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

这项研究将应用Kolmogorov-Arnold-Moser(KAM)理论和非定常流动中一般非定常双曲点的稳定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或LCS。KAM理论与流形结构有着密切的联系。应用领域包括：(I)喷流、输运障碍和地球--S海洋和平流层中的潜在涡度障碍之间的联系；(Ii)与海洋环流有关的大尺度气候系统气候学，以及这些结构和大气环流的主要欧拉特征之间的联系；(Iii)海洋大尺度CS的生物应用，包括有害藻华、浮游生物斑块和对观测到的生物地理边界的理解；以及(Iv)涉及动力系统方法来研究波在随机的非均匀介质中传播的问题。这项工作将在几个方面潜在地对社会有益。首先，它解决了海洋中财产的运输问题。应用包括鱼苗的运输、浮游生物的分布，包括有害的藻华、毒素和污染物。其中一些物质对人类健康有潜在的重大影响。鱼苗的运输与渔业种群的管理和海洋保护区的设计有关。海上搜救行动也与海洋运输密切相关。其次，关于平流层传输障碍的发现将对全球变暖产生影响，因为这种障碍强烈影响温室气体(包括臭氧)在大气中的分布。此外，如果考虑采取地球工程措施来对抗温室气体引起的全球变暖，了解这些障碍也是至关重要的。第三，我们的工作具有潜在的工业应用价值。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.