CHAOTIC MIXING AND TRANSPORT IN WAVE SYSTEMS AND THE ATMOSPHERE

Abstract

In this paper, we presented some results on chaotic mixing and transport in dynamical systems, particularly in wave systems and in the atmosphere. In wave systems, we studied chaotic mixing and transport in a perturbed traveling wave and in a perturbed stationary wave. We found that there is a fundamentally difference between the two cases. There can exist invariant KAM tori in perturbed traveling waves, and no such invariant KAM tori can be identified in perturbed stationary waves. The characteristics of mixing and transport in wave systems much depend on the perturbation structure. In some cases for both perturbed traveling waves and perturbed stationary waves, there exist porous barriers preventing rapid mixing and transport. The mechanism of chaotic mixing and transport is stretching and folding, which is shown to be the horseshoe mapping topologically. Global chaotic mixing and transport on the isentropic surface of the atmosphere was also studied. To better understand the generation of fine structure in the smooth velocity field and to gain an insight to chaotic mixing and transport, a color visualization method has been applied. Using different color pallets, we were able to emphasize different parts of the domain and obtained a wealth of information about the process. Several statistical characteristics methods have been introduced, including fractal dimension and the Lyapunov exponent. The probability distribution function and multifractal were also mentioned with application to chaotic mixing and transport. We then turned our attention to dynamically active mixing and transport. We discussed chaotic wave packet mixing and transport, describing 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. We showed that passive wave mixing and transport is described by the Lagrangian trajectory of the basic flow whereas dynamically active wave mixing and transport is described by the Lagrangian trajectory of wave packet. We found that there are two mechanisms in dynamically active wave mixing and transport. The first is advection by the medium flow; the second is the dispersion process, which is directly related to energy dispersion of the waves. Armed with the concept of dynamically active mixing and transport, we simulated the cloud pattern in the tropospheric atmosphere using an evaporation-transport-condensation model, and found it to be strikingly similar to cloud pictures taken by satellite.