dorsal/arxiv
View SchemaAdvection and diffusion in a three dimensional chaotic flow
| Authors | X. Z. Tang, A. H. Boozer |
|---|---|
| Categories | |
| ArXiv ID | physics/9803025 |
| URL | https://arxiv.org/abs/physics/9803025 |
Abstract
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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"abstract": "The advection-diffusion equation is studied via a global Lagrangian\ncoordinate transformation. The metric tensor of the Lagrangian coordinates\ncouples the dynamical system theory rigorously into the solution of this class\nof partial differential equations. If the flow has chaotic streamlines, the\ndiffusion will dominate the solution at a critical time, which scales\nlogarithmically with the diffusivity. The subsequent rapid diffusive relaxation\nis completed on the order of a few Lyapunov times, and it becomes more\nanisotropic the smaller the diffusivity. The local Lyapunov time of the flow is\nthe inverse of the finite time Lyapunov exponent. A finite time Lyapunov\nexponent can be expressed in terms of two convergence functions which are\nresponsible for the spatio-temporal complexity of both the advective and\ndiffusive transports. This complexity gives a new class of diffusion barrier in\nthe chaotic region and a fractal-like behavior in both space and time. In an\nintegrable flow with shear, there also exist fast and slow diffusion. But\nunlike that in a chaotic flow, a large gradient of the scalar field across the\nKAM surfaces can be maintained since the fast diffusion in an integrable flow\nis strictly confined within the KAM surfaces.",
"arxiv_id": "physics/9803025",
"authors": [
"X. Z. Tang",
"A. H. Boozer"
],
"categories": [
"physics.flu-dyn",
"chao-dyn",
"math-ph",
"math.MP",
"nlin.CD"
],
"title": "Advection and diffusion in a three dimensional chaotic flow",
"url": "https://arxiv.org/abs/physics/9803025"
},
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