= Solution
In a homogeneous ambient fluid, $N=0$ and $R=\rho_0$ is constant. Use kinematic fluxes or, equivalently, the primitive form of the <unsteady top-hat line-plume balances>:
$$
b_t+(bw)_z=\alpha w,\qquad
(bw)_t+(bw^2)_z=bg',\qquad
(bg')_t+(bg'w)_z=0.
$$
Here $g'=g(\rho_0-\rho)/R>0$ is the positive buoyant <reduced gravity>. Seek a nonsteady separable <power-law ansatz> with both time differentiation and vertical transport retained. Matching powers in the volume equation forces $b\propto z$ with no time dependence. Matching the two momentum terms then gives $w\propto z/\tau$, and the <buoyancy> force has $g'\propto z/\tau^2$, where $\tau=t+t_*>0$ is a shifted age. Thus write
$$
b=Bz,\qquad w=A\frac z\tau,\qquad g'=C\frac z{\tau^2}.
$$
Substitution determines the constants rather than just the exponents. The volume equation gives $2BA=\alpha A$, so for a nonzero <buoyant plume> $B=\alpha/2$. The momentum equation gives $C=-A+3A^2$, while the <buoyancy> equation gives $C(-2+3A)=0$. The buoyant nontrivial branch has $A=2/3$ and $C=2/3$. Consequently the <separable decaying top-hat line plume> is
$$
\boxed{b=\frac{\alpha z}{2},\qquad
w=\frac{2z}{3\tau},\qquad
g'=\frac{2z}{3\tau^2},\qquad
\rho=R\left(1-\frac{2z}{3g\tau^2}\right).}
$$
The <Boussinesq approximation> requires $2z/(3g\tau^2)\ll1$. The corresponding <mass density>-weighted fluxes, to this order, are
$$
\boxed{Q=\frac{R\alpha z^2}{3\tau},\qquad
M=\frac{2R\alpha z^3}{9\tau^2},\qquad
F=\frac{2R\alpha z^3}{9\tau^3}.}
$$
For a direct check, $Q^2/M=R\alpha z/2$ is time independent and $FQ/M=R\alpha z^2/(3\tau^2)$. Hence $(Q^2/M)_t+Q_z=2R\alpha z/(3\tau)=R\alpha M/Q$, $Q_t+M_z=R\alpha z^2/(3\tau^2)=FQ/M$, and $(FQ/M)_t+F_z=0$. This verifies all three balances and the required width coefficient.
This nonsteady branch is not the steady, continuously supplied, constant-flux <line plume>. The latter has constant $w$ and half-width $b=\alpha z$ in the same top-hat convention. Also, the unshifted <power law> has $Q,F\to0$ as $z\to0$ and is singular at $\tau=0$. It is therefore an interior separable solution, not by itself a solution with a nonzero continuously operating point-source flux at the origin. A finite-width source boundary, a virtual origin, or a matching region is needed to connect it to physical source and initial data. For example, replacing $z$ by $z+z_*$ supplies a finite base width and time-varying positive base fluxes when $z_*>0$. No source or initial flux history is specified in the question to select that matching; the requested power-law width follows from the local balances above.
Back to article page
