Chen Nan, Li Yuchen, Lunasin Evelyn
Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA.
Department of Mathematics, United States Naval Academy, Annapolis, Maryland 21402, USA.
Chaos. 2021 Oct;31(10):103123. doi: 10.1063/5.0057421.
Complex nonlinear turbulent dynamical systems are ubiquitous in many areas of research. Recovering unobserved state variables is an important topic for the data assimilation of turbulent systems. In this article, an efficient continuous-in-time data assimilation scheme is developed, which exploits closed analytic formulas for updating the unobserved state variables. Therefore, it is computationally efficient and accurate. The new data assimilation scheme is combined with a simple reduced order modeling technique that involves a cheap closure approximation and noise inflation. In such a way, many complicated turbulent dynamical systems can satisfy the requirements of the mathematical structures for the proposed efficient data assimilation scheme. The new data assimilation scheme is then applied to the Sabra shell model, which is a conceptual model for nonlinear turbulence. The goal is to recover the unobserved shell velocities across different spatial scales. It is shown that the new data assimilation scheme is skillful in capturing the nonlinear features of turbulence including the intermittency and extreme events in both the chaotic and the turbulent dynamical regimes. It is also shown that the new data assimilation scheme is more accurate and computationally cheaper than the standard ensemble Kalman filter and nudging data assimilation schemes for assimilating the Sabra shell model with partial observations.
复杂的非线性湍流动力系统在许多研究领域中普遍存在。恢复未观测到的状态变量是湍流系统数据同化的一个重要课题。在本文中，我们开发了一种高效的连续时间数据同化方案，该方案利用封闭的解析公式来更新未观测到的状态变量。因此，它在计算上既高效又准确。新的数据同化方案与一种简单的降阶建模技术相结合，该技术涉及廉价的封闭近似和噪声放大。通过这种方式，许多复杂的湍流动力系统能够满足所提出的高效数据同化方案的数学结构要求。然后，将新的数据同化方案应用于Sabra壳模型，该模型是一个非线性湍流的概念模型。目标是恢复不同空间尺度上未观测到的壳速度。结果表明，新的数据同化方案能够巧妙地捕捉湍流的非线性特征，包括混沌和湍流动力区域中的间歇性和极端事件。研究还表明，对于用部分观测值同化Sabra壳模型，新的数据同化方案比标准集合卡尔曼滤波器和推估数据同化方案更准确且计算成本更低。
