Long time correlations in Lagrangian dynamics: a key to intermittency in turbulence

New aspects of turbulence are uncovered if one considers flow motion from the perspective of a fluid particle (known as the Lagrangian approach) rather than in terms of a velocity field (the Eulerian viewpoint). Using a new experimental technique, based on the scattering of ultrasounds, we have obtained a direct measurement of particle velocities, resolved at all scales, in a fully turbulent flow. It enables us to approach intermittency in turbulence from a dynamical point of view and to analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the 'walk' have random uncorrelated directions but a magnitude that is extremely long-range correlated in time. Theoretically, we study a Langevin equation that incorporates these features and we show that the resulting dynamics accounts for the observed one- and two-point statistical properties of the Lagrangian velocity fluctuations. Our approach connects the intermittent statistical nature of turbulence to the dynamics of the flow.Comment: 4 pages, 4 figure

Oscillations and secondary bifurcations in nonlinear magnetoconvection

Complicated bifurcation structures that appear in nonlinear systems governed by partial differential equations (PDEs) can be explained by studying appropriate low-order amplitude equations. We demonstrate the power of this approach by considering compressible magnetoconvection. Numerical experiments reveal a transition from a regime with a subcritical Hopf bifurcation from the static solution, to one where finite-amplitude oscillations persist although there is no Hopf bifurcation from the static solution. This transition is associated with a codimension-two bifurcation with a pair of zero eigenvalues. We show that the bifurcation pattern found for the PDEs is indeed predicted by the second-order normal form equation (with cubic nonlinearities) for a Takens-Bogdanov bifurcation with Z2 symmetry. We then extend this equation by adding quintic nonlinearities and analyse the resulting system. Its predictions provide a qualitatively accurate description of solutions of the full PDEs over a wider range of parameter values. Replacing the reflecting (Z2) lateral boundary conditions with periodic [O(2)] boundaries allows stable travelling wave and modulated wave solutions to appear; they could be described by a third-order system

Intermittency of velocity time increments in turbulence

We analyze the statistics of turbulent velocity fluctuations in the time domain. Three cases are computed numerically and compared: (i) the time traces of Lagrangian fluid particles in a (3D) turbulent flow (referred to as the "dynamic" case); (ii) the time evolution of tracers advected by a frozen turbulent field (the "static" case), and (iii) the evolution in time of the velocity recorded at a fixed location in an evolving Eulerian velocity field, as it would be measured by a local probe (referred to as the "virtual probe" case). We observe that the static case and the virtual probe cases share many properties with Eulerian velocity statistics. The dynamic (Lagrangian) case is clearly different; it bears the signature of the global dynamics of the flow.Comment: 5 pages, 3 figures, to appear in PR

Imperfect Homoclinic Bifurcations

Experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system.Comment: 8 pages, 11 figures, submitted to PR

Multifractal characterization of stochastic resonance

We use a multifractal formalism to study the effect of stochastic resonance in a noisy bistable system driven by various input signals. To characterize the response of a stochastic bistable system we introduce a new measure based on the calculation of a singularity spectrum for a return time sequence. We use wavelet transform modulus maxima method for the singularity spectrum computations. It is shown that the degree of multifractality defined as a width of singularity spectrum can be successfully used as a measure of complexity both in the case of periodic and aperiodic (stochastic or chaotic) input signals. We show that in the case of periodic driving force singularity spectrum can change its structure qualitatively becoming monofractal in the regime of stochastic synchronization. This fact allows us to consider the degree of multifractality as a new measure of stochastic synchronization also. Moreover, our calculations have shown that the effect of stochastic resonance can be catched by this measure even from a very short return time sequence. We use also the proposed approach to characterize the noise-enhanced dynamics of a coupled stochastic neurons model.Comment: 10 pages, 21 EPS-figures, RevTe

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

Comprehensive structural classification of ligand binding motifs in proteins

Comprehensive knowledge of protein-ligand interactions should provide a useful basis for annotating protein functions, studying protein evolution, engineering enzymatic activity, and designing drugs. To investigate the diversity and universality of ligand binding sites in protein structures, we conducted the all-against-all atomic-level structural comparison of over 180,000 ligand binding sites found in all the known structures in the Protein Data Bank by using a recently developed database search and alignment algorithm. By applying a hybrid top-down-bottom-up clustering analysis to the comparison results, we determined approximately 3000 well-defined structural motifs of ligand binding sites. Apart from a handful of exceptions, most structural motifs were found to be confined within single families or superfamilies, and to be associated with particular ligands. Furthermore, we analyzed the components of the similarity network and enumerated more than 4000 pairs of ligand binding sites that were shared across different protein folds.Comment: 13 pages, 8 figure

Chaos in magnetoconvection

The partial differential equations (PDEs) for two-dimensional incompressible convection in a strong vertical magnetic field have a codimension-three bifurcation when the parameters are chosen so that the bifurcations to steady and oscillatory convection coincide and the limit of narrow rolls is taken. The third-order set of ordinary differential equations (ODEs) that govern the behaviour of the PDEs near this bifurcation are derived using perturbation theory. These ODEs are the normal form of the codimension-three bifurcation; as such, they prove to be an excellent predictor of the behaviour of the PDEs. This is the first time that a detailed comparison has been made between the chaotic behaviour of a set of PDEs and that of the corresponding set of model ODEs, in a parameter regime where the ODEs are expected to provide accurate approximations to solutions of the PDEs. Most significantly, the transition from periodic orbits to a chaotic Lorenz attractor predicted by the ODEs is recovered in the PDEs, making this one of the few situations in which the nature of chaotic oscillations observed numerically in PDEs can be established firmly. Including correction terms obtained from the perturbation calculation enables the ODEs to track accurately the bifurcations in the PDEs over an appreciable range of parameter values. Numerical calculations suggest that the T-point (where there are heteroclinic connections between a saddle point and a pair of saddle-foci), which is associated with the transition from a Lorenz attractor to a quasi-attractor in the normal form, is also found in the PDEs. Further numerical simulations of the PDEs with square rolls confirm the existence of chaotic oscillations associated with a heteroclinic connection between a pair of saddle-foci