Boussinesq top-hat plume in a stratified ambient ($Q_z=2\alpha \sqrt{M},M_z=FQ/M,F_z=-N^{2}Q$)

With common factors of $\pi$ removed, steady kinematic fluxes satisfy $Q_z=2\alpha \sqrt{M}$, $M_z=FQ/M$ and $F_z=-N^{2}Q$. The Batchelor entrainment hypothesis supplies volume, the integrated buoyancy force supplies momentum, and the environmental mass density gradient reduces buoyancy flux. This uses the Boussinesq approximation.

For $N^2=K{z}^{\beta }$ and positive pure-power fluxes, the similarity solution exponents are $q=3+\beta /2$, $m=4+\beta$ and $f=4+3\beta /2$. Increasing volume and momentum and decreasing positive buoyancy flux require $-4<\beta <-8/3$. At fixed nonzero $K$ the endpoint coefficient limits are singular.

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.

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.

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.

The linearized relative-flux matrix is $J=[(-q,q/2,0);(m,-2m,m);(f,0,-f)]$. Its characteristic polynomial has constant term $qmf<0$ for physical positive-buoyancy similarity, so at least one real eigenvalue is positive. The growing mode behaves as $z^{\lambda }$ and makes the similarity trajectory unstable with increasing height, rather than with physical time.