= Solution
<Batchelor entrainment> models a turbulent plume as drawing ambient fluid inward at a speed proportional to its characteristic vertical velocity. In a top-hat axisymmetric plume, let
$$
Q=\pi b^2w,\qquad
M=\pi b^2w^2,\qquad
F=Qg'
$$
be volume, momentum, and buoyancy flux. The Boussinesq approximation uses a common density in inertia and volume conservation while retaining the small density difference in $g'=g(\rho_0-\rho)/\rho_0$. It requires $g'\ll g$ and becomes inaccurate for very hot source fluid or near openings with large density changes.
For a point source in an unstratified lower layer, the integral plume equations are
$$
\frac{dQ}{dz}=E\sqrt M,\qquad
\frac{dM}{dz}=\frac{FQ}{M},\qquad
\frac{dF}{dz}=0,
\qquad E=2\alpha\sqrt\pi,
$$
where $\alpha$ is the entrainment coefficient. Their pure-plume solution is
$$
\boxed{
Q_1(z)=C_QF_1^{1/3}z^{5/3},
\qquad
M_1(z)=C_MF_1^{2/3}z^{4/3},
\qquad
g'_{10}(z)=\frac{F_1}{Q_1(z)}
},
$$
with
$$
C_M=\left(\frac{9E}{20}\right)^{2/3},
\qquad
C_Q=\frac43C_M^2.
$$
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