Statistical Laws and Dune Ridges in Southern Africa

In a recent note, Mabbutt and Sullivan (1968) investigated the nature and impor? tance of Y-junctions in dune ridge systems in Central Australia. From the descriptions in their paper, and also those of Mabbutt (1968) and Madigan (1936), it appears clear that the sand-ridges in question are very similar to those in the southern Kalahari. The dunes around the Molopo River before it reaches the large pan of Abiquas Puts (27? 20/S, 20? io/E) correspond to Mabbutt's 'parallel ridges* and 'ridges with intersecting trends*. They are mainly between 10 and 25 m high, with a spacing of around 300 m. Their surfaces are lightly vegetated, with the vegetation occurring mainly along the crests, though the sand on the upper parts of the dunes is still capable of movement. Rainfall totals are around 17 cm per annum. The dunes were discussed in an early paper by Lewis (1936), whilst the Kalahari dunes in general have been mapped and described by Grove (1969). The sand has a mean size of 2-2 0 (approxi? mately 0*21 mm). Between the dunes there are many small pans or defiation basins, some of which may now be smaller than in the past. Some old strand lines seem to be evident, for example, around Abiquas Puts itself, whilst recently the Molopo channel has become crossed and constricted by small dunes. The Y-junctions in the Molopo area are of great interest as they appear to be random in distribution, and to be associa? ted with a network of dunes that obeys Horton*s Law of Stream Numbers (Horton, 1945). Using air photographs made available by the South African trigonometrical survey in Pretoria, I mapped the dune systems, and was able to identify 'basins* consisting of discrete dune networks which form, topologically, simply connected trees. Hence it is of interest to explore the formal analogy between these dune systems and river systems. After delineating the 'basins* the dune segments were ordered after the method of Strahler (1952), and dune order was plotted against numbers of dunes of each order on semi-logarithmic paper. A straightline inverse relationship was estab? lished. These dunes therefore obey Horton*s Law of Stream Numbers which states 'the number of streams of different orders in a given drainage basin tend closely to approximate an inverse geometric series in which the first term is unity and the ratio is the bifurcation ratio* (Horton, 1945, p. 291). Subsequent to Horton*s work, the law has been found to apply to a large number of diverse phenomena: alpine glaciers, bovine livers, some trees, and some road networks (Woldenburg, 1968). Because of the statistical nature and 'remarkable generality* of Horton*s Law of Stream Numbers, Shreve (1966) suggested that the law arises from the statistics of a large number of randomly merging stream channels. To test the applicability of this suggestion for dunes, I tested the distribution of dune junetions for randomness by plotting the nodes, and then counting the number of nodes in square quadrats (Kershaw, 1964). In no case did the number of nodes in any one quadrat exceed six. The mean density of nodes per quadrat was obtained (m), and then, following Kershaw (p. 99), from the Poisson distribution, e_m, me_m, m2e~m/2! , m3e~m/3! ,. ,. ,. ,