Analysis of models for large-scale geophysical flows

对模拟大气和海洋流动的高度非线性方程进行严格分析是一项非常困难的任务，通常超出了当前数学工具的能力。对于完全控制方程，著名的可压缩的Navier-Stokes方程(忽略粘性时称为Euler方程)肯定是这样的，其解太复杂，即使是数值计算也是如此。事实上，在实践中，为天气预报等应用程序提供信息的建模是基于控制方程的平均版本或简化的简化。虽然这样的方程被普遍用于模拟复杂的物理现象和执行数值近似，但模型的可解性和计算的近似的有效性在很大程度上取决于启发式，而不是严格的数学基础。这一建议涉及一种特定的方程系统，即半地转系统，它模拟了无粘性地球物理流的大尺度动力学。这一特殊模型的重要性在于，作为一种渐近约化，该系统预计将比实践中使用的其他约化更准确地逼近整个模型。此外，当某些参数，例如地球自转系数被认为是可变的时，这种简化的有效性也持续存在。因此，该模型比其解被假定为接近统一参考状态的模型能够更准确地逼近流动的大尺度动力学。在数学上，半地转系统支持奇异解，因此它可以严格地捕捉到锋面形成等现象。这一点很重要，因为这个系统的物理推导是由建立大气锋面形成模型的需要精确指导的。半地转模型的数学兴趣重新引起了人们的兴趣，因为发现了实践者所熟知的变量的特定变化，将其转化为一个可以用现代变分分析和最优输送理论进行严格分析的系统。在过去二十年中，这些领域的活动取得了非常重要的成果和进展，这取决于精细和复杂的数学工具。这个项目的主要目标是适应和翻译这些技术，并利用最近的新见解，在越来越现实的情况下获得关于半地转系统解的存在和唯一性的结果。所提出的研究还旨在证明系统作为欧拉方程的约化的有效性和渐近阶，从而为基于这些方程的数值模拟和物理模拟奠定了坚实的基础。

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.