  We explore the transport mechanisms of heat in two- and three-dimensional
turbulent convection flows by means of the long-term evolution of Lagrangian
coherent sets. They are obtained from the spectral clustering of trajectories
of massless fluid tracers that are advected in the flow. Coherent sets result
from trajectories that stay closely together under the dynamics of the
turbulent flow. For longer times, they are always destroyed by the intrinsic
turbulent dispersion of material transport. Here, this constraint is overcome
by the application of evolutionary clustering algorithms that add a time memory
to the coherent set detection and allow individual trajectories to leak in or
out of evolving clusters. Evolutionary clustering thus also opens the
possibility to monitor the splits and mergers of coherent sets. These rare
dynamic events leave clear footprints in the evolving eigenvalue spectrum of
the Laplacian matrix of the trajectory network in both convection flows. The
Lagrangian trajectories reveal the individual pathways of convective heat
transfer across the fluid layer. We identify the long-term coherent sets as
those fluid flow regions that contribute least to heat transfer. Thus, our
evolutionary framework defines a complementary perspective on the slow dynamics
of turbulent superstructure patterns in convection flows that were recently
discussed in the Eulerian frame of reference. The presented framework might be
well suited for studies in natural flows which are typically based on sparse
information from drifters and probes.
