Linked cavity drainage system ($N=N_K(Q),N_K^{\prime }(Q)<0$)

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.