 Solution

Glacier sliding is motion of the ice base relative to its bed, distinct from internal ice deformation. Near a melting bed, basal water, pressure-melting/refreezing around obstacles and creep past bed roughness can permit motion; a cold frozen bed can suppress it. A simple sliding law is increasing in basal traction, for example $u_b\propto {\tau }_b^m$ on the positive-traction branch, with resistance also affected in richer models by roughness and basal water pressure.

Slow ice motion has negligible inertial acceleration. In a long shallow glacier, leading vertical stress is hydrostatic and longitudinal stress gradients are neglected relative to transverse shear gradients. Resolving gravity downslope and normally yields the two stated balances. This approximates a two-dimensional slab/valley centreline; lateral drag, rapidly varying geometry and strong longitudinal-stress regions need additional terms.

With a stress-free surface,Put $K=\rho g\sin \alpha$ and $D=1-\cot \alpha h_x$, assuming positive driving shear for the stated power law. Integrating from the bed givesThe surface speed is $u_s=u_b+A(KD{)}^n{h}^{n+1}/(n+1)$. Mass conservation, with $a$ measured as ice-equivalent thickness accumulation, is $\boxed{h_t+q_x=a}$.

Choose $h=H\hat{h}$, $x=l\hat{x}$, $u_b=U{\hat{u}}_b$ and $t=(l/U)\hat{t}$, with $U=(KH/R{)}^m$. Scale accumulation with $HU/l$. The dimensionless basal law and parameters areThe resulting flux is exactly $h{u}_b+\delta (1-\mu h_x{)}^n{h}^{n+2}/(n+2)$.

When internal deformation is small and slope corrections are uniformly small, $u_b\simeq h^m$ andFor a steady accumulation profile and prescribed upstream flux $q_{\mathrm{in}}$, $\boxed{h_{\ast }(x)=[q_{\mathrm{in}}+{\int }_0^{x}a(\xi )d\xi {]}^{1/(m+1)}}$. For constant $a$ and zero upstream flux this is $(ax{)}^{1/(m+1)}$ on the region of positive flux.

A small thickness disturbance $\zeta$ satisfies ${\zeta }_t+[c(x)\zeta {]}_x=0$ with $c=(m+1)h_{\ast }^m$. Hence it travels down-glacier along $dx/dt=c>0$, with amplitude changing as $d\zeta /dt=-c^{\prime }\zeta$. Since surface speed is $u_s\simeq u_b=h_{\ast }^m$ in this sliding-dominated limit,The thickness wave therefore outruns individual ice particles. This is the sliding-dominated glacier kinematic wave.