[Paper Review] Chaotic mean wind in turbulent thermal convection and long-term correlations in solar activity

A. Bershadskii|ArXiv.org|Aug 27, 2009

Solar and Space Plasma Dynamics11 references3 citations

TL;DR

This paper demonstrates that the mean wind in turbulent thermal convection exhibits chaotic dynamics with a positive largest Lyapunov exponent and exponentially decaying correlation functions, indicating long-term memory. It further shows that the daily sunspot number exhibits identical statistical properties—exponential decay and positive Lyapunov exponent—suggesting that solar activity's long-term correlations are an imprint of chaotic mean winds in the solar convection zone.

ABSTRACT

It is shown that correlation function of the mean wind velocity in a turbulent thermal convection (Rayleigh number $Ra \sim 10^{11}$) exhibits exponential decay with a very long correlation time, while corresponding largest Lyapunov exponent is certainly positive. These results together with the reconstructed phase portrait indicate presence of a chaotic component in the examined mean wind. Telegraph approximation is also used to study relative contribution of the chaotic and stochastic components to the mean wind fluctuations and an equilibrium between these components has been studied. Since solar activity is based on the thermal convection processes, it is reasoned that the observed solar activity long-term correlations can be an imprint of the mean wind chaotic properties. In particular, correlation function of the daily sunspots number exhibits exponential decay with a very long correlation time and corresponding largest Lyapunov exponent is certainly positive, also relative contribution of the chaotic and stochastic components follows the same pattern as for the convection mean wind.

Motivation & Objective

• To investigate the dynamical origin of long-term correlations in solar activity.
• To determine whether the observed long-term correlations in solar activity stem from chaotic dynamics in the solar convection zone.
• To compare statistical properties of mean wind in laboratory thermal convection with those of solar activity.
• To distinguish between chaotic and stochastic components in mean wind and solar activity fluctuations using telegraph approximation.
• To establish a physical link between turbulent thermal convection and the generation of large-scale solar activity patterns.

Proposed method

• Analyzed autocorrelation functions of mean wind velocity from Rayleigh-Bénard convection experiments (Ra ~ 10^11) to identify exponential decay with long correlation time τ₀ ~ 600 s.
• Calculated the largest Lyapunov exponent (λ_max) using the Wolf algorithm on the same time series to detect deterministic chaos.
• Reconstructed the phase portrait from noise-reduced time series to visualize chaotic attractor structure.
• Applied telegraph approximation to quantify the relative contributions of chaotic and stochastic components in mean wind fluctuations.
• Analyzed daily sunspot number (SSN) data from 1850–1944 and 1964–2008 using maximum entropy method to assess correlation functions and Lyapunov exponents.
• Used semi-log scales to identify exponential decay in SSN correlation functions and compared decay rates and Lyapunov exponents with those of the convection mean wind.

Experimental results

Research questions

• RQ1 Does the mean wind in turbulent thermal convection exhibit chaotic dynamics despite a turbulent background?
• RQ2 What is the relationship between the long correlation time of the mean wind and its chaotic nature?
• RQ3 Can the long-term correlations observed in solar activity be explained by chaotic behavior in the solar convection zone?
• RQ4 How do the statistical properties of the mean wind (correlation function, Lyapunov exponent) compare to those of the daily sunspot number?
• RQ5 What is the relative contribution of chaotic versus stochastic components in the dynamics of the mean wind and solar activity?

Key findings

• The autocorrelation function of the mean wind velocity in turbulent convection exhibits exponential decay with a long correlation time τ₀ ≈ 600 s, significantly longer than the circulation period T_c ≈ 30 s.
• The largest Lyapunov exponent for the mean wind is positive (λ_max ≈ 0.0023 s⁻¹), confirming the presence of deterministic chaos.
• The phase portrait reconstructed from noise-reduced data reveals a chaotic attractor, supporting the chaotic nature of the mean wind dynamics.
• The daily sunspot number (SSN) correlation function also shows exponential decay with a long correlation time τ₀ ≈ 660 days (1850–1944) and τ₀ ≈ 670 days (1964–2008), consistent with the convection mean wind.
• The largest Lyapunov exponent for the daily SSN data is positive (λ_max ≈ 0.0047 d⁻¹), indicating chaotic dynamics in solar activity at daily to annual timescales.
• The normalized correlation time τ₀/T_c ≈ 20 for the convection mean wind and ≈ 3 for solar activity, suggesting a common underlying mechanism tied to large-scale flow dynamics.