Subglacial drainage

Subglacial drainage transports water beneath a glacier. The water pressure affects basal sliding through the effective pressure in subglacial drainage. Röthlisberger channel drainage concentrates discharge in melt-enlarged conduits; a linked cavity drainage system distributes it through connected spaces maintained by sliding and creep.

The water pressure at a glacier-bed interface affects the effective normal pressure $N$, the difference between ice overburden pressure $p_i$ and water pressure $p_w$. Higher water pressure at fixed overburden reduces effective compression of the bed contact and can alter cavitation and basal resistance. It is not identical to basal shear stress, and a sliding law depending only on shear suppresses this independent hydraulic influence.

A glacial lake outburst flood is rapid release of water stored in or behind glacial ice or its associated dam. Melt-driven enlargement of a subglacial channel can create positive feedback between discharge, frictional heating and further channel opening. Lake depletion and pressure-dependent material creep closure then alter the event. Channel models explain one mechanism and should not be taken as a universal description of every glacial flood.

At fixed total discharge $Q_R+Q_K$, remove a small amount $\epsilon$ from a channel and send it through connected cavities. The equilibrium channel effective pressure drops by $N_R^{\prime }\epsilon$, while the cavity pressure difference drops by $|N_K^{\prime }|\epsilon$. Under the displayed inequality the cavity system leaves a common effective pressure higher than the reduced-flux channel can sustain. Excess creep closure then decreases channel discharge further, amplifying the perturbation.

Sliding opens water-filled cavities behind bed obstacles and ice creep closes them; narrow connections allow drainage between cavities. In a decreasing effective-pressure/discharge relation, a route carrying extra flux needs higher water pressure. That pressure diverts water to neighboring routes, favoring a distributed network instead of concentration into one channel.

A linear storage law gives $\kappa Q_{\tau }+Q_x=M(\tau )$ with zero upstream water flux. Integration along characteristic curves yields the displayed boundary-generated solution. For $M=1-\delta \cos \tau$, $J=\tau -\delta \sin \tau$ and $J^{-1}(\theta )=\theta +\delta \sin \theta +O({\delta }^2)$. A fixed-flux contour therefore obeysIts extrema have phase $\tau =\kappa Q/2(\mathrm{mod}\pi )$ when the first-order amplitude is nonzero. The phase lag arises from water storage and downstream propagation, even if the ice thickness is approximately frozen during one fast seasonal cycle.

Frictional dissipation of water flow melts a channel's ice walls, while ice creep under effective pressure in subglacial drainage closes them. The resulting equilibrium has increasing effective pressure with discharge in the basic channel theory. A route gaining water can lower its water pressure and attract further discharge, widening at the expense of neighbors. This positive redistribution feedback produces arterial drainage.

A lake supplies a melt-opened subglacial channel while its changing hydrostatic head changes effective pressure and material creep closure. In the displayed scaled model the positive equilibrium is $S={\mu }^{3/4}$, $N={\mu }^{1/12}$. Its Jacobian has trace ${\mu }^{1/4}/3>0$ and determinant $4{\mu }^{7/6}$. The instability becomes oscillatory when $\mu >1/1728$. Growing linear oscillations signal departure from steady drainage; they do not prove unbounded physical lake discharge after nonlinear effects and lake depletion become important.

Effective pressure is ice-overburden pressure minus water pressure beneath the ice. It controls creep closure of channels and cavities; smaller positive $N$ weakens that closure and favors separation of ice from bed obstacles. It is therefore central to both basal sliding and subglacial drainage.