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.
The ubiquity and importance of thermal convection in technical and natural settings, particularly in the astroand geophysical sciences, is very well known (e.g., [3][4][5][6]). The primary aim of studies of turbulent Rayleigh-Bénard convection between two horizontal surfaces has been to determine the Nusselt number, Nu, defined as the ratio of total heat flux to conductive heat flux, as a function of the Rayleigh number Ra, the Prandtl number Pr, and the aspect ratio of the convection cell Γ. The principal control parameter is the Rayleigh number Ra = gα∆T H 3 /νκ, which is the ratio of buoyancy to viscous forces, where g is the acceleration due to gravity, α the thermal expansion coefficient of the fluid, ν the kinematic viscosity, κ the thermal diffusivity, and ∆T is the temperature difference across the fluid layer of depth H. The Prandtl number, Pr = ν/κ, is considered a property of the fluid and the aspect ratio of the cell, Γ, defined as the ratio of its width to height, is considered a property of the experiment. For Ra ≫ 1 and fixed Pr and Γ, a relation is usually sought in the form of a power law: Nu = A(Pr, Γ)Ra γ .
Experiments, simulations, theory and analysis have focused on the nature of the power law at higher and higher Ra. Central here is how the boundary layers (BLs) at the cold upper and warm lower surfaces of the cell interact with the core flow in the interior; for a succinct history and description see Doering [6] and references therein.
An important component of this interaction is the “mean wind”, which is the large-scale-circulation of the flow superimposed on the background turbulence. The mean wind has a scale of order the experimental cell, and it is the transitions in the mean wind measured in a cylindrical cell by Sreenivasan et al. [2] that we focus upon here. Whether these transitions are wholesale reversals, that is flow cessation and reversal, or azimuthal rotations, we refer to them as reversals in the sense of the sign of the vertical velocity (see Figure 1 and the end of §I).
It is important to appreciate the nature of these experiments. Prior to measuring the time series of reversals, Sreenivasan et al. [2] held their apparatus at constant experimental conditions for nearly a month. They then measured abrupt reversals of the mean wind as shown in their Figure 1 (reproduced here as Figure 1), where they plot an increment of their time series of the mean wind for Ra = 1.5 × 10 11 . Finally, to generate reliable statistics, they recorded reversals continuously for 5.5 days. Experiments were performed for Ra between about 10 8 and 10 13 , but the most extensive for Ra = 1.5 × 10 11 , where they focused their analysis.
The velocity of the mean wind was about V M = 7 cm/s, the circumference of the apparatus was about 200 cm, so on average the fluid traversed the cell in 30 s, which is long relative to the transition between the two mean wind directions seen in Figure 1, which spans about 330 traversal times with the entire time series spanning some 16,000 traversal times. In contrast, high resolution direct numerical simulations of the full Oberbeck-Boussinesq equations are run for 100-1000 turnover times and, depending on resolution and dimensionality, can take 10-20 days using parallel HPC methods on GPUs (See e.g., [7]). In this context the numerical challenge was addressed by Benzi and Verzicco [8] who effectively decreased Pr by artificially increasing thermal fluctuations, thereby allowing them to generate statistics for a more modest value of Ra = 6 × 10 5 . They interpreted the transitions in terms of a bistable stochastic system, as did Sreenivasan et al. [2]. Another approach, using socalled augmented Lorenz equations, treats the random transitions in the rotation of a gas turbine [9], akin to the Malkus-Howard chaotic waterwheel, as an analogue to the wind reversals in the experiments of Sreenivasan et al. [2]. Here we study the mathematical aspects of these wind reversal experiments using a stochastic variant of the canonical reduced model for convection due to Lorenz, which is derived directly from the Oberbeck-Boussinesq equations [1,10].
Experiments with a larger array of probes have characterized changes in the vertical velocity as either due to wholesale reversal (i.e., cessation and reversal) or azimuthal rotation [11]. However, Brown and Ahlers [11] define an average rate of occurrence of reorientations, and note that it (a) depended strongly on the definition of the reorientation parameters, and (b) was extremely sensitive to minor changes of the apparatus. Therefore, while we recognize the existence of reversals and rotations, we remain agnostic concerning their relative occurrence in the experiments of Sreenivasan et al. [2] and, as noted above, we refer to the transitions in the vertical velocity in Figure 1 as reversals.
rt-term is that elocity lower le that r Rayers of yleigh elation liable; uld be wind resent a few he obght be e cona torus large
This content is AI-processed based on open access ArXiv data.