Shallow-ice flux with linear basal drag ($q=-\frac{\rho g}{\mu }(h^3/3+h^2/\beta )h_x$)

On a horizontal bed, the shallow-ice approximation with a linear basal drag law gives $u=-(\rho g/\mu )h_x(h/\beta +hz-z^2/2)$ and $q=-(\rho g/\mu )(h^3/3+h^2/\beta )h_x$. With no ice accumulation or ice ablation, mass conservation is $h_t+q_x=0$. Internal shear supplies the cubic mobility and basal sliding the quadratic mobility.

A locally steady glacier nose moving at speed $v>0$ has $q=vh$ by mass conservation. At distance $s$ behind its edge, the linear basal drag law then gives $s=(\rho g/\mu v)(h^3/9+h^2/(2\beta ))$. The thin nose has $h\propto s^{1/2}$ from basal sliding; a thicker outer nose has $h\propto s^{1/3}$ from internal deformation. A thick bulk therefore still has a sliding-controlled inner tip. The divergent idealized edge slope limits the immediate validity of the shallow-ice approximation.

The two mobilities in the shallow-ice flux with linear basal drag are equal at thickness $H_{\ast }=3/\beta$. A conserved half-volume $V_g$ gives length $L_{\ast }=V_g/H_{\ast }$ and time $T_{\ast }=3\mu L_{\ast }^2/(\rho g{H}_{\ast }^3)$. Nondimensional thick profiles spread as ${\tau }^{1/5}$; thin sliding profiles spread as ${\tau }^{1/4}$. Matching the centre of the mass-normalized early planar volume-conserving nonlinear-diffusion similarity to unit depth estimates ${\tau }_{\mathrm{tr}}=3/(10I_3^2)\simeq 0.42385$. The numerical estimate depends on the matching convention and the initial-profile relaxation.