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Title: Fickian dispersion is anomalous

Abstract

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 aremore » introduced to examine multi-scaling behavior. Several examples of these ideas are presented and discussed.« less

Authors:
 [1];  [2]
  1. Purdue Univ., West Lafayette, IN (United States)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1236687
Report Number(s):
LA-UR-15-20679
Journal ID: ISSN 0022-1694; PII: S0022169415004497
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Hydrology
Additional Journal Information:
Journal Volume: 531; Journal Issue: P1; Journal ID: ISSN 0022-1694
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; non-Fickian; anomalous; transport; scaling; renormalization

Citation Formats

Cushman, John H., and O’Malley, Dan. Fickian dispersion is anomalous. United States: N. p., 2015. Web. doi:10.1016/j.jhydrol.2015.06.036.
Cushman, John H., & O’Malley, Dan. Fickian dispersion is anomalous. United States. doi:10.1016/j.jhydrol.2015.06.036.
Cushman, John H., and O’Malley, Dan. Mon . "Fickian dispersion is anomalous". United States. doi:10.1016/j.jhydrol.2015.06.036. https://www.osti.gov/servlets/purl/1236687.
@article{osti_1236687,
title = {Fickian dispersion is anomalous},
author = {Cushman, John H. and O’Malley, Dan},
abstractNote = {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.},
doi = {10.1016/j.jhydrol.2015.06.036},
journal = {Journal of Hydrology},
number = P1,
volume = 531,
place = {United States},
year = {2015},
month = {6}
}

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