= Solution
\b[Sliding and cavities.] <Regelation> is melting under increased <pressure> and refreezing when the <pressure> is released. For <ice> near its melting point, the <Clapeyron equation> gives a lower melting <temperature> at greater <pressure>. Against an obstacle, <ice> melts on the high-<pressure> upstream face, the <water> passes around it, and it refreezes on the lower-<pressure> downstream face. Transport of the released <latent heat> back towards the melting face sustains the process. Together with viscous deformation around obstacles, this permits <basal sliding>; <regelation> is especially effective for small obstacles, for which heat conduction distances are short.
<Subglacial cavitation> occurs where the contact <pressure> predicted behind an obstacle falls below the <fluid pressure>[water pressure]: the <ice> separates from the bed and a <water>-filled cavity opens. The <effective pressure in subglacial drainage> is
$$
\boxed{N=p_i-p_w,}
$$
the <ice> overburden <pressure> minus <fluid pressure>[water pressure]. Smaller $N$ facilitates larger cavities and reduces the resisting <ice>–bed contact. The exponent $n$ comes from <Glen's flow law>, with <strain rate> proportional to the $n$th power of <deviatoric stress>, commonly $n\simeq3$. Its homogeneity makes $N^n$ the natural <velocity> scale, yielding $\tau/N=f(u/N^n)$ after including the fixed bed and <ice>-rheology scales.
Without cavities, <material creep> gives a drag law with $f(x)\propto x^{1/n}$. Cavities lower the resistance relative to that extrapolation. On a bounded-slope bed the drag is bounded in proportion to $N$; depending on bed geometry, the curve may level off or pass through a maximum and weaken as contact diminishes. The illustrated weakening curve is a qualitative possibility, not a universal formula for all beds.
\b[Channels versus distributed drainage.] In <Röthlisberger channel drainage>, frictional dissipation of flowing <water> supplies heat to melt the channel walls, while <material creep> of <ice> closes them under effective <pressure>. An area balance has the schematic form
$$
\dot S_{\mathrm{ch}}=\frac{Q\Psi}{\rho_i L_f}-\mathcal A N^nS_{\mathrm{ch}},
$$
where $\Psi$ is hydraulic-potential drop per length, $L_f$ is <latent heat> per mass, and the positive constant $\mathcal A$ incorporates <ice> <material creep> and geometry. <Water>-flow resistance relates the flux to channel size and potential gradient. Balancing melting against closure produces the stated increasing relation $N_R(Q)$.
A channel gaining flux can maintain lower <fluid pressure>[water pressure], since larger $N$ means smaller $p_w$. It draws still more <water> from neighboring routes, melts wider, and causes competing small conduits to lose flux and close. This <positive feedback> concentrates flow into an arterial network.
A <linked cavity drainage system> consists of <water>-filled lee-side cavities connected by narrow passages. Sliding over bed obstacles opens cavities; <ice> <material creep> tends to close them. The given decreasing relation $N_K(Q)$ means that carrying more <water> requires higher <fluid pressure>[water pressure]. A route gaining flux then has a higher <pressure> and diverts <water> towards neighboring routes. This negative redistribution feedback supports many connected routes and a distributed system.
\b[Coexistence instability.] At a coexistence point let $N_R(Q_R)=N_K(Q_K)$ and hold $Q=Q_R+Q_K$ fixed. A loss $\Delta Q_R=-\varepsilon$ from the channel gives $\Delta Q_K=\varepsilon$. The changes in the two equilibrium effective pressures are
$$
\Delta N_R=-N_R'(Q_R)\varepsilon,\qquad\Delta N_K=N_K'(Q_K)\varepsilon=-|N_K'(Q_K)|\varepsilon.
$$
Consequently
$$
N_R(Q_R-\varepsilon)-N_K(Q_K+\varepsilon)\simeq-\bigl[N_R'(Q_R)-|N_K'(Q_K)|\bigr]\varepsilon.
$$
If the bracket is positive, the cavity system's common effective <pressure> exceeds what the reduced-flux channel can support against <material creep> closure. The channel shrinks, loses further flux, and closes down. Thus the <channel-cavity drainage instability> condition is
$$
\boxed{N_R'(Q_R)>|N_K'(Q_K)|.}
$$
\Image[/past-exam-of-the-mathematics-and-the-environment-course-of-the-university-of-oxford/2011-cavitation-drainage.png]
{title=Qualitative cavity-modified sliding and the different feedbacks of channel and cavity drainage}
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