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For the Boussinesq top-hat plume in a stratified ambient with $N^2=C{z}^p$, the exact pure-power solution has the displayed radius and velocity, and $g^{\prime }=-2C{z}^{p+1}/(3p+8)$. 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.

Let ${\rho }_r$ be a constant reference density close to the ambient density, and let ${\rho }_p$ denote the mean plume density. The Boussinesq approximation permits full-cross-section kinematic fluxesThus $Q$ is the specific mass flux, equal to volume flux at this order; $M$ is the specific vertical momentum flux; and $F$ is the buoyancy flux. Physical mass and momentum fluxes are ${\rho }_{r}Q$ and ${\rho }_{r}M$. These definitions include the full circular area. If factors of $\pi$ are removed in a course's flux convention, divide $Q,M,F$ by $\pi$ throughout rather than mixing normalizations. The PDF's source density is ${\rho }_0$, distinct from ambient ${\rho }_a(z)$; the TeX changes the source symbol incorrectly.

For the top-hat plume model, let the plume radius be $r(z)$ and let $w,b$ be uniform inside it. ThenThe Batchelor entrainment hypothesis supplies the necessary closure: the inward edge velocity is $v_e={\alpha }_{e}w$, with constant entrainment coefficient ${\alpha }_e>0$. Entrainment across a cylindrical slice gives $Q^{\prime }=2\pi r{\alpha }_{e}w=2{\alpha }_e\sqrt{\pi }\sqrt{M}$. The entrained ambient has negligible upward momentum. The vertical buoyancy force on the slice is $\pi r^{2}b=F/w$, hence $M^{\prime }=FQ/M$.

For the buoyancy balance, let $R={\int }_A{\rho }_{p}wdA$ be the transported density scalar. Entrainment gives $R^{\prime }={\rho }_a{Q}^{\prime }$. Differentiate $F=g({\rho }_{a}Q-R)/{\rho }_r$ to get $F^{\prime }=g{\rho }_a^{\prime }Q/{\rho }_r$. Therefore the closed flux equations areHere $N$ is the buoyancy frequency when the ambient is stably stratified. These are the Boussinesq top-hat plume in a stratified ambient equations in the explicitly stated full-area convention. Stable stratification consumes positive buoyancy flux as the plume rises. The top-hat assumption alone would not determine $Q^{\prime }$ without an entrainment law.

For uniform ambient density, $F=F_0$ is constant. The source volume $Q_0$ becomes negligible when accumulated entrainment ${\int }_0^{z}c\sqrt{M(s)}ds$ is much greater than $Q_0$. In a jet-like region, $M\simeq M_0$, so the criterion is $z\gg Q_0/(c\sqrt{M_0})$. In a buoyancy-dominated far field with $Q\simeq C_Q{z}^{5/3}$, it is $z\gg (Q_0/C_Q{)}^{3/5}$, with $C_Q$ determined below. These are the two source-volume lengths of a turbulent plume. They are not the jet length, which measures the persistence of imposed source momentum. A nonzero source volume can also appear as a plume virtual origin; it is not valid to discard it at every distance simply because the source is geometrically small.

For $Q_0=M_0=0$, $F_0>0$, seek $Q=C_Q{z}^{5/3}$, $M=C_M{z}^{4/3}$. The two flux equations give $C_Q=(3c/5)\sqrt{C_M}$ and $C_M^{3/2}=9c{F}_0/20$. Thus the pure-plume solution isFor completeness its directly measurable top-hat fields areThis is an axisymmetric pure plume. In this regime the explicit source-volume criterion isFor $Q_0=F_0=0$, $M_0>0$, the buoyancy vanishes and momentum is constant. The pure-jet solution isThis is the jet limit of the top-hat plume model. Both ideal zero-volume sources have singular speeds at $z=0$; they model regions outside a finite source where entrainment has already made the source-volume correction small.

When $M_0$ and $F_0$ are both positive, use the jet solution near the source and the pure-plume solution far above it, with an appropriate virtual-origin correction. To quantify this transition, eliminate $z$:The conserved quantity is the plume flux-balance invariant. With negligible $Q_0$, comparison of its two terms and $Q\simeq c\sqrt{M_0}z$ gives a transition height of order $M_0^{3/4}/(\sqrt{c}\sqrt{F_0})$. The exact connection, including finite $Q_0$, is the finite-source forced-plume quadraturewith $M$ then recovered from the invariant. A positive integrand gives the physical increasing-volume branch. As $Q$ grows, the source constant is negligible relative to $Q^2$, proving attraction to pure plume similarity. One matches the two asymptotic regimes or uses this quadrature; the jet and plume solutions cannot be linearly superposed because their governing flux equations are nonlinear.

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.

Use the steady Boussinesq top-hat plume in a stratified ambient, with negligible ambient vertical momentum and no externally added heat or particles. The Batchelor entrainment hypothesis supplies inward velocity $\alpha w$ across circumference $2\pi b$. Volume conservation gives $\pi Q_z=2\pi b\alpha w$, and $bw=\sqrt{M}$, henceThe momentum flux increases by the integrated buoyancy force. Writing $g^{\prime }=g\Delta \rho /{\rho }_o$, the force per reference density and per $\pi$ is $g^{\prime }{b}^2$. Since $F=g^{\prime }Q$ and $b^2=Q^2/M$, this givesFor the buoyancy flux, entrained material has the local environmental mass density. Let ${\rho }_p$ be the plume's averaged density. Its scalar mass conservation is $({\rho }_{p}Q{)}_z={\rho }_a{Q}_z$. Since $F=(g/{\rho }_o)({\rho }_a-{\rho }_p)Q$, differentiating cancels both entrainment terms and givesIn stable density stratification, ${\rho }_a^{\prime }<0$ and a rising plume loses buoyancy flux. These are kinematic fluxes with the common factor $\pi {\rho }_o$ removed, so no additional factor of $\pi$ belongs in the three differential equations.

For prescribed $N^2=-G^2<0$, the steady Boussinesq top-hat plume in a stratified ambient has kinematic fluxes $Q={\alpha }^{2}G{z}^3/9$, $M={\alpha }^2{G}^2{z}^4/36$, and $F={\alpha }^2{G}^3{z}^4/36$. Balancing powers in $Q_z=2\alpha \sqrt{M}$, $M_z=FQ/M$, and $F_z=G^{2}Q$ gives exponents $(3,4,4)$ and these positive amplitudes. The ambient's static instability is prescribed without allowing it to drive ambient motion; the Boussinesq approximation additionally requires small density differences over the modeled height range.