A finite-size (or scale) Lyapunov exponent (FSLE), lambdaa(x), is presented in a statistical mechanical framework and employed to characterize mixing in a variety of laboratory and computational fluid mechanics experiments. The FSLE is the exponential rate at which two particles separate from a distance x to ax. Laboratory particle tracking experiments are used to study penetrative convection and flow in porous media while computational experiments are used to study Lévy processes and deterministic diffusion. The apparent scaling relation lambaa(x) approximately Cax(-beta(a)) of the FSLE holds over intermediate initial separations where the laboratory experiment data is most accurate and asymptotically for the computational experiments. The dependence of the exponent beta on a decreases with increasing a. In the matched index porous system, Ca is also a function of mean fluid velocity. The exponent beta is alpha when the Lévy process is alpha-stable and in this case beta is independent of a.
[1] Super-diffusive mixing in geophysics occurs in atmospheric turbulence, near surface currents in oceans, and macro-pore flow in the subsurface to name three of many areas. Models of super-diffusion have been around for almost a century, yet here we put forth a new perspective on the topic which clarifies and substantially expands on classical approaches. Eighty years after it was introduced, we trivially derive Richardson's scaling law for atmospheric super-diffusion, where the mean square separation of two tracer particles goes as t 3 , by assuming the Lagrangian velocity can be represented as a Brownian process. Next we generalize in the spirit of Mandelbrot's intermittency to other types of flows by employing Lagrangian velocities represented as a-stable Lévy processes. For a specific flow field, we show how to obtain the stability parameter, a, from tracer experiments and the finite-size Lyapunov exponent.
Over some range of scales natural porous media often display a fractal Eulerian velocity or conductivity field. If one assumes the fractal conductivity gives rise to fractal drift velocities, then particle paths may be studied in the framework of stochastic differential equations (SODEs). On the microscale, trajectories are modeled as solutions to a SODE with Markovian, stationary, ergodic drift subject to a fluctuating Lévy force. The Lévy force allows for self-motile particles such as flagellated microbes. On the mesoscale the trajectories are modeled as solutions to a SODE with Lévy (fractal) drift and diffusion arising from the microscale asymptotics. On the macroscale the process is driven by the asymptotics of the mesoscale drift without diffusion. Asymptotic scaling laws and dispersion equations are presented.
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