Collaboration in Mathematical Geosciences: Nonintegrable Hamiltonian Systems in Geophysical Fluid Dynamics

本研究将应用Kolmogorov-Arnold-Moser （KAM）理论的结果以及非定常流动中一般非平稳双曲点的稳定流形和不稳定流形的结构。在流体动力学应用中，这些流形通常被称为拉格朗日相干结构（LCSs）。KAM理论与流形结构密切相关。应用有：(i)地球上的喷流、输运屏障和位势涡障之间的联系？大洋和平流层；（ii）与海洋环流有关的LCS气候学以及这些结构与环流的主要欧拉特征之间的联系；（iii）海洋生态系统的生物学应用，包括涉及有害藻华、浮游生物斑块的问题，以及了解观察到的生物地理界限；（iv）涉及随机非均匀介质中波传播的动力系统方法的问题。这项工作将在几个方面对社会有潜在的好处。首先，它解决了海洋财产的运输问题。应用包括运输鱼苗，浮游生物分布，包括有害藻华，毒素和污染物。其中一些物质可能对人类健康产生重大影响。鱼苗的运输关系到渔业资源的管理和海洋保护区的设计。海上搜救行动也与海洋运输密切相关。第二，关于平流层运输屏障的发现将对全球变暖产生影响，因为这种屏障强烈影响大气中温室气体（包括臭氧）的分布。此外，如果考虑采取地球工程措施来抵消温室气体引起的全球变暖，了解这些障碍是至关重要的。第三，我们的研究成果有产业化的潜力。在大多数涉及运输的工业应用中，目标是有效地混合两种或两种以上的流体。

This study will apply results of Kolmogorov-Arnold-Moser (KAM) theory and the structure of stable and unstable manifolds of generally nonstationary hyperbolic points in nonsteady flows. These manifolds are often referred to as Lagrangian coherent structures, or LCSs, in fluid dynamical applications. KAM theory and manifold structure are intimately linked. Applications are: (i) the connections between jets, transport barriers and potential vorticity barriers in the Earth?s oceans and stratosphere; (ii) LCS climatology associated with the general circulation of the ocean and the connection between these structures and the predominant Eulerian features of the general circulation; (iii) biological applications of oceanic LCSs including problems involving harmful algal blooms, plankton patchiness, and understanding observed biogeographical boundaries; and (iv) problems involving a dynamical systems approach to wave propagation in random inhomogeneous media.The work will be potentially beneficial to the society in several ways. First, it addresses transport of properties in the ocean. Applications include transport of fish larvae, plankton distributions including harmful algal blooms, toxins, and pollutants. Some of these substances have potentially significant human health implications. The transport of fish larvae is relevant to management of fishery stocks and the design of marine reserves. Search and rescue operations at sea are also intimately linked to ocean transport. Second, the findings on transport barriers in the stratosphere will have implications for global warming because such barriers strongly influence the distribution of greenhouse gases (including ozone) in the atmosphere. Also, it is critically important to understand such barriers, if considering geo-engineering measures to counteract greenhouse-induced global warming. Third, there are potential industrial applications of our work. In most industrial applications involving transport, the objective is to efficiently mix two or more fluids.