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Fractal river networks of Southern Africa

(corresponding author), CIGCES, Department of Geology, University of Cape Town, Rondebosch 7701, South Africa, jacek{at}cigces.uct.ac.za

Maarten J. de Wit

CIGCES, Department of Geology, University of Cape Town, Rondebosch 7701, South Africa, maarten{at}cigces.uct.ac.za

Fractals and scaling laws abound in nature, and it is said that geometry of river networks and basins is an epitome of this. This study investigates how, in the southern section of the tectonically unique African continent, scaling parameters and deviations from ‘perfect fractal patterns’ relate to parameters like geomorphology through which the river flows, and the underlying geology. A number of river network scaling laws and scaling parameters have been put forward, but it has been suggested that all river networks can be divided into universality classes represented by just 2 of these scaling parameters. One of these is the fractal dimension of individual streams, usually labelled d and having a value of ~1.1. The other parameter, Hack’s exponent h, expresses the dependence of stream length (l) on drainage area (a) via Hack’s Law l = ca^h. There is no universal value for h. Different networks often have different values for h, and inside a given network the parameter is often observed to change with scale. We use the natural laboratory of networks in southern Africa to investigate the variations in Hack’s exponent and find evidence to confirm the existence of scaling regimes. We attempt to explain these variations in scaling using the regime model of Dodds and Rothman (2000). At the smallest scale we find that non-convergent mountain streams exist in different settings, but their spacing is determined by underlying rock type. In this type of drainage a~l, and hence h 1. Once streams begin to converge, the value of h drops, and is inversely correlated to the roughness of the underlying topography. This trend stops once basin sizes reach a threshold value, above which basins may be self-similar. This threshold varies in individual networks. In the smoothest topographies it occurs as low as 400 km², but can occur as high as 1400 km² in other networks. While we have identified a number of guidelines for correlating scaling parameters with basin settings, there exist significant variations around these guidelines which we can only attribute to randomness, or small variations in the initial conditions during the initial formation of the river basins.