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Volume 1,     Number 4,     Fall 1993

 

CHAOTIC WAVE PACKET MIXING AND TRANSPORT
HUIJUN YANG

Abstract. In this paper we further developed wave packet theory and proposed a new concept of chaotic wave packet mixing and transport to describe two types of wave mixing and transport processes, i.e., dynamically passive wave mixing and dynamically active wave mixing. When the path of the wave packet is chaotic, we have chaotic wave packet mixing and transport. The mechanism of passive tracer mixing, or dynamically passive wave mixing, is just advection by the medium flow. In a dynamically active wave system such as a dispersive wave system, however, there are two mechanisms operating the mixing and transport process. The first one is the advection by the medium flow; the second one is the dispersion process, which is directly related to energy dispersion of the waves. Hence, the passive wave mixing and transport is described by the Lagrangian trajectory of the basic flow whereas the dynamically active wave mixing and transport is described by the Lagrangian trajectory of the wave packet.
    We characterized the chaotic mixing and transport by a variety of means including the initially small blob experiments, correlation dimension and Lyapunov exponents. In the passive wave mixing, we superimposed a perturbation wave on the basic wave and investigated the dependence of perturbation wave velocity (or frequency) on the mixing and transport. This dependence is rather complicated. The complexity of geometry of structure of mixing and transport increases with the frequency. In a dispersive wave system, by varying the dispersive parameter, we studied the role of dispersion mechanism in the wave mixing and transport. We found that turning the dispersion mechanism on will affect the mixing and transport greatly. In our example, it is found that the greater the dispersive parameter, the more mixing and transport the dispersive wave system becomes. The dispersion mechanism enhances the mixing and transport. Furthermore, the dispersion mechanism results in mixing and transport on a large scale, whereas advection by the medium flow results in mixing and transport on a small scale. The governing dynamical system constitutes a Hamiltonian system. When the basic flow is periodic in time, this Hamiltonian has more than two degrees of freedom, which results in Arnold diffusion and enhances mixing and transport in the system. Implications of the results in general waves are discussed.

 

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