= Solution
Use the kinematic fluxes $q=Q/(\pi\rho_0)$, $m=M/(\pi\rho_0)$ and $f_0=F_0/(\pi\rho_0)$. In the steady <Boussinesq approximation>, the <non-Boussinesq top-hat plume equations> reduce to
$$
q'=2\alpha\sqrt m,\qquad m'=\frac{f_0q}{m},\qquad f=f_0.
$$
Eliminating $z$ gives $m^{5/2}=5f_0q^2/(8\alpha)$, where the integration constant vanishes for a <pure plume> from a point source. Write $b=az$ and $w=Kz^{-1/3}$. The two balances give $a=6\alpha/5$ and $K^3=3f_0/(4a^2)=25f_0/(48\alpha^2)$. The <Boussinesq point-source plume> is therefore
$$
\boxed{b(z)=\frac{6\alpha}{5}z,\qquad
w(z)=\left(\frac{25f_0}{48\alpha^2z}\right)^{1/3},\qquad
g'(z)=\frac43K^2z^{-5/3},\qquad
\rho(z)=\rho_0\left(1-\frac{g'(z)}g\right).}
$$
Here $q=a^2Kz^{5/3}$ and $m=a^2K^2z^{4/3}$ tend to zero at the source, while $f_0$ remains positive. The plume <Froude number> is independent of height:
$$
\boxed{C_p^2=\frac{w^2}{g'b}=\frac5{8\alpha}.}
$$
The ideal point source is a far-field similarity idealization. Since $g'\propto z^{-5/3}$, the <Boussinesq approximation> fails near $z=0$ and is valid only when $g'/g\ll1$. In terms of $K$, this requires $z\gg[4K^2/(3g)]^{3/5}$; the density formula must not be extrapolated into its unphysical negative-density region.
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