# 2024 CSDMS meeting-078

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# Fractal Dimensions of River Networks and Hack's Law

**Jo Martin**,

*CU Boulder Boulder Colorado, United States.*jo.martin@colorado.edu

Fractal geometry is a branch of mathematics pioneered by Benoit Mandelbrot in the 1970's with the goal of finding a mathematically rigorous way to define the geometry found in nature, including what he saw in river networks. Since then, much work on the geometry and structure of river networks has involved fractal method, from passing mention to assumed fractal characteristic's to trying to tie older geomorphic parameters to Mandelbrot's fractal math. However results on the fractal dimensions of river networks have been contradictory and not always well matched to theoretical explanations of fractal geometry. For example, in a 1988 work, Tarboton et al. found that the measured fractal dimension of river networks transitioned from close to 1 at small scales to close to 2 at large scales. They attributed this to switching from a regime where fractal dimension was dominated by Sinuosity to one where it was dominated by the branching characteristics of rivers. Neither of these matches Mandelbrot's prediction of a fractal dimension of 1.2 for river networks, which he derived from a Hack exponent of 0.6, used in the relation between stream length and basin area, which would likely be influenced by river branching. More recent unpublished calculation of the fractal dimension of large North American river basins found a dimension close 1.1, which conveniently would correspond to a Hack exponent of 0.55 which matches more recent empirical work on Hack's law. To better understand the connection between fractal dimension and Hack's Law, in this poster I present work comparing the fractal dimension of modeled river networks to physical ones, and look at what theoretical parameters may explain them variability in measured fractal dimensions of river networks.