 Separable decaying line-plume similarity

The unsteady Boussinesq triangular-profile line-plume balances admit a buoyant similarity solution $Q=\alpha {\rho }_0{z}^2/\tau$, $M=(2/3)\alpha {\rho }_0{z}^3/{\tau }^2$ and $F=(2/3)\alpha {\rho }_0{z}^3/{\tau }^3$, with $\tau =t+t_{\ast }>0$. It reconstructs $b=\alpha z$, $W=z/\tau$ and reduced gravity $z/{\tau }^2$. Origins can be shifted, but the zero-time unshifted solution is singular. Source and initial data are required to select the physical member and region of validity.