= Solution
The flux solution gives $w=M/Q=sz^{-1/3}$, while $b^2=Q^2/(\rho M)$. Thus
$$
\boxed{b(z)=dz\left[1+(z_b/z)^{5/3}\right]^{1/2},\qquad
g'(z)=g\frac{\rho_0-\rho}{\rho_0}
=\frac{g}{1+(z/z_b)^{5/3}}.}
$$
At $z=z_b$, $\rho=\rho_0/2$, $T=2T_0$ and $g'=g/2$. This height marks the transition between a strongly non-Boussinesq near-source plume and the dilute far-field plume. For $z\gg z_b$, $b\sim dz$ and $g'\sim g(z_b/z)^{5/3}=(4s^2/3)z^{-5/3}$, recovering the Boussinesq solution. For $z\ll z_b$, $b\sim d z_b^{5/6}z^{1/6}$, $g'\to g$ and $\rho\propto z^{5/3}$.
The ideal point-source limit still has a divergent velocity and temperature; it is not a literal resolved finite burner. In particular, zero source mass flux does not imply zero physical hot-gas volume flux: $wb^2$ tends to $B/g$ in this limiting solution. Source size, compressibility and heat transfer impose additional near-source limits on the model.
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