This project studies several mathematical aspects of fluid mechanics. Computational tests will be conducted for numerical methods, known as vortex methods, for time-dependent, inviscid, incompressible fluid flow. The test problems used are vortex rings with swirl; they can be computed reliably by other researchers using variational methods. The results may serve as a guide in the application of vortex methods and may suggest possible improvements in the methods. An analysis of the errors in related methods for fluid interfaces will be initiated, in the hope of better understanding their design. In such cases a moving interface is tracked, along with information determining its velocity. A convergence proof will be attempted along the lines of earlier analysis for vortex methods. Work on the construction of exact solutions for the equations of three-dimensional water waves, representing interacting long waves, of the type modeled by the Korteweg-de Vries equation, will be continued. Near-resonances with waves in the usual scaling have led to delicate analysis. Finally, the relationship between the equations describing large-scale atmospheric and oceanic motions and their singular limiting form, the quasi-geostrophic equations, will be investigated. The nature of this approximation is closely related to the generation of unwanted small scales in time or space in the solutions of the full system. In many situations where the nature of fluid flow is of practical importance, such as the flow past an object or combustion, quantitative prediction is difficult because of the large number of variables involved. The development of reliable methods for numerical simulation of realistic flows offers the hope that a greater number of test cases could be studied with less effort than through experiments alone. Vortex methods are a special class of methods which have particular advantages in important but difficult cases. An interface occurs where one kind of fluid motion meets another. Analysis and numerical simulation can predict whether or not desired configurations can be maintained. The proposed work might suggest improvements in the numerical methods used for these special problems. Nonlinear waves of the type described in the third topic have been found to be remarkably stable in water and in other physical settings; they are completely different from the waves occurring in linear theory. Little is known about their fully nonlinear interaction. The last topic is a mathematical investigation of the fundamental and long-standing question of how the equations for large-scale circulating flow in the atmosphere or oceans can be reduced to a simpler system. One cannot hope to solve the full system accurately, and much effort has been devoted to finding ways to predict the main features by calculating a smaller number of variables.
