Geophysical Models: Regularity, Justification and Long-time Behavior

研究人员Titi 9704632和他的同事研究了复杂地球物理流体问题近似的有效性。地球物理流体动力学面临的基本问题是，仅基于基本物理原理的数学描述，将被称为“基本方程”，通常计算成本高得令人望而却步，而且难以分析研究。因此，更易管理的近似模型，应该代表感兴趣的时间和长度尺度上的物理系统的行为，被用于对现实世界进行预测。自然，这种更简单的模型的有效性问题就出现了。该项目的第一个主题是证明两个典型的浅水模型的合理性，这两个模型称为“湖”和“大湖”方程，它们描述了在重力影响下，在特征速度很小和表面振幅很小的限制下，浅水盆地中无粘性、不可压缩的流体在空间上缓慢变化的底部、自由上表面和垂直侧壁中的长期运动。预计浅水近似只能在有限的时间间隔内被证明是合理的，而时间间隔的长度可以估计。为了验证流体动力学模型的长时间行为，人们必须比较它们吸引不变集的统计性质，而不是比较单个解。要做到这一点，有必要把重点放在包含某种耗散机制的模型上。本项目的第二个主题涉及作为一种可能的“平滑”机制的减震-一种比通常认为的粘度弱的机制。从技术上讲，问题是线性阻尼是否足以断言存在光滑的甚至是解析的全局吸引子。这是对非线性薛定谔方程的首次研究。虽然不是地球物理模型，但它是一个重要的原型，S已经有了部分结果。其次，考虑了墨西哥湾流的Stommel-Charney模式，其中科里奥利力有望作为第二种正则化机制发挥重要作用。最后，研究了多孔介质中的Benard对流模型。这里的温度是扩散的，而流体速度只是线性衰减的。计算机对大范围或全球范围的现象的预测，例如天气或气候预测，需要在预测的准确性和可用的计算资源之间进行折衷。因此，设计既高效又可信的近似方案和模型是一个主要的科学问题。该项目的第一个方面涉及浅水模型的有效性。例如，海洋是浅的，因为它们的宽度比深度大得多。浅薄是通常被利用来获得不那么复杂的模型的特征之一。该项目的目的是在两个例子的背景下证明这一程序的合理性，这两个例子足够简单，可以进行详细研究，但又包含了在更一般的情况下发现的大多数困难。该项目的第二个方面涉及模型在弱阻尼存在时的长时间行为。当对一个物理系统进行长时间的建模时，能量输入(“驾驶”)和能量损失(“减震”)之间的平衡变得非常重要。该项目重点研究了阻尼的一种特殊的数学表现形式，即所谓的弱阻尼，特别是它在很长一段时间内如何影响模型。

Titi 9704632 The investigator and his colleague study the validity of approximations to complex geophysical fluid problems. The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which will be called the ``Primitive Equations,'' is often prohibitively expensive computationally, and hard to study analytically. More manageable approximate models, which should represent the behavior of the physical system on time and length scales of interest, are therefore used for making predictions about the real world. Naturally, the question of validity of such simpler models arises. The first theme of this project is the justification of two prototypical models of shallow water, called the "Lake" and the "Great Lake" equations, which describe the long-time motion of an inviscid, incompressible fluid contained in a shallow basin with a slowly spatially varying bottom, a free upper surface and vertical side walls, under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude. It is expected that the shallow water approximation can be justified only for limited intervals of time, the length of which can be estimated. In order to validate the long time behavior of fluid dynamical models, one has to compare the statistical properties of their attracting invariant sets, rather than compare individual solutions. To do so, it is necessary to focus on models that include some mechanism of dissipation. The second theme of this project concerns damping as a possible ``smoothing'' mechanism---a mechanism that is weaker than the viscosity normally considered. Technically, the question is whether linear damping is sufficient to assert existence of a smooth or even analytic global attractor. This is first investigated for the nonlinear Schrodinger equation. Although not a geophysical model, it is an important prototype for which partial result s are already available. Next, the Stommel-Charney model for the Gulf stream is considered, where the Coriolis force is expected to play an important role as a second regularization mechanism. Lastly, a model for Benard convection in a porous medium is investigated. Here the temperature is diffusive while the fluid velocity is only linearly damped. Computer predictions of phenomena on large or global scales, for example weather or climate forecasts, need to compromise between accuracy of the predictions and available computing resources. It is therefore a major scientific concern to devise approximation schemes and models that are efficient as well as trustworthy. The first aspect of this project concerns the validity of models of shallow water. Oceans, for example, are shallow in the sense that they are much wider than they are deep. The shallowness is one of the features that are usually exploited to obtain less complex models. The project aims at the justification of this procedure in the context of two examples that are simple enough to allow detailed study, and yet contain most of the difficulties found in more generic situations. The second aspect of the project concerns the long time behavior of models in the presence of weak damping. When a physical system is modeled for long periods of time, the balance of energy input (``driving'') and energy loss (``damping'') becomes very important. The project focuses on a particular mathematical manifestation of damping, the so-called weak damping, and in particular on how it influences the model for large times.