Numerical study of chaos based on a shell model

Area of Science:

• Fluid Dynamics
• Nonlinear Dynamics
• Computational Physics

Background:

• Turbulence driven by thermal instability, specifically Rayleigh-Benard convection, is a complex phenomenon.
• Understanding the cascade and chaotic behavior in strong turbulence requires robust computational models.

Purpose of the Study:

• Introduce and analyze a novel shell model for simulating turbulence.
• Investigate the characteristics of turbulence, including chaos, in high Rayleigh number regimes.
• Compare the shell model's performance and results against the established Gledzer-Ohkitani-Yamada (GOY) model.

Main Methods:

• Development of a shell model equation to describe turbulent cascade and chaos.
• Numerical simulation of the shell model for strong turbulence.
• Analysis of turbulence characteristics: energy, spectra, probability distributions, and Lyapunov exponents.
• Examination of the impact of numerical integration errors on simulation accuracy.

Main Results:

• The shell model successfully describes cascade and chaos in strong turbulence.
• Evaluated quantities include energy, standard deviation, wave and frequency spectra, and Lyapunov exponent distributions.
• A key finding is the insensitivity of the maximum Lyapunov exponent to truncation errors.
• No clear correlation was found between numerical accuracy and the accuracy of evaluated quantities due to error-nonlinear interactions.

Conclusions:

• The introduced shell model provides a viable framework for studying turbulent phenomena.
• The insensitivity of the maximum Lyapunov exponent to numerical errors is a significant observation for computational fluid dynamics.
• Further research can explore the model's applicability to other turbulent systems.