Röthlisberger channel drainage ($N=N_R(Q),N_R^{\prime }(Q)>0$)

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.