Lagrangian coherent sets in turbulent Rayleigh-B\'enard convection

Coherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh--B\'{e}nard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number ${\rm Ra}=10^6$ and at a Prandtl number ${\rm Pr}=10$. Three different Lagrangian analysis techniques based on graph Laplacians -- distance spectral trajectory clustering, time-averaged diffusion maps and finite-element based dynamic Laplacian discretization -- are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time-averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport.