Publications and Preprints

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.