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Register a new userHamiltonian Maps and Transport in Structured Fluids
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Authors
Jeffrey B. Weiss
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Structures such as waves, jets, and vortices have a dramatic impact on the transport properties of a flow. Passive tracer transport in incompressible two-dimensional flows is described by Hamiltonian dynamics, and, for idealized structures, the system is typically integrable. When such structures are perturbed, chaotic trajectories can result which can significantly change the transport properties. It is proposed that the transport due to the chaotic regions can be efficiently calculated using Hamiltonian mappings designed specifically for the structure of interest. As an example a new map is constructed, appropriate for studying transport by propagating isolated vortices. It is found that a perturbed vortex will trap fluid parcels for varying lengths of time, and that the distribution of such trapping times has slopes which are independent of the amplitudes of both the vortex and the perturbation.
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Chaotic transport is a subject of paramount importance in a variety of problems in plasma physics, specially those related to anomalous transport and turbulence. On the other hand, a great deal of information on chaotic transport can be obtained from simple dynamical systems like two-dimensional area-preserving (symplectic) maps, where powerful mathematical results like KAM theory are available. In this work we review recent works on transport barriers in area-preserving maps, focusing on systems which do not obey the so-called twist property. For such systems KAM theory no longer holds everywhere and novel dynamical features show up as non-resistive reconnection, shearless curves and shearless bifurcations. After presenting some general features using a standard nontwist mapping, we consider magnetic field line maps for magnetically confined plasmas in tokamaks.
We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofers metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincar{e} recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.
Linking thermodynamic variables like temperature $T$ and the measure of chaos, the Lyapunov exponents $lambda$, is a question of fundamental importance in many-body systems. By using nonlinear fluid equations in one and three dimensions, we prove that in thermalised flows $lambda propto sqrt{T}$, in agreement with results from frustrated spin systems. This reveals an underlying universality and provides evidence for recent conjectures on the thermal scaling of $lambda$. We also reconcile seemingly disparate effects -- equilibration on one hand and pushing systems out-of-equilibrium on the other -- of many-body chaos by relating $lambda$ to $T$ through the dynamical structures of the flow.
A method to predict the emergence of different kinds of ordered collective behaviors in systems of globally coupled chaotic maps is proposed. The method is based on the analogy between globally coupled maps and a map subjected to an external drive. A vector field which results from this analogy appears to govern the transient evolution of the globally coupled system. General forms of global couplings are considered. Some simple applications are given.
Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example, where waves originate from a source exhibiting a back-and-forth movement in radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves (``superspiral). Using the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonous growth or decay as well as saturation of these modulations away from the source depending on the perturbation frequency. Our findings allow a consistent interpretation of recent experimental observations concerning superspirals and their decay to spatio-temporal chaos.
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