Projection Operators and Fractal Dispersion
It is common in statistical physics to upscale phenomena via projection operators. Specifically one tries to isolate fast variables and to selectively project them out of the dynamics at higher scales. Here we show how this method may be used to obtain constitutive theories for. dispersive processes in geophysical problems with evolving spatial/temporal heterogeneity. The resultant constitutive model is space-time nonlocal and it reduces to an α-stable (Levy) law with an appropriate choice of the kernel in the integro-differential flux. We examine the role of long range correlations and study conditions under which such processes may be renormalized. Two examples are presented involving nanofilm dispersion and Darcy-scale dispersion.
KeywordsPorous Medium Projection Operator Stochastic Differential Equation Constitutive Theory Conservative Tracer
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- Cushman, J.H., and Ginn, T.R., 2000. The fractional ADE is a classical mass balance with convolution-Fickian flux. Water Resour. Res (to appear).Google Scholar
- Moroni, M., and Cushman, J.H., 1999. Anomalous dispersion of conservative tracers: Theory and 3-DPTV experiments. In Stochastic Methods in Subsurface Hydrology, Ed. R.S. Govindaraju.Google Scholar
- Moroni, M., and Cushman, J.H., 2000. 3-DPTV Comparison with Statistical Mechanical Theories of Steady Conservative Tracer Transport in Porous Media. Phys. Fluids (to appear).Google Scholar