Linear and Nonlinear Data Assimilation in Turbulent Systems

湍流在天气和气候动力学中起着基础性作用，而天气和气候动力学是影响环境稳定、农业生产、民用基础设施等重要领域的主要因素。湍流是高度混乱的，因此预测其行为的现代方法是基于模拟。准确模拟湍流的一个主要困难是确定流动的初始状态。例如，天气预报模型通常需要输入天气的当前状态。然而，天气状态只在某些点上测量，如气象站或气象卫星的位置。数据同化弥补了对初始状态缺乏完整知识的不足。它将传入的数据合并到方程式中，将模拟驱动到正确的解。该项目的目标是开发创新的计算和数学方法，以测试、改进和推广一类有前途的湍流数据同化新算法。这项工作的结果提高了科学家的预测能力，产生了新的数学和计算工具，并帮助学生在具有现实世界影响的挑战新领域中进行教育。一名学生参与了该项目的工作。该项目侧重于主要研究领域，旨在使一种新的数据同化工具尽可能地对流体动力学和地球物理的研究人员有用。首先，对新的非线性版本数据同化算法进行了深入的分析和计算研究，并对其收敛速度进行了仔细的估计。其次，在不可压缩的流体Navier-Stokes方程的背景下，首次使用新算法进行了三维数值模拟，并与前沿的数据同化方法进行了详细的比较。最后，将该方法扩展到多物理环境，以包括由热对流驱动的流体和具有磁性的流体。一名学生参与该项目的工作。

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.