## 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 »

- Authors:

- Purdue Univ., West Lafayette, IN (United States)
- 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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