Probing turbulent superstructures in Rayleigh-Benard convection by Lagrangian trajectory clusters

We analyze large-scale patterns in three-dimensional turbulent convection in a horizontally extended square convection cell by Lagrangian particle trajectories calculated in direct numerical simulations. A simulation run at a Prandtl number Pr = 0.7, a Rayleigh number Ra = 10(5) , and an aspect ratio F = 16 is therefore considered. These large-scale structures, which are denoted as turbulent superstructures of convection, are detected by the spectrum of the graph Laplacian matrix. Our investigation, which follows Hadjighasem et al. [Phys. Rev. E 93, 063107 (2016)], builds a weighted and undirected graph from the trajectory points of Lagrangian particles. Weights at the edges of the graph are determined by a mean dynamical distance between different particle trajectories. It is demonstrated that the resulting trajectory clusters, which are obtained by a subsequent k-means clustering, coincide with the superstructures in the Eulerian frame of reference. Furthermore, the characteristic times tau(L) and lengths lambda(L)(U) of the superstructures in the Lagrangian frame of reference agree very well with their Eulerian counterparts, tau and lambda(U) , respectively. This trajectory-based clustering is found to work for times t less than or similar to tau approximate to tau(L) . Longer time periods t greater than or similar to tau(L) require a change of the analysis method to a density-based trajectory clustering by means of time-averaged Lagrangian pseudotrajectories, which is applied in this context for the first time. A small coherent subset of the pseudotrajectories is obtained in this way consisting of those Lagrangian particles that are trapped for long times in the core of the superstructure circulation rolls and are thus not subject to ongoing turbulent dispersion.