Accurate control of hyperbolic trajectories in any dimension

The unsteady (nonautonomous) analog of a hyperbolic fixed point is a hyperbolic trajectory, whose importance is underscored by its attached stable and unstable manifolds, which have relevance in fluid flow barriers, chaotic basin boundaries, and the long-term behavior of the system. We develop a method for obtaining the unsteady control velocity which forces a hyperbolic trajectory to follow a user-prescribed variation with time. Our method is applicable in any dimension, and accuracy to any order is achievable.We demonstrate and validate our method by (1) controlling the fixed point at the origin of the Lorenz system, for example, obtaining a user-defined nonautonomous attractor, and (2) the saddle points in a droplet flow, using localized control which generates global transport.Sanjeeva Balasuriya, Kathrin Padberg-Gehl

Controlling the unsteady analogue of saddle stagnation points

It is well known that saddle stagnation points are crucial flow organizers in steady (autonomous) flows due to their accompanying stable and unstable manifolds. These have been extensively investigated experimentally, numerically, and theoretically in situations related to macroand micromixers in order to either restrict or enhance mixing. Saddle points are also important players in the dynamics of mechanical oscillators, in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior. The entity analogous to a saddle point in an unsteady (nonautonomous) flow is a timevarying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time. Within the context of nearly steady flows, the unsteady velocity perturbation required to ensure that such a hyperbolic (saddle) trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained via a heuristic approach. An expression for the error in the hyperbolic trajectory's motion is also derived. This provides a new tool for the control of both fluid transport and mechanical oscillators. The method is applied to two examples-a four-roll mill and a Duffing oscillator-and the performance of the control strategy is shown to be excellent in both instances. © 2013 Society for Industrial and Applied Mathematics.Sanjeeva Balasuriya and Kathrin Padberg-Gehl

Spectral early-warning signals for sudden changes in time-dependent flow patterns

Lagrangian coherent sets are known to crucially determine transport and mixing processes in non-autonomous flows. Prominent examples include vortices and jets in geophysical fluid flows. Coherent sets can be identified computationally by a probabilistic transfer-operator-based approach within a set-oriented numerical framework. Here, we study sudden changes in flow patterns that correspond to bifurcations of coherent sets. Significant changes in the spectral properties of a numerical transfer operator are heuristically related to critical events in the phase space of a time-dependent system. The transfer operator approach is applied to different example systems of increasing complexity. In particular, we study the 2002 splitting event of the Antarctic polar vortex

Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators

The detection of regions in the ocean that are coherent over an extended period of time is a fundamental problem in many oceanic applications. For instance such regions are important for studying the transport of marine species and for the distribution of nutrients. In this study we demonstrate the efficacy of transfer operators in detecting and analysing such structures. We focus first on the detection of the Weddell and Ross Gyre for the four seasons spanning December 2003–November 2004 within the 3-D oceanic domain south of 30° S, and show distinct seasonal differences in both the three-dimensional structure and the persistence of the gyres. Further, we demonstrate a new technique based on the discretised transfer operators to calculate the mean residence time of water within parts of the gyres and determine pathways of water leaving and entering the gyres

Network-based study of Lagrangian transport and mixing

Transport and mixing processes in fluid flows are crucially influenced by coherent structures and the characterization of these Lagrangian objects is a topic of intense current research. While established mathematical approaches such as variational methods or transfer-operator-based schemes require full knowledge of the flow field or at least high-resolution trajectory data, this information may not be available in applications. Recently, different computational methods have been proposed to identify coherent behavior in flows directly from Lagrangian trajectory data, that is, numerical or measured time series of particle positions in a fluid flow. In this context, spatio-temporal clustering algorithms have been proven to be very effective for the extraction of coherent sets from sparse and possibly incomplete trajectory data. Inspired by these recent approaches, we consider an unweighted, undirected network, where Lagrangian particle trajectories serve as network nodes. A link is established between two nodes if the respective trajectories come close to each other at least once in the course of time. Classical graph concepts are then employed to analyze the resulting network. In particular, local network measures such as the node degree, the average degree of neighboring nodes, and the clustering coefficient serve as indicators of highly mixing regions, whereas spectral graph partitioning schemes allow us to extract coherent sets. The proposed methodology is very fast to run and we demonstrate its applicability in two geophysical flows – the Bickley jet as well as the Antarctic stratospheric polar vortex

Eulerian and Lagrangian Perspectives on Turbulent Superstructures in Rayleigh-Bénard Convection

Large-scale computations in combination with new mathematical analysis tools make studies of the large-scale patterns, which are termed turbulent superstructures, in extended turbulent convection flows now accessible. Here, we report recent analyses in the Eulerian and Lagrangian frames of reference that reveal the characteristic spatial and temporal scales of the patterns as a function of Prandtl number, the dimensionless number which relates momentum to temperature diffusion in the working fluid

Molecular crowding creates traffic jams of kinesin motors on microtubules

Despite the crowdedness of the interior of cells, microtubule-based motor proteins are able to deliver cargoes rapidly and reliably throughout the cytoplasm. We hypothesize that motor proteins may be adapted to operate in crowded environments by having molecular properties that prevent them from forming traffic jams. To test this hypothesis, we reconstituted high-density traffic of purified kinesin-8 motor protein, a highly processive motor with long end-residency time, along microtubules in a total internal-reflection fluorescence microscopy assay. We found that traffic jams, characterized by an abrupt increase in the density of motors with an associated abrupt decrease in motor speed, form even in the absence of other obstructing proteins. To determine the molecular properties that lead to jamming, we altered the concentration of motors, their processivity, and their rate of dissociation from microtubule ends. Traffic jams occurred when the motor density exceeded a critical value (density-induced jams) or when motor dissociation from the microtubule ends was so slow that it resulted in a pileup (bottleneck-induced jams). Through comparison of our experimental results with theoretical models and stochastic simulations, we characterized in detail under which conditions density- and bottleneck-induced traffic jams form or do not form. Our results indicate that transport kinesins, such as kinesin-1, may be evolutionarily adapted to avoid the formation of traffic jams by moving only with moderate processivity and dissociating rapidly from microtubule ends