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Basal sliding allows ice to pass over bed obstacles by enhanced deformation and regelation: pressure-induced melting on the upstream side and refreezing downstream transfer ice around small obstacles. Cavitation opens water-filled gaps on the downstream side. Their size and the remaining ice-bed contact depend on effective pressure in subglacial drainage $N=p_{\mathrm{ice}}-p_{\mathrm{water}}$ as well as basal shear stress $\tau$. At fixed $\tau$, reducing $N$ generally enlarges cavities and reduces resistance, giving a sliding law $u(\tau ,N)$ with $u_N<0$ in this regime.

A linked cavity drainage system connects these cavities through passages along the bed. Under the prescribed decreasing relation $N(Q)$, the chain rule gives $\boxed{u_Q=u_N{N}^{\prime }(Q)>0}$. This conclusion uses that hydraulic relation; it need not apply to a different drainage regime.

With $t=\epsilon \tau$, the ice continuity equation becomes $H_{\tau }+\epsilon (uH{)}_x=\epsilon s^{\prime }(x)$. For bounded spatial gradients over $\tau =O(1)$, $H$ is independent of the fast seasonal time to leading order. Meanwhile $S^{\prime }(Q)=\epsilon \kappa$ gives $S_t=\kappa Q_{\tau }$, soLet $J(\tau )={\int }_0^{\tau }M(r)dr$. Direct differentiation proves thatsatisfies the equation and $Q(0,\tau )=0$. This is the boundary-generated solution, or a periodic-forcing solution with the forcing continued into its past. An independently prescribed initial profile would also need matching in the region whose characteristic curves have not yet reached the upstream boundary.

If $M>0$, $J$ is invertible and rearranging gives $x=[\tau -J^{-1}(J(\tau )-\kappa Q)]/\kappa$. For $M=1-\delta \cos \tau$ with $|\delta |<1$,Substitution, followed by the sine-difference identity, givesFor fixed $Q$ and nonzero first-order amplitude, the extrema occur at $\tau =\kappa Q/2+n\pi +O(\delta )$. Since $x=Q+O(\delta )$, the seasonal phase lag in linked cavity drainage gives the stated phase relation $\boxed{\tau \approx \kappa x/2(\mathrm{mod}\pi )}$. Maxima and minima alternate; when $\sin (\kappa Q/2)=0$, the first-order approximation is flat and does not locate individual extrema.

Sliding and cavities. Regelation is melting under increased pressure and refreezing when the pressure is released. For ice near its melting point, the Clapeyron equation gives a lower melting temperature at greater pressure. Against an obstacle, ice melts on the high-pressure upstream face, the water passes around it, and it refreezes on the lower-pressure downstream face. Transport of the released latent heat back towards the melting face sustains the process. Together with viscous deformation around obstacles, this permits basal sliding; regelation is especially effective for small obstacles, for which heat conduction distances are short.

Subglacial cavitation occurs where the contact pressure predicted behind an obstacle falls below the water pressure: the ice separates from the bed and a water-filled cavity opens. The effective pressure in subglacial drainage isthe ice overburden pressure minus water pressure. Smaller $N$ facilitates larger cavities and reduces the resisting ice–bed contact. The exponent $n$ comes from Glen's flow law, with strain rate proportional to the $n$th power of deviatoric stress, commonly $n\simeq 3$. Its homogeneity makes $N^n$ the natural velocity scale, yielding $\tau /N=f(u/N^n)$ after including the fixed bed and ice-rheology scales.

Without cavities, material creep gives a drag law with $f(x)\propto x^{1/n}$. Cavities lower the resistance relative to that extrapolation. On a bounded-slope bed the drag is bounded in proportion to $N$; depending on bed geometry, the curve may level off or pass through a maximum and weaken as contact diminishes. The illustrated weakening curve is a qualitative possibility, not a universal formula for all beds.

Channels versus distributed drainage. In Röthlisberger channel drainage, frictional dissipation of flowing water supplies heat to melt the channel walls, while material creep of ice closes them under effective pressure. An area balance has the schematic formwhere $\Psi$ is hydraulic-potential drop per length, $L_f$ is latent heat per mass, and the positive constant $\mathcal{A}$ incorporates ice material creep and geometry. Water-flow resistance relates the flux to channel size and potential gradient. Balancing melting against closure produces the stated increasing relation $N_R(Q)$.

A channel gaining flux can maintain lower water pressure, since larger $N$ means smaller $p_w$. It draws still more water from neighboring routes, melts wider, and causes competing small conduits to lose flux and close. This positive feedback concentrates flow into an arterial network.

A linked cavity drainage system consists of water-filled lee-side cavities connected by narrow passages. Sliding over bed obstacles opens cavities; ice material creep tends to close them. The given decreasing relation $N_K(Q)$ means that carrying more water requires higher water pressure. A route gaining flux then has a higher pressure and diverts water towards neighboring routes. This negative redistribution feedback supports many connected routes and a distributed system.

Coexistence instability. At a coexistence point let $N_R(Q_R)=N_K(Q_K)$ and hold $Q=Q_R+Q_K$ fixed. A loss $\Delta Q_R=-\epsilon$ from the channel gives $\Delta Q_K=\epsilon$. The changes in the two equilibrium effective pressures areConsequentlyIf the bracket is positive, the cavity system's common effective pressure exceeds what the reduced-flux channel can support against material creep closure. The channel shrinks, loses further flux, and closes down. Thus the channel-cavity drainage instability condition is