**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.