Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection

Abstract

We explore numerically the high-dimensional spatiotemporal chaos of Rayleigh-Bénard convection using covariant Lyapunov vectors. We integrate the three-dimensional and time-dependent Boussinesq equations for a convection layer in a shallow square box geometry with an aspect ratio of 16 for very long times and for a range of Rayleigh numbers. We simultaneously integrate many copies of the tangent space equations in order to compute the covariant Lyapunov vectors. The dynamics explored has fractal dimensions of 20≲D_{λ}≲50, and we compute on the order of 150 covariant Lyapunov vectors. We use the covariant Lyapunov vectors to quantify the degree of hyperbolicity of the dynamics and the degree of Oseledets splitting and to explore the temporal and spatial dynamics of the Lyapunov vectors. Our results indicate that the chaotic dynamics of Rayleigh-Bénard convection is nonhyperbolic for all of the Rayleigh numbers we have explored. Our results yield that the entire spectrum of covariant Lyapunov vectors that we have computed are tangled as indicated by near tangencies with neighboring vectors. A closer look at the spatiotemporal features of the Lyapunov vectors suggests contributions from structures at two different length scales with differing amounts of localization.

Lyapunov spectrum of forced homogeneous isotropic turbulent flows

“…Further, the spectrum of dissipative LV was found to be localized in Fourier space. Other work with the Rayleigh-Bénard convection showed that the energy spectra of the CLV were not independent of the Lyapunov indices [36]. Given the present result, it can be expected that the first few CLV also have a non- localized energy spectrum.…”

“…It has been reported [36,60] in different chaotic systems that only a small part of the physical space was responsible for most of the perturbation growth in chaotic systems. This property is called the localization.…”

Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow

“…However, many chaotic fluid systems are not uniform hyperbolic. For a Kolmogorov flow simulated with 224 DOFs (Inubushi et al 2012) and for a 3D Boussinesq equations simulated with 5 × 10 5 DOFs (Xu & Paul 2016), researchers found for both cases that the angles between CLVs can be very small. Currently, there are not many results regarding hyperbolicity for CFD simulated 3D Navier-Stokes fluid systems.…”

Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators

“…CLVs have since attracted a large amount of scientific interest and proved to be a fruitful source of insight, especially into phase-space structures of highdimensional dynamical systems. In particular, CLVs have been used to study e. g. dynamics of rigid disk systems [32,33], chaotic motion in spatially extended systems [34,35], stability properties of geophysical models [36] and collective chaos in systems of coupled oscillators [8,9,37].…”