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

Abstract

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh-Bénard convection cell via a Koopman eigenfunction analysis. A datadriven 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.

Data-driven Koopman operator approach for computational neuroscience

“…We now describe how the Koopman eigenfunctions z j can be used to decompose the observed ECoG signal h n into coherent spatiotemporal patterns. The reconstruction method is similar to the procedure used in SSA [34] and NLSA [25][26][27][28].…”

“…In the Koopman operator literature, A j (0) is referred to as the Koopman mode associated with the observable F [7,49], though note that here we will use A j (q) with several values of q to perform reconstruction. In particular, following [25][26][27][28]34], we define the spatiotemporal pattern associated with the pair (z j , A j ) as the function F j :…”

“…These oscillations are commonly divided into 5-8 bands based on their frequency. The oscillations vary from slow, e.g., Delta (1)(2)(3)(4) and Theta (4)(5)(6)(7)(8), to medium like Alpha (8)(9)(10)(11)(12), to quickly oscillating modes such as Beta (12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30) and Gamma . Electrophysiologic recordings, which allow researchers to capture distinct oscillations, are one of the most affordable and readily-accessible sources of data on brain activation patterns.…”

Proper orthogonal decomposition analysis and modelling of large-scale flow reorientations in a cubic Rayleigh–Bénard cell

“…Random low-frequency reorientations between these quasi-stable states have been observed in experiments (Vasiliev et al 2016) and numerical simulations (Foroozani et al 2017) which consist in a rotation of the LSC from one diagonal plane to the other. No cessation-led reorientation events have been reported yet, although the study of Giannakis et al (2018) suggests that reversals (reorientations of an angle π) can happen.…”

Quantum mechanics and data assimilation

“…The eigenvectors φ j then provide representations of the basis elements φ j,N (see Section V C). Elsewhere, we have demonstrated the feasibility of this implementation for computing eigenfunctions from high-dimensional datasets of moderate sample number, (d, N ) = O(10 6 , 10 4 ) [51], or datasets of moderate dimension and high sample number, (d, N ) = O(10 2 , 10 6 ) [26]. In the latter case, it should be possible to speed up the kernel matrix calculation using tree-based [52] or randomized [53] approximate nearest-neighbor algorithms, though we have not explored such options in the present work.…”