The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times

Abstract

We present a physical model to explain the behavior of long-term, time series measurements of chloride, a natural passive tracer, in rainfall and runoff in catchments [Kirchner et al., Nature 403(524), 2000]. A spectral analysis of the data shows the chloride concentrations in rainfall to have a white noise spectrum, while in streamflow, the spectrum exhibits a fractal $1/f$ scaling. The empirically derived distribution of tracer travel times $h(t)$ follows a power-law, indicating low-level contaminant delivery to streams for a very long time. Our transport model is based on a continuous time random walk (CTRW) with an event time distribution governed by $ψ(t)\sim A_βt^{-1-β}$. The CTRW using this power-law $ψ(t)$ (with $0<β<1$) is interchangeable with the time-fractional advection-dispersion equation (FADE) and has accounted for the universal phenomenon of anomalous transport in a broad range of disordered and complex systems. In the current application, the events can be realized as transit times on portions of the catchment network. The travel time distribution is the first passage time distribution $F(t; l)$ at a distance $l$ from a pulse input (at $t=0$) at the origin. We show that the empirical $h(t)$ is the catchment areal composite of $F(t; l)$ and that the fractal $1/f$ spectral response found in many catchments is an example of the larger class of transport phenomena cited above. The physical basis of $ψ(t)$, which determines $F(t; l)$, is the origin of the extremely long chemical retention times in catchments.