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The characteristic roots satisfyThey are a nonreal conjugate pair precisely when ${\mu }^{1/2}/9-16{\mu }^{7/6}<0$, or ${\mu }^{2/3}>1/144$. ThereforeIn lake-coupled Röthlisberger drainage instability, above this value the real part is ${\mu }^{1/4}/6>0$, so nonzero linear perturbations are sinusoidal oscillations with an exponentially growing envelope. Their angular frequency is $[4{\mu }^{7/6}-{\mu }^{1/2}/36{]}^{1/2}$.

Physically the feedback represents repeated lake filling and pressure increase, channel enlargement by melting, rapid drainage, pressure reduction and material creep closure. Large drainage events are glacial lake outburst floods, or jökulhlaups. The unbounded amplitude is a conclusion of the linearized equations, not a proof of infinite physical discharge or an indefinitely growing nonlinear solution. Once perturbations are comparable with the equilibrium scales, nonlinear dynamics and constraints such as lake depletion must be retained. At the threshold the two real roots coincide; below it the instability is nonoscillatory.