### Abstract

Turbulent dynamical systems with a large phase space and a high degree of instabilities are ubiquitous in climate science and engineering applications. Statistical uncertainty quantification (UQ) to the response to the change in forcing or uncertain initial data in such complex turbulent systems requires the use of imperfect models due to the lack of both physical understanding and the overwhelming computational demands of Monte Carlo simulation with a large-dimensional phase space. Thus, the systematic development of reduced low-order imperfect statistical models for UQ in turbulent dynamical systems is a grand challenge. This paper applies a recent mathematical strategy for calibrating imperfect models in a training phase and accurately predicting the response by combining information theory and linear statistical response theory in a systematic fashion. A systematic hierarchy of simple statistical imperfect closure schemes for UQ for these problems is designed and tested which are built through new local and global statistical energy conservation principles combined with statistical equilibrium fidelity. The forty mode Lorenz 96 (L-96) model which mimics forced baroclinic turbulence is utilized as a test bed for the calibration and predicting phases for the hierarchy of computationally cheap imperfect closure models both in the full phase space and in a reduced three-dimensional subspace containing the most energetic modes. In all of phase spaces, the nonlinear response of the true model is captured accurately for the mean and variance by the systematic closure model, while alternative methods based on the fluctuation-dissipation theorem alone are much less accurate. For reduced-order model for UQ in the three-dimensional subspace for L-96, the systematic low-order imperfect closure models coupled with the training strategy provide the highest predictive skill over other existing methods for general forced response yet have simple design principles based on a statistical global energy equation. The systematic imperfect closure models and the calibration strategies for UQ for the L-96 model serve as a new template for similar strategies for UQ with model error in vastly more complex realistic turbulent dynamical systems.

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

Pages (from-to) | 233-285 |

Number of pages | 53 |

Journal | Journal of Nonlinear Science |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2016 |

### Fingerprint

### Keywords

- Information metric
- Linear response theory
- Low-order statistical closure models
- Turbulent systems

### ASJC Scopus subject areas

- Applied Mathematics
- Modeling and Simulation
- Engineering(all)

### Cite this

**Improving Prediction Skill of Imperfect Turbulent Models Through Statistical Response and Information Theory.** / Majda, Andrew J.; Qi, Di.

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 26, no. 1, pp. 233-285. https://doi.org/10.1007/s00332-015-9274-5

}

TY - JOUR

T1 - Improving Prediction Skill of Imperfect Turbulent Models Through Statistical Response and Information Theory

AU - Majda, Andrew J.

AU - Qi, Di

PY - 2016/2/1

Y1 - 2016/2/1

N2 - Turbulent dynamical systems with a large phase space and a high degree of instabilities are ubiquitous in climate science and engineering applications. Statistical uncertainty quantification (UQ) to the response to the change in forcing or uncertain initial data in such complex turbulent systems requires the use of imperfect models due to the lack of both physical understanding and the overwhelming computational demands of Monte Carlo simulation with a large-dimensional phase space. Thus, the systematic development of reduced low-order imperfect statistical models for UQ in turbulent dynamical systems is a grand challenge. This paper applies a recent mathematical strategy for calibrating imperfect models in a training phase and accurately predicting the response by combining information theory and linear statistical response theory in a systematic fashion. A systematic hierarchy of simple statistical imperfect closure schemes for UQ for these problems is designed and tested which are built through new local and global statistical energy conservation principles combined with statistical equilibrium fidelity. The forty mode Lorenz 96 (L-96) model which mimics forced baroclinic turbulence is utilized as a test bed for the calibration and predicting phases for the hierarchy of computationally cheap imperfect closure models both in the full phase space and in a reduced three-dimensional subspace containing the most energetic modes. In all of phase spaces, the nonlinear response of the true model is captured accurately for the mean and variance by the systematic closure model, while alternative methods based on the fluctuation-dissipation theorem alone are much less accurate. For reduced-order model for UQ in the three-dimensional subspace for L-96, the systematic low-order imperfect closure models coupled with the training strategy provide the highest predictive skill over other existing methods for general forced response yet have simple design principles based on a statistical global energy equation. The systematic imperfect closure models and the calibration strategies for UQ for the L-96 model serve as a new template for similar strategies for UQ with model error in vastly more complex realistic turbulent dynamical systems.

AB - Turbulent dynamical systems with a large phase space and a high degree of instabilities are ubiquitous in climate science and engineering applications. Statistical uncertainty quantification (UQ) to the response to the change in forcing or uncertain initial data in such complex turbulent systems requires the use of imperfect models due to the lack of both physical understanding and the overwhelming computational demands of Monte Carlo simulation with a large-dimensional phase space. Thus, the systematic development of reduced low-order imperfect statistical models for UQ in turbulent dynamical systems is a grand challenge. This paper applies a recent mathematical strategy for calibrating imperfect models in a training phase and accurately predicting the response by combining information theory and linear statistical response theory in a systematic fashion. A systematic hierarchy of simple statistical imperfect closure schemes for UQ for these problems is designed and tested which are built through new local and global statistical energy conservation principles combined with statistical equilibrium fidelity. The forty mode Lorenz 96 (L-96) model which mimics forced baroclinic turbulence is utilized as a test bed for the calibration and predicting phases for the hierarchy of computationally cheap imperfect closure models both in the full phase space and in a reduced three-dimensional subspace containing the most energetic modes. In all of phase spaces, the nonlinear response of the true model is captured accurately for the mean and variance by the systematic closure model, while alternative methods based on the fluctuation-dissipation theorem alone are much less accurate. For reduced-order model for UQ in the three-dimensional subspace for L-96, the systematic low-order imperfect closure models coupled with the training strategy provide the highest predictive skill over other existing methods for general forced response yet have simple design principles based on a statistical global energy equation. The systematic imperfect closure models and the calibration strategies for UQ for the L-96 model serve as a new template for similar strategies for UQ with model error in vastly more complex realistic turbulent dynamical systems.

KW - Information metric

KW - Linear response theory

KW - Low-order statistical closure models

KW - Turbulent systems

UR - http://www.scopus.com/inward/record.url?scp=84953638007&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84953638007&partnerID=8YFLogxK

U2 - 10.1007/s00332-015-9274-5

DO - 10.1007/s00332-015-9274-5

M3 - Article

VL - 26

SP - 233

EP - 285

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 1

ER -