 Extensional screening length of a slumping column

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.