Catchment-scale advection and dispersion as a mechanism for fractal scaling in stream tracer concentrations

Time series of chemical tracers in rainfall and streamflow can be used to probe the internal workings of catchments. We have recently proposed that catchments act as fractal filters for inert chemical tracers like chloride, converting ‘white noise’ rainfall chemistry inputs into fractal ‘1/f noise’ chemical time series in runoff [Nature 403 (2000) 524]. This implies that catchments have long-tailed travel-time distributions, and thus retain soluble contaminants for unexpectedly long timespans. Here we show that these long-tailed travel-time distributions, and the fractal tracer time series that they imply, can be generated by advection and dispersion of spatially distributed rainfall inputs as they travel toward a channel. Tracer pulses that land close to the stream reach it promptly, with relatively little dispersion. Tracer pulses that land farther upslope must travel farther to reach the stream, and undergo more dispersion. The tracer signal in the stream will be the integral of the contributions from each point along the length of the hillslope, with a peak at short lag times (reflecting tracers landing near the stream) and a long tail (reflecting tracers landing farther from the stream). Here we integrate the advection–dispersion equation for rainfall tracers landing at all points on a simple model hillslope, and show that it yields fractal tracer behavior, as well as a travel-time distribution nearly equivalent to that found empirically [Nature 403 (2000) 524]. However, it does so only when the dispersion length scale approaches the length of the hillslope, implying that subsurface transport is dominated by large conductivity contrasts related to macropores, fracture networks, and similar large-scale heterogeneities in subsurface conductivity. Thus, the 1/f scaling observed at our study sites indicates that these catchments are dominated by flowpaths that exhibit macro-dispersion over the longest possible length scales.