A Lagrangian analysis of advection-diffusion equation for a three dimensional chaotic flow

The advection-diffusion equation is studied via a global Lagrangian coordin ate transformation. The metric tensor of the Lagrangian coordinates couples the dynamical system theory rigorously into the solution of this class of partial differential equations. If the flow has chaotic streamlines, the di ffusion will dominate the solution at a critical time, which scales logarit hmically with the diffusivity. The subsequent rapid diffusive relaxation is completed on the order of a few Lyapunov times, and it becomes more anisot ropic the smaller the diffusivity. The local Lyapunov time of the flow is t he inverse of the finite time Lyapunov exponent. A finite time Lyapunov exp onent can be expressed in terms of two convergence functions which are resp onsible for the spatio-temporal complexity of both the advective and diffus ive transports. This complexity gives a new class of diffusion barrier in t he chaotic region and a fractal-like behavior in both space and time. In an integrable flow with shear, there also exist fast and slow diffusion. But unlike that in a chaotic flow, a large gradient of the scalar field across the KAM surfaces can be maintained since the fast diffusion in an integrabl e flow is strictly confined within the KAM surfaces. (C) 1999 American Inst itute of Physics. [S1070-6631(99)02106-6].