= Solution
For a steady upward plume with prescribed $N^2=-G^2<0$, the <kinematic plume fluxes> satisfy
$$
Q_z=2\alpha\sqrt M,\qquad M_z=\frac{FQ}{M},\qquad F_z=G^2Q.
$$
The positive sign in the last equation expresses the gain of <buoyancy flux> as the plume rises through the prescribed unstable ambient. Seek a positive <power law> solution $Q=qz^p$, $M=mz^r$, $F=fz^s$, with $z>0$. Matching powers gives
$$
p-1=\frac r2,\qquad r-1=s+p-r,\qquad s-1=p.
$$
The last relation gives $s=p+1$ and the first $r=2p-2$. Substitution into the middle relation yields $4p-5=2p+1$, hence
$$
p=3,\qquad r=s=4.
$$
The amplitude equations are
$$
3q=2\alpha\sqrt m,\qquad 4m=\frac{fq}{m},\qquad4f=G^2q.
$$
From the first relation $m=9q^2/(4\alpha^2)$, while the last two give $m^2=G^2q^2/16$. Taking positive amplitudes gives $m=Gq/4$, and consequently
$$
\boxed{Q=\frac{\alpha^2G}{9}z^3,\qquad
M=\frac{\alpha^2G^2}{36}z^4,\qquad
F=\frac{\alpha^2G^3}{36}z^4.}
$$
Reconstructing the radius, speed and <reduced gravity> gives the <top-hat plume in constant unstable stratification>:
$$
\boxed{b=\frac Q{\sqrt M}=\frac{2\alpha z}{3},\qquad
w=\frac MQ=\frac{Gz}{4},\qquad
g'=\frac FQ=\frac{G^2z}{4}.}
$$
As a direct check, $M_z=\alpha^2G^2z^3/9=FQ/M$ and $F_z=\alpha^2G^3z^3/9=G^2Q$, while $Q_z=\alpha^2Gz^2/3=2\alpha\sqrt M$.
The corresponding ambient gradient is $\rho_{0z}=\rho_rG^2/g$. Thus $\rho(z)=\rho_0(z)-\rho_rG^2z/(4g)$; for a linear ambient this is $\rho_0(0)+3\rho_rG^2z/(4g)$. The plume is locally lighter than the ambient even though both densities increase upward in this prescribed unstable stratification.
The coordinate $z$ can be measured from a virtual origin. This exact <power law> branch has fluxes tending to zero there and describes <buoyancy> generated by the maintained unstable ambient; it does not impose arbitrary nonzero source fluxes at that origin. The given idealization holds the ambient quiescent despite its static instability. Finally, the <Boussinesq approximation> limits the modeled height range to small relative <density> changes, here $G^2z/g\ll1$. Within these assumptions, the displayed positive branch proves the requested existence.
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