Seasonal phase lag in linked cavity drainage ($Q=[J(\tau )-J(\tau -\kappa x)]/\kappa ,J^{\prime }=M$)

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.