This dissertation addresses the general problem of transport in chaotic systems. Typical fluid problem of the kind is the advection and diffusion of a passive scalar. The magnetic field evolution in a chaotic conducting media is an example of the chaotic transport of a vector field. In kinetic theory, the collisional relaxation of a distribution function in phase space is also an advection-diffusion problem, but in a higher dimensional space.;In a chaotic flow neighboring points tend to separate exponentially in time, exp({dollar}\omega t{dollar}) with {dollar}\omega{dollar} the Liapunov exponent. The characteristic parameter for the transport of a scalar in a chaotic flow is {dollar}\Omega\ \equiv\ \omega L\sp2/D{dollar} where L is the spatial scale and D is the diffusivity. For {dollar}\Omega\ \gg\ 1{dollar}, the scalar is advected with the flow for a time {dollar}t\sb{lcub}a{rcub}\ \equiv{dollar} ln(2{dollar}\Omega{dollar})/2{dollar}\omega{dollar} and then diffuses during the relatively short period 1/{dollar}\omega{dollar} centered on the time {dollar}t\sb{lcub}a{rcub}{dollar}. This rapid diffusion occurs only along the field line of the {dollar}\rm \ s\sb\infty{dollar} vector, which defines the stable direction for neighboring streamlines to converge. Diffusion is impeded at the sharp bends of an {dollar}\rm \ s{dollar} line because of a peculiarly small finite time Lyapunov exponent, hence a class of diffusion barriers is created inside a chaotic sea. This result comes from a fundamental relationship between the finite time Lyapunov exponent and the geometry of the {dollar}\rm \ s{dollar} lines, which we rigorously show in 2D and numerically validated for 3D flows.;The evolution of a general 3D magnetic field in a highly conducting chaotic media is also related to the spatial-temporal dependence of the finite time Lyapunov exponent. The Ohmic dissipation in a chaotic plasma will become a dominate process despite a small plasma resistivity. We show that the Ohmic heating in a chaotic plasma occurs in current filaments or current sheets. The particular form is determined by the time dependence of spatial gradient of the finite time Lyapunov exponent along a direction in which neighboring point neither diverge nor converge.
| Identifer | oai:union.ndltd.org:wm.edu/oai:scholarworks.wm.edu:etd-3792 |
| Date | 01 January 1996 |
| Creators | Tang, Xian Zhu |
| Publisher | W&M ScholarWorks |
| Source Sets | William and Mary |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Dissertations, Theses, and Masters Projects |
| Rights | © The Author |
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