Quantifying Spatiotemporal Chaos in Rayleigh-B\'enard Convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-B\'enard 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

Stability Properties of Nonhyperbolic Chaotic Attractors under Noise

We study local and global stability of nonhyperbolic chaotic attractors contaminated by noise. The former is given by the maximum distance of a noisy trajectory from the noisefree attractor, while the latter is provided by the minimal escape energy necessary to leave the basin of attraction, calculated with the Hamiltonian theory of large fluctuations. We establish the important and counterintuitive result that both concepts may be opposed to each other. Even when one attractor is globally more stable than another one, it can be locally less stable. Our results are exemplified with the Holmes map, for two different sets of parameter, and with a juxtaposition of the Holmes and the Ikeda maps. Finally, the experimental relevance of these findings is pointed out.Comment: Phys.Rev. Lett., to be publishe

The low dimensional dynamical system approach in General Relativity: an example

In this paper we explore one of the most important features of the Galerkin method, which is to achieve high accuracy with a relatively modest computational effort, in the dynamics of Robinson-Trautman spacetimes.Comment: 7 pages, 5 figure

Languages and Learning at Key Stage 2: A Longitudinal Study Final Report

In 2006, The Open University, the University of Southampton and Canterbury Christ Church University were commissioned by the then Department for Education and Skills (DfES), now Department for Children, Schools and Families (DCSF) to conduct a three-year longitudinal study of languages learning at Key Stage 2 (KS2). The qualitative study was designed to explore provision, practice and developments over three school years between 2006/07 and 2008/09 in a sample of primary schools and explore children’s achievement in oracy and literacy, as well as the possible broader cross-curricular impact of languages learning

The Play’s the Thing: Experimentally Examining the Social and Cognitive Effects of School Field Trips to Live Theater Performances

Field trips to see theater performances are a long-standing educational practice, however, there is little systematic evidence demonstrating educational benefits. This article describes the results of five random assignment experiments spanning two years where school groups were assigned by lottery to attend a live theater performance, or for some groups, watch a movie-version of the same story. We find significant educational benefits from seeing live theater, including higher levels of tolerance, social perspective taking, and stronger command of the plot and vocabulary of those plays. Students randomly assigned to watch a movie did not experience these benefits. Our findings also suggest that theater field trips may cultivate the desire among students to frequent the theater in the future

Some relations between Lagrangian models and synthetic random velocity fields

We propose an alternative interpretation of Markovian transport models based on the well-mixedness condition, in terms of the properties of a random velocity field with second order structure functions scaling linearly in the space time increments. This interpretation allows direct association of the drift and noise terms entering the model, with the geometry of the turbulent fluctuations. In particular, the well known non-uniqueness problem in the well-mixedness approach is solved in terms of the antisymmetric part of the velocity correlations; its relation with the presence of non-zero mean helicity and other geometrical properties of the flow is elucidated. The well-mixedness condition appears to be a special case of the relation between conditional velocity increments of the random field and the one-point Eulerian velocity distribution, allowing generalization of the approach to the transport of non-tracer quantities. Application to solid particle transport leads to a model satisfying, in the homogeneous isotropic turbulence case, all the conditions on the behaviour of the correlation times for the fluid velocity sampled by the particles. In particular, correlation times in the gravity and in the inertia dominated case, respectively, longer and shorter than in the passive tracer case; in the gravity dominated case, correlation times longer for velocity components along gravity, than for the perpendicular ones. The model produces, in channel flow geometry, particle deposition rates in agreement with experiments.Comment: 54 pages, 8 eps figures included; contains additional material on SO(3) and on turbulent channel flows. Few typos correcte

Extraction of coherent structures in a rotating turbulent flow experiment

The discrete wavelet packet transform (DWPT) and discrete wavelet transform (DWT) are used to extract and study the dynamics of coherent structures in a turbulent rotating fluid. Three-dimensional (3D) turbulence is generated by strong pumping through tubes at the bottom of a rotating tank (48.4 cm high, 39.4 cm diameter). This flow evolves toward two-dimensional (2D) turbulence with increasing height in the tank. Particle Image Velocimetry (PIV) measurements on the quasi-2D flow reveal many long-lived coherent vortices with a wide range of sizes. The vorticity fields exhibit vortex birth, merger, scattering, and destruction. We separate the flow into a low-entropy ``coherent'' and a high-entropy ``incoherent'' component by thresholding the coefficients of the DWPT and DWT of the vorticity fields. Similar thresholdings using the Fourier transform and JPEG compression together with the Okubo-Weiss criterion are also tested for comparison. We find that the DWPT and DWT yield similar results and are much more efficient at representing the total flow than a Fourier-based method. Only about 3% of the large-amplitude coefficients of the DWPT and DWT are necessary to represent the coherent component and preserve the vorticity probability density function, transport properties, and spatial and temporal correlations. The remaining small amplitude coefficients represent the incoherent component, which has near Gaussian vorticity PDF, contains no coherent structures, rapidly loses correlation in time, and does not contribute significantly to the transport properties of the flow. This suggests that one can describe and simulate such turbulent flow using a relatively small number of wavelet or wavelet packet modes.Comment: experimental work aprox 17 pages, 11 figures, accepted to appear in PRE, last few figures appear at the end. clarifications, added references, fixed typo

Statistical analysis of coherent structures in transitional pipe flow

Numerical and experimental studies of transitional pipe flow have shown the prevalence of coherent flow structures that are dominated by downstream vortices. They attract special attention because they contribute predominantly to the increase of the Reynolds stresses in turbulent flow. In the present study we introduce a convenient detector for these coherent states, calculate the fraction of time the structures appear in the flow, and present a Markov model for the transition between the structures. The fraction of states that show vortical structures exceeds 24% for a Reynolds number of about Re=2200, and it decreases to about 20% for Re=2500. The Markov model for the transition between these states is in good agreement with the observed fraction of states, and in reasonable agreement with the prediction for their persistence. It provides insight into dominant qualitative changes of the flow when increasing the Reynolds number.Comment: 11 pages, 26 (sub)figure