= Solution
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 in subglacial drainage>[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 $KSN^3$. 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 $\rho_wg\sin\alpha$ 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>, $\rho_wg\sin\alpha\,Q$. It assumes that this dissipated <power> is available for melting rather than substantial sensible heating or other <energy> losses. The model thus couples \b[melt opening, pressure-driven <material creep> closure and hydraulic discharge].
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