= Solution
Let $\mathcal B_0>0$ be the source <buoyancy flux> per unit length. The conversion from the heat-source strength depends on its units: if $F$ is a kinematic temperature-volume flux, $\mathcal B_0=g\beta F$; if it is a physical heat power per unit length, $\mathcal B_0=g\beta F/(\rho_0c_p)$, where $c_p$ is <specific heat capacity at constant pressure>. This explicit convention avoids treating heat power as a <buoyancy flux> without conversion.
For uniform ambient $N^2=0$, so $\mathcal B=\mathcal B_0$. The ideal pure line-source limit has negligible initial volume and <momentum> transport. The flux equations admit $w=w_*$ constant, $q=2\alpha w_*z$ and $b=\alpha z$. Substituting into the <momentum> equation gives $2\alpha w_*^2=2bG$, and the conserved <buoyancy flux> then fixes $2\alpha w_*^3=\mathcal B_0$. Thus
$$
\boxed{2b=2\alpha z,\qquad w=w_* =\left(\frac{\mathcal B_0}{2\alpha}\right)^{1/3},\qquad g'=G=\frac{w_*^2}{z}=\frac{\mathcal B_0^{2/3}}{(2\alpha)^{2/3}z}.}
$$
This verifies all three differential equations, not just their dimensional exponents. The plume grows in width while maintaining constant upward <velocity>; its <temperature> anomaly decreases like $z^{-1}$. The zero-height singularity is the ideal point-source limit. A finite source or nonzero <momentum> input introduces a <plume virtual origin> or a forced near-source region; the heat strength alone cannot uniquely determine that finite-source region.
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