The large-scale circulation in Rayleigh-B\'enard convection: A dynamical system subjected to the fury of turbulence

Guenter Ahlers

{Department of Physics and iQCD, University of California, Santa Barbara,

Understanding turbulent Rayleigh-B\'enard convection (RBC) in a fluid heated from below  remains one  of the challenging problems in nonlinear physics. A major component of the dynamics of this system is a large-scale circulation (LSC). The LSC plays an important role in many natural phenomena, including atmospheric and oceanic convection where it impacts on climate and weather, convection  in the outer core of Earth where it is responsible for the generation of the magnetic field, and convection in the outer 1/3 of the radius of the Sun where it is the major heat transport mechanism and determines the surface temperature (and thus also our well being on Earth).  We studied  turbulent RBC experimentally under idealized laboratory conditions, in  cylindrical samples of aspect ratio $\Gamma \equiv D / L \simeq 1$ ($D$ is the diameter and $L$ the height). There  the LSC consists of a single convection roll, with both down-flow and up-flow near the side wall but at azimuthal locations $\theta_0$ that differ by $\pi$.

An interesting aspect of the LSC dynamics is an as yet unexplained lateral twisting oscillation on a relatively fast time scale of the (on average) near-vertical circulation plane. On a longer time  scale the orientation $\theta_0$ of the circulation plane, under the influence of the turbulent background fluctuations, undergoes azimuthal  diffusion. Although {\it a priori} one would expect $\theta_0$ to have a uniform distribution $p(\theta_0)$ because the sample is rotationally invariant, we find that $p(\theta_0)$ has a maximum in a westerly direction that can be attributed to an interaction with Earth's Coriolis force. Another important feature consists of rare relatively fast re-orientation events of $\theta_0$. Re-orientations can occur by rotation through angles $\Delta \theta$ with a monotonically decreasing  probability distribution $p(\Delta \theta)$, and even more rarely by cessations (where the LSC stops temporarily) with   a uniform $p(\Delta\theta)$. Reorientations have Poissonian statistics in time.

A model inspired by the equations of motion (the Navier-Stokes equations), consisting of two stochastic ordinary differential equations, will be presented. Its stochastic terms represent the action of the turbulent background fluctuations (the fury of turbulence!) on the LSC. All parameters of the model are either measured independently or estimated from theory. The model explains semi-quantitatively the Coriolis-force effect as well as the statistics and properties of re-orientations.