### Abstract

Statistical bounds controlling the total fluctuations in mean and variance about a basic steady-state solution are developed for the truncated barotropic flow over topography. Statistical ensemble prediction is an important topic in weather and climate research. Here, the evolution of an ensemble of trajectories is considered using statistical instability analysis and is compared and contrasted with the classical deterministic instability for the growth of perturbations in one pointwise trajectory. The maximum growth of the total statistics in fluctuations is derived relying on the statistical conservation principle of the pseudo-energy. The saturation bound of the statistical mean fluctuation and variance in the unstable regimes with non-positive-definite pseudo-energy is achieved by linking with a class of stable reference states and minimizing the stable statistical energy. Two cases with dependence on initial statistical uncertainty and on external forcing and dissipation are compared and unified under a consistent statistical stability framework. The flow structures and statistical stability bounds are illustrated and verified by numerical simulations among a wide range of dynamical regimes, where subtle transient statistical instability exists in general with positive short-time exponential growth in the covariance even when the pseudo-energy is positive-definite. Among the various scenarios in this paper, there exist strong forward and backward energy exchanges between different scales which are estimated by the rigorous statistical bounds.

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

Pages (from-to) | 1-53 |

Number of pages | 53 |

Journal | Journal of Nonlinear Science |

DOIs | |

State | Accepted/In press - Apr 23 2018 |

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### Keywords

- Statistical energy conservation
- Statistical stability analysis
- Topographic barotropic equations

### ASJC Scopus subject areas

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

### Cite this

**Rigorous Statistical Bounds in Uncertainty Quantification for One-Layer Turbulent Geophysical Flows.** / Qi, Di; Majda, Andrew J.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Rigorous Statistical Bounds in Uncertainty Quantification for One-Layer Turbulent Geophysical Flows

AU - Qi, Di

AU - Majda, Andrew J.

PY - 2018/4/23

Y1 - 2018/4/23

N2 - Statistical bounds controlling the total fluctuations in mean and variance about a basic steady-state solution are developed for the truncated barotropic flow over topography. Statistical ensemble prediction is an important topic in weather and climate research. Here, the evolution of an ensemble of trajectories is considered using statistical instability analysis and is compared and contrasted with the classical deterministic instability for the growth of perturbations in one pointwise trajectory. The maximum growth of the total statistics in fluctuations is derived relying on the statistical conservation principle of the pseudo-energy. The saturation bound of the statistical mean fluctuation and variance in the unstable regimes with non-positive-definite pseudo-energy is achieved by linking with a class of stable reference states and minimizing the stable statistical energy. Two cases with dependence on initial statistical uncertainty and on external forcing and dissipation are compared and unified under a consistent statistical stability framework. The flow structures and statistical stability bounds are illustrated and verified by numerical simulations among a wide range of dynamical regimes, where subtle transient statistical instability exists in general with positive short-time exponential growth in the covariance even when the pseudo-energy is positive-definite. Among the various scenarios in this paper, there exist strong forward and backward energy exchanges between different scales which are estimated by the rigorous statistical bounds.

AB - Statistical bounds controlling the total fluctuations in mean and variance about a basic steady-state solution are developed for the truncated barotropic flow over topography. Statistical ensemble prediction is an important topic in weather and climate research. Here, the evolution of an ensemble of trajectories is considered using statistical instability analysis and is compared and contrasted with the classical deterministic instability for the growth of perturbations in one pointwise trajectory. The maximum growth of the total statistics in fluctuations is derived relying on the statistical conservation principle of the pseudo-energy. The saturation bound of the statistical mean fluctuation and variance in the unstable regimes with non-positive-definite pseudo-energy is achieved by linking with a class of stable reference states and minimizing the stable statistical energy. Two cases with dependence on initial statistical uncertainty and on external forcing and dissipation are compared and unified under a consistent statistical stability framework. The flow structures and statistical stability bounds are illustrated and verified by numerical simulations among a wide range of dynamical regimes, where subtle transient statistical instability exists in general with positive short-time exponential growth in the covariance even when the pseudo-energy is positive-definite. Among the various scenarios in this paper, there exist strong forward and backward energy exchanges between different scales which are estimated by the rigorous statistical bounds.

KW - Statistical energy conservation

KW - Statistical stability analysis

KW - Topographic barotropic equations

UR - http://www.scopus.com/inward/record.url?scp=85045841393&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045841393&partnerID=8YFLogxK

U2 - 10.1007/s00332-018-9462-1

DO - 10.1007/s00332-018-9462-1

M3 - Article

AN - SCOPUS:85045841393

SP - 1

EP - 53

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

ER -