Episodic Erosion With a Power Law Probability Density, and the Accumulation of Cosmogenic Nuclides

Abstract

In steady state, the concentration n of a cosmogenic nuclide in a surface can be used to study the local erosion rate. When the erosion is due to episodic spalling events, this concentration is not constant in time but has a statistical distribution which reflects the underlying spallation process. In this paper, we study a model in which slabs of rock of depth w are removed in instantaneous events; the time intervals t between these events are random variables. The probability density for t, p(t), is taken to have a Pareto distribution, with a power law decrease at large values of t. This case is interesting to study, since the Pareto distribution can lead to the occurrence of a wide range of time intervals, which can lead to a broad distribution of measured values of n. We derive analytic expressions for the average and the variance of n in steady state and compare our results to previous work, which took p(t) to decrease exponentially with t. We also show results from simulations of the Pareto model, which allow us to gain insight into the time-dependent behavior of the nuclide concentration.