= Solution
For a rising light plume use $g'=g(\rho-\rho_p)/\rho_0>0$ in the <Boussinesq approximation>, with ambient density as the inertial reference. The full cross-sectional kinematic <volume flux>, <momentum flux> and <buoyancy flux> of a <top-hat plume> are
$$
\boxed{Q=\pi b^2w,\quad M=\pi b^2w^2,\quad B=\pi b^2wg'=Qg'.}
$$
Here momentum flux per unit reference density is meant, not momentum per unit volume. The <Batchelor entrainment hypothesis> says ambient fluid crosses the plume edge at mean inward speed $\alpha w$, where $\alpha$ is the dimensionless <entrainment coefficient>. Volume added per unit height is $2\pi b\alpha w$. With $a_e=2\alpha\sqrt\pi$, the integrated plume balances in uniform ambient density are
$$
Q'=a_e\sqrt M,\qquad M'=BQ/M,\qquad B'=0.
$$
Seek a <pure plume> with no persistent source volume or momentum scale. Equating powers of $z$ gives $Q\propto z^{5/3}$ and $M\propto z^{4/3}$; constant buoyancy gives $B=B_0$. Equating coefficients yields, relative to the ideal source or a fitted <plume virtual origin>,
$$
\boxed{M=\left(\frac{9a_eB_0}{20}\right)^{2/3}z^{4/3},\quad
Q=\frac{3a_e}{5}\left(\frac{9a_eB_0}{20}\right)^{1/3}z^{5/3},\quad B=B_0.}
$$
Thus $Q\sim B_0^{1/3}z^{5/3}$, $M\sim B_0^{2/3}z^{4/3}$ and $b=6\alpha z/5$. A finite nozzle or nonzero source momentum requires a near-source transition rather than this singular point-source solution at arbitrarily small $z$.
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