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Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems
1CSIRO Environment, Aspendale, Melbourne 3195, Australia.
Entropy (Basel, Switzerland)
|February 25, 2023
Summary
This study analyzes dynamical vectors for geophysical fluid models, revealing how singular vectors (SVs) connect to finite-time normal modes (FTNMs). It establishes asymptotic convergence between covariant Lyapunov vectors (CLVs) and FTNMs for improved instability characterization.
Area of Science:
- Geophysical fluid dynamics
- Dynamical systems theory
- Numerical weather prediction
Background:
- Instability characterization in geophysical fluid dynamics is crucial for ensemble prediction systems.
- Various dynamical vectors, including covariant Lyapunov vectors (CLVs), orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors, and finite-time normal modes (FTNMs), are used to describe system dynamics and instability.
- Understanding the relationships between these vectors is key to improving prediction models.
Purpose of the Study:
- To analyze dynamical vectors characterizing instability for geophysical fluid dynamical models.
- To examine the relationships between CLVs, OLVs, SVs, Floquet vectors, and FTNMs in periodic and aperiodic systems.
- To establish the asymptotic convergence of CLVs and FTNMs and propose efficient numerical methods.
Main Methods:
- Analysis of relationships between CLVs, OLVs, SVs, Floquet vectors, and FTNMs.
- Application of the Oseledec theorem to connect OLVs and CLVs.
- Investigation of covariant properties and phase-space independence of vectors.
- Examination of conditions for validity, including ergodicity and boundedness.
- Development of efficient numerical methods for calculating leading CLVs.
Main Results:
- Singular vectors (SVs) equate with unit norm finite-time normal modes (FTNMs) at critical times in the FTNM coefficient phase-space.
- Covariant Lyapunov vectors (CLVs) are connected to FTNMs in the long-time limit, approaching orthonormal Lyapunov vectors (OLVs).
- Asymptotic convergence of CLVs and FTNMs is established using their covariant properties and phase-space independence.
- Efficient numerical methods for calculating leading CLVs are proposed.
- Norm-independent finite-time versions of Kolmogorov-Sinai entropy production and Kaplan-Yorke dimension are presented.
Conclusions:
- The study establishes a theoretical framework connecting various dynamical vectors, particularly SVs and CLVs to FTNMs.
- The findings provide a deeper understanding of instability characterization in geophysical fluid dynamics.
- Efficient methods for calculating CLVs and new formulations for entropy and dimension offer practical advancements for prediction models.
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