Tabak 9701751 In this CAREER grant, the investigator pursues two lines of research on the nonlinear dynamics of the Atmosphere and the Ocean: one focused on turbulent cascades in dispersive systems, particularly the Munk-Garret scale distribution of energy in the Ocean, and the other focused on equatorial dynamics. Both projects study resonance among geophysical waves, in very different scales: the former considers energy transfer through the self-similar inertial range, from the long waves beyond which the system is forced, to the short waves below which dissipative mechanisms take over, and the latter concentrates on the very long waves, comparable to the radius of the Earth, where forcing plays an important role. In fact, the resonance between long equatorial waves could be considered as a starting point for the resonant energy transfer toward shorter waves. Once the scales become relatively small, detailed information on the nature of the forces is lost, and a statistically self-similar regime arises, further carrying the energy through a wide range of scales toward the very short waves. The educational component of this proposal involves developing a curriculum which blends applied mathematics and geophysical fluid dynamics. Particular efforts are devoted to integrate experimental and numerical work into both graduate and undergraduate education. This is facilitated by the building of a Laboratory for Fluid Dynamics at the Courant Institute, scheduled to open in the Spring of 1997. Two new graduate courses in the mathematical modeling of geophysical waves are developed, one introductory and the other more advanced, the latter focused on nonlinear mechanisms of energy transfer among geophysical waves. Numerical modeling and desk-top experimentation are integrated into advanced undergraduate mathematical courses, with the goal of introducing undergraduate students to the lure and potential of interdisciplinary work. The Ocean and Atmospheric Sciences have reached a degree of maturity such that the accurate prediction of the weather and even of longer term climatological changes appears to be within reach. A basic understanding of many of the fundamental processes underlying the prevailing winds and currents has been developed over the last few decades; and the computational power brought about by the computer revolution makes it possible to run global models with relatively fine grids. However, the dynamics of the weather and climate has a tremendous complexity, with phenomena taking place in a wide range of spatial and temporal scales. This complexity makes it hopeless to resolve all the relevant phenomena without appealing to strong simplifying assumptions. Applied Mathematics provides powerful tools that may help clarify the validity of the various assumptions and shed light on many phenomena not yet fully understood. The two proposed subjects of this research are examples of fields where this contribution should be particularly fruitful. The dynamics of equatorial waves is known to strongly affect the global weather and climate; phenomena such as the El Nino Southern Oscillation and its global effects exemplify this. In order to study phenomena of this kind, one needs to go beyond the time scale of the order of days of the waves, to the months or years where small effects accumulate to yield substantial changes in the amplitude and behavior of the waves. Asymptotic multiple-scale analysis is an ideal applied mathematical tool to achieve this. As for the other line of research, understanding the transfer of energy among scales in the Ocean and Atmosphere is fundamental to predict the long-time effects that a change in forcing, such as the one brought about by the release of chemicals in the Atmosphere and the Ocean, may produce on our weather. Involving students in these lines of research, and bringing research-related results and methods into their classroom education, also helps develop a generation of scientists better equipped to understand the difficult and important problems of atmosphere and ocean interactions.
