Department of Mathematics and Climate, Atmospheric and Oceanic Sciences, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
Proc Natl Acad Sci U S A. 2013 Aug 20;110(34):13705-10. doi: 10.1073/pnas.1313065110. Epub 2013 Aug 5.
A framework for low-order predictive statistical modeling and uncertainty quantification in turbulent dynamical systems is developed here. These reduced-order, modified quasilinear Gaussian (ROMQG) algorithms apply to turbulent dynamical systems in which there is significant linear instability or linear nonnormal dynamics in the unperturbed system and energy-conserving nonlinear interactions that transfer energy from the unstable modes to the stable modes where dissipation occurs, resulting in a statistical steady state; such turbulent dynamical systems are ubiquitous in geophysical and engineering turbulence. The ROMQG method involves constructing a low-order, nonlinear, dynamical system for the mean and covariance statistics in the reduced subspace that has the unperturbed statistics as a stable fixed point and optimally incorporates the indirect effect of non-Gaussian third-order statistics for the unperturbed system in a systematic calibration stage. This calibration procedure is achieved through information involving only the mean and covariance statistics for the unperturbed equilibrium. The performance of the ROMQG algorithm is assessed on two stringent test cases: the 40-mode Lorenz 96 model mimicking midlatitude atmospheric turbulence and two-layer baroclinic models for high-latitude ocean turbulence with over 125,000 degrees of freedom. In the Lorenz 96 model, the ROMQG algorithm with just a single mode captures the transient response to random or deterministic forcing. For the baroclinic ocean turbulence models, the inexpensive ROMQG algorithm with 252 modes, less than 0.2% of the total, captures the nonlinear response of the energy, the heat flux, and even the one-dimensional energy and heat flux spectra.
这里开发了一种用于湍流动力系统的低阶预测统计建模和不确定性量化的框架。这些降阶、修正拟线性高斯（ROMQG）算法适用于存在未扰系统中显著线性不稳定性或线性非正常动力学以及能量守恒非线性相互作用的湍流动力系统，这些相互作用将能量从不稳定模式转移到稳定模式，在稳定模式中发生耗散，导致统计稳态；这种湍流动力系统在地球物理和工程湍流中普遍存在。ROMQG 方法涉及构建一个低阶、非线性动力系统，用于降低子空间中的均值和协方差统计，该系统具有未扰统计作为稳定的固定点，并在系统校准阶段以最佳方式纳入未扰系统的非高斯三阶统计的间接影响。这种校准过程是通过仅涉及未扰平衡的均值和协方差统计的信息来实现的。ROMQG 算法的性能在两个严格的测试案例中进行了评估：40 模式的 Lorenz 96 模型模拟中纬度大气湍流和两层斜压海洋湍流模型，自由度超过 125000。在 Lorenz 96 模型中，仅使用单个模式的 ROMQG 算法可以捕获对随机或确定性强迫的瞬态响应。对于斜压海洋湍流模型，具有 252 个模式的廉价 ROMQG 算法，不到总模式的 0.2%，可以捕获能量、热通量甚至一维能量和热通量谱的非线性响应。
