Lyapunov exponents for small aspect ratio Rayleigh-Bénard convection

Characterization of stability of dynamic particle ensemble systems using topological data analysis

Holes are ubiquitous structures in phase space, and their time evolution could indicate an instability in the dynamics of the system. However, the properties of these holes are difficult to study directly due to their theoretical complexity and lack of computational tools. This study proposes the use of persistent homology (PH), a technique from topological data analysis, as a computational tool for analyzing the properties of these phase-space holes, or more formally the H1 homology class according to PH. Initially, by using a toy data set, it is shown that the time evolution and the growth rate of a H1 class in phase space could be obtained by PH. For further validation, PH is applied to particle ensemble systems, such as the Hamiltonian flow and the two-stream instability (TSI). Both the stable case, where no H1 forms, and the unstable case, where H1 forms, were analyzed. It was shown that PH can distinguish between the stable and unstable cases purely from the phase-space time evolution plots. In unstable TSI, the PH also distinguished the transition of the H1 class from linear to non-linear growth. The growth rate, thus, obtained is in excellent agreement with the growth rate of the particle energy in the TSI system.

Using covariant Lyapunov vectors to quantify high-dimensional chaos with a conservation law

We explore the high-dimensional chaos of a one-dimensional lattice of diffusively coupled tent maps using the covariant Lyapunov vectors (CLVs). We investigate the connection between the dynamics of the maps in the physical space and the dynamics of the covariant Lyapunov vectors and covariant Lyapunov exponents that describe the direction and growth (or decay) of small perturbations in the tangent space. We explore the tangent space splitting into physical and transient modes and find that the splitting persists for all of the conditions we explore. In general, the leading CLVs are highly localized in space and the CLVs become less localized with increasing Lyapunov index. We consider the dynamics with a conservation law whose strength is controlled by a parameter that can be continuously varied. Our results indicate that a conservation law delocalizes the spatial variation of the CLVs. We find that when a conservation law is present, the leading CLVs are entangled with fewer of their neighboring CLVs than in the absence of a conservation law.

Correlations between the leading Lyapunov vector and pattern defects for chaotic Rayleigh-Bénard convection

We probe the effectiveness of using topological defects to characterize the leading Lyapunov vector for a high-dimensional chaotic convective flow field. This is accomplished using large-scale parallel numerical simulations of Rayleigh-Bénard convection for experimentally accessible conditions. We quantify the statistical correlations between the spatiotemporal dynamics of the leading Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. We use the ideas of precision and recall to build a statistical description of each pattern diagnostic's ability to describe the spatial variation of the leading Lyapunov vector. The precision of a diagnostic indicates the probability that it will locate a region where the Lyapunov vector is larger than a threshold value. The recall of a diagnostic indicates its ability to locate all of the possible spatial regions where the Lyapunov vector is above threshold. By varying the threshold used for the Lyapunov vector magnitude, we generate precision-recall curves which we use to quantify the complex relationship between the pattern diagnostics and the spatiotemporally varying magnitude of the leading Lyapunov vector. We find that pattern diagnostics which include information regarding the flow history outperform pattern diagnostics that do not. In particular, an emerging target defect has the highest precision of all of the pattern diagnostics we have explored.

Chaotic Rayleigh-Bénard convection with finite sidewalls

We explore the role of finite sidewalls on chaotic Rayleigh-Bénard convection. We use large-scale parallel spectral-element numerical simulations for the precise conditions of experiment for cylindrical convection domains. We solve the Boussinesq equations for thermal convection and the conjugate heat transfer problem for the energy transfer at the solid sidewalls of the cylindrical domain. The solid sidewall of the convection domain has finite values of thickness, thermal conductivity, and thermal diffusivity. We compute the Lyapunov vectors and exponents for the entire fluid-solid coupled problem. We quantify the chaotic dynamics of convection over a range of thermal sidewall boundary conditions. We find that the dynamics become less chaotic as the thermal conductivity of the sidewalls increases as measured by the value of the fractal dimension of the dynamics. The thermal conductivity of the sidewall is a stabilizing influence; the heat transfer between the fluid and solid regions is always in the direction to reduce the fluid motion near the sidewalls. Although the heat interaction for strongly conducting sidewalls is only about 1% of the heat transfer through the fluid layer, it is sufficient to reduce the fractal dimension of the dynamics by approximately 25% in our computations.

Spatiotemporal dynamics of the covariant Lyapunov vectors of chaotic convection

We explore the spatiotemporal dynamics of the spectrum of covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection. We use the inverse participation ratio to quantify the amount of spatial localization of the covariant Lyapunov vectors. The covariant Lyapunov vectors are found to be spatially localized at times when the instantaneous covariant Lyapunov exponents are large. The spatial localization of the Lyapunov vectors often occurs near defect structures in the fluid flow field. There is an overall trend of decreasing spatial localization of the Lyapunov vectors with increasing index of the vector. The spatial localization of the covariant Lyapunov vectors with positive Lyapunov exponents decreases an order of magnitude faster with increasing vector index than all of the remaining vectors that we have computed. We find that a weighted covariant Lyapunov vector is useful for the visualization and interpretation of the significant connections between the Lyapunov vectors and the flow field patterns.

Effects of nonlinear gradient terms on the defect turbulence regime in weakly dissipative systems

We investigate the behavior of traveling waves in a defect turbulence regime with the periodic boundary conditions by using the lowest-order complex Ginzburg-Landau equation (CGLE), and we show the effect of the nonlinear gradient terms in the system. It is found that the nonlinear gradient terms which appear at the same order as the quintic term can change the behavior of the wave patterns. The presence of the nonlinear gradient terms can cause major changes in the behavior of the solution. They can be considered like the stabilizing terms. The system which was initially unstable or chaotic can become stable by including the nonlinear gradient terms.

Two-dimensional localized chaotic patterns in parametrically driven systems

We study two-dimensional localized patterns in weakly dissipative systems that are driven parametrically. As a generic model for many different physical situations we use a generalized nonlinear Schrödinger equation that contains parametric forcing, damping, and spatial coupling. The latter allows for the existence of localized pattern states, where a finite-amplitude uniform state coexists with an inhomogeneous one. In particular, we study numerically two-dimensional patterns. Increasing the driving forces, first the localized pattern dynamics is regular, becomes chaotic for stronger driving, and finally extends in area to cover almost the whole system. In parallel, the spatial structure of the localized states becomes more and more irregular, ending up as a full spatiotemporal chaotic structure.

Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection

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.

Chaotic dynamics of a convection roll in a highly confined, vertical, differentially heated fluid layer

Direct numerical simulation is used to study the air flow between two vertical plates maintained at different temperatures. The periodic dimensions of the plates are small so as to accommodate only one flow structure, which consists of a convection roll with oblique vorticity braids. At lower Rayleigh numbers, the roll and the braids grow and shrink alternatively following a cyclical process. As the Rayleigh number is increased, the flow becomes temporally chaotic through a period-doubling cascade. Windows corresponding to multiperiodic regimes and interior crises are observed. As the Rayleigh number is further increased, the structure intermittently switches between two vertical positions, which is seen to correspond to an "attractor-merging" crisis. The chaotic flow dynamics are characterized and the corresponding physical mechanisms are identified. We show that some of the flow key features, such as the chaotic oscillation and intermittency, can be captured by a low-order model.

Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

Modal spectra extracted from nonequilibrium fluid patterns in laboratory experiments on Rayleigh-Bénard convection

We describe a method to extract from experimental data the important dynamical modes in spatiotemporal patterns in a system driven out of thermodynamic equilibrium. Using a novel optical technique for controlling fluid flow, we create an experimental ensemble of Rayleigh-Bénard convection patterns with nearby initial conditions close to the onset of secondary instability. An analysis of the ensemble evolution reveals the spatial structure of the dominant modes of the system as well as the corresponding growth rates. The extracted modes are related to localized versions of instabilities found in the ideal unbounded system. The approach may prove useful in describing instability in experimental systems as a step toward prediction and control.

Extensive chaos in Rayleigh-Bénard convection

Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.