PLAYFAIR'S LAW OF BERYLLIUM-10

Playfair (1802) remarked on the "nice adjustment" of the branches of river systems. Given his interest in the erosive work of rivers, he may also have wondered whether the sediment carried by each branch was "nicely adjusted" to the long-term erosion rate of its catchment. Two hundred years on, we can examine this question with cosmogenic Be-10.

Consider a simple, "well-adjusted" drainage network, in which each contributing catchment yields sediment in proportion to its long-term erosion rate. It can be shown that the mixed sediment downstream of every junction in such a network must obey the relationship:

A/N = A₁/N₁ + A₂/N₂

where A and N represent drainage area and Be-10 concentration (in quartz), respectively. The left-hand side of the equation refers to the mixed sediment and total contributing area, and the right-hand side to the two contributing catchments. In a more complex basin where the Be-10 production rate (P) differs between catchments, the analogous quantity AP/N grows cumulatively downstream. In either case, the rule is obeyed as long as Be-10 concentrations accurately predict the sediment fluxes from the contributing catchments. By analogy with Playfair's Law, if either catchment supplies sediment "at too high or too low a level", the relationship breaks down.

This suggests a simple means of testing the widely used Be-10 method for measuring catchment-scale erosion rates. The test - based on comparing Be-10 concentrations above and below stream junctions - can be applied to different stream orders, and to sediment of different grain sizes. The first could be used to determine the catchment scale at which sporadic sediment fluxes, reflecting infrequent events such as landsliding at the tips of a drainage network, average out to reflect the long-term, basin-wide erosion rate. The second would help characterise differences in transport and mixing behavior between grain sizes, and indicate which are most likely to provide accurate erosion rate estimates at a given catchment scale.

References: Playfair, John (1802) Illustrations of the Huttonian Theory of the Earth. Printed for Cadell and Davies, London and William Creech, Edinburgh.