Advection and diffusion in a three dimensional chaotic flow
msra(1998)
摘要
The advection-diffusion equation is studied via a global Lagrangian
coordinate 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
diffusion will dominate the solution at a critical time, which scales
logarithmically with the diffusivity. The subsequent rapid diffusive relaxation
is completed on the order of a few Lyapunov times, and it becomes more
anisotropic the smaller the diffusivity. The local Lyapunov time of the flow is
the inverse of the finite time Lyapunov exponent. A finite time Lyapunov
exponent can be expressed in terms of two convergence functions which are
responsible for the spatio-temporal complexity of both the advective and
diffusive transports. This complexity gives a new class of diffusion barrier in
the 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 integrable flow
is strictly confined within the KAM surfaces.
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关键词
hamiltonian chaos,diffusion barriers,mixing i. introduction,fractal,finite time lyapunov exponent,advection-diffusion equation,ˆ line,advection diffusion equation,three dimensional,fluid dynamics,partial differential equation,scalar field,coordinate transformation,dynamic systems theory
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