  We introduce an idealised model for overland flow generated by rain falling
on a hill-slope. Our prime motivation is to show how the coalescence of runoff
streams promotes the total generation of runoff. We show that, for our model,
as the rate of rainfall increases in relation to the soil infiltration rate,
there is a distinct phase-change. For low rainfall (the subcritical case) only
the bottom of the hill-slope contributes to the total overland runoff, while
for high rainfall (the supercritical case) the whole slope contributes and the
total runoff increases dramatically. We identify the critical point at which
the phase-change occurs, and show how it depends on the degree of coalescence.
When there is no stream coalescence the critical point occurs when the rainfall
rate equals the average infiltration rate, but when we allow coalescence the
critical point occurs when the rainfall rate is less than the average
infiltration rate, and increasing the amount of coalescence increases the total
expected runoff.
