  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 $\psi(t)\sim A_{\beta}t^{-1-\beta}$. The
CTRW using this power-law $\psi(t)$ (with $0<\beta<1$) 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 $\psi(t)$, which
determines $F(t; l)$, is the origin of the extremely long chemical retention
times in catchments.
