Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / A / 1

For Röthlisberger channel drainage, $S$ is the water-filled cross-sectional area of the subglacial channel, $Q$ its volume discharge, $m$ the mass of ice melted per unit channel length per unit time, and $N=p_i-p_w$ the effective pressure, namely ice overburden pressure minus water pressure. The first balance converts melt mass to area growth through division by ice density ${\rho }_i$, and subtracts material creep closure $KS{N}^3$. The cubic pressure dependence comes from the adopted ice-flow law; $K$ includes its material and geometric factors.

The second relation balances the downslope hydraulic driving gradient ${\rho }_{w}g\sin \alpha$ against the adopted frictional pressure loss. It is an instantaneous discharge-area law, giving $Q=(\sin \alpha /f{)}^{1/2}{S}^{4/3}$ for positive downstream flow. The final balance equates latent heat $mL$ needed to melt the channel walls with the gravitational power dissipated per unit length, ${\rho }_{w}g\sin \alpha Q$. It assumes that this dissipated power is available for melting rather than substantial sensible heating or other energy losses. The model thus couples melt opening, pressure-driven material creep closure and hydraulic discharge.

With constant lake surface area, raising its level by $dh$ changes its volume by $S_{L}dh$. The symbol $M_L$ in the proposed balance is an effective filling rate measured as lake-level rise per unit time, so the volume inflow is $S_L{M}_L$. Volume conservation givesThus the lake rises when its inflow exceeds channel discharge and falls in the opposite case. If an inflow were instead specified directly as a volume rate $I_L$, its contribution would be $I_L/S_L$; keeping these units distinct is necessary for the interpretation of $M_L$.

Hydrostatic pressure at the entrance is $p_w=p_{\mathrm{ref}}+{\rho }_{w}gh$, with a constant reference pressure. Therefore $N=p_i-p_{\mathrm{ref}}-{\rho }_{w}gh$, and constant overburden pressure givesA filling lake raises water pressure and lowers effective pressure, reducing material creep closure. A draining lake has the reverse effect. These signs supply the pressure feedback in the coupled drainage model.

Write dimensional variables as their scales times dimensionless variables, with positive scales $S_0,Q_0,m_0,N_0,t_0$. For compactness defineA consistent explicit choice isThese identities give $t_0{m}_0/({\rho }_i{S}_0)=1$, $K{t}_0{N}_0^3=1$, $f{Q}_0^2/(\sin \alpha S_0^{8/3})=1$ and ${\rho }_{w}g{t}_0{Q}_0/(S_L{N}_0)=1$. Substitution therefore leaves the unit coefficients in all four balances, and the sole supply parameter is $\mu =M_L{S}_L/Q_0$. Expanding its scales yieldsAfter dropping scale hats, eliminate the instantaneous quantities to obtain the useful two-dimensional systemThese scales assume a positive slope, positive friction coefficient and nonzero lake area; they nondimensionalize exactly the hydraulic law printed in the source.

For $\mu >0$, the positive equilibrium isSet $S=S_{\ast }+\epsilon S_1$ and $N=N_{\ast }+\epsilon N_1$ and retain first-order terms. The linear system isDifferentiating its first row and using the second givesIn particular the trace is positive and determinant is $4{\mu }^{7/6}>0$. The steady drainage state is linearly unstable for every positive supply, whether its instability is oscillatory or nonoscillatory.

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.