= Solution
Two source scales obtained from <dimensional analysis> are
$$
\boxed{L_M=\frac{M_0^{3/4}}{B_0^{1/2}},\qquad L_Q=\frac{Q_0^{3/5}}{B_0^{1/5}}}.
$$
The <jet length> $L_M$ measures how far source <momentum> remains important relative to buoyancy-generated <momentum>. The <source-volume length of a turbulent plume> $L_Q$ measures how far imposed source volume remains important relative to entrained volume. Numerical constants and <entrainment coefficient> factors depend on the chosen normalization. From the equations here the useful exact volume-growth scale is $\ell_Q=[Q_0^3/(20\alpha^4B_0)]^{1/5}$; the pure-similarity height whose flux equals $Q_0$ is $5\ell_Q/3$. A momentum crossover likewise contains a factor of order $\alpha^{-1/2}L_M$. The source radius $Q_0/\sqrt{M_0}$ is another useful geometrical combination, but it does not replace the two buoyancy/source crossover comparisons.
Integrate the differential relation from part (a) with finite source values:
$$
M^{5/2}-M_0^{5/2}=\frac{5B_0}{8\alpha}(Q^2-Q_0^2).
$$
Define
$$
\boxed{c=1-\frac{8\alpha M_0^{5/2}}{5B_0Q_0^2}}.
$$
This <plume flux-balance invariant> rearranges to
$$
\boxed{\frac{8\alpha M^{5/2}}{5B_0Q_0^2}+c=\frac{Q^2}{Q_0^2}}.
$$
Use it to express $\sqrt M=[5B_0Q_0^2/(8\alpha)]^{1/5}[(Q/Q_0)^2-c]^{1/5}$, and substitute into $Q'=2\alpha\sqrt M$. The coefficient satisfies $(2\alpha/Q_0)^5[5B_0Q_0^2/(8\alpha)]=20\alpha^4B_0/Q_0^3$, giving
$$
\boxed{\frac1{Q_0}\frac{dQ}{dz}=\left(\frac{20\alpha^4B_0}{Q_0^3}\right)^{1/5}\left(\frac{Q^2}{Q_0^2}-c\right)^{1/5}}.
$$
With positive source fluxes, $c<1$. For $c<0$ the source has excess <momentum> relative to <pure plume balance>, so it is a momentum-dominated <forced plume>. For $c=0$ it is already in <pure plume balance>, and the exact solution is the pure similarity translated to a finite <plume virtual origin>. For $0<c<1$ the source has deficient <momentum> relative to its buoyancy and volume fluxes, a <lazy plume>. Here “forced” in the broad sense merely means finite source fluxes; this sign classification distinguishes its jet-like and lazy branches. At large height, finite $cQ_0^2/Q^2$ vanishes and every positive branch approaches <pure plume balance>.
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