Theoretical Statistics and Mathematics Unit, ISI Delhi

Hack's law in a drainage network model: a Brownian web approach

by Rahul Roy, Kumarjit Saha and Anish Sarkar

Hack (1957), while studying the
drainage system in the Shenandoah valley and the adjacent mountains of Virginia,
observed a power law relation $l \sim a^{0.6}$
between the length $l$ of a stream from its
source to a divide and the area $a$ of the basin that collects the
precipitation contributing to the stream as tributaries. We study the
tributary structure of Howard's drainage network model of headward growth and branching studied
by Gangopadhyay $et \;al$.
We show that the exponent of Hack's law is $2/3$ for Howard's model.
Our study is based on a scaling of the process whereby the limit of the watershed area of a stream
is area of a Brownian excursion process.
To obtain this we define a dual of the model and show that under diffusive scaling,
both the original network and its dual converge jointly to the standard
Brownian web and its dual.

isid/ms/2015/08 [fulltext]

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