Embid 9704423 The investigator studies mathematical problems with applications in geophysical fluid dynamics and reactive flows. The two main topics of research are: 1. Averaging over fast gravity waves for geophysical flows. 2. Front wave propagation in reactive flows. The investigator combines formal asymptotic analysis and rigorous mathematical theory of physical models, together with the numerical analysis of simplified problems. In the first topic he studies the limiting dynamics of several geophysical fluid dynamical systems that include the rotating shallow water equations and the rotating Boussinesq equations for stably stratified flow. In the physically relevant regimes of low Froude number or low Rossby number the equations evolve with two separated scales; the slow modes are associated with quasi-geostrophic flows, and the fast modes correspond to gravity waves. The investigator derives simplified equations for the limiting dynamics of geophysical flows at low Froude or low Rossby number by averaging over the fast gravity waves. These limiting equations are applicable even when the flow is not in geostrophic balance initially. The investigator studies the resulting simplified dynamics equations, which include in natural and systematic manner the important effects of three-wave resonant interactions of three fast gravity waves, or two gravity waves and a vortical mode. In the second topic he studies the propagation of fronts in the presence of an underlying advective velocity field. This velocity field involves several space-time scales, describing large variations both at the integral scale, as well as rapid fluctuations at a scale intermediate between the integral scale and the front thickness. The investigator works on the derivation and analysis of effective large scale equations for the propagation of fronts in the physical limit of fast reaction, slow diffusion, and fast velocity fluctuations at the intermediate scale. He applies these results to study the interaction of either premixed or diffusive flames with the turbulent velocity field, and to study moist convection in clouds. The research on the limiting dynamics of geophysical flows and the role of the fast gravity waves has potential applications in atmospheric and oceanic sciences. As the demand for more accurate numerical weather predictions increases, it is necessary to gain better understanding of the effect of the unavoidable fast gravity waves that pollute all calculations. Similarly, the need for long range forecasting also requires a better understanding of the accumulative effects of the fast gravity waves. Additionally, this work is relevant in the area of stratified turbulence, where the simplified dynamics equations can help to explain some of the universal features observed in the velocity spectrum in the atmosphere and the ocean. The research on front wave propagation has potential applications in combustion design, guiding the engineer in evaluating the reliability of the simplified advection algorithms currently used in flame front propagation. This work is also relevant to cloud physics. Clouds are a key ingredient in the radiative balance of the earth and also a key variable in the study of weather forecasting and global climate. Currently, in large-scale numerical simulations of the atmosphere, the important effects of the small-scale cloud physics cannot be modelled in detail. This work gives insight into ways of incorporating the small-scale effects of the interaction of moisture and convective currents in clouds into these large-scale calculations.
