Random and deterministic wave-vortex interactions in fluid dynamics

这是对两个学科领域中波和涡之间的非线性相互作用的理论和数值研究：与气候科学有关的大气海洋流体动力学，以及与托卡马克动力学和核聚变有关的等离子体物理。这些相互作用问题的共同之处是流动的不同动力成分之间存在巨大的内在尺度分离，这使得直接的数值模拟无望，而且由于小尺度成分只是部分可见的，这也给系统动力学引入了随机成分。因此，这项研究涉及使用渐近分析、随机建模和数值计算相结合的方法对各种多维非线性偏微分方程组进行数学研究，所有这些都着眼于与气候科学和核聚变相关的现实世界的最终应用。具体地说，这项研究包括三个项目。第一种考虑了海洋中一种特殊的小尺度内重力波及其与海底附近平均流的相互作用，第二种考虑了在强迫和耗散存在的情况下深海内波诱导的随机粒子运动，第三种考虑了托卡马克等离子体约束问题中强漂移波和强纬向流相互作用所引起的波浪-湍流拼图。与气候科学相关的大气海洋动力学包含了大量的活跃动力学尺度，从与声波有关的毫米和毫秒到与行星尺度有关的几十年，以及与大尺度海洋环流有关的几十年。所有这些尺度通过各种各样的非线性相互作用动态地耦合在一起，这就是基本数学理论可以做出决定性贡献的地方，即通过将这些相互作用减少为更容易处理和理解的组件，然后可以更容易地进行模拟。类似的评论也适用于与聚变研究和托卡马克装置的行为相关的等离子体动力学。因此，这一理论研究的结果可能会被提供给下一代大气海洋天气和气候预报系统的未来设计，这具有重要的社会意义，它们还应该加强为托卡马克核聚变装置设计提供信息的基本原则。

This is a theoretical and numerical study of nonlinear interactions between waves and vortices in two subject areas: atmosphere ocean fluid dynamics relevant to climate science, and plasma physics relevant to tokamak dynamics and nuclear fusion. Common to these interaction problems are the huge inherent scale separations between the different dynamical components of the flow, which render a direct numerical simulation hopeless and which also introduce a stochastic component to the system dynamics because the small-scale components are only partially observable. Hence the study involves the mathematical investigation of various systems of multidimensional nonlinear PDEs using a combination of asymptotic analysis, stochastic modeling, and numerical computation, all with an eye towards ultimate applications in the real world that are relevant to climate science and nuclear fusion. Specifically, the study consists of three projects. The first considers a special kind of small-scale internal gravity wave in the ocean and its interactions with mean currents near the sea floor, the second considers random particle motion induced by deep-ocean internal waves in the presence of forcing and dissipation, and the third investigates the wave-turbulence jigsaw puzzle posed by interactions between strong drift waves and strong zonal flows in the problem of plasma confinement in tokamaks.Atmosphere ocean dynamics relevant to climate science contains a vast span of active dynamical scales, from millimeters and milliseconds associated with sound waves to planetary scales and decades associated with large-scale ocean circulations, for example. All these scales are dynamically coupled together through a wide variety of nonlinear interactions and this is where fundamental mathematical theory can make a decisive contribution, namely by reducing these interactions into more tractable and understandable components that can then be simulated with significantly greater ease. Analogous comments apply to plasma dynamics that is relevant to fusion research and the behavior of tokamak devices. Hence, results from this theoretical study may feed into the future design of next generation atmosphere ocean forecasting systems for weather and climate, which is of great societal importance, and they should also strengthen the fundamental principles that feed into the design of nuclear fusion devices in tokamaks.