$$
Then the remaining similarity laws take the compact form
$$
Q_B(z)=C_PB^{1/3}z^{5/3},
\qquad
g'_B(z)=C_P^{-1}B^{2/3}z^{-5/3}.
$$
The second relation also follows immediately from conservation of <buoyancy flux>, $B=Q_Bg'_B$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 3 / c / Solution

For a top-hat axisymmetric pure plume of radius $r$, speed $U$, and reduced gravity $g'$, the conserved fluxes are $Q=\pi r^2U$, $M=\pi r^2U^2$, and $B=\pi r^2Ug'$. The <Batchelor entrainment hypothesis> gives $Q\propto B^{1/3}z^{5/3}$.

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