Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems
Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems
复制标题
用于预测复杂湍流动力系统中的统计响应和不确定性量化的降阶模型策略
作者：
A. Majda;D. Qi
在科学和工程中的许多复杂系统中，具有高维相空间和大量不稳定性的湍流动力系统是普遍存在的。在含有大量正Lyapunov指数的湍流系统中存在奇怪的吸引子，导致小不确定性的快速增长，自然需要对湍流系统的演化进行概率刻画。在湍流动力系统中，不确定性量化是一个巨大的挑战，其目标是获得统计估计，例如关键物理量对外部强迫参数变化或不确定初始数据的非线性响应中的均值和方差的变化。当代研究的一个中心问题是发展一种系统的方法，能够恢复自然系统在统计平衡中的关键特征(模型保真度)，并改进不完美的模型预测技能，以应对各种外部扰动(模型敏感性)。本文讨论了构造具有统计精度的降阶模型的一般数学框架，该模型能够捕捉一类一般类型的有阻尼项和受迫的复杂湍流动力系统在具有最大能量的主方向上的统计变异性。这些方法是在一类具有二次非线性的通用湍流动力系统下发展起来的，这类系统在应用数学和工程中的许多应用中具有代表性。在指导性随机三元模型上验证了降阶模型一般框架的有效性。本文还综述了湍流射流和涡旋组合在大气和海洋两层斜压湍流中的最新应用。
Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering. The existence of a strange attractor in the turbulent systems containing a large number of positive Lyapunov exponents results in a rapid growth of small uncertainties, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification in turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in their nonlinear responses to changes in external forcing parameters or uncertain initial data. One central issue in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity). A general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems is discussed here. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed.