River networks of southern Africa: Scaling laws governing their geometry and deviations from scaling

Africa is tectonically and hypsometrically different to all other continents and therefore provides a unique natural laboratory for the study of river network geometry. Southern Africa is one of the more unusual geomorphological regions of this continent, hence our interest in the geometry of the rivers draining this region. The 14 river basins analyzed here come from the larger networks of the Orange and Limpopo rivers, as well as from the southernmost section of the continent where rivers meander through an exhumed belt of folded Paleozoic mountains. Basic scaling laws, such as Horton’s laws and Hack’s law, hold for the rivers studied here, but the parameters governing these relationships vary between networks. A strong inverse correlation between Horton’s ratio of stream numbers and mean source stream slope is observed. We believe this is the first time such a relation has been reported. Horton’s ratio of basin areas is proportional to that of stream numbers but always greater than it; this contradicts existing models that equate the two parameters. An expression for the ratio of stream lengths in terms of the ratio of stream numbers is derived theoretically and verified for the available data. The parameter governing Hack’s law, h, varies not only between networks but also with scale inside each network. Existing models provide up to four scaling regimes. Given the spatial resolution of this study, the two regimes at the middle scales are visible here. These are termed the short-range and randomness regimes, with a crossover occurring at a drainage area of approximately 800 km2 . The values for h in the smaller short-range regime have an inverse correlation with the roughness of the topography. Observed values for h imply basins appear more elongated with increasing area; this would be more pronounced in rivers draining a smooth topography. In the randomness regime the values of h are lower, mostly between 0.5 and 0.6, with a weak positive correlation to topography roughness. This range of h corresponds to self-similar basins, but a more detailed analysis of individual stream fractality is necessary before such a conclusion can be made.