9704906 Jones Dynamical systems offers a body of modern mathematical techniques with the potential for application to a number of problems in current science and technology. The principal investigator plans to develop techniques that are specifically geared to solving important problems in optical pulse propagation and mixing attendant to mesoscale ocean structures. In optical communications, lasers, and fiber couplers a key issue is the support of multiple pulses. These pulses may be separated in time, by polarization, frequency or live on distinct but coupled fibers. Based on needed extensions of the current mathematical technology, techniques will be developed that promise a major contribution to these issues through stability calculations of pulses found by homoclinic bifurcation theory and other related methods. For mixing in ocean structures, perturbations of the quasigeostrophic models will be used to assess the efficacy of the added mechanisms to promoting Lagrangian transport. This has already been successful in studying the effects of viscosity and, using this analysis as a starting point, the plan is to incorporate other effects and ascertain their role in what is now known to be a significant phenomenon in ocean transport, namely the advection of fluid particles due to the splitting of separatrices. The importance of the work in this project lies both in the significance of the applications and the promise of building a body of techniques that can be used in other areas. The capacity of optical communication systems is limited by the ability of the fiber to carry, simultaneously, multiple pieces of information. Many scenarios have been proposed but most require the resolution of difficult mathematical problems concerning the stability of these pulses and assessment of their interactions. Envisioned in the work of this project is the development of techniques for addressing these difficult questions. The second application to be considered is quite different. To understand the distribution of fluid properties, such as salinity and temperature, in regions of the ocean governed by well-known currents, such as the Gulf Stream, is central in understanding, for instance, the origin and progress of weather patterns. The key to determining these distributions are the transport mechanisms associated with well-defined structures within these currents. The flows are, however, so complex that the elucidation of these transport scenarios remains a formidable challenge. The work under this project will be to implement analytic approaches aimed at isolating the underlying physical mechanisms responsible for certain observed types of transport and thus lead to more effective predictability of fluid redistribution near such currents and other ocean structures.
