Probabilistic formulation of conservation of cosmogenic nuclides: Effect of surface elevation fluctuations on approach to steady state

Cosmogenic nuclides have provided a powerful tool for determining rates of landscape erosion. Once a steady-state profile of cosmogenic nuclides is reached, the erosion rate becomes inversely proportional to the concentration of cosmogenic nuclides in the eroded sediment. Erosional landscapes are, however, subject to varying degrees of elevation fluctuations due to gullying, landsliding and bioturbation, for example. Here elevation fluctuation is described in terms of a probability density of deviation from mean elevation. A probabilistic formulation for the conservation of cosmogenic nuclides is derived so as to include the effect of the fluctuations. At steady state, the relation between the mean erosion rate and the concentration of cosmogenic radionucides in the eroded material becomes identical to that which would prevail in the absence of fluctuations. The fluctuations, however, can significantly lengthen the time required to reach steady state. Here the approach to steady state is studied in terms of (a) an impulsive increase in erosion rate starting from an antecedent steady state, and (b) a constant erosion rate but the initial absence of any cosmogenic nuclides. The problem allows an analytical solution. The effect of the fluctuations increases monotonically with increase in the ratio of the standard deviation of elevation fluctuations to the e-folding penetration length of cosmic rays. Of prime importance to the practitioner is whether or not the concentration of cosmogenic nuclides in the eroded material has reached its steady-state value. The central result of this paper is a simple plot allowing estimation of the time required to reach within a specified tolerance of the steady-state value of this concentration. The time increases markedly with increase in the ratio defined above. The current formulation of the model is most applicable to weathering- or transport-limited systems, e.g. badlands or landscapes dominated by deep-seated landslides. Copyright