Chaotic Transport Research Articles

Mechanism of destruction of transport barriers in geophysical jets with Rossby waves

The mechanism of destruction of a central transport barrier in a dynamical model of a geophysical zonal jet current in the ocean or the atmosphere with two propagating Rossby waves is studied. We develop a method for computing a central invariant curve which is an indicator of existence of the barrier. Breakdown of this curve under a variation in the Rossby wave amplitudes and onset of chaotic cross-jet transport happen due to specific resonances producing stochastic layers in the central jet. The main result is that there are resonances breaking the transport barrier at unexpectedly small values of the amplitudes that may have serious impact on mixing and transport in the ocean and the atmosphere. The effect can be found in laboratory experiments with azimuthal jets and Rossby waves in rotating tanks under specific values of the wave numbers that are predicted in the theory.

Directed chaotic transport in the tokamap with mixed phase space

On the basis of the tokamap, characteristic features of magnetic field lines and zeroth-order guiding-center particle motion in the whole body of a magnetically confined plasma, e.g., a tokamak plasma, are investigated. It is shown that the tokamap exhibits a poloidal transport that can be considered as a Hamiltonian ratchet. In a situation with partially chaotic magnetic field lines the locking of the averaged poloidal velocity occurs to a value that does not depend precisely on the initial conditions. The so-called sum rule predicts the mean velocity in agreement with the observed magnitude. Possible consequences for the onset of poloidal plasma rotation in ergodized plasmas are elucidated.

Chaotic transport across two-dimensional jet streams

A dynamically consistent model of a jet stream modulated by Rossby waves is used to analyze the onset of chaotic transport of a passive tracer across the jet (cross-jet transport). For odd Rossby wavenumbers, a technique is developed to find a central invariant curve whose breakup signifies the onset of cross-jet transport. Amplitude and resonance mechanisms of breakup of this curve are investigated. For even-odd pairs of wavenumbers, an alternative technique is proposed for detecting cross-jet transport, based on an overlap of northern and southern stochastic layers. In this case, the Rossby-wave amplitudes required for global crossjet transport to occur are larger than those in the case of odd wavenumbers. The predicted results should be tested in laboratory experiments simulating jet streams modulated by Rossby waves.

Chaotic transport and chronology of complex asteroid families

We present a transport model that describes the orbital diffusion of asteroids in chaotic regions of the 3-D space of proper elements. Our goal is to use a simple random-walk model to study the evolution and derive accurate age estimates for dynamically complex asteroid families. To this purpose, we first compute local diffusion coefficients, which characterize chaotic diffusion in proper eccentricity (e_p) and inclination (I_p), in a selected phase-space region. Then, a Monte-Carlo-type code is constructed and used to track the evolution of random walkers (i.e. asteroids), by coupling diffusion in (e_p,I_p) with a drift in proper semi-major axis (a_p) induced by the Yarkovsky/YORP thermal effects. We validate our model by applying it to the family of (490) Veritas, for which we recover previous estimates of its age (~8.3 Myr). Moreover, we show that the spreading of chaotic family members in proper elements space is well reproduced in our random-walk simulations. Finally, we apply our model to the family of (3556) Lixiaohua, which is much older than Veritas and thus much more affected by thermal forces. We find the age of the Lixiaohua family to be 155+/-36 Myr.

Stochastic webs and quantum transport in superlattices: an introductory review

Stochastic webs were discovered, first by Arnold for multi-dimensional Hamiltonian systems, and later by Chernikov et al. for the low-dimensional case. Generated by weak perturbations, they consist of thread-like regions of chaotic dynamics in phase space. Their importance is that, in principle, they enable transport from small energies to high energies. In this introductory review, we concentrate on low-dimensional stochastic webs and on their applications to quantum transport in semiconductor superlattices subject to electric and magnetic fields. We also describe a recently-suggested modification of the stochastic web to enhance chaotic transport through it and we discuss its possible applications to superlattices.

Effects of Dissipation and Noise on Chaotic Transport in Superlattices

Effects of Dissipation and Noise on Chaotic Transport in Superlattices

Proliferation of atomic wave packets at the nodes of a standing light wave

A quantum analysis is presented of the motion and internal state of a two-level atom in a strong standing-wave light field. Coherent evolution of the atomic wave-packet, atomic dipole moment, and population inversion strongly depends on the ratio between the detuning from atom-field resonance and a characteristic atomic frequency. In the basis of dressed states, atomic motion is represented as wave-packet motion in two effective optical potentials. At exact resonance, coherent population trapping is observed when an atom with zero momentum is centered at a standing-wave node. When the detuning is comparable to the characteristic atomic frequency, the atom crossing a node may or may not undergo a transition between the potentials with probabilities that are similar in order of magnitude. In this detuning range, atomic wave packets proliferate at the nodes of the standing wave. This phenomenon is interpreted as a quantum manifestation of chaotic transport of classical atoms observed in earlier studies. For a certain detuning range, there exists an interval of initial momentum values such that the atom simultaneously oscillates in an optical potential well and moves as a ballistic particle. This behavior of a wave packet is a quantum analog of a classical random walk of an atom, when it enters and leaves optical potential wells in a seemingly irregular manner and freely moves both ways in a periodic standing light wave. In a far-detuned field, the transition probability between the potentials is low, and adiabatic wave-packet evolution corresponding to regular classical motion of an atom is observed.

Transport properties in nontwist area-preserving maps

Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds.

Space and time resolving molecular acoustics as a tool to visualize epoxy formation at a planar hardener–resin interface

Scanning Brillouin microscopy visualizes the temporal evolution of a heterogeneous hypersonic profile during chemical reactions and transport processes around a planar phase boundary between an epoxy resin and hardener. The hypersonic investigations are interpreted in terms of structural formation and annihilation in the layered sample and are compared to results obtained for a well mixed bulk sample of the same composition of reactants. Chaotic transport channels are made responsible for the creation of local glassy heterogeneities not observable in the bulk material on a macroscopic level.

The N-vortex problem on a sphere: geophysical mechanisms that break integrability

We describe the dynamical system governing the evolution of a system of point vortices on a rotating spherical shell, highlighting features which break what would otherwise be an integrable problem. The importance of the misalignment of the center-of-vorticity vector associated with a cluster of point vortices with the axis of rotation is emphasized as a crucial factor in the interpretation of dynamical features for many flow configurations. We then describe two important physical mechanisms which break what would otherwise be an integrable problem—the interactions between the local center-of-vorticity vectors of more than one region of concentrated vorticity, and the coupling between the center-of-vorticity vector and the background vorticity field which supports Rossby waves. Focusing on the Polar vortex splitting event of September 2002, we describe simple (i.e., low dimensional) mechanisms that can trigger instabilities whose subsequent development cause the onset of chaotic advection and global particle transport. At the linear level, eigenvalues that oscillate between elliptic and hyperbolic configurations initiate the pinch-off process of a passive patch representing the Polar vortex. At the nonlinear level, the evolution and topological bifurcations of the streamline patterns are responsible for its further splitting, stretching, and subsequent transport over the sphere. We finish by briefly describing how to incorporate conservation of potential vorticity and the development of a model governing the probability density function associated with the point vortex system.

Vortices in time-periodic shear flow

Vortices emerging in geophysical turbulence may experience deformations due to the non-uniform ambient flow induced by neighbouring vortices. At first approximation this ambient flow is modeled by a linear shear flow. It is well known from previous studies that the vortex may be (partially) destructed through removal of weak vorticity at the vortex edge—a process referred to as ‘stripping’. While most previous studies considered a stationary external shear flow, we have examined the behaviour of the vortex embedded in a linear shear flow whose strength changes harmonically in time. Aspects of the vortex dynamics and the (chaotic) transport of tracers have been studied by both laboratory experiments and numerical simulations based on a simple kinematical model.

Control-oriented approaches to anticipating synchronization of chaotic deterministic ratchets

In virtue of techniques derived from nonlinear control system theory, we establish conditions under which one could obtain anticipating synchronization between two periodically driven deterministic ratchets that are able to exhibit directed transport with a finite velocity. Criteria are established in order to guarantee the anticipating synchronization property of such systems as well as characterize phase space dynamics of the ratchet transporting behaviors. These results allow one to predict the chaotic direct transport features of particles on a ratchet potential using a copy of the same system that performs as a slave, which are verified through numerical simulation.

Chaotic transport in deterministic sine-Gordon soliton ratchets

We investigate homogeneous and inhomogeneous sine-Gordon ratchet systems in which a temporal symmetry and the spatial symmetry, respectively, are broken. We demonstrate that in the inhomogeneous systems with ac driving the soliton dynamics is chaotic in certain parameter regions, although the soliton motion is unidirectional. This is qualitatively explained by a one-collective-coordinate theory which yields an equation of motion for the soliton that is identical to the equation of motion for a single particle ratchet which is known to exhibit chaotic transport in its underdamped regime. For a quantitative comparison with our simulations we use a two-collective-coordinate (2CC) theory. In contrast to this, homogeneous sine-Gordon ratchets with biharmonic driving, which breaks a temporal shift symmetry, do not exhibit chaos. This is explained by a 2CC theory which yields two ODEs: one is linear, the other one describes a parametrically driven oscillator which does not exhibit chaos. The latter ODE can be solved by a perturbation theory which yields a hierarchy of linear equations that can be solved exactly order by order. The results agree very well with the simulations.

A multimodal public transport planning guidance for sustainable transport in developing countries

Many chaotic public transport systems in developing countries remain unchanged. The question is why we do not have faith in changing the system of public transport in terms of services and performance? Because many local governments think that modern system of public transport is difficult to start with, or it needs much money that they could not afford to support, or, as usual – it is no need to hurry up changing the system. This concept is misleading. In fact if the guidance to improve the public transport system is available then sustainable transport can be achieved and unnecessary expenses caused by pollution and deteriorating city can be avoided. Sustainable transport can be achieved by reducing fuel used for transportation for example reduce car use and switch to public transport use, decrease the excessive public transport to the efficient and effective public transport, switch unimodal to multimodal public transport system. Many examples can be adapted, but they still need adjustment, which might take longer than previously doing the complete survey. Therefore, it is desirable to write guidance for the initiative of establishing the multimodal public transport (MMPT) system.

Chaotic particle sedimentation in a rotating flow with time-periodic strength

Particle sedimentation in the vicinity of a fixed horizontal vortex with time-dependent intensity can be chaotic, provided gravity is sufficient to displace the particle cloud while the vortex is off or weak. This "stretch, sediment, and fold" mechanism is close to the so-called blinking vortex effect, which is responsible for chaotic transport of perfect tracers, except that in the present case the vortex motion is replaced by gravitational settling. In the present work this phenomenon is analyzed for heavy Stokes particles moving under the sole effect of gravity and of a linear drag. The vortex is taken to be a fixed isolated point vortex, the intensity of which varies under the effect of either boundary conditions or volume force. When the unsteadiness of the vortex is weak and the free-fall velocity is of the order of the fluid velocity, and the particle response time is small, the particle motion equation can be written asymptotically as a perturbed Hamiltonian system, the phase portrait of which displays a homoclinic trajectory. A homoclinic bifurcation is therefore likely to occur, and the contribution of particle inertia to the occurrence of this bifurcation is analyzed asymptotically by using Melnikov's method.

Phase synchronization in tilted inertial ratchets as chaotic rotators

The phenomenon of phase synchronization for a particle in a periodic ratchet potential is studied. We consider the deterministic dynamics in the underdamped case where the inertia plays an important role since the dynamics can become chaotic. The ratchet potential is tilted due to a constant external force and is rocking by an external periodic forcing. This potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of the external periodic forcing; this oscillator then acquires an intrinsic frequency that can be locked with the frequency of the external driving. We introduced an instantaneous linear phase, using a set of discrete time markers, and the associated average frequency, and show that this frequency can be synchronized with the frequency of the driving. We calculate Arnold tongues in a two-dimensional parameter space and discuss their implications for the chaotic transport in ratchets. We show that the local maxima in the current correspond to the borders of these Arnold tongues; in this way we established a link between optimal transport in ratchets and phase synchronization.

Reduction of chaotic particle transport driven by drift waves in sheared flows

Investigations of chaotic particle transport by drift waves propagating in the edge plasma of tokamaks with poloidal zonal flow are described. For large aspect ratio tokamaks, the influence of radial electric field profiles on convective cells and transport barriers, created by the nonlinear interaction between the poloidal flow and resonant waves, is investigated. For equilibria with edge shear flow, particle transport is seen to be reduced when the electric field shear is reversed. The transport reduction is attributed to the robust invariant tori that occur in nontwist Hamiltonian systems. This mechanism is proposed as an explanation for the transport reduction in Tokamak Chauffage Alfvén Brésilien [R. M. O. Galvão et al., Plasma Phys. Controlled Fusion 43, 1181 (2001)] for discharges with a biased electrode at the plasma edge.

Theory of dissipative chaotic atomic transport in an optical lattice

We study dissipative transport of spontaneously emitting atoms in a one-dimensional (1D) standing-wave laser field in the regimes where the underlying deterministic Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic wave-function method is applied to simulate semiclassically the atomic dynamics with coupled internal and translational degrees of freedom. It is shown in numerical experiments and confirmed analytically that chaotic atomic transport can take the form either of ballistic motion or a random walking with specific statistical properties. The character of spatial and momentum diffusion in the ballistic atomic transport is shown to change abruptly in the atom-laser detuning regime where the Hamiltonian dynamics is irregular in the sense of dynamical chaos. We find a clear correlation between the behavior of the momentum diffusion coefficient and Hamiltonian chaos probability which is a manifestation of chaoticity of the fundamental atom-light interaction in the diffusivelike dissipative atomic transport. We propose to measure a linear extent of atomic clouds in a 1D optical lattice and predict that, beginning with those values of the mean cloud's momentum for which the probability of Hamiltonian chaos is close to 1, the linear extent of the corresponding clouds should increase sharply. A sensitive dependence of statistical characteristics of dissipative transport on the values of the detuning allows us to manipulate the atomic transport by changing the laser frequency.

Finite Larmor radius effects on nondiffusive tracer transport in a zonal flow

Finite Larmor radius (FLR) effects on nondiffusive transport in a prototypical zonal flow with drift waves are studied in the context of a simplified chaotic transport model. The model consists of a superposition of drift waves from the linearized Hasegawa–Mima equation and a zonal shear flow perpendicular to the density gradient. High frequency FLR effects are incorporated by gyroaveraging the E×B velocity. Transport in the direction of the density gradient is negligible and we therefore focus on transport parallel to the zonal flows. A prescribed asymmetry produces strongly asymmetric non-Gaussian probability distribution functions (PDFs) of particle displacements, with Lévy flights in one direction only. For k⊥ρth=0, where k⊥ is the characteristic wavelength of the flow and ρth is the thermal Larmor radius, a transition is observed in the scaling of the second moment of particle displacements: σ2∼tγ. The transition separates ballistic motion (γ≈2) at intermediate times from superdiffusion (γ=1.6) at larger times. This change of scaling is accompanied by the transition of the PDF of particle displacements from algebraic decay to exponential decay. However, FLR effects seem to eliminate this transition. In all cases, the Lagrangian velocity autocorrelation function exhibits nondiffusive algebraic decay, C∼τ−κ, with κ=2−γ to a good approximation. The PDFs of trapping and flight events show clear evidence of algebraic scaling with decay exponents depending on the value of k⊥ρth. The shape and spatiotemporal self-similar anomalous scaling of the PDFs of particle displacements are reproduced accurately with a neutral (α=β), asymmetric, effective fractional diffusion model, where α and β are the orders of the spatial and temporal fractional derivatives, respectively.

Adiabatic divergence of the chaotic layer width and acceleration of chaotic and noise-induced transport

We show that, in spatially periodic Hamiltonian systems driven by a time-periodic coordinate-independent (AC) force, the upper energy of the chaotic layer grows unlimitedly as the frequency of the force goes to zero. This remarkable effect is absent in any other physically significant systems. It gives rise to the divergence of the rate of the spatial chaotic transport. We also generalize this phenomenon for the presence of a weak noise and weak dissipation. We demonstrate for the latter case that the adiabatic AC force may greatly accelerate the spatial diffusion and the reset rate at a given threshold.