Statistical energy conservation principle for inhomogeneous turbulent dynamical systems

• Published in: Proceedings of the National Academy of Sciences - PNAS Vol. 112; no. 29; pp. 8937 - 8941
• Main Author: Majda, Andrew J.
• Format: Journal Article
• Language: English
• Published: United States National Academy of Sciences 21.07.2015; National Acad Sciences
• Subjects: Anisotropy; Dynamical systems; Energy conservation; Energy dissipation; Oceanography; Physical Sciences; Theorems; Turbulence models; fluctuations; mean; interaction statistical symmetries
• ISSN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.1510465112

Understanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science with important societal impact. In such inhomogeneous turbulent dynamical systems there is a large dimensional phase space with a large dimension of unstable directions where a large-scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other. These complex features strongly impact practical prediction and uncertainty quantification. A systematic energy conservation principle is developed here in aTheoremthat precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations. This statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence. This result applies to general inhomogeneous turbulent dynamical systems including the above applications. TheTheoreminvolves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below.Corollary 1andCorollary 2illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds, provided that the negative symmetric dissipation matrix is diagonal in a suitable basis. Implications of the energy principle for low-order closure modeling and automatic estimates for the single point variance are discussed below.