Extensional screening length of a slumping column ($\hat{z}=\sqrt{h_0^3/(2\lambda a_0)}$)

For a uniform initial column, annular return flow around a slumping column competes with axial extensional stress over $\hat{z}=\sqrt{h_0^3/(2\lambda a_0)}$. With $G=\Delta \rho g/(3\mu )$, choose velocity $\hat{w}=G{\hat{z}}^2$ and time $\hat{t}=\hat{z}/\hat{w}$. Equivalently $\hat{z}=R/\sqrt{\lambda {\alpha }_0}$, where $\varphi =a^2/R^2$ and $\alpha (\varphi )=2\sqrt{\varphi }/(1-\sqrt{\varphi }{)}^3$ in the leading thin-annulus model. This length describes the vertical range of the basal constraint: a much taller column has a localized basal boundary layer, while a much shorter column responds predominantly through extension.

Using the extensional screening length of a slumping column, an initially uniform column of length $\Lambda$ obeys $W_{ZZ}=1+W$, $W(0)=0$ and $W_Z(\Lambda )=0$. Thus $W=\cosh (\Lambda -Z)/\cosh \Lambda -1$ and its initial area-fraction growth is ${\varphi }_T={\varphi }_0\sinh (\Lambda -Z)/\cosh \Lambda$. For $\Lambda \gg 1$, the velocity is $e^{-Z}-1$ outside exponentially small end corrections; for $\Lambda \ll 1$, it is $Z^2/2-\Lambda Z$. The first limit balances excess weight with annular hydraulic drag, the second with extensional viscous stress.