Vered Rom-Kedar

An analytical study of transport, mixing and chaos in an unsteady vortical flow

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikovs technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

Homoclinic tangles-classification and applications

Here we develop the topological approximation method which gives a new description of the mixing and transport processes in chaotic two-dimensional time-periodic Hamiltonian flows. It is based upon the structure of the homoclinic tangle, and supplies a detailed solution to a transport problem for this class of systems, the characteristics of which are typical to chaotic, yet not ergodic dynamical systems. These characteristics suggest some new criteria for quantifying transport and mixing-hence chaos-in such systems. The results depend on several parameters, which are found by perturbation analysis in the near integrable case, and numerically otherwise. The strength of the method is demonstrated on a simple model. We construct a bifurcation diagram describing the changes in the homoclinic tangle as the physical parameters are varied. From this diagram we find special regions in the parameter space in which we approximate the escape rates from the vicinity of the homoclinic tangle, finding nontrivial self-similar solutions as the forcing magnitude tends to zero. We compare the theoretical predictions with brute force calculations of the escape rates, and obtain satisfactory agreement.

Transport rates of a class of two-dimensional maps and flows

Abstract A method is developed for estimating the transport rates of phase space areas for a class of two-dimensional diffeomorphisms and flows. The class of diffeomorphisms we considered are defined by the topological structure of their stable and unstable manifolds, and hence are universal. We show how to estimate the transport rates for a class of diffeomorphisms found by Easton and for an extension of this class of diffeomorphisms which is found via a “perturbation” of the topology of the stable and unstable manifolds. This is done by introducing symbolic dynamics and transfer matrices which in turn relate transport phenomena in phase space to Markov processes in a precise manner. In addition to the transport rates, we use the transfer matrices to obtain estimates for the topological entropy, averaged stretching rates, and the elongation rate of the unstable manifold. The flows we consider are two-dimensional, time-periodic flows which can be reduced via Poincare section to the extended family of maps. We develop an analytical method, based on Chirikovs Whisker map, to classify a given flow according to the structure of its manifolds in its Poincare section. This allows the techniques developed here for maps to be directly applied to time-periodic flows.

Universal properties of chaotic transport in the presence of diffusion

The combined, finite time effects of molecular diffusion and chaotic advection on a finite distribution of scalar are studied in the context of time periodic, recirculating flows with variable stirring frequency. Comparison of two disparate frequencies with identical advective fluxes indicates that diffusive effects are enhanced for slower oscillations. By examining the geometry of the chaotic advection in both high and low frequency limits, the flux function and the width of the stochastic zone are found to have a universal frequency dependence for a broad class of flows. Furthermore, such systems possess an adiabatic transport mechanism which results in the establishment of a “Lagrangian steady state,” where only the asymptotically invariant core remains after a single advective cycle. At higher frequencies, transport due to chaotic advection is confined to exchange along the perimeter of the recirculating region. The effects of molecular diffusion on the total transport are different in these two cases...

Big islands in dispersing billiard-like potentials

Abstract We derive a rigorous estimate of the size of islands (in both phase space and parameter space) appearing in smooth Hamiltonian approximations of scattering billiards. The derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials. Thus, universality classes for the local behavior are found. Moreover, for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter ϵ→0, islands of polynomial size in ϵ appear. This suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling, for example, scattering molecules, may be of physically noticeable effect.

Islands of accelerator modes and homoclinic tangles.

Islands are divided according to their phase space structure-resonant islands and tangle islands are considered. It is proved that in the near-integrable limit these correspond to two distinct sets, hence that in general their definitions are not trivially equivalent. It is demonstrated and proved that accelerator modes of the standard map and of the web map are necessarily of the tangle island category. These islands have an important role in determining transport-indeed it has been demonstrated in various works that stickiness to these accelerator modes may cause anomalous transport even for initial conditions starting in the ergodic component. (c) 1999 American Institute of Physics.

Beyond all orders: Singular perturbations in a mapping

SummaryWe consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parameterε. Asε → 0+ these maps asymptote to flows which attain a heteroclinic connection. We show that for smallε the heteroclinic connection breaks up and that the splitting between its components scales withε likeεγexp[-β/ε]. We estimateβ using the singularities of theε → 0+ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior of the functions in the complex plane, are supported by our numerical calculations.

Transport in two-dimensional maps: concepts, examples, and a comparison of the theory of Rom-Kedar and Wiggins with the Markov model of MacKay, Meiss, Ott, and percival

Abstract We consider the problem of transport across partial barriers in the phase space of two-dimensional maps. We study examples illustrating the recent transport theory of Rom-Kedar and Wiggins and provide a detailed comparison with the Markov model of transport across partial barriers developed by MacKay, Meiss, and Percival and Meiss and Ott.

Soft Billiards with Corners

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiards corner point. We prove that smooth Hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are “acceptable.” The criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners, which is expressed in terms of a scattering function. We define such an asymptotic scattering function and prove the existence of it, explain how can it be calculated and predict some of its properties. In particular, we show that it is non-monotone for some potentials in some phase space regions. We prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there. We then prove that if the scattering function is extremal, the smooth system has elliptic periodic orbits limiting to the corner polygon, and, furthermore, that the return map near these periodic orbits is conjugate to a small perturbation of the Hénon map and therefore has elliptic islands. We find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit.

Bistability and Bacterial Infections

Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patients neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection.