HR: 10:35h
AN: H32C-02 INVITED     [Abstracts]
TI: Scaling of Ln(Permeability) in Sediments and Velocity Distributions in Turbulence: The Possibility of an Analogy.
AU: * Molz, F J
EM: fredi@clemson.edu
AF: Clemson University, Environmental Engineering and Geology, 342 Computer Court, Anderson, SC 29625 United States
AU: Kozubowski, T J
EM: tkozubow@unr.edu
AF: University of Nevada, Reno, Dept. of Mathematics and Statistics, Reno, NV 89557 United States
AU: Miller, R S
EM: RM@clemson.edu
AF: Clemson University, Environmental Engineering and Geology, 342 Computer Court, Anderson, SC 29625 United States
AU: Podgorski, K
EM: podgorsk@math.iupui.edu
AF: Indiana-Purdue University, Dept. of Mathematical Sciences, Indianapolis, IN 46202 United States
AB: The theory of non-stationary stochastic processes with stationary increments gives rise to stochastic fractals. When such fractals are used to represent measurements of (assumed stationary) physical properties, such as ln(k) increments in sediments or velocity increments "delta(v)" in turbulent flows, the resulting measurements exhibit scaling, either spatial, temporal or both. (In the present context, such scaling refers to systematic changes in the statistical properties of the increment distributions, such as variance, with the lag size over which the increments are determined.) Depending on the class of probability density functions (PDFs) that describe the increment distributions, the resulting stochastic fractals will display different properties. Until recently, the stationary increment process was represented using mainly Gaussian, Gamma or Levy PDFs. However, measurements in both sediments and fluid turbulence indicate that these PDFs are not commonly observed. Based on recent data and previous studies referenced and discussed in Meerschaert et al. (2004) and Molz et al. (2005), the measured increment PDFs display an approximate double exponential (Laplace) shape at smaller lags, and this shape evolves towards Gaussian at larger lags. A model for this behavior based on the Generalized Laplace PDF family called fractional Laplace motion, in analogy with its Gaussian counterpart - fractional Brownian motion, has been suggested (Meerschaert et al., 2004) and the necessary mathematics elaborated (Kozubowski et al., 2005). The resulting stochastic fractal is not a typical self-affine monofractal, but it does exhibit monofractal-like scaling in certain lag size ranges. To date, it has been shown that the Generalized Laplace family fits ln(k) increment distributions and reproduces the original 1941 theory of Kolmogorov when applied to Eulerian turbulent velocity increments. However, to make a physically self-consistent application to turbulence, one must adopt a Lagrangian viewpoint, and the details of this approach are still being developed. The potential analogy between turbulent delta(v) and sediment delta[ln(k)] is intriguing, and perhaps offers insight into the underlying chaotic processes that constitute turbulence and may result also in the pervasive heterogeneity observed in most natural sediments. Properties of the new Laplace fractal are presented, and potential applications to both sediments and fluid turbulence are discussed.
DE: 1828 Groundwater hydraulics
DE: 1832 Groundwater transport
DE: 1869 Stochastic hydrology
DE: 4440 Fractals and multifractals
DE: 4475 Scaling: spatial and temporal (1872, 3270, 4277)
SC: Hydrology [H]
MN: Fall Meeting 2005