Lake-coupled Röthlisberger drainage instability ($S_t=S^{4/3}-S{N}^3,N_t=S^{4/3}-\mu$)

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.