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Runoff on rooted trees

Part of: Special processes

Published online by Cambridge University Press:  11 December 2019

Owen Dafydd Jones
Owen Dafydd Jones*
Affiliation:
Cardiff University
*
*Postal address: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK.
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Abstract

We introduce an idealised model for overland flow generated by rain falling on a hillslope. 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 hillslope 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.

Keywords

Runoffoverland flowcoalescencephase transitionrecursive distributional equationdistributional fixed point equationvoter modelBienaymé–Galton–Watson processparking function

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1065 - 1085
Copyright
© Applied Probability Trust 2019 

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