= Amplitudes and physical branches of a power-law stratified plume
{title2=$b=4\alpha z/(p+6),\quad w^2=-2Cz^{p+2}/[(p+4)(3p+8)]$}
For the <Boussinesq top-hat plume in a stratified ambient> with $N^2=Cz^p$, the exact pure-power solution has the displayed radius and velocity, and $g'=-2Cz^{p+1}/(3p+8)$. With positive entrainment, radius, upward speed and reduced gravity, stable $C>0$ requires $-4<p<-8/3$, whereas unstable $C<0$ requires $p>-8/3$. The stable branch has divergent buoyancy flux at zero height, so it is not a finite-flux point-source solution. The unstable branch has zero limiting source buoyancy flux and extracts buoyancy from its maintained unstable ambient. The cases $p=-6,-4,-8/3$ are singular at nonzero $C$; $C=0$ instead admits the ordinary <Boussinesq point-source plume>.
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