= Solution
Use pi-normalized <kinematic plume fluxes>: physical volume, momentum and buoyancy fluxes are $\pi Q$, $\pi M$ and $\pi F$. For a <top-hat plume model> in a homogeneous ambient, constant <entrainment coefficient> $\alpha$ gives
$$
\frac{dQ}{dz}=2\alpha\sqrt M,
\qquad \frac{dM}{dz}=\frac{FQ}{M},
\qquad \frac{dF}{dz}=0.
$$
Eliminating height between the first two equations yields $d(M^{5/2})/dQ=5FQ/(4\alpha)$. Hence the <plume flux-balance invariant> is $M^{5/2}-5FQ^2/(8\alpha)$. <Pure plume balance> means that this invariant vanishes, including at a finite-volume source:
$$
\boxed{M_s^{5/2}=\frac{5F_sQ_s^2}{8\alpha}.}
$$
It does not mean that $Q_s$ and $M_s$ must individually be zero. The balanced solution has a <plume virtual origin> below the physical source:
$$
Q(z)=C F_s^{1/3}(z+z_v)^{5/3},
\quad M(z)=\left(\frac{9\alpha F_s}{10}\right)^{2/3}(z+z_v)^{4/3},
\quad C=\frac{6\alpha}{5}\left(\frac{9\alpha}{10}\right)^{1/3},
$$
where $z_v=[Q_s/(CF_s^{1/3})]^{3/5}>0$. For given nonzero source fluxes, the displayed balance is a compatibility condition; they cannot be chosen independently and still describe an exactly balanced source.
Back to article page
