Linear and Nonlinear Data Assimilation in Turbulent Systems

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Turbulent flows play a fundamental role in weather and climate dynamics, which are major factors impacting environmental stability, agricultural production, civil infrastructure, and other important areas. Turbulence is highly chaotic, and therefore modern approaches to predicting its behavior are based on simulations. A major difficulty in accurately simulating turbulent flows is the problem of determining the initial state of the flow. For example, weather prediction models typically require the present state of the weather as input. However, the state of the weather is only measured at certain points, such as at the locations of weather stations or weather satellites. Data assimilation makes up for the lack of complete knowledge of the initial state. It incorporates incoming data into the equations, driving the simulation to the correct solution. The objective of this project is to develop innovative computational and mathematical methods to test, improve, and extend a promising new class of algorithms for data assimilation in turbulent flows. Results of this work increase predictive capabilities of scientists, produce new mathematical and computational tools, and help educate students in challenging new areas with real-world impacts. A student participates in the work of the project.The project focuses on major areas of research aimed at making a new data assimilation tool as useful as possible to researchers in fluid dynamics and geophysics. Firstly, an in-depth analytical and computational study of new nonlinear versions of the data assimilation algorithm is carried out, and its convergence rates are carefully estimated. Secondly, the investigator carries out the first 3D simulations using the new algorithm in the context of the incompressible Navier-Stokes equations of fluids, and makes a detailed comparison of the method with cutting-edge data assimilation methods. Finally, the method is extended to multi-physics settings to include fluids driven by heat convection and fluids with magnetic properties. A student participates in the work of the project.