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

Abstract

[1] We present a physical model to explain the behavior of longterm, 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 y(t) $ A b t À1Àb . The CTRW using this power-law y(t) (with 0 < b < 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 y(t), which determines F(t;l), is the origin of the extremely long chemical retention times in catchments.

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

“…Another field of relevance is the tracer diffusion in heterogeneous assemblies of distributed obstacles [57] mimicking features of the cell cytoplasm [8] and diffusion on chemically and mesoscopically periodically patterned solid-liquid interfaces [58]. On a macroscopic scale, water diffusion in subsurface hydrology applications is to be mentioned [12], as well as tracer motion in porous heterogeneous media [59]. For the latter there likely exists a distance-dependent diffusivity within each pore constructing a network governing the diffusion of water and contaminants in soil specimen [12].…”

“…In such cases the tracer diffusion becomes characterized by a non-uniform, position-dependent diffusivity D(x). Similarly, spatially varying transport characteristics are ubiquitous in contaminant dispersion in subsurface water aquifers [12]. In the field of stochastic dynamics anomalous diffusion in spatially random media, disordered energy landscapes, * Electronic address: a.cherstvy@gmail.com † Electronic address: rmetzler@uni-potsdam.de weakly chaotic systems, and dynamic maps received considerable attention [13][14][15][16][17][18][19][20].…”

Residence time distributions for hydrologic systems: Mechanistic foundations and steady-state analytical solutions

“…Secondly, understanding the mechanisms that result in the observed heavy-tailing and early arrivals times that seem consistent across many observations should continue. Some current work has related this to work on solute transport in heterogeneous systems (Engdahl et al, 2012;Green et al, 2014;Scher et al, 2002) and flow system dynamics at multiple scales (Cardenas, 2008;Kirchner et al, 2001). However, these studies by no means constitute an exhaustive list of mechanisms that can influence RTDs.…”

“…Physical situations where Gamma distributions may be appropriate are when dispersivities are as large as the hillslope length arising from a wide variety of flow paths. Possible mechanisms for that include mass transfer between mobile and immobile zones with long retention times (Russian et al, 2013;Scher et al, 2002), large contrasts of conductivity at all scales resulting in preferential flow (Lindgren et al, 2004), and complex flow patterns caused by local hillslope topography or catchment geometry (Cardenas, 2007;McGuire et al, 2005). The parameter might be used to specifically express a multi-scale aspect of the transport in the system.…”

Fractal patterns in riverbed morphology produce fractal scaling of water storage times

“…Other mechanisms that can produce broad travel time distributions include subsurface heterogeneity and nested flow paths in homogeneous systems [Kirchner et al, 2001;Scher et al, 2002;Cardenas, 2007]. Therefore, fractal topography may not be a necessary condition for anomalous hyporheic transport.…”