Koopman analysis of the long-term evolution in a turbulent convection cell

TL;DR

This paper uses Koopman eigenfunction analysis with machine learning techniques to study the long-term evolution and stable states of turbulent convection in a cubic cell, revealing detailed flow structures and state transitions.

Contribution

It introduces a data-driven Koopman analysis approach employing diffusion kernels to identify and analyze large-scale flow states in turbulent convection.

Findings

Identification of four stable large-scale circulation states.

Recapture of known flow switching behaviors and flow structures.

Discovery of oscillatory eigenfunction pairs linked to specific flow states.

Abstract

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh-B\'{e}nard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as a thousand convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.