= Solution
From the definitions of mass and momentum fluxes,
$$
\boxed{w=M/Q,\qquad
\rho=\frac{\rho_0}{1+\rho_0B/(gQ)},\qquad
b^2=\frac{Q^2}{\rho M}.}
$$
The density relation follows by solving the conserved-buoyancy identity in the preceding part. Hence $wb\sqrt{\rho}=\sqrt M$ and
$$
\boxed{Q'=2\alpha\sqrt{M\rho_0},\qquad
M'=g(\rho_0-\rho)b^2=\frac{B\rho_0Q}{M}.}
$$
The second equality uses $B=g(\rho_0-\rho)Q/(\rho_0\rho)$ and $b^2=Q^2/(\rho M)$.
Set $q=Q/\rho_0$ and $m=M/\rho_0$. The equations become $q'=2\alpha\sqrt m$, $m'=Bq/m$, exactly the same flux system as for the pi-suppressed Boussinesq pure plume. For the heated point source take zero initial mass and momentum flux and the finite conserved heat/buoyancy supply. Define
$$
d=\frac{6\alpha}{5},\qquad s=\left(\frac{25B}{48\alpha^2}\right)^{1/3}.
$$
The resulting mass and momentum fluxes are
$$
\boxed{Q=\rho_0d^2s\,z^{5/3},\qquad M=\rho_0d^2s^2\,z^{4/3}.}
$$
Substituting $Q$ into the density relation gives the <non-Boussinesq pure-plume density transition>
$$
\boxed{\rho(z)=\frac{\rho_0}{1+(z_b/z)^{5/3}},\qquad
z_b=\left[\frac{B}{gd^2s}\right]^{3/5}
=\left[\frac{4s^2}{3g}\right]^{3/5}.}
$$
The two expressions for $z_b$ agree because $B=4d^2s^3/3$. Density increases from the very dilute near-source regime toward the ambient value as entrainment increases the mass flux.
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