Stochastic Lorenz dynamics and wind reversals in Rayleigh-B\'enard Convection

TL;DR

This paper investigates the stochastic Lorenz equations as a simplified model for understanding wind reversals in Rayleigh-Bénard convection, revealing multifractal behavior and turbulence-like intermittency through long-term numerical simulations.

Contribution

It establishes a connection between stochastic Lorenz dynamics and experimental wind reversals, demonstrating multifractality and turbulence characteristics in a low-dimensional model.

Findings

Probability distribution of lobe switching times shows non-Gaussian, multifractal behavior.

Simulations match laboratory measurements in the Gaussian frequency range.

Multifractality arises from multiplicative intermittency similar to turbulence.

Abstract

The Lorenz equations [1] are a severe Galerkin-truncation of the Oberbeck-Boussinesq (OB) equations describing Rayleigh-B\'enard 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.