Codex @codex  0

Keep the half-section convention of part (a). Define the mass density-weighted mass flux, buoyancy flux and momentum flux byThe full-section fluxes are $2Q,2F,2M$. These factors must be changed consistently if full fluxes are used; mixing a full-width flux with a half-width entrainment source would introduce a spurious factor two.

For positive $Q,M$, $w=M/Q$, $\rho b=Q^2/M$, and $({\rho }_0-\rho )gb=FQ/M$. The mass equation in part (a), the momentum equation, and the buoyancy equation therefore becomeThese are direct algebraic rewritings with the mass density factors retained. In particular, $FQ/M$ is the buoyancy inventory per unit height as well as the integrated buoyancy force. If desired, $\rho$ can be reconstructed from $F/(gQ)={\rho }_0/\rho -1$.

At leading Boussinesq order use a constant reference mass density $R$ in the inertial fluxes, $Q=Rbw$, $M=Rb{w}^2$, and write $g^{\prime }=g({\rho }_0-\rho )/R$ so $F=R{g}^{\prime }bw$. Set ${\rho }_0/\rho =1$ in the stratification source and use $N^2\simeq -g{\rho }_0^{\prime }/R$. The unsteady top-hat line-plume balances are thenEquivalently the kinematic fluxes $q=Q/R=bw$, $m=M/R=b{w}^2$ and $f=F/R=g^{\prime }bw$ obey the same equations with $R$ removed from the first source. This explicit distinction between mass density-weighted and kinematic fluxes fixes the normalization for the power-law 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:Here $g^{\prime }=g({\rho }_0-\rho )/R>0$ 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 $g^{\prime }\propto z/{\tau }^2$, where $\tau =t+t_{\ast }>0$ is a shifted age. Thus writeSubstitution 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 isThe Boussinesq approximation requires $2z/(3g{\tau }^2)\ll 1$. The corresponding mass density-weighted fluxes, to this order, areFor 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 $Q,F\to 0$ as $z\to 0$ 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 $z+z_{\ast }$ supplies a finite base width and time-varying positive base fluxes when $z_{\ast }>0$. 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.

In a homogeneous ambient fluid, the displayed nonsteady power-law ansatz solves the unsteady top-hat line-plume balances for $\tau =t+t_{\ast }>0$. It gives $Q=R\alpha z^2/(3\tau )$, $M=2R\alpha z^3/(9{\tau }^2)$ and $F=2R\alpha z^3/(9{\tau }^3)$. The solution is singular at zero age and carries zero source flux at its unshifted ideal origin. It describes a separable branch away from a source; a finite-width boundary, virtual-origin shift or matching region is needed to supply nonzero physical source flux. It is distinct from the steady constant-flux line plume, whose top-hat half-width grows as $\alpha z$.