Geophysical Models: Regularity, Justification and Long-time Behavior

研究者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，即所谓的“原始方程”，通常在计算上非常昂贵，而且很难进行分析研究。更易于管理的近似模型，应该表示感兴趣的物理系统在时间和长度尺度上的行为，因此用于对现实世界进行预测。自然，这种简单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浅水的原型模型，称为“湖”和“大湖”方程，它们描述了在重力的影响下，在很小的特征速度和很小的表面振幅的极限下，底部空间变化缓慢、自由的上表面和垂直的侧壁的浅盆地中包含的无粘性、不可压缩流体的长期运动。预计浅水近似只能在有限的时间间隔内成立，而这段时间的长度是可以估计的。为了验证流体动力学模型的长时间行为，人们必须比较其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括一些耗散机制的模型上。该项目的第二个主题是将阻尼作为一种可能的“平滑”机制——一种比通常认为的粘度更弱的机制。从技术上讲，问题是线性阻尼是否足以断言光滑或甚至解析全局吸引子的存在。首先对非线性薛定谔方程进行了研究。虽然不是地球物理模型，但它是一个重要的原型，已经有了部分结果。接下来，考虑了墨西哥湾流的Stommel-Charney模型，其中科里奥利力预计将作为第二正则化机制发挥重要作用。最后，研究了多孔介质中的贝纳德对流模型。这里温度是扩散的，而流体速度只是线性阻尼。对大尺度或全球尺度现象的计算机预测，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折衷。因此，设计既有效又值得信赖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，因为它的宽度远远大于深度。浅度是通常用来获得较不复杂模型的特征之一。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，可以进行详细的研究，但却包含了在更一般情况下发现的大多数困难。项目的第二个方面涉及模型在弱阻尼存在下的长期行为。当一个物理系统长时间建模时，能量输入（“驱动”）和能量损失（“阻尼”）的平衡变得非常重要。该项目侧重于阻尼的特定数学表现，即所谓的弱阻尼，特别是它如何在大时间内影响模型。

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.