Chaotic Measures as an Alternative to Spectral Measures for Analysing Turbulent Flow
Abstract
:1. Introduction
Advantages of Chaotic Measures Applied to Turbulence
2. Chaotic Measures of Turbulence
2.1. Lyapunov Exponents
2.2. KS Entropy
3. Theoretical Predictions
3.1. Theory of Lorenz
3.2. Theory of Ruelle
3.3. Multifractal Corrections
3.4. Predictions for Attractor Dimension and KS Entropy
3.5. Two Dimensional Turbulence
4. Results from Simulations
4.1. Assessing Theoretical Predictions
4.1.1. Ruelle’s Prediction
4.1.2. KS Entropy and the Lyapunov Spectrum
4.2. Predictability
4.2.1. Error Spectrum
4.2.2. Cascades of Error
4.2.3. Linear Growth of Error
4.2.4. FTLE Statistics
4.3. Efficient Simulation
4.4. Insights
4.4.1. Theoretical Extensions
4.4.2. Changes in System Dimension
4.4.3. Importance of Vortex Stretching
4.4.4. MHD
5. Discussion and Conclusions
- Turbulence is a multiscale phenomena both spatially and temporally. This is true even in chaos, where the different timescales interact to lead to a long-term large length scale predictability co-existing with increasingly rapid error growth at the smallest scales.
- The Eulerian chaos and vortex stretching appear to be fundamentally linked, and may contribute to the changes seen in different dimensions.
- The Lyapunov spectrum may emerge as a practical measure for a complete description of the fluid flow.
- Chaotic measures, especially the maximum Lyapunov exponent, are more stable than spectral quantities and show spatial features that are hard to discern from spectral quantities alone.
- By using chaotic measures, we can quantify simulations in a model-free way that does not rely on a priori unknown spectral quantities and thus improve computational efficiency.
- Chaotic measures can be used to detect phase transitions.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Ho, R.D.J.G.; Clark, D.; Berera, A. Chaotic Measures as an Alternative to Spectral Measures for Analysing Turbulent Flow. Atmosphere 2024, 15, 1053. https://doi.org/10.3390/atmos15091053
Ho RDJG, Clark D, Berera A. Chaotic Measures as an Alternative to Spectral Measures for Analysing Turbulent Flow. Atmosphere. 2024; 15(9):1053. https://doi.org/10.3390/atmos15091053
Chicago/Turabian StyleHo, Richard D. J. G., Daniel Clark, and Arjun Berera. 2024. "Chaotic Measures as an Alternative to Spectral Measures for Analysing Turbulent Flow" Atmosphere 15, no. 9: 1053. https://doi.org/10.3390/atmos15091053