Stochastic Lorenz dynamics and wind reversals in Rayleigh-Bénard Convection

TL;DR

This paper addresses mean-wind reversals in Rayleigh-Bénard convection by proposing a stochastic Lorenz surrogate with additive Gaussian noise in the Z-equation, yielding a transformed system that preserves key RBC dynamics. Long-time simulations reveal non-Gaussian lobe-switch timings with multifractal statistics, while band-limited data display Brownian-second-moment behavior, linking micro-scale turbulence to macro-scale reversals. A Cantor-set cascade provides analytic support for the observed multifractality, demonstrating that intermittent, multiplicative processes shape the reversal timing statistics. The results suggest the stochastic Lorenz model is a faithful, low-dimensional surrogate for RBC reversals and offers a practical framework for probing boundary-layer interactions in high-Rayleigh-number convection.

Abstract

The Lorenz equations [1] are a severe Galerkin-truncation of the Oberbeck-Boussinesq (OB) equations describing Rayleigh-Bénard convection (RBC). Here we examine the mathematical connections between the chaotic lobe-switching behavior of a stochastic form of the Lorenz equations, that model the interaction between the thermal boundary layers and the core circulation, and the mean wind reversals in the experiments of Sreenivasan et al. [2]. Long-time numerical simulations of these stochastic equations, not easily accessible with the OB equations, yield a probability distribution for lobe inter-switch timings that exhibits non-Gaussian, multifractal behavior. In the Gaussian frequency range the simulations mirror the laboratory measurements and the classical Hurst exponent and quadratic variation show Brownian second-moment statistics. Further scrutiny reveals a non-linear cumulant generating function, or moment-exponent function, and thus multifractality. A simple generalized two-scale Cantor-cascade analysis reproduces these properties, showing that multiplicative intermittency, characteristic of turbulence, strongly influences the statistics. This demonstrates that this stochastic Lorenz system is a faithful, low-dimensional surrogate for mean-wind reversals in RBC.

Figures (16)

• Figure 1: For $\mathrm{Ra}=1.5 \times 10^{11}$, the wind velocity as a function of time (for a total of about 3 hours) reproduced from Fig. 1 of Sreenivasan et al. Sreenivasan2002. They measured this by correlating two neighboring temperature probes on the center plane of the convection cell outside of the sidewall boundary layers. They highlighted that the wind switches direction abruptly with no apparent order. The arrows on the right indicate the magnitude of the wind during its persistence in one direction. See Sreenivasan2002 for more details of the experiment.
• Figure 2: The time series of $X$ after being back transformed from a realization of Eqs. \ref{['eq:SLT']} by a non-adaptive Euler-Maruyama scheme. The simulation time is $600/\chi$, the reduced Rayleigh number is $\rho=14$, and the noise amplitude is $\hat{\alpha}=5$. Compare with Fig. \ref{['fig:a']}.
• Figure 3: The PSD versus the frequencies (cycles per switch) of the process $G_n$ with a light blue line fitted as a visual for the decay. The bottom-left inset displays the raw $\log_{10}$(PDF) of the same process, with the indigo curve being the expected quadratic fit for a true Gaussian. The black line graph above the PDF is the full time series data of $G_n$.
• Figure 4: The power law exponent, $\widetilde{\beta}$, across a wide range of time-scales (inverse frequencies). Window sizes for exponent approximation are 0.75 decades wide, with step sizes between tiles being 0.1 decades wide.
• Figure 5: The $\log_{10}$(PDF) of $\widetilde{G}_n$ under the same parameter regime---only the values of $G_n$ within the $\left[10^{-4},10^{-2} \right]$ cycles per switch frequency range (in frequency space) are kept.