Geophysical Models: Regularity, Justification and Long-time Behavior

Titi 9704632 研究者和他的同事研究复杂的地球物理流体问题的近似的有效性。 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“原始方程”，通常在计算上非常昂贵，并且难以分析研究。 因此，更易于管理的近似模型应该代表物理系统在感兴趣的时间和长度尺度上的行为，用于对真实的世界进行预测。 自然，出现了这种简单模型的有效性问题。 该项目的第一个主题是证明两个浅水原型模型的合理性，称为“湖”和“大湖”方程，它们描述了包含在具有缓慢空间变化的底部、自由上表面和垂直侧壁的浅水盆地中的无粘性不可压缩流体的长时间运动，在重力的影响下，在小的特征速度和非常小的表面振幅的限制下。 据估计，浅水近似只能在有限的时间间隔内证明是合理的，其长度可以估计。 为了验证流体动力学模型的长期行为，必须比较它们的吸引不变集的统计特性，而不是比较单个解。 要做到这一点，有必要把重点放在包括一些耗散机制的模型上。 本项目的第二个主题涉及阻尼作为一种可能的“平滑”机制-一种比通常考虑的粘度弱的机制。 从技术上讲，问题是线性阻尼是否足以断言存在一个光滑的甚至是解析的全局吸引子。 这是首次研究的非线性薛定谔方程。 虽然它不是一个地球物理模型，但它是一个重要的原型，其部分结果已经可用。 接下来，Stommel-Charney模型的墨西哥湾流，科里奥利力预计将发挥重要作用，作为第二个正规化机制。 最后，研究了多孔介质中的Benard对流模型。 在这里，温度是扩散的，而流体速度只是线性阻尼。 计算机对大尺度或全球尺度现象的预测，例如天气或气候预报，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，设计有效且可靠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。 浅度是通常用于获得不太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，这两个例子很简单，可以进行详细的研究，但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在弱阻尼存在下模型的长期行为。 当一个物理系统被长时间建模时，能量输入（“驱动”）和能量损失（“阻尼”）的平衡变得非常重要。 该项目的重点是阻尼的一种特殊的数学表现形式，即所谓的弱阻尼，特别是它如何影响模型的大时间。

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.