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Register a new userTransport barriers in symplectic maps
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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.
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Single trajectories of magnetic line motion indicate the persistence of a central protected plasma core, surrounded by a chaotic shell enclosed in a double-sided transport barrier : the latter is identified as being composed of two Cantori located on two successive most-noble numbers values of the perturbed safety factor, and forming an internal transport barrier (ITB). Magnetic lines which succeed to escape across this barrier begin to wander in a wide chaotic sea extending up to a very robust barrier (as long as L<1) which is identified mathematically as a robust KAM surface at the plasma edge. In this case the motion is shown to be intermittent, with long stages of pseudo-trapping in the chaotic shell, or of sticking around island remnants, as expected for a continuous time random walk.
Nowadays, divertors are used in the main tokamaks to control the magnetic field and to improve the plasma confinement. In this article, we present analytical symplectic maps describing Poincare maps of the magnetic field lines in confined plasmas with a single null poloidal divertor. Initially, we present a divertor map and the tokamap for a diverted configuration. We also introduce the Ullmann map for a diverted plasma, whose control parameters are determined from tokamak experiments. Finally, an explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent toroidal equilibrium magnetic surfaces. In this method, we also give examples of onset of chaotic field lines at the plasma edge due to resonant perturbations.
Barriers have been identified in magnetically confined plasmas reducing the particle transport and improving the confinement. One of them, the primary shearless barriers are associated to extrema of non-monotonic plasma profiles. Previously, we identified these barriers in a model described by a map that allows the integration of charged particles motion in drift waves for a long time scale. In this work, we show how the existence of these robust barriers depends on the fluctuation amplitude and on the electric shear. Moreover, we also find control parameter intervals for which these primary barriers onset and break-up are recurrent. Another noticeable feature, in these transitions, is the appearance of a layer of particle trajectory stickiness after the shearless barrier break-up or before its onset. Besides the mentioned primary barriers, we also observe sequences of secondary shearless barriers, not reported before, created and destroyed by a sequence of bifurcations as the main control parameters, the fluctuation amplitude and electric shear, are varied. Furthermore, in these bifurcations, we also find hitherto unknown double and triple secondary shearless barriers which constitute a noticeable obstacle to the chaotic transport.
We present theory and experiments demonstrating the existence of invariant manifolds that impede the motion of microswimmers in two-dimensional fluid flows. One-way barriers are apparent in a hyperbolic fluid flow that block the swimming of both smooth-swimming and run-and-tumble emph{Bacillus subtilis} bacteria. We identify key phase-space structures, called swimming invariant manifolds (SwIMs), that serve as separatrices between different regions of long-time swimmer behavior. When projected into $xy$-space, the edges of the SwIMs act as one-way barriers, consistent with the experiments.
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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