= Quintic density-profile similarity in an unstable tube
{title2=$\bar\rho-\rho_0\propto z^5(1-t/t_*)^{-2}$}
Ignoring end boundaries, the <buoyancy-driven turbulent density diffusion> model admits $\bar\rho=\rho_0[1+A(t)Cz^5]$ for $C>0$. Since $(\partial_z\bar\rho)^{3/2}\propto A^{3/2}z^6$, the amplitude obeys $A'=6\,5^{3/2}KW^2\sqrt{gC}\,A^{3/2}$. With $A(0)=1$, $A=(1-t/t_*)^{-2}$ and $t_*=[3\,5^{3/2}KW^2\sqrt{gC}]^{-1}$. 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.
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