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Normalize volume, momentum and buoyancy by a positive plume similarity in power-law stratification and use $\xi =\ln (z/z_{\ast })$. The normalized equations are ${\hat{Q}}_{\xi }=q(\sqrt{\hat{M}}-\hat{Q})$, ${\hat{M}}_{\xi }=m(\hat{F}\hat{Q}/\hat{M}-\hat{M})$ and ${\hat{F}}_{\xi }=f(\hat{Q}-\hat{F})$. They form an autonomous differential equation.

Write $s=\tilde{z}$ and use full positive coefficients, $\tilde{Q}=A{s}^{\theta }$, $\tilde{M}=D{s}^{\mu }$, $\tilde{B}=C{s}^{\psi }$. Comparing exponents in the three plume equations givesSolving these equations yieldsThe coefficient equations are $\theta A=\sqrt{D}$, $\mu D=CA/D$ and $\psi C=-{\lambda }^{\beta }A$. ThusFor a rising plume with $Q,M,B>0$, the first equation requires $\theta >0$, the momentum equation $\mu >0$, and the decreasing-buoyancy equation $\psi <0$. Together these requireThey are the physical plume similarity in power-law stratification branches, not positive-plume solutions for arbitrary powers. The endpoint coefficients are singular; the endpoints cannot be included by simply substituting into these formulas. At the prescribed positive source coordinate $s_s=1/\lambda$, the matched source buoyancy isfinite and positive for every finite $\lambda >0$ in the open range. In contrast, the same branch has divergent buoyancy at the mathematical origin $s=0$. A source at positive height makes the branch physically usable. Merely saying a power is finite at a nonzero coordinate would not itself determine this range; positivity and the plume balances are essential.

At large $s$, $Q$ and $M$ increase without bound, $B$ decreases to zero, radius $Q/\sqrt{\pi M}$ grows proportionally to height, and mean upward velocity $M/Q\propto s^{1+\beta /2}$ decreases but remains positive. The integrated buoyancy loss is finite because $\theta +\beta =\psi -1<-1$. The weakening ambient stratification never makes this matched plume negatively buoyant at a finite height; its upward momentum continues to grow even while its speed decreases through entrainment. Consequently its maximum rise height is infinite in this ideal model. This particular exact branch requires its three source fluxes to match the displayed coefficients; it is not a proof that every source in the same ambient escapes.

Use the steady axisymmetric top-hat plume model, with plume radius $b(z)$, uniform mean upward speed $w(z)>0$ and reduced gravity $g^{\prime }(z)=g[{\rho }_a(z)-{\rho }_p(z)]/{\rho }_0$. 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 areThe 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 areThese equations stop describing an upward light plume once $w$ vanishes, and their light-plume interpretation requires $g^{\prime }>0$.

For nonzero $C$, seek plume similarity in power-law stratification in the form $b=\beta z$, $w=A{z}^n$ and $g^{\prime }=G{z}^m$. 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 $3n=2+n+p$, soChoosing $A>0$ gives the exact power-law family, including its normalization: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 $Q=b^{2}w$, $M=b^2{w}^2$ and $B=b^{2}w{g}^{\prime }$, their values areThe 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 $g^{\prime }$ and real positive $w$ further require $p>-4$ and $-C/(3p+8)>0$, givingFor 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 $p\le -1$, 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 $g^{\prime }$: 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: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.

Define $K=N_s^2{z}_s^{-\beta }>0$. The buoyancy frequency definition gives $F_z=-KQ{z}^{\beta }$. The powers in a plume similarity in power-law stratification satisfySolving givesA rising positive-buoyancy turbulent plume entrains, so $Q_z>0$; its momentum flux increases, so $M_z>0$; its buoyancy flux decreases in the stable ambient, so $F_z<0$. With positive coefficients, this requires $q>0$, $m>0$ and $f<0$. The strongest restrictions areIn this interval $1<q<5/3$, $0<m<4/3$ and $-2<f<0$.

Coefficient matching makes the positive similarity solution explicit:soIt is an ideal scaling for $z>0$, often a far-field description. As $z\to 0$, the prescribed ambient stratification and this model's buoyancy flux are singular; a finite physical source would require a regularized near-source region.

Take logarithmic height $\xi =\ln (z/z_{\ast })$ with an arbitrary fixed length $z_{\ast }$, so $d/d\xi =zd/dz$. This is dimensionally meaningful and has the same derivatives as the abbreviated $\ln z$. For example,The exponent vanishes and the coefficient is $q$, by the plume similarity in power-law stratification. Likewise the momentum coefficient is $m$ and the buoyancy coefficient is $f$. The logarithmic-height plume equations are thereforeThe reference similarity solution becomes the equilibrium $(1,1,1)$ of an autonomous differential equation. Normalizing each flux by its own height-dependent similarity value is essential; simply differentiating the unnormalized fluxes would miss the three subtraction terms.