Stochastic Lorenz dynamics and the mean wind reversals in Rayleigh-Bénard Convection

The Lorenz equations are a severe Galerkin-truncation of the Oberbeck-Boussinesq equations describing Rayleigh-Bénard convection (RBC), or the buoyancy-driven flow between parallel isothermal plates. Here, we use the chaotic lobe-switching behavior of the stochastic Lorenz equations as an analogue of the mean wind reversals in the experiments of Sreenivasan et al. (Phys Rev E 65, 056306, 2002). The boundary layers are central to the transport in RBC, and we use the stochastic Lorenz system to model the associated small-scale turbulence embedded within a large-scale convective roll. Forcing the Z-equation with additive Gaussian white noise (GWN) generates ergodic invariant measures, for all parameters, while transparently manipulating the boundary layers. Long-time numerical simulations yield a probability distribution for lobe inter-switch timings that exhibits multifractal behavior. Filtering that signal to frequencies with Brownian power spectral density roll-off creates a Gaussian distribution, mirroring laboratory measurements. Multifractal analysis reveals a bent mass-scaling exponent spectrum and non-linear generalized dimensions, while the classical Hurst exponent gives Brownian second-moment statistics. A simple Cantor-cascade reproduces these values, showing that multiplicative intermittency strongly influences the statistics. This demonstrates that a GWN-forced Lorenz system is a faithful, low-dimensional surrogate for mean-wind reversals in RBC.