= Friction-dominated grounding-line evolution law
{title2=$[\alpha\rho_w/\rho-H_x]\dot x_G$}
For a bed depth $b=\alpha x$, a thin <subglacial till> layer of thickness $l$ is idealized by <Newtonian till lubrication>, with <dynamic viscosity> $\lambda\mu$ and basal drag $\lambda\mu u/l$. Neglecting longitudinal stress divergence in the bulk yields $u=-(l/\lambda)(g/\nu)Hh_x$. Retaining the <unbuttressed Newtonian grounding-line stress> condition as boundary data and differentiating <ice-sheet flotation> along the moving <grounding line> gives
$$
\left(\alpha\frac{\rho_w}{\rho}-H_x\right)\dot x_G=\frac l\lambda\frac g\nu Hh_x(h_x+\alpha)-\frac{g'H^2}{8\nu}.
$$
Indeed, <mass conservation> gives $H_t=-uH_x-Hu_x$, $H_x=h_x+\alpha$, and <ice-sheet flotation> gives $H_t=(\alpha\rho_w/\rho-H_x)\dot x_G$. This is a formal local friction closure, not a uniformly valid removal of membrane stress near every grounding line. If the coefficient of $\dot x_G$ vanishes, it is an implicit compatibility condition and cannot be divided to obtain a finite speed.
Back to article page
