Kathrin Padberg-Gehle

Almost-Invariant and Finite-Time Coherent Sets: Directionality, Duration, and Diffusion

Regions in the phase space of a dynamical system that resist mixing over a finite-time duration are known as almost-invariant sets (for autonomous dynamics) or coherent sets (for nonautonomous or time-dependent dynamics). These regions provide valuable information for transport and mixing processes; almost-invariant sets mitigate transport between their interior and the rest of phase space, and coherent sets are good transporters of ‘mass’ precisely because they move about with minimal dispersion (e.g. oceanic eddies are good transporters of water that is warmer/cooler/saltier than the surrounding water). The most efficient approach to date for the identification of almost-invariant and coherent sets is via transfer operators. In this chapter we describe a unified setting for optimal almost-invariant and coherent set constructions and introduce a new coherent set construction that is suited to tracking coherent sets over several finite-time intervals. Under this unified treatment we are able to clearly explain the fundamental differences in the aims of the techniques and describe the differences and similarities in the mathematical and numerical constructions. We explore the role of diffusion, the influence of the finite-time duration, and discuss the relationship of time directionality with hyperbolic dynamics. All of these issues are elucidated in detailed case studies of two well-known systems.

A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data

We present a numerical method to identify regions of phase space that are approximately retained in a mobile compact neighbourhood over a finite time duration. Our approach is based on spatio-temporal clustering of trajectory data. The main advantages of the approach are the ability to produce useful results (i) when there are relatively few trajectories and (ii) when there are gaps in observation of the trajectories as can occur with real data. The method is easy to implement, works in any dimension, and is fast to run.

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 macro- and 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 time-varying 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...

Nonautonomous control of stable and unstable manifolds in two-dimensional flows

Abstract We outline a method for controlling the location of stable and unstable manifolds in the following sense. From a known location of the stable and unstable manifolds in a steady two-dimensional flow, the primary segments of the manifolds are to be moved to a user-specified time-varying location which is near the steady location. We determine the nonautonomous perturbation to the vector field required to achieve this control, and give a theoretical bound for the error in the manifolds resulting from applying this control. The efficacy of the control strategy is illustrated via a numerical example.

Role of critical points of the skin friction field in formation of plumes in thermal convection.

The dynamics in the thin boundary layers of temperature and velocity is the key to a deeper understanding of turbulent transport of heat and momentum in thermal convection. The velocity gradient at the hot and cold plates of a Rayleigh-Bénard convection cell forms the two-dimensional skin friction field and is related to the formation of thermal plumes in the respective boundary layers. Our analysis is based on a direct numerical simulation of Rayleigh-Bénard convection in a closed cylindrical cell of aspect ratio Γ=1 and focused on the critical points of the skin friction field. We identify triplets of critical points, which are composed of two unstable nodes and a saddle between them, as the characteristic building block of the skin friction field. Isolated triplets as well as networks of triplets are detected. The majority of the ridges of linelike thermal plumes coincide with the unstable manifolds of the saddles. From a dynamical Lagrangian perspective, thermal plumes are formed together with an attractive hyperbolic Lagrangian coherent structure of the skin friction field. We also discuss the differences from the skin friction field in turbulent channel flows from the perspective of the Poincaré-Hopf index theorem for two-dimensional vector fields.

Set-oriented numerical computation of rotation sets

We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of \begin{document}

Preface:Special issue on the occasion of the 4th International Workshop on Set-Oriented Numerics (SON 13, Dresden, 2013)

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

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Finite-time entropy: A probabilistic approach for measuring nonlinear stretching

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BIOLOGISTICS AND THE STRUGGLE FOR EFFICIENCY: CONCEPTS AND PERSPECTIVES

\end{document} -pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as \begin{document}