CMG: Nonintegrable Hamiltonian Systems in Ocean Dynamics

摘要：0417425CMG：海洋动力学中的不可积哈密顿系统本研究集中于海洋物理中的两组问题：(I)非定常二维不可压缩流动中的粒子轨迹(拉格朗日)动力学；(Ii)波在非均匀运动介质中的传播。在这两个案例中要调查的问题都是出于海洋学问题，而不是纯粹的数学问题。然而，预计对数学细节的更多关注将导致对一些非常实际的问题的新见解。解决上述两类问题的传统方法严重依赖于严格随机方法。从动力系统的角度来看，这种方法是不现实的，也是不必要的限制。与问题(I)有关的研究主题包括：随机与混沌动力学；示踪剂斑块和反常扩散；背景流对轨迹稳定性的重要性；涡度约束对轨迹稳定性的重要性；绝热不变性；以及扰动Taylor-Couette流中混合的详细研究。问题(Ii)中特别关注的是波在随机非均匀介质中传播理论(WPRIM)的发展；关于这个问题的初步工作表明，它在一些基本方面不同于传统的(均匀背景)WPRM问题。更广泛的影响。这项拟议的工作是地球科学家和数学家之间的合作。它将有助于启动两位有前途的年轻科学家的职业生涯，他们是代表不足的群体的成员。地球科学家和数学家之间的合作将提高数学严谨性的水平，以提高所开展的地球物理激励研究的数学水平，并开发用于数学课程的地球物理激励教材。后者在本科生阶段尤其重要，以指出所教授材料的实际重要性。将开发基于网络的教具。拟议的活动为进一步教育研究生提供了几种可能性；至少有一种可能性将参与拟议的工作。这项拟议的工作还具有许多有益于社会的实际应用。拉格朗日动力学工作的主要应用是在大气、海洋或其他自然水体中混合(例如污染物扩散)。拉格朗日可预测性问题与污染物扩散问题和海上搜救行动有关。了解示踪剂浓度统计，包括示踪剂斑块，在生物学应用中可能是至关重要的。与Taylor-Couette流动中的混合有关的拟议工作具有直接的工业应用，例如，用于化学品或药物的混合；在后一种情况下，未能完全混合可能导致对人体有毒的物质的局部浓度。WPRIM工作的主要应用是反问题(层析成像、无损评估)和通信。波(声学、弹性、电磁学)在这两种地球物理应用中都得到了广泛的应用。几乎所有这类地球物理应用中的环境都具有不均匀背景的特点；因此，拟议的工作对于理解波场统计和信号一致性损失具有重要意义。

Abstract: 0417425CMG: Nonintegrable Hamiltonian Systems in Ocean DynamicsThis study focuses on two sets of problems in ocean physics in which the underlying dynamics are those of a nonintegrable Hamiltonian system: (i) particle trajectory (Lagrangian) dynamics in unsteady two-dimensional incompressible flows; and (ii) wave propagation in inhomogeneous moving media. The issues to be investigated in both cases are motivated by oceanographic questions rather than purely mathematical issues. However, it is anticipated that greater attention to mathematical detail than has heretofore been applied will lead to new insights into some very practical issues.Intellectual Merit. Traditional approaches to the above two classes of problem rely heavily on strictly stochastic methods. From a dynamical systems perspective such an approach is unrealistic and unnecessarily restrictive. Topics relating to problem (i) to be investigated include: stochastic vs. chaotic dynamics; tracer patchiness and anomalous diffusion; the importance of the background flow on trajectory stability; the importance of vorticity constraints on trajectory stability; adiabatic invariance; and a detailed investigation of mixing in perturbed Taylor-Couette flows. Of particular concern in problem (ii) is the development of a theory of wave propagation in random inhomogeneous media (WPRIM); preliminary work on this problem suggests that it differs in some fundamental respects from the traditional (homogeneous background) WPRM problem. Broader Impacts. The proposed work is a collaboration between geoscientists and a mathematician. It will serve to launch the careers of two promising, young scientists who are members of underrepresented groups. The collaboration between the geoscientists and mathematicians will improve the level of mathematical rigorof the geophysically motivated research performed and develop geophysically motivated teaching material for use in mathematics courses. The latter is particularly important at the undergraduate level to point out the practical importance of the material being taught. Web-based teaching aids will be developed. The proposed activity offers several possibilities to further the education of graduate students; at least one will be involved in the proposed work. The proposed work also has numerous practical applications that are beneficial to society. The principal application of the Lagrangian dynamics work is to mixing (e.g. pollutant dispersal) in the atmosphere, the ocean, or other natural bodies of water. The issue of Lagrangian predictability is relevant to pollutant dispersal issues and search and rescue operations at sea. Understanding tracer concentration statistics, including tracer patchiness, can be critically important in biological applications. The proposed work relating to mixing in Taylor-Couette flows has immediate industrial applications, for example to the mixing of chemicals or drugs; in the latter case, failure to thoroughly mix could lead to a localized concentration of a substance that is toxic to humans. The principal application of the WPRIM work is to inverse problems (tomography, nondestructive evaluation) and communication. Waves (acoustics, elastic, electromagnetic) are widely used in geophysical applications of both types. The environment in almost all such geophysical applications is characterized by an inhomogeneous background; hence the importance of the proposed work for understanding wavefield statistics and loss of signal coherence.