Analysis of models for large-scale geophysical flows

点击翻译按钮获取中文摘要

The rigorous analysis of the highly nonlinear equations that model atmospheric and oceanic flows is a very difficult task, generally beyond the power of current mathematical tools. This is certainly true of the full governing equations, the famous compressible Navier-Stokes equations (called Euler equations when viscosity is neglected), whose solution is too complicated to compute, even numerically. Indeed, in practice the modelling that informs applications, such as forecasting the weather, is based on averaged versions or simplified reductions of the governing equations. While such equations are used ubiquitously to model complex physical phenomena and to perform numerical approximations, both the solvability of the models and the validity of the approximations computed rests largely on heuristics rather than on rigorous mathematical ground. This proposal concerns a particular system of equations, the semi-geostrophic system, that models the large-scale dynamics of inviscid geophysical flows.The importance of this particular model rests on the fact that, as an asymptotic reduction, the system is expected to be a more accurate approximation to the full model than other reductions used in practice. In addition the validity of this reduction persists also when certain parameters, for example the earth rotation coefficient, are taken to be variable. For this reason, the model can approximate the large-scale dynamics of the flow more accurately than models whose solutions are assumed close to a uniform reference state. Mathematically, the semi-geostrophic system supports singular solutions, thus it can capture rigorously phenomena such as front formation. This is important in view of the fact that the physical derivation of this system was guided precisely by the need to model the formation of atmospheric fronts.The mathematical interest in the semi-geostrophic model has been revived by the discovery that a specific change of variables, well known to practitioners, transforms it into a system that can be analysed rigorously by using modern techniques of variational analysis and optimal transport theory. Activity in these areas in the past twenty years has seen very important results and advances, depending on delicate and sophisticated mathematical tools. The overarching aim of this project is to adapt and translate these techniques and, using new recent insights, to obtain results on the existence and uniqueness of solutions of the semigeostrophic system in increasingly realistic cases. The research proposed also aims at proving the validity and asymptotic order of the system as a reduction of the Euler equations, thus putting on rigorous foundations the numerical and physical modelling based on these equations.