Milewski 9704606 We propose to apply numerical and asymptotic methods to study several problems of physical relevance in fluid mechanics and wave propagation. In particular we propose to study, together with several collaborators: The resonance phenomena of equatorial Kelvin waves and of certain waves in stratified flows; The strong interactions of solitary waves on a beach, and of the rotational-irrotational components in free surface flows; The diffraction of nonlinear long waves due to abrupt changes in the lateral boundaries. These projects are relevant both in developing efficient psudo--spectral numerical methods and in understanding resonance effects in a large class of geophysical wave propagation problems. We also continue ongoing collaboration on Singularities in two- and three-dimensional free surface gravity flows and similarity solutions of jets impinging on rigid walls; The evolution of hexagonal patterns in reaction-diffusion equations with applications to the dynamics of stabilized flame fronts. In this proposal we use mathematical and computational methods to study problems of physical and geophysical importance in fluid mechanics. Specifically we work on 1) Equatorial and mid-latitude waves in the atmosphere. These waves are thought have an impact on the global climate and a mathematical understanding of their interaction with other waves and with topographical features is therefore of practical importance. 2) the diffraction and amplification of waves approaching a beach. Here we attempt to predict mathematically the size and pattern of ocean waves near a beach. The mathematical predictions can then be compared to real data from ocean waves during a storm. 3) Shapes of the free surfaces of fluids. We study different types of singular free surface flow (such as sharp wave coasts) and the shapes of fluid jets impinging on objects. 4) The evolution of patterns in flame-fronts. Here, we study the propagation and stability of flame fronts in a fuel-air mixture. The mathematical formulation and efficient numerical method used here is applicable to a wide class of physical problems.
