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

Di Qi, Andrew J. Majda

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1-53
Number of pages53
JournalJournal of Nonlinear Science
DOIs
StateAccepted/In press - Apr 23 2018

Fingerprint

Geophysical Flows
Uncertainty Quantification
Turbulent Flow
Trajectories
Energy
Flow structure
Fluctuations
Topography
Conservation
Ensemble
Uncertainty
Statistics
Trajectory
Computer simulation
Exponential Growth
Steady-state Solution
Climate
Weather
Positive definite
Forcing

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.

In: Journal of Nonlinear Science, 23.04.2018, p. 1-53.

Research output: Contribution to journalArticle

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