Random and deterministic wave-vortex interactions in fluid dynamics

这是一个波和涡旋之间非线性相互作用的理论和数值研究，涉及两个主题领域：与气候科学相关的大气海洋流体动力学和与托卡马克动力学和核聚变相关的等离子体物理。这些相互作用问题的共同之处在于流动的不同动力成分之间存在巨大的固有尺度分离，这使得直接的数值模拟变得毫无希望，并且由于小规模成分只能部分观察到，因此也为系统动力学引入了随机成分。因此，该研究涉及使用渐近分析、随机建模和数值计算相结合的多维非线性偏微分方程的各种系统的数学研究，所有这些都着眼于与气候科学和核聚变相关的现实世界中的最终应用。具体来说，研究包括三个项目。第一个研究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，第二个研究了在强迫和耗散存在下由深海内波引起的随机粒子运动，第三个研究了托卡马克等离子体约束问题中强漂移波和强带状流相互作用造成的波湍流拼图。与气候科学相关的大气海洋动力学包含广泛的动态尺度，例如，从与声波相关的毫米和毫秒到与大尺度海洋环流相关的行星尺度和几十年尺度。所有这些尺度都是通过各种各样的非线性相互作用动态耦合在一起的，这就是基础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简化为更易于处理和理解的组件，然后可以更轻松地模拟。类似的评论也适用于与聚变研究和托卡马克装置的行为有关的等离子体动力学。因此，这项理论研究的结果可以用于未来设计下一代天气和气候的大气海洋预报系统，这具有重要的社会意义，它们也应该加强用于托卡马克核聚变装置设计的基本原则。

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.