Separable decaying top-hat line plume ($b=\alpha z/2,w=2z/(3\tau ),g^{\prime }=2z/(3{\tau }^2)$)

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 $\alpha z$.