Geometric Structures, Lobe Dynamics, and Lagrangian Transport in Flows with Aperiodic Time-Dependence, with Applications to Rossby Wave Flow |
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Authors: | N Malhotra S Wiggins |
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Institution: | (1) Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, USA, US |
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Abstract: | Summary. In this paper we develop the mathematical framework for studying transport in two-dimensional flows with aperiodic time dependence
from the geometrical point of view of dynamical systems theory. We show how the notion of a hyperbolic fixed point, or periodic
trajectory, and its stable and unstable manifolds generalize to the aperiodically time-dependent setting. We show how these
stable and unstable manifolds act as mediators of transport, and we extend the technique of lobe dynamics to this context.
We discuss Melnikov's method for two classes of systems having aperiodic time dependence. We develop a numerical method for
computing the stable and unstable manifolds of hyperbolic trajectories in two-dimensional flows with aperiodic time dependence.
The theory and the numerical techniques are applied to study the transport in a kinematic model of Rossby wave flow studied
earlier by Pierrehumbert 1991a]. He considered flows with periodic time dependence, and we continue his study by considering
flows having quasi-periodic, wave-packet, and purely aperiodic time dependencies. These numerical simulations exhibit a variety
of new transport phenomena mediated by the stable and unstable manifolds of hyperbolic trajectories that are unique to the
case of aperiodic time dependence.
Received April 10, 1997; revision received August 1, 1997 and accepted October 7, 1997 |
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