Fundamentally discrete stochastic model for wind ripple dynamics

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Abstract
We present a discrete, stochastic model for wind formed ripples in sand, which are observed to increase in size through mergers and seemingly approach an asymptotic spatial scale. The model is shown to predict (1) a logarithmic increase in pattern scale with time, (2) a proportionality between the microscopic discrete scale (sand grain diameter) and the macroscopic pattern scale (ripple height), and (3) a lack of scale separation. The implications are that growth and apparent stabilization of scale both can be explained by a single mechanism, and that the evolution of wind ripples and other physical systems of this type cannot be modeled by either deterministic methods or spatial continuum methods.