This article presents a topological data analysis of the Lagrangian orbits occurring in natural convection flows inside cylindrical containers heated from below. The fluid motions are calculated via the numerical integration of the mass, momentum, and energy conservation equations using the Fourier-Chebyshev pseudospectral method. The study focuses on steady-state flows, and their qualitative properties are described in terms of the Rayleigh number. The Lagrangian orbits are calculated using the Eulerian velocity fields and their features are analyzed with Poincaré maps placed at the horizontal central plane of the cylinder. The qualitative features of the Lagrangian orbits calculated are closely related to those predicted by Hamiltonian theory, with the phase space being the real space in our case and considering the Rayleigh number as the driving parameter. The orbits are embedded in nested tori surfaces for low Rayleigh numbers ($\mathrm{Ra}=8\times {10}^3$). Some orbits seem to be dense in the tori and others form sets of points divided into segments in the Poincaré maps. At larger Rayleigh numbers ($\mathrm{Ra}={10}^4$, $2\times {10}^5$) the orbits form increasingly complicated structures, including loop islands in the Poincaré maps. Also, large areas in the maps show irregular or chaotic distributions of points. Topological data analysis (TDA) was used to define parameters that quantify the geometrical properties of the points distributions in the Poincaré maps. The number of segments in the sets of points and the occurrence of holes were determined with the 0- and 1-persistent homologies, respectively. Pairs of relatively prime numbers are used to characterize quasiperiodic orbits; the pairs are naturally identified with torus knots. It is found that the ratio of toroidal to the poloidal number of turns follows a simple rule for each Rayleigh number explored. The most important advantage of using TDA is the quantification and accurate identification of topological features found in the Poincaré maps.

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• Rayleigh-Bénard convection
• Thermal convection
• Hamiltonian systems
• Lagrangian particle tracking
• Topological tools