Fractal-based scaling and scale-invariant dispersion of peak concentrations of crop protection chemicals in rivers

A new regulatory approach is needed to characterize peak pesticide concentrations in surface waters over a range of watershed scales. Methods now in common use rely upon idealized edge-of-field scenarios that ignore scaling effects. Although some watershed-scale regulatory models are available, their complexity generally prevents them from being used during the pesticide registration decision process, even though nearly all exposure to both humans and aquatic organisms can occur only at this scale. The theory of fractal geometry offers a simpler method for addressing this regulatory need. Mandelbrot described rivers as space-filling curves (Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: New York, 1983), a class of fractal objects implying two useful properties we exploit in this work. The first is a simple power-law relationship in which log-log plots of maximum daily concentrations as a function of watershed area tend to be linear with a negative slope. We demonstrate that the extrapolation of such plots down to smaller watersheds agrees with edge-of-field concentrations predicted using the Pesticide Root Zone Model, but only when the modeling results are properly adjusted for use intensity within the watershed. We also define a second useful property, scale-invariant dispersion, in which concentrations are well described by a single analytical solution to the convective-dispersion equation, regardless of scale. Both of these findings make it possible to incorporate the effect of watershed scale directly into regulatory assessments.