Basal sliding

Basal sliding is motion of glacier ice relative to its bed. A sliding law relates the basal velocity to basal shear stress, effective pressure, roughness, or other bed properties.

For a shallow glacier with basal traction proportional to thickness and sliding law $u_b\propto {\tau }_b^m$, negligible internal-deformation flux gives $u_b\propto h^m$. After scaling, the thickness flux is $h^{m+1}$. Linearization around a steady thickness $h_{\ast }(x)$ gives ${\zeta }_t+[c(x)\zeta {]}_x=0$, $c=(m+1)h_{\ast }^m$. Thickness disturbances therefore propagate faster than the surface material speed $u_s\simeq h_{\ast }^m$ by factor $m+1$.

A steady accumulation profile and upstream discharge $q_{\mathrm{in}}$ give $h_{\ast }^{m+1}=q_{\mathrm{in}}+{\int }_0^{x}a(\xi )d\xi$. Along a disturbance ray, $dx/dt=c$ and $d\zeta /dt=-c^{\prime }\zeta$, so a spatially varying background also changes amplitude. The surface-speed ratio is specific to sliding-dominated flow; internal deformation and significant surface-slope corrections change the flux derivative.

A water-filled cavity opens where the normal contact pressure behind a bed obstacle would otherwise fall below subglacial water pressure. Reduced effective pressure in subglacial drainage enlarges cavities and changes the resisting contact geometry. For a fixed bounded-slope bed, the basal shear stress is bounded by a bed-dependent multiple of effective pressure; a sliding curve may saturate or weaken as cavities grow.

Pressure changes the melting temperature of ice according to the Clapeyron equation. Ice pushed against an obstacle can melt on its high-pressure side and refreeze downstream where pressure is lower. Conduction of the released latent heat back towards the melting region permits motion around the obstacle. Small obstacles favor this heat-transfer mechanism; larger ones can be bypassed mainly by ice deformation.

A linear basal drag law has resisting traction $-\mu \beta u_b$, with $\beta >0$ of dimension inverse length. It idealizes lubrication by subglacial till and is equivalent to a Navier slip boundary condition of slip length $1/\beta$ when the ice is a Newtonian fluid. It is a constitutive idealization, not a universal sliding law.

Subglacial till is a mixture of rock fragments and fine sediment beneath ice. A wet deformable layer can lubricate basal sliding. Its constitutive equation must be specified; Newtonian till lubrication is one idealization, rather than a claim that all subglacial till is a Newtonian fluid.

A thin layer of subglacial till idealized as a Newtonian fluid of dynamic viscosity $\lambda \mu$ and thickness $l$ undergoes Couette flow between stationary bedrock and ice moving at speed $u$. Its resisting basal shear stress has magnitude ${\tau }_b=\lambda \mu u/l$ and acts opposite to the sliding motion. Combining this drag with the hydrostatic driving force gives the bulk approximation used in the friction-dominated grounding-line evolution law.