Dynamics and Transportation Mechanism in Two-Coupled Large-Scale Circulations ---A Novel Theoretical Approach

The large-scale circulation (LSC) in Rayleigh-Bénard Convection has been extensively studied including two stochastic models which treat diffusion of the LSC strength in a potential well [1,2] and other deterministic models with chaotic solutions [3,4]. Therein, the rich dynamics and inherent physics of LSC can be well-described by stochastic models proposed by Brown and Ahlers [5]. However, the majority of LSC found in nature entails ceaseless interaction with adjacent LSC, such as ocean circulation and general atmospheric circulation which always exist as multiple flows coupled with each other. Thus, existing models are inadequate to describe the coupling mechanism when it comes to multiple flows.

Based on single-roll LSC model BA07 [4], taking volume average of physically important terms rooted in Navier-Stocks (NS) equations, we formulate a modified one to account for double-layer LSC dynamics and coupling mechanism, i.e., thermal coupling and viscous coupling, in two-layer fluid system. The updated model [6] contains four stochastic ordinary differential equations to describe two degrees of freedom of each roll. We numerically calculate these equations and obtain the probability distribution function (P.D.F) of relatively azimuthal orientation. The results of numerical calculation on the stochastic model are in good accordance with previous experiment results in two-layer RBC [7], which shows a peak around zero degrees and a small one close to Pi. It can be concluded that, in a two-layered system consisting of heavy fluid in the lower layer and a light fluid as the upper layer, the rolls prefer to stay in the thermal coupled mode, which may have broad implications in a wide-spread of physics problems.