= Separable decaying top-hat line plume
{title2=$b=\alpha z/2,\quad w=2z/(3\tau),\quad g'=2z/(3\tau^2)$}
In a homogeneous ambient fluid, the displayed nonsteady <power-law ansatz> solves the <unsteady top-hat line-plume balances> for $\tau=t+t_*>0$. 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 $\alpha z$.
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