= Solution
Define $K=N_s^2z_s^{-\beta}>0$. The <buoyancy frequency> definition gives $F_z=-KQz^\beta$. The powers in a <plume similarity in power-law stratification> satisfy
$$
q-1=\frac m2,\qquad m-1=f+q-m,\qquad f-1=q+\beta.
$$
Solving gives
$$
\boxed{q=3+\frac\beta2,\qquad m=4+\beta,\qquad f=4+\frac{3\beta}{2}.}
$$
A 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 are
$$
\boxed{-4<\beta<-\frac83.}
$$
In this interval $1<q<5/3$, $0<m<4/3$ and $-2<f<0$.
Coefficient matching makes the positive <similarity solution> explicit:
$$
qQ_o=2\alpha\sqrt{M_o},\qquad mM_o^2=F_oQ_o,\qquad fF_o=-KQ_o,
$$
so
$$
\boxed{M_o=-\frac{4\alpha^2K}{fmq^2},\qquad Q_o=\frac{2\alpha\sqrt{M_o}}q,\qquad F_o=-\frac{KQ_o}f.}
$$
It is an ideal scaling for $z>0$, often a far-field description. As $z\to0$, the prescribed ambient stratification and this model's <buoyancy flux> are singular; a finite physical source would require a regularized near-source region.
Back to article page
