= Solution
Use the steady axisymmetric <top-hat plume model>, with plume radius $b(z)$, uniform mean upward speed $w(z)>0$ and <reduced gravity> $g'(z)=g[\rho_a(z)-\rho_p(z)]/\rho_0$. The <Boussinesq approximation> uses constant reference density $\rho_0$ in inertia and retains the small density difference only in <buoyancy>. Assume a quiescent hydrostatic ambient, a slender fully <turbulent plume>, negligible molecular transport and an <entrainment> <velocity> $\alpha w$, with constant positive <entrainment coefficient> $\alpha$. Define the ambient <buoyancy frequency> by $N^2=-(g/\rho_0)d\rho_a/dz$; positive $N^2$ is stable stratification.
The integrated volume, momentum and <buoyancy> balances of a <Boussinesq top-hat plume in a stratified ambient> are
$$
\boxed{\frac d{dz}(b^2w)=2\alpha bw,\qquad
\frac d{dz}(b^2w^2)=b^2g',\qquad
\frac d{dz}(b^2wg')=-N^2b^2w.}
$$
The first follows from side <entrainment>, the second from the integrated <buoyancy> force, and the third accounts for the changing ambient density. Common factors of $\pi$ are cancelled. Equivalently, the requested radius, speed and reduced-gravity equations are
$$
\boxed{b'=2\alpha-\frac{bg'}{2w^2},\qquad
w'=\frac{g'}w-\frac{2\alpha w}b,\qquad
(g')'=-N^2-\frac{2\alpha g'}b.}
$$
These equations stop describing an upward light plume once $w$ vanishes, and their light-plume interpretation requires $g'>0$.
For nonzero $C$, seek <plume similarity in power-law stratification> in the form $b=\beta z$, $w=Az^n$ and $g'=Gz^m$. The first balance gives $\beta=2\alpha/(n+2)$. Momentum gives $m=2n-1$ and $G=2(n+1)A^2$. The <buoyancy> balance then requires $3n=2+n+p$, so
$$
n=\frac{p+2}{2},\quad m=p+1,\quad
\beta=\frac{4\alpha}{p+6},\quad G=-\frac{2C}{3p+8},\quad
A^2=-\frac{2C}{(p+4)(3p+8)}.
$$
Choosing $A>0$ gives the exact power-law family, including its normalization:
$$
\boxed{b=\frac{4\alpha}{p+6}z,\qquad
w=\left[\frac{-2C}{(p+4)(3p+8)}\right]^{1/2}z^{(p+2)/2},\qquad
g'=-\frac{2C}{3p+8}z^{p+1}.}
$$
These are the <amplitudes and physical branches of a power-law stratified plume>. No source flux was specified, so an arbitrary finite-source solution is not determined by $C,p$ alone; the displayed family is an exact similarity solution, rather than the general solution for arbitrary initial fluxes.
With the specific flux convention $Q=b^2w$, $M=b^2w^2$ and $B=b^2wg'$, their values are
$$
\boxed{Q=\beta^2A z^{3+p/2},\qquad
M=\beta^2A^2z^{4+p},\qquad
B=\beta^2AGz^{4+3p/2}.}
$$
The physical <volume flux> is $\pi Q$, mass flux is $\rho_0\pi Q$, <momentum flux> is $\rho_0\pi M$, and kinematic <buoyancy flux> is $\pi B$. Thus the “specific mass flux” is the mass flux divided by reference density, with or without the explicitly stated factor $\pi$.
The parameter restrictions follow directly from positivity. Positive radius needs $p>-6$. Positive $g'$ and real positive $w$ further require $p>-4$ and $-C/(3p+8)>0$, giving
$$
\boxed{C>0:\ -4<p<-8/3;\qquad C<0:\ p>-8/3.}
$$
For stable $C>0$, <buoyancy flux> decreases with height, while volume and momentum fluxes increase. However, $B\propto z^{4+3p/2}$ diverges at $z=0$ throughout this interval. Also the ambient density gradient and plume speed are singular there. Thus this stable similarity branch can describe a region away from a virtual source, but cannot represent a finite-buoyancy-flux point source extending all the way to $z=0$. For ordinary regular stable stratifications, including positive $p$, a finite-source rising plume instead loses <buoyancy> and eventually reaches neutral <buoyancy> and a finite rise limit; the pure-power ansatz does not supply such a solution.
For unstable $C<0$ and $p>-8/3$, the <buoyancy flux> grows with height and tends to zero at the origin. The plume draws its increasing <buoyancy> from an unstable ambient rather than a finite positive source <buoyancy flux>. Such an ambient must be maintained or treated locally, since it is itself convectively unstable. If $p\leq-1$, its density profile is singular at the source, while for $p>-1$ its increasing density contrast eventually violates the <Boussinesq approximation> at sufficiently large height. These are formal local similarity solutions with corresponding physical limits. For $C<0$ and $-6<p<-4$, the algebra also gives a real positive speed but negative $g'$: this is a heavy upward flow, not the specified light plume.
At nonzero $C$, the values $p=-6,-4,-8/3$ do not give finite coefficients. Respectively they would require zero entrained volume-flux growth, zero momentum-flux growth despite nonzero <buoyancy>, or constant <buoyancy flux> despite nonzero ambient stratification. They are not omitted regular endpoints.
When $C=0$, the ambient is unstratified regardless of the nominal value of $p$. The nontrivial positive-buoyancy point-source solution is the <Boussinesq point-source plume>:
$$
\boxed{b=\frac{6\alpha}{5}z,\qquad
w=A_0z^{-1/3},\qquad
g'=\frac43A_0^2z^{-5/3},\qquad
A_0^3=\frac{25B_0}{48\alpha^2}.}
$$
Here $B_0>0$ is the constant specific <buoyancy flux> with $\pi$ removed. Its $Q\propto z^{5/3}$ and $M\propto z^{4/3}$ vanish at the point source, while $B=B_0$. Its near-source singularity is the usual ideal point-source limit; a finite vent replaces that limit by finite source data or a virtual origin.
Back to article page
