Fickian dispersion is anomalous

Journal Article · Mon Jun 22 00:00:00 EDT 2015 · Journal of Hydrology

DOI: https://doi.org/10.1016/j.jhydrol.2015.06.036 · OSTI ID:1236687

Cushman, John H. ^[1]; O’Malley, Dan ^[2]

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1. Purdue Univ., West Lafayette, IN (United States)
2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

The thesis put forward here is that the occurrence of Fickian dispersion in geophysical settings is a rare event and consequently should be labeled as anomalous. What people classically call anomalous is really the norm. In a Lagrangian setting, a process with mean square displacement which is proportional to time is generally labeled as Fickian dispersion. With a number of counter examples we show why this definition is fraught with difficulty. In a related discussion, we show an infinite second moment does not necessarily imply the process is super dispersive. By employing a rigorous mathematical definition of Fickian dispersion we illustrate why it is so hard to find a Fickian process. We go on to employ a number of renormalization group approaches to classify non-Fickian dispersive behavior. Scaling laws for the probability density function for a dispersive process, the distribution for the first passage times, the mean first passage time, and the finite-size Lyapunov exponent are presented for fixed points of both deterministic and stochastic renormalization group operators. The fixed points of the renormalization group operators are p-self-similar processes. A generalized renormalization group operator is introduced whose fixed points form a set of generalized self-similar processes. Finally, power-law clocks are introduced to examine multi-scaling behavior. Several examples of these ideas are presented and discussed.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE
• Grant/Contract Number: AC52-06NA25396
• OSTI ID: 1236687
• Report Number(s): LA-UR--15-20679; PII: S0022169415004497
• Journal Information: Journal of Hydrology, Journal Name: Journal of Hydrology Journal Issue: P1 Vol. 531; ISSN 0022-1694
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English