The effect of small-scale inhomogeneities on ozone depletion in the Arctic

Universal alignment in turbulent pair dispersion

Countless processes in nature and industry, from rain droplet nucleation to plankton interaction in the ocean, are intimately related to turbulent fluctuations of local concentrations of advected matter. These fluctuations can be described by considering the change of the separation between particle pairs, known as pair dispersion, which is believed to obey a cubic in time growth according to Richardson's theory. Our work reveals a universal, scale-invariant alignment between the relative velocity and position vectors of dispersing particles at a mean angle that we show to be a universal constant of turbulence. We connect the value of this mean angle to Richardson's traditional theory and find agreement with data from a numerical simulation and a laboratory experiment. While the Richardson's cubic regime has been observed for small initial particle separations only, the constancy of the mean angle manifests throughout the entire inertial range of turbulence. Thus, our work reveals the universal nature of turbulent pair dispersion through a geometrical paradigm whose validity goes beyond the classical theory, and provides a framework for understanding and modeling transport and mixing processes.

Front propagation in a regular vortex lattice: Dependence on the vortex structure

We investigate the dependence on the vortex structure of the propagation of fronts in stirred flows. For this, we consider a regular set of vortices whose structure is changed by varying both their boundary conditions and their aspect ratios. These configurations are investigated experimentally in autocatalytic solutions stirred by electroconvective flows and numerically from kinematic simulations based on the determination of the dominant Fourier mode of the vortex stream function in each of them. For free lateral boundary conditions, i.e., in an extended vortex lattice, it is found that both the flow structure and the front propagation negligibly depend on vortex aspect ratios. For rigid lateral boundary conditions, i.e., in a vortex chain, vortices involve a slight dependence on their aspect ratios which surprisingly yields a noticeable decrease of the enhancement of front velocity by flow advection. These different behaviors reveal a sensitivity of the mean front velocity on the flow subscales. It emphasizes the intrinsic multiscale nature of front propagation in stirred flows and the need to take into account not only the intensity of vortex flows but also their inner structure to determine front propagation at a large scale. Differences between experiments and simulations suggest the occurrence of secondary flows in vortex chains at large velocity and large aspect ratios.

Front propagation in a vortex lattice: dependence on boundary conditions and vortex depth

We experimentally address the propagation of reaction-diffusion fronts in vortex lattices by combining, in a Hele-Shaw cell and at low Reynolds number, forced electroconvective flows and an autocatalytic reaction in solution. We consider both vortex chains and vortex arrays, the former referring to mixed free/rigid boundary conditions for vortices and the latter to free boundary conditions. Varying the depth of the fluid layer, we observe no variation of the mean front velocities for vortex arrays and a noticeable variation for vortex chains. This questions the two-dimensional character of front propagation in low Reynolds number vortex lattices, as well as the mechanisms of this dependence.

Competitive autocatalytic reactions in chaotic flows with diffusion: prediction using finite-time Lyapunov exponents

We investigate chaotic advection and diffusion in autocatalytic reactions for time-periodic sine flow computationally using a mapping method with operator splitting. We specifically consider three different autocatalytic reaction schemes: a single autocatalytic reaction, competitive autocatalytic reactions, which can provide insight into problems of chiral symmetry breaking and homochirality, and competitive autocatalytic reactions with recycling. In competitive autocatalytic reactions, species B and C both undergo an autocatalytic reaction with species A such that [Formula: see text] and [Formula: see text]. Small amounts of initially spatially localized B and C and a large amount of spatially homogeneous A are advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that local finite-time Lyapunov exponents (FTLEs) can accurately predict the final average concentrations of B and C after the reaction completes. The species that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If B and C start in regions with similar FTLEs, their average concentrations at the end of the reaction will also be similar. When a recycling reaction is added, the system evolves towards a single species state, with the FTLE often being useful in predicting which species fills the entire domain and which is depleted. The FTLE approach is also demonstrated for competitive autocatalytic reactions in journal bearing flow, an experimentally realizable flow that generates chaotic dynamics.

Fluctuations of a passive scalar in a turbulent mixing layer

The turbulent flow originating downstream of the Kelvin-Helmholtz instability in a mixing layer has great relevance in many applications, ranging from atmospheric physics to combustion in technical devices. The mixing of a substance by the turbulent velocity field is usually involved. In this paper, a detailed statistical analysis of fluctuations of a passive scalar in the fully developed region of a turbulent mixing layer from a direct numerical simulation is presented. Passive scalar spectra show inertial ranges characterized by scaling exponents -4/3 and -3/2 in the streamwise and spanwise directions, in agreement with a recent theoretical analysis of passive scalar scaling in shear flows [Celani et al., J. Fluid Mech. 523, 99 (2005)]. Scaling exponents of high-order structure functions in the streamwise direction show saturation of intermittency with an asymptotic exponent ζ_{∞}=0.4 at large orders. Saturation of intermittency is confirmed by the self-similarity of the tails of the probability density functions of the scalar increments at different scales r with the scaling factor r^{-ζ_{∞}} and by the analysis of the cumulative probability of large fluctuations. Conversely, intermittency saturation is not observed for the spanwise increments and the relative scaling exponents agree with recent results for homogeneous isotropic turbulence with mean scalar gradient. Probability density functions of the scalar increments in the three directions are compared to assess anisotropy.

Fast chemical reaction in two-dimensional Navier-Stokes flow: initial regime

This paper studies an infinitely fast bimolecular chemical reaction in a two-dimensional biperiodic Navier-Stokes flow. The reactants in stoichiometric quantities are initially segregated by infinite gradients. The focus is placed on the initial stage of the reaction characterized by a well-defined one-dimensional material contact line between the reactants. Particular attention is given to the effect of the diffusion κ of the reactants. This study is an idealized framework for isentropic mixing in the lower stratosphere and is motivated by the need to better understand the effect of resolution on stratospheric chemistry in climate-chemistry models. Adopting a Lagrangian straining theory approach, we relate theoretically the ensemble mean of the length of the contact line, of the gradients along it, and of the modulus of the time derivative of the space-average reactant concentrations (here called the chemical speed) to the joint probability density function of the finite-time Lyapunov exponent λ with two times τ and τ[over ̃]. The time 1/λ measures the stretching time scale of a Lagrangian parcel on a chaotic orbit up to a finite time t, while τ measures it in the recent past before t, and τ[over ̃] in the early part of the trajectory. We show that the chemical speed scales like κ(1/2) and that its time evolution is determined by rare large events in the finite-time Lyapunov exponent distribution. The case of smooth initial gradients is also discussed. The theoretical results are tested with an ensemble of direct numerical simulations (DNSs) using a pseudospectral model.

Are the fractal skeletons the explanation for the narrowing of arteries due to cell trapping in a disturbed blood flow?

We show that common circulatory diseases, such as stenoses and aneurysms, generate chaotic advection of blood particles. This phenomenon has major consequences on the way the biochemical particles behave. Chaotic advection leads to a peculiar filamentary particle distribution, which in turn creates a favorable environment for particle reactions. Furthermore, we argue that the enhanced stretching dynamics induced by chaos can lead to the activation of platelets, particles involved in the thrombus formation. In particular, we vary the size of both stenoses and aneurysms, and model them under resting and exercising conditions. We show that the filamentary particle distribution, governed by the fractal skeleton, depends on the size of the vessel wall irregularity, and investigate how it varies under resting or exercising conditions.

Fractal structures in stenoses and aneurysms in blood vessels

Recent advances in the field of chaotic advection provide the impetus to revisit the dynamics of particles transported by blood flow in the presence of vessel wall irregularities. The irregularity, being either a narrowing or expansion of the vessel, mimicking stenoses or aneurysms, generates abnormal flow patterns that lead to a peculiar filamentary distribution of advected particles, which, in the blood, would include platelets. Using a simple model, we show how the filamentary distribution depends on the size of the vessel wall irregularity, and how it varies under resting or exercise conditions. The particles transported by blood flow that spend a long time around a disturbance either stick to the vessel wall or reside on fractal filaments. We show that the faster flow associated with exercise creates widespread filaments where particles can get trapped for a longer time, thus allowing for the possible activation of such particles. We argue, based on previous results in the field of active processes in flows, that the non-trivial long-time distribution of transported particles has the potential to have major effects on biochemical processes occurring in blood flow, including the activation and deposition of platelets. One aspect of the generality of our approach is that it also applies to other relevant biological processes, an example being the coexistence of plankton species investigated previously.

Propagation of a chemical wave front in a quasi-two-dimensional superdiffusive flow

Pattern formation in reaction-diffusion systems is an important self-organizing mechanism in nature. Dynamics of systems with normal diffusion do not always reflect the processes that take place in real systems when diffusion is enhanced by a fluid flow. In such reaction-diffusion-advection systems diffusion might be anomalous for certain time and length scales. We experimentally study the propagation of a chemical wave occurring in a Belousov-Zhabotinsky reaction subjected to a quasi-two-dimensional chaotic flow created by the Faraday experiment. We present a novel analysis technique for the local expansion of the active wave front and find evidence of its superdiffusivity. In agreement with these findings the variance σ(2)(t)∝t(γ) of the reactive wave grows supralinear in time with an exponent γ>2. We study the characteristics of the underlying flow with microparticles. By statistical analysis of particle trajectories we derive flight time and jump length distributions and find evidence that tracer-particles undergo complex trajectories related to Lévy statistics. The propagation of active and passive media in the flow is compared.

Chaotic advection in blood flow

In this paper we argue that the effects of irregular chaotic motion of particles transported by blood can play a major role in the development of serious circulatory diseases. Vessel wall irregularities modify the flow field, changing in a nontrivial way the transport and activation of biochemically active particles. We argue that blood particle transport is often chaotic in realistic physiological conditions. We also argue that this chaotic behavior of the flow has crucial consequences for the dynamics of important processes in the blood, such as the activation of platelets which are involved in the thrombus formation.

Front propagation in a laminar cellular flow: shapes, velocities, and least time criterion

We experimentally investigate the propagation of chemical fronts in steady laminar cellular flows at large Péclet numbers and large Damköhler numbers. Fronts are generated in an aqueous solution by an autocatalytic oxydoreduction reaction. They propagate in a channel in which a chain of counter-rotative parallel vortices is induced by electroconvection. We first accurately determine the form, the dynamics and the mean velocity of these fronts in the whole Hele-Shaw regime of the flow. We then address the modeling of the evolution of their mean velocity with the flow amplitude. The structure of the front wakes yields us to reject an effective reaction-diffusion wave as a relevant model for large-scale front propagation. On the other hand, analysis of the role of front heads brings us to introduce a kinematic model at the vortex scale for uncovering the front dynamics. This model addresses the propagation of the front leading point in a chain of vortices whose field is modeled by a two-dimensional solid rotation complemented by a boundary layer. Interestingly, it sensitively relies on the effective trajectory followed by the front leading point. To account for this, a competition is worked out among a one-parameter family of potential trajectories. The actual trajectory is then selected as the fastest one with quite a good agreement with measurements and observations. In particular, the measured effective front velocities are well recovered from the model, including their intrinsic dependence on the boundary layer width. Accordingly, effective front propagation in a laminar steadily stirred medium is thus understood from an optimization principle similar to the Fermat principle of ray propagation in heterogeneous media.

Chemical-wave dynamics in a vertically oscillating fluid layer

Classical Faraday experiments were conducted on the oscillatory chemical Belousov-Zhabotinsky (BZ) reaction. The vertical periodic modulation of the acceleration induces flows in the system that change the BZ dynamics, and thus the patterns exhibited. The resulting reaction-diffusion-advection system exhibits four different types of pattern for increasing stirring amplitude: deformed targets and spiral waves, filamentary patterns arranged in large-scale vortices, advection phase waves, and finally front annihilation where the medium becomes homogeneous. A wave period analysis of the forced system has been carried out. Contrary to what is expected, i.e., a continuous increase of the wave period with increasing forcing, the period changes dramatically at the boundaries between pattern domains.

Stopping and initiation of a chemical pulse at the interface of excitable media with different diffusivity

The dynamics of a chemical pulse at the interface of excitable media having different diffusion properties is presented experimentally using the Belousov-Zhabotinsky reaction system. When a chemical pulse propagates from a larger diffusion region to a smaller diffusion region, it stops and forms a steady excitation band at the interface. Furthermore, a pulse train stimulated by this band appears subsequently. A simple one-dimensional model with discontinuous diffusion is proposed, and a numerical simulation is performed that shows good agreement with the experiment. Unidirectional pulse propagation across the interface is strongly suggested on the basis of the proposed model.

Fast chemical reaction and multiple-scale concentration fields in singular vortices

Two species involved in a simple, fast reaction tend to become segregated in patches composed of a single of these reactants. These patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs, fed by diffusion. Stirred by advection, this boundary and the concentration fields within the patches may tend to present multiple-scale characteristics. Based on this segregated state, this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it, when concentrations presenting initial self-similar fluctuations are advected by a singular vortex. First the two sources of singularity, i.e., the self-similar initial conditions and the singular vortex, are considered separately. On the one hand, self-similar initial conditions are imposed to a diffusion-reaction system, for one- and two-dimensional cases. On the other hand, an imposed singular vortex advects initially on/off concentration fields, in combination with diffusion and reaction. This problem is addressed analytically, by characterizing the boundary by a box-counting dimension and the concentration fields by a Hölder exponent, and numerically, by direct numerical simulations of the advection-diffusion-reaction equations. Second, the way the two sources hang together shows that, depending on the self-similar properties of the initial concentration fields, the vortex promotes the chemical activity close to its inner smoothed-out core or close to the outer region where the boundary starts to spiral. For all the considered situations, the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws.

Bifurcations of flame filaments in chaotically mixed combustion reactions

We study the behaviour of steady-state solutions of a two-component flame filament system subject to chaotic mixing. This system exhibits a saddle-node bifurcation at a critical Damköhler number. We analyze the system through a one-dimensional phenomenological lamellar model. We present a nonperturbative technique, which allows us to describe the behaviour of the reduced lamellar model near the saddle node bifurcation. The influence of the Lewis number on the solution behaviour is investigated. We present a simple empirical formula for the wave speed valid for large Damköhler and large Lewis numbers. This formula allows us to describe the solution far away from the bifurcation. Numerical simulations show good agreement with the results.

Persistent localized states for a chaotically mixed bistable reaction

We describe the evolution of a bistable chemical reaction in a closed two-dimensional chaotic laminar flow, from a localized initial disturbance. When the fluid mixing is sufficiently slow, the disturbance may spread and eventually occupy the entire fluid domain. By contrast, rapid mixing tends to dilute the initial state and so extinguish the disturbance. Such a dichotomy is well known. However, we report here a hitherto apparently unremarked intermediate case, a persistent highly localized disturbance. Such a localized state arises when the Damköhler number is great enough to sustain a "hot spot," but not so great as to lead to global spread. We show that such a disturbance is located in the neighborhood of an unstable periodic orbit of the flow, and we describe some limited aspects of its behavior using a reduced, lamellar model.

Resonant pattern formation in active media driven by time-dependent flows

The effect of a time-dependent flow in an oscillatory chemical system supporting front propagation is studied. Resonant target patterns depend on the strength and frequency of the time-dependent flow. The flow time scale needed to entrain the system to the resonant target period of oscillation depends on the closeness to the natural oscillation frequency of the medium. The flow strength needed to obtain these patterns is interpreted in terms of mixing optimization, and we give conditions for the flow that guarantee the best mixing with the Bernoulli property.

The Role of Pair Dispersion in Turbulent Flow

Mixing and transport in turbulent flows-which have strong local concentration fluctuations-are essential in many natural and industrial systems including reactions in chemical mixers, combustion in engines and burners, droplet formation in warm clouds, and biological odor detection and chemotaxis. Local concentration fluctuations, in turn, are intimately tied to the problem of the separation of pairs of fluid elements. We have measured this separation rate in an intensely turbulent laboratory flow and have found, in quantitative agreement with the seminal predictions of Batchelor, that the initial separation of the pair plays an important role in the subsequent spreading of the fluid elements. These results have surprising consequences for the decay of concentration fluctuations and have applications to biological and chemical systems.

The impact of diffusion and stirring on the dynamics of interacting populations

We investigate the combined effects of diffusion and stirring on the dynamics of interacting populations which have spatial structure. Specifically we consider the marine phytoplankton and zooplankton populations, and model them as an excitable medium. The results are applicable to other biological and chemical systems. Under certain conditions the combination of diffusion and stirring is found to enhance the excitability, and hence population growth of the system. Diffusion is found to play an important role: too much and initial perturbations are smoothed away, too little and insufficient mixing takes place before the reaction is over. A key time-scale is the mix-down time, the time it takes for the spatial scale of a population to be reduced to that of a diffusively controlled filament. If the mix-down time is short compared to the reaction time-scale, then excitation of the system is suppressed. For intermediate values of the mix-down time the peak population can attain values many times that of a population without spatial structure. We highlight the importance of the spatial scale of the initial disturbance to the system.

Bifurcations in reaction-diffusion systems in chaotic flows

We study the behavior of reacting tracers in a chaotic flow. In particular, we look at an autocatalytic reaction and at a bistable system which are subjected to stirring by a chaotic flow. The impact of the chaotic advection is described by a one-dimensional phenomenological model. We use a nonperturbative technique to describe the behavior near a saddle node bifurcation. We also find an approximation of the solution far away from the bifurcation point. The results are confirmed by numerical simulations and show good agreement.

Reactive particles in random flows

We study the dynamics of chemically or biologically active particles advected by open flows of chaotic time dependence, which can be modeled by a random time dependence of the parameters on a stroboscopic map. We develop a general theory for reactions in such random flows, and derive the reaction equation for this case. We show that there is a singular enhancement of the reaction in random flows, and this enhancement is increased as compared to the nonrandom case. We verify our theory in a model flow generated by four point vortices moving chaotically.

Universality in active chaos

Many examples of chemical and biological processes take place in large-scale environmental flows. Such flows generate filamental patterns which are often fractal due to the presence of chaos in the underlying advection dynamics. In such processes, hydrodynamical stirring strongly couples into the reactivity of the advected species and might thus make the traditional treatment of the problem through partial differential equations difficult. Here we present a simple approach for the activity in inhomogeneously stirred flows. We show that the fractal patterns serving as skeletons and catalysts lead to a rate equation with a universal form that is independent of the flow, of the particle properties, and of the details of the active process. One aspect of the universality of our approach is that it also applies to reactions among particles of finite size (so-called inertial particles).

Reactive dynamics of inertial particles in nonhyperbolic chaotic flows

Anomalous kinetics of infective (e.g., autocatalytic) reactions in open, nonhyperbolic chaotic flows are important for many applications in biological, chemical, and environmental sciences. We present a scaling theory for the singular enhancement of the production caused by the universal, underlying fractal patterns. The key dynamical invariant quantities are the effective fractal dimension and effective escape rate, which are primarily determined by the hyperbolic components of the underlying dynamical invariant sets. The theory is general as it includes all previously studied hyperbolic reactive dynamics as a special case. We introduce a class of dissipative embedding maps for numerical verification.

New scaling law for the decay exponent of bimolecular reactions in unbounded transitional flows

We propose a new scaling law for global kinetics of the stoichiometric reaction A+B-->P in unsteady, transitional flows. We find in the nonlinear flow regime the decay as approximately t(-alpha) where alpha is related to a space-time scaling parameter psi as alpha proportional, variant psi(m), for the considered parameter range m=0.067. In the linear flow regime, we find that the maximum is alpha approximately 2/3 for psi approximately 1. The proposed scaling law should be useful for linking dynamical subgrid processes with reaction kinetics in a variety of transitional flow systems.

Chaotic mixing induced transitions in reaction-diffusion systems

We study the evolution of a localized perturbation in a chemical system with multiple homogeneous steady states, in the presence of stirring by a fluid flow. Two distinct regimes are found as the rate of stirring is varied relative to the rate of the chemical reaction. When the stirring is fast localized perturbations decay towards a spatially homogeneous state. When the stirring is slow (or fast reaction) localized perturbations propagate by advection in form of a filament with a roughly constant width and exponentially increasing length. The width of the filament depends on the stirring rate and reaction rate but is independent of the initial perturbation. We investigate this problem numerically in both closed and open flow systems and explain the results using a one-dimensional "mean-strain" model for the transverse profile of the filament that captures the interplay between the propagation of the reaction-diffusion front and the stretching due to chaotic advection. (c) 2002 American Institute of Physics.

Small-scale structure of nonlinearly interacting species advected by chaotic flows

We study the spatial patterns formed by interacting biological populations or reacting chemicals under the influence of chaotic flows. Multiple species and nonlinear interactions are explicitly considered, as well as cases of smooth and nonsmooth forcing sources. The small-scale structure can be obtained in terms of characteristic Lyapunov exponents of the flow and of the chemical dynamics. Different kinds of morphological transitions are identified. Numerical results from a three-component plankton dynamics model support the theory, and they serve also to illustrate the influence of asymmetric couplings. (c) 2002 American Institute of Physics.

Diffusivity dependence of ozone depletion over the midnorthern latitudes

The mixing and reaction properties of advected chemicals (and passive scalars) are determined by the fractal dimension D of the interface between the chemicals. We show that the scaling of the amount m of reacted chemicals with diffusivity kappa is m(0)-m(kappa) proportional, proportional to kappa(1-D/2) in the two-dimensional case. This relation is valid in a range of times and diffusivities where the diffusive length scales of the chemicals are within the range of scales where the chemical interface has a well-defined fractal dimension. We apply the relation to the problems of chlorine deactivation and ozone depletion over the midnorthern latitudes. We determine numerically the fractal dimension of an interface advected by stratospheric winds. This allows us, first, to explain the diffusivity dependence of chlorine deactivation and ozone depletion that was previously observed in numerical simulations (Tan et al., J. Geophys. Res., [Atmos.] 103, 1585 (1998)) and, second, to extrapolate the results of such simulations down to realistically low diffusivities.

Scalar turbulence

The advection of a passive substance by a turbulent flow is important in many natural and engineering settings. The concentration of such a substance can exhibit complex dynamic behaviour that shows many phenomenological parallels with the behaviour of the turbulent velocity field. Yet the statistical properties of this so-called 'passive scalar' turbulence are decoupled from those of the underlying velocity field. Passive scalar turbulence has recently yielded to mathematical analysis, and such progress may ultimately lead to a better understanding of the still intractable problem of fluid turbulence itself.

Multifractal structure of chaotically advected chemical fields

The structure of the concentration field of a decaying substance produced by chemical sources and advected by a smooth incompressible two-dimensional flow is investigated. We focus our attention on the nonuniformities of the Holder exponent of the resulting filamental chemical field. They appear most evidently in the case of open flows where irregularities of the field exhibit strong spatial intermittency as they are restricted to a fractal manifold. Nonuniformities of the Holder exponent of the chemical field in closed flows appears as a consequence of the nonuniform stretching of the fluid elements. We study how this affects the scaling exponents of the structure functions, displaying anomalous scaling, and relate the scaling exponents to the distribution of local Lyapunov exponents of the advection dynamics. Theoretical predictions are compared with numerical experiments.

Chaotic advection, diffusion, and reactions in open flows

We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows. First, the problem of passive advection is considered, and its fractal and chaotic nature is pointed out. Next, we study the effect of weak molecular diffusion or randomness of the flow. Finally, we investigate the influence of passive advection on chemical or biological activity superimposed on open flows. The nondiffusive approach is shown to carry some features of a weak diffusion, due to the finiteness of the reaction range or reaction velocity. (c) 2000 American Institute of Physics.

Structure function of passive scalars in two-dimensional turbulence

The structure function of a scalar theta(x,t), passively advected in a two-dimensional turbulent flow u(x,t), is discussed by means of the fractal dimension delta1g of the passive-scalar graph. A relation between delta1g, the scaling exponent zeta(theta)(1) of the scalar structure function D1(theta)r, and the structure function D2(r) of the underlying flow field is derived. Different from the three-dimensional (3D) case, the 2D structure function also depends on an additional parameter, characteristic of the driving of the passive scalar. In the enstrophy inertial subrange a mean-field approximation for the velocity structure function gives a scaling of the passive scalar graph with delta1g<2 for intermediate and large values of the Prandtl number Pr. In the energy inertial subrange a model for the energy spectrum and thus D2(r) gives a passive-scalar graph scaling with exponent delta1g=5/3. Finally, we discuss an application to recent observations of scalar dispersion in nonuniversal 2D flows.

Chemical or biological activity in open chaotic flows

We investigate the evolution of particle ensembles in open chaotic hydrodynamical flows. Active processes of the type A+B-->2B and A+B-->2C are considered in the limit of weak diffusion. As an illustrative advection dynamics we consider a model of the von Kármán vortex street, a time-periodic two-dimensional flow of a viscous fluid around a cylinder. We show that a fractal unstable manifold acts as a catalyst for the process, and the products cover fattened-up copies of this manifold. This may account for the observed filamental intensification of activity in environmental flows. The reaction equations valid in the wake are derived either in the form of dissipative maps or differential equations depending on the regime under consideration. They contain terms that are not present in the traditional reaction equations of the same active process: the decay of the products is slower while the productivity is much faster than in homogeneous flows. Both effects appear as a consequence of underlying fractal structures. In the long time limit, the system locks itself in a dynamic equilibrium state synchronized to the flow for both types of reactions. For particles of finite size an emptying transition might also occur leading to no products left in the wake.