### Abstract

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 a Theorem that 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. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below. Corollary 1 and Corollary 2 illustrate 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.

Original language | English (US) |
---|---|

Pages (from-to) | 8937-8941 |

Number of pages | 5 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 112 |

Issue number | 29 |

DOIs | |

State | Published - Jul 21 2015 |

### Fingerprint

### Keywords

- Fluctuations
- Interaction statistical symmetries
- Mean

### ASJC Scopus subject areas

- General

### Cite this

**Statistical energy conservation principle for inhomogeneous turbulent dynamical systems.** / Majda, Andrew J.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Statistical energy conservation principle for inhomogeneous turbulent dynamical systems

AU - Majda, Andrew J.

PY - 2015/7/21

Y1 - 2015/7/21

N2 - 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 a Theorem that 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. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below. Corollary 1 and Corollary 2 illustrate 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.

AB - 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 a Theorem that 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. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below. Corollary 1 and Corollary 2 illustrate 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.

KW - Fluctuations

KW - Interaction statistical symmetries

KW - Mean

UR - http://www.scopus.com/inward/record.url?scp=84937848931&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937848931&partnerID=8YFLogxK

U2 - 10.1073/pnas.1510465112

DO - 10.1073/pnas.1510465112

M3 - Article

AN - SCOPUS:84937848931

VL - 112

SP - 8937

EP - 8941

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 29

ER -