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Hack [Studies of longitudinal stream profiles in Virginia and Maryland(1957). Report], while studying the drainage system in the Shenandoah valleyand the adjacent mountains of Virginia, observed a power law relation $l\sima^{0.6}$ between the length $l$ of a stream from its source to a divide and thearea $a$ of the basin that collects the precipitation contributing to thestream as tributaries. We study the tributary structure of Howard's drainagenetwork model of headward growth and branching studied by Gangopadhyay, Roy andSarkar [Ann. Appl. Probab. 14 (2004) 1242-1266]. We show that the exponent ofHack's law is $2/3$ for Howard's model. Our study is based on a scaling of theprocess whereby the limit of the watershed area of a stream is area of aBrownian excursion process. To obtain this, we define a dual of the model andshow that under diffusive scaling, both the original network and its dualconverge jointly to the standard Brownian web and its dual.

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