Detecting the order of chaotic Lagrangian orbits in convective flows through their topological properties

Abstract: Natural convection is a fundamental mechanism of heat and mass transport that plays a crucial role in both natural phenomena and technological applications. It governs large-scale processes such as atmospheric circulation and ocean currents, and is also essential in engineering contexts, including crystal growth, energy systems, and thermal management.

In this talk, we numerically investigate the Lagrangian orbits generated by a three-dimensional convective flow in a cubic domain, restricted to regimes in which the flow remains in a steady state. These orbits are modeled as finite point clouds in R^3, enabling the characterization of their geometric structure via persistent homology. We use the Rayleigh number as a control parameter of the flow. For low Rayleigh numbers, the orbits are organized into families of nested tori. As the Rayleigh number increases, a second family of nested tori emerges, and the two families are separated by a chaotic region. We show that a topology-based metric allows one to detect an intrinsic ordering of the orbits within this chaotic region according to their shape, revealing a smooth evolution despite the underlying dynamical complexity. In particular, within the chaotic region, we identify an orbit whose topological properties are analogous to those of a trivalent 2-stratifold, highlighting the richness of the transition between ordered and chaotic dynamics.

geometric topology

Audience: researchers in the topic

GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology