Analyzing thermal convection in a two-dimensional circular annulus via spatio-temporal Koopman decomposition

Abstract

This paper deals with the analysis of numerically obtained spatio-temporal data for thermal convection in a two-dimensional circular annulus. These data are post-processed using a recent method called the spatio-temporal Koopman decomposition, which applies to the frequent case in which the underlying dynamics exhibit oscillatory (possibly growing or decaying) behavior in both time and one of the spatial directions (the azimuthal coordinate for the present problem). When this holds, the method decomposes the data into Fourier-like series in both the distinguished spatial direction and time. In the general case, the obtained series account for, not only the involved temporal frequencies and spatial wavenumbers, but also the spatial and temporal growth rates. In the simpler situation of attractors showing spatially periodic behavior, the spatial and temporal growth rates vanish (or are very small). In this case, the analysis of the wavenumber–frequency pairs that are present allows for uncovering the spatio-temporal structure of the flow in the circular annulus. The analysis focuses on periodic and quasi-periodic attractors, which exhibit spatio-temporal symmetries that are identified by the method. In addition, for quasi-periodic attractors, the method gives semi-analytic descriptions for the tori densely covered by particular trajectories. Although the paper concentrates on the thermal convection problem in an annulus, it will become clear that the method applies to other related dynamics as well.