  The dynamics in a confined turbulent convection flow is dominated by multiple
long-lived macroscopic circulation states, which are visited subsequently by
the system in a Markov-type hopping process. In the present work, we analyze
the short transition paths between these subsequent macroscopic system states
by a data-driven learning algorithm that extracts the low-dimensional
transition manifold and the related new coordinates, which we term collective
variables, in the state space of the complex turbulent flow. We therefore
transfer and extend concepts for conformation transitions in stochastic
microscopic systems, such as in the dynamics of macromolecules, to a
deterministic macroscopic flow. Our analysis is based on long-term direct
numerical simulation trajectories of turbulent convection in a closed cubic
cell at a Prandtl number $Pr = 0.7$ and Rayleigh numbers $Ra = 10^6$ and $10^7$
for a time lag of $10^5$ convective free-fall time units. The simulations
resolve vortices and plumes of all physically relevant scales resulting in a
state space spanned by more than 3.5 million degrees of freedom. The transition
dynamics between the large-scale circulation states can be captured by the
transition manifold analysis with only two collective variables which implies a
reduction of the data dimension by a factor of more than a million. Our method
demonstrates that cessations and subsequent reversals of the large-scale flow
are unlikely in the present setup and thus paves the way to the development of
efficient reduced-order models of the macroscopic complex nonlinear dynamical
system.
