Quintic density-profile similarity in an unstable tube ($\bar{\rho }-{\rho }_0\propto z^5(1-t/t_{\ast }{)}^{-2}$)

Ignoring end boundaries, the buoyancy-driven turbulent density diffusion model admits $\bar{\rho }={\rho }_0[1+A(t)C{z}^5]$ for $C>0$. Since $({\partial }_z\bar{\rho }{)}^{3/2}\propto A^{3/2}{z}^6$, the amplitude obeys $A^{\prime }=65^{3/2}K{W}^2\sqrt{gC}{A}^{3/2}$. With $A(0)=1$, $A=(1-t/t_{\ast }{)}^{-2}$ and $t_{\ast }=[35^{3/2}K{W}^2\sqrt{gC}{]}^{-1}$. Its formal growth is not a physical finite-time singularity of a closed mixing tube: ignored end fluxes, loss of the Boussinesq approximation and the finite energy budget invalidate the continuation.