Lagrangian single-particle, multi-particle and topological analyses in turbulent Rayleigh-B\'enard convection

We present three-dimensional direct numerical simulations of turbulent Rayleigh-B\'enard convection (RBC) in the Lagrangian frame of reference for Rayleigh numbers $10^5 \leq Ra \leq 10^{10}$ and a Prandtl number $Pr=0.7$ in a plane layer at an aspect ratio $L:L:H=4:4:1$ with a horizontal length $L$ and height $H$. We use particle accelerations, Lagrangian heat transfer, $Q$-$R$ invariant topology, Lagrangian particle pair dispersion, scale-dependent Lagrangian eddy viscosity, and principal component analysis (PCA) of dense particle clouds to characterise convective transport along material trajectories. By computing particle accelerations at the integration time step and controlling spectral element method signatures, we obtain robust acceleration statistics and recover Heisenberg-Yaglom behaviour. Lagrangian heat transfer is extremely intermittent: individual massless Lagrangian particles can carry convective heat fluxes up to $500$ times the global Eulerian mean, although higher-order heat flux moments decrease toward Gaussian values with increasing $Ra$. The analysis of velocity gradient invariants in the $Q$-$R$ plane along trajectories identifies a distinct topological footprint of dust-devil-like convective vortices in the quadrant of $Q>0$, $R<0$, associated with vortex stretching, plume detachment, and intense localised heat transfer. Global unconditioned pair dispersion exhibits neither extended Richardson nor Bolgiano-Obukhov scaling plateaus. Rather, scale-dependent eddy viscosity and conditioned PCA of dense particle clouds reveal that buoyancy- and shear-driven dispersion are temporally organised: rapid plume-driven ejection produces a short $t^5$-like episode, followed by sustained Richardson-like $t^3$-scaling. Thus, Lagrangian topology and cloud geometry provide mechanism-resolving diagnostics for active-scalar turbulence beyond RBC-specific global scaling laws.