Random and deterministic wave-vortex interactions in fluid dynamics

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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.