And Inertia-Induced Inhomogeneity in the Distribution of Small Particles in ﬂuid ﬂows Julyan H

CHAOS VOLUME 12, NUMBER 2 JUNE 2002

Noise- and inertia-induced inhomogeneity in the distribution of small particles in fluid flows Julyan H. E. Cartwrighta) Laboratorio de Estudios Cristalogra´ficos, CSIC, E-18071 Granada, Spain Marcelo O. Magnascob) Mathematical Physics Laboratory, Rockefeller University, P.O. Box 212, 1230 York Avenue, New York, New York 10021 Oreste Piroc) Institut Mediterrani d’Estudis Avançats, CSIC–UIB, E-07071 Palma de Mallorca, Spain ͑Received 2 November 2001; accepted 25 March 2002; published 20 May 2002͒ The dynamics of small spherical neutrally buoyant particulate impurities immersed in a two-dimensional fluid flow are known to lead to particle accumulation in the regions of the flow in which vorticity dominates over strain, provided that the Stokes number of the particles is sufficiently small. If the flow is viewed as a Hamiltonian dynamical system, it can be seen that the accumulations occur in the nonchaotic parts of the phase space: the Kolmogorov–Arnold–Moser tori. This has suggested a generalization of these dynamics to Hamiltonian maps, dubbed a bailout embedding. In this paper we use a bailout embedding of the standard map to mimic the dynamics of neutrally buoyant impurities subject not only to drag but also to fluctuating forces modeled as white noise. We find that the generation of inhomogeneities associated with the separation of particle from fluid trajectories is enhanced by the presence of noise, so that they appear in much broader ranges of the Stokes number than those allowing spontaneous separation. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1480441͔

Impurities suspended in a fluid flow are frequently ob- I. INTRODUCTION served to be distributed inhomogeneously. Even in very When impurities have a different density to the fluid, it is chaotic flows, particulate impurities arrange themselves intuitively clear that they will be expelled from rapidly ro- in extraordinarily structured distributions, in apparent tating regions of the flow—for heavy particles—or attracted contradiction to the high mixing efficiency expected from to the centers of these regions—for light particles—because the characteristics of the basic flow. To give just one ex- of centrifugal effects.7 However, it was recently demon- ample, in the particular instance of geophysical fluids, the strated that neutrally buoyant particles also tend to settle in filamentary structure, or patchiness, often displayed by the rotation ͑vorticity͒-dominated regions of a flow, but by a plankton populations in the oceans is a puzzling problem more subtle mechanism involving the separation between the currently under intense investigation.1–3 Several mecha- fluid and particle trajectories that can occur in the opposite 8 nisms to produce this type of inhomogeneity have been regions, i.e., in the areas of the flow dominated by strain. studied, and include dynamical aspects of the flow as well However, this mechanism is only relevant when the flow as the reactive properties of the considered impurities. gradients are of the order of the particle drag coefficient, a condition that may not be fulfilled in some physically inter- The basic idea in many of these mechanisms is that the esting situations. We show here by means of a minimal particle loss due either to the flow—in open flows—or to model that this condition may be relaxed if a small amount the chemical or population dynamics of the particles—in of noise is added to the forces acting on the impurity. closed flows—is minimized on some manifolds associated Our approach is qualitative, in the sense that instead of 4–6 with the hyperbolic character of the flow. In this paper considering a specific flow and the precise particle dynamics we explore an alternative purely dynamical mechanism induced by it, we describe the system with an iterative map for inhomogeneity with nonreactive particles in bounded whose evolution contains the basic features of both the fluid flows. We show that particle inertial effects combined flow and the particle dynamics: flow volume preservation, with fluctuating forces are capable of producing inhomo- together with particle separation in the hyperbolic regions. geneity even in cases in which the impurity and fluid den- The reason for moving to a discrete system is that in the map sities match exactly. the phenomena that we describe may be understood more intuitively, while translating the results back to the flow case is immediate. The strategy of attempting to understand fluid-

͒ dynamical phenomena by using iterated maps amenable to a Electronic mail: [email protected] b͒Electronic mail: [email protected] the powerful artillery of dynamical-systems theory has c͒Electronic mail: [email protected] proved successful on several different occasions. For ex-

Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp 490 Chaos, Vol. 12, No. 2, 2002 Cartwright, Magnasco, and Piro

ample, the structures of the chaotic advection induced by Here v represents the velocity of the particle, u that of the ␳ ␳ time-periodic three-dimensional incompressible flows were fluid, p the density of the particle, f the density of the fluid predicted by studying the qualitatively equivalent dynamics it displaces, ␯ the kinematic viscosity of the fluid, a, the of three-dimensional volume-preserving maps.9,10 Later, radius of the particle, and g, gravity. The terms on the right- these structures were confirmed in realistic flows.11–13 Other hand side of Eq. ͑1͒ represent, respectively, the force exerted examples are the treatment of the propagation of combustion by the undisturbed flow on the particle, buoyancy, Stokes fronts in laminar flows by a qualitative map approach,14 and drag, the added mass due to the boundary layer of fluid mov- the description of the formation of plankton population struc- ing with the particle,19,20 and the Basset–Boussinesq tures due to inhomogeneities of the nutrient sources.15 Re- force21,22 that depends on the history of the relative accelera- markably, in the instance we discuss in this paper, the pro- tions of particle and fluid. The terms in a2ٌ2u are the cedure is also useful from the point of view of dynamical- Faxeń23 corrections. The Maxey–Riley equation is derived systems theory, as it has suggested a new technique—bailout under the assumptions that the particle radius and its Rey- embedding—for the control of Hamiltonian chaos.16 nolds number are small, as are the velocity gradients around The plan of the paper is as follows. First, we briefly the particle. review the classical model for the forces acting on a small First let us consider a minimal model for a neutrally ␳ ϭ␳ ͑ ͒ spherical particle moving relative to the fluid in which it is buoyant particle. For this we set p f in Eq. 1 . We con- immersed, and, concentrating on the case of a neutrally sider the Faxeń corrections and the Basset–Boussinesq term buoyant impurity, we trace the construction of a minimal to be negligible. We now rescale space, time, and velocity by model that makes evident the separation of particle and fluid scale factors L, TϭL/U, and U, to arrive at trajectories in the regions in which the flow presents strong strain ͑Sec. II͒. On the basis of this model, we make a gen- dv Du Ϫ 1 dv Du ϭ ϪSt 1͑vϪu͒Ϫ ͩ Ϫ ͪ , ͑4͒ eralization that allows us to build a discrete mapping that dt Dt 2 dt Dt represents the Lagrangian evolution of the fluid parcels as ϭ 2 ␯ well as the dynamics of the particle ͑Sec. III͒. While this where St is the particle Stokes number St 2a U/(9 L) ϭ 2 map displays particle–fluid separation when the bailout pa- 2/9(a/L) Ref ,Ref being the fluid Reynolds number. The ͑ ͒ Ӷ rameter ␥, a function of the Stokes number, is relatively assumptions involved in deriving Eq. 1 require that St 1 ͑ ͒ small, we show in Sec. IV that a small amount of noise, in Eq. 4 . added to the dynamics of the particle to separate it continu- If we substitute the expressions for the derivatives in ͑ ͒ ͑ ͒ ͑ ͒ ally from the flow, enhances the impact of the hyperbolic Eqs. 2 and 3 into Eq. 4 , we obtain ␥ regions far beyond the values of required for separation in d the deterministic case. Section V extends and formalizes ͑ Ϫ ͒ϭϪ͑͑ Ϫ ͒" ͒ Ϫ␥͑ Ϫ ͒ ͑ ͒ v u v u “ u v u , 5 these ideas with analytical arguments. Our conclusions are to dt be found in Sec. VI. where we have written ␥ϭ2/3StϪ1. We may then write A ϭvϪu, whence II. MAXEY–RILEY EQUATIONS AND MINIMAL MODEL dA ϭϪ͑Jϩ␥I͒"A, ͑6͒ The equation of motion for a small, spherical particle in dt an incompressible fluid we term the Maxey–Riley equation,8,17,18 which may be written as where J is the velocity gradient matrix—we now concentrate on two-dimensional flows uϭ(u ,u )— dv Du 9␯␳ a2 x y ͪ ␳ ϭ␳ ϩ͑␳ Ϫ␳ ͒ Ϫ f ͩ Ϫ Ϫ ٌ2 ץ ץ p f p f g 2 v u u dt Dt 2a 6 xux yux Jϭͩ ͪ . ͑7͒ u ץ u ץ ␳ 2 f dv D a x y y y Ϫ ͩ Ϫ ͫuϩ ٌ2uͬͪ 2 dt Dt 10 If we diagonalize this matrix, and heuristically assume that the dependence on time of the diagonalizing transformation 9␳ ␯ t 1 d a2 Ϫ f ͱ ͵ Ϫ Ϫ ٌ2 ␨ is unimportant, we obtain ␲ ␨ ͩ v u uͪ d , 2a 0 ͱtϪ␨ d 6 ␭Ϫ␥ dAD 0 ͑1͒ ϭͩ ͪ "A , ͑8͒ dt 0 Ϫ␭Ϫ␥ D where the derivative Du/Dt is along the path of a fluid ele- ␭ Ͼ␥ ␭ ment so if Re( ) , AD may grow exponentially. Now satisfies Ϫ␭ ϭ ␭2Ϫ ϩ ϭ -u det(J I) 0, so tr J det J 0. Since the flow is inץ Du ϭ ϭ Ϫ␭2ϭ ץϩ ץ ϭ ϩ͑u" ͒u, ͑2͒ compressible, xux yuy tr J 0, thence det J. 2 ץϪ ץ t “ ␻2ϭץ Dt Given squared vorticity ( xuy yux) , and squared ץ2ϭ 2ϩ 2 ϭ whereas the derivative du/dt is taken along the trajectory of strain s s1 s2, where the normal component is s1 xux ץϩ ץϭ ץϪ the particle yuy and the shear component is s2 yux xuy , we may write uץ du ϭ ϩ͑v" ͒u. ͑3͒ t “ Qϭ␭2ϭϪdet Jϭ͑s2Ϫ␻2͒/4, ͑9͒ץ dt

Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp Chaos, Vol. 12, No. 2, 2002 Small particles in fluid flows 491 where Q is the Okubo–Weiss parameter.24,25 If QϾ0, ␭2 of the above-described impurity dynamics. Given a map Ͼ ␭ ϭ 0, and is real, deformation dominates, as around hyper- xnϩ1 T(xn)—x being a point in a space of arbitrary bolic points, whereas if QϽ0, ␭2Ͻ0, and ␭ is complex, dimension—a bailout embedding is the second-order recur- rotation dominates, as near elliptic points. rence Equation ͑6͒, together with dx/dtϭAϩu, define a dis- Ϫ ͑ ͒ϭ ͑ ͒͑ Ϫ ͑ ͒͒ ͑ ͒ sipative dynamical system xnϩ2 T xnϩ1 K xn xnϩ1 T xn , 11 d␰/dtϭF͑␰,t͒ ͑10͒ where K(x) is chosen such that ͉K(x)͉Ͼ1 over the unwanted set of orbits, so that they become unstable in the embedding. " ϭϪ ␥ with constant divergence “ F 2 in the four dimensional In this discrete system, almost any expression written for the ␰ϭ phase space (x,y,Ax ,Ay), so that while small values of ordinary differential equation, Eq. ͑6͒, translates to some- ␥ St—large values of —allow for large values of the diver- thing close to an exponential; in particular, stability eigenval- ␥ gence, large values of St—small values of —force the di- ues have to be negative in the ordinary-differential-equation vergence to be small. The Stokes number is the relaxation case to represent stability, while they have to be smaller than time of the particle back onto the fluid trajectories compared one in absolute value in the map case. In order to simulate ͑ ␥͒ to the time scale of the flow—with larger St smaller , the the dynamics of particles, the operator Ϫ(Jϩ␥I) in the con- particle has more independence from the fluid flow. From tinuous system should translate into the particular choice Eq. ͑8͒, about areas of the flow near hyperbolic stagnation points with QϾ␥2, particle and flow trajectories separate ͑ ͒ϭ Ϫ␥ ͑ ͒ K x e “T 12 exponentially. The result is that the particle can overcome Stokes drag and abandon the fluid trajectories in the neigh- in the map setting. borhood of the saddle points, to finally end up in a regular To represent qualitatively a chaotic two-dimensional in- region of the flow. This effect implies that particles tend to compressible base flow we choose a classical testbed of stay away from the regions of strongest strain. Hamiltonian systems, the area preserving standard map in- In earlier work8 it was shown that when the flow is troduced by Chirikov and Taylor: chaotic—for example for two-dimensional time-periodic in- T:͑x ,y ͒→͑x ϩ ,y ϩ ͒, ͑13͒ compressible flows—it is a consequence of this phenomenon n n n 1 n 1 that particles asymptotically settle on invariant tori, and, in where general, explore the ordered regions of the base flow. In the following, we investigate further this behavior in a qualita- k x ϩ ϭx ϩ sin͑2␲y ͒, tive but more general framework based on iterative map n 1 n 2␲ n modeling of the dynamics. ͑14͒ ϭ ϩ y nϩ1 y n xnϩ1 ,

III. DISCRETE DYNAMICS DESCRIPTION and k is the parameter controlling integrability. Recall that, in general, the dynamics defined by this map present a mix- An examination of the dynamical system defining our ture of quasiperiodic motions occurring on the Kolmogorov– minimal model for the behavior of neutrally buoyant par- Arnold–Moser ͑KAM͒ tori, and chaotic ones, depending on ticles shows that it is composed of some dynamics within the initial conditions. As the value of k is increased, the some other larger set of dynamics. Equation ͑4͒ can be seen region dominated by chaotic trajectories pervades more and as an equation for a variable A that defines another equation more of the phase space, except for increasingly small is- of motion, vϭu, for a fluid element, when A is zero. The lands of KAM quasiperiodicity. Our qualitative description inner dynamics is that representing the Lagrangian trajecto- of the impurity dynamics, the bailout embedding of the stan- ries of the flow, and in two dimensions it is a Hamiltonian dard map, is given by the coupled second-order iterative sys- system. We may say that this Hamiltonian system—the fluid tem defined by Eqs. ͑11͒–͑14͒. flow—is embedded in a larger dynamical system—the fluid– Due to the area-preserving nature of the standard map, particle system—this time dissipative, whose trajectories the two eigenvalues of the derivative matrix must multiply to may or may not converge to zero. If they do, the system ends one. If they are complex, this means that both have an abso- up on the same trajectories as those of the smaller embedded lute value of one, while if they are real, generically one of system, but in general this need not be the case. The concept them will be larger than one and the other smaller. We can of some dynamics embedded within some other dynamics then separate the phase space into elliptic and hyperbolic may be exploited within the framework of dynamical- regions corresponding to each of these two cases. If a trajec- systems theory to design techniques to reject unwanted tra- tory of the original map lies entirely in the elliptic regions, jectories of the original dynamics by making them unstable the overall factor exp(Ϫ␥) damps any small perturbation in the embedding, paralleling what the particles do in the away from it in the embedded system. But for chaotic trajec- fluid-dynamical case. We have dubbed this idea, which has tories that inevitably visit some hyperbolic regions, there ex- obvious applications to control and targeting, a bailout em- ists a value of ␥ such that, for ␥ smaller than this, perturba- bedding of a dynamical system.16 tions away from a standard-map trajectory are amplified Trading generality for clarity, in this section we present instead of dying out in the embedding. As a consequence, an example of a bailout embedding for discrete-time dynami- these trajectories are expelled from the chaotic regions fi- cal sytems that closely represents all the qualitative aspects nally to settle in the elliptic KAM islands.

Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp 492 Chaos, Vol. 12, No. 2, 2002 Cartwright, Magnasco, and Piro

etc., renders the system ergodic with only a uniform invariant measure. Thus, the distribution of ﬂuid parcels is expected to be uniform with or without the presence of noise. The situation is, however, very different if the noise is ap- plied to the dynamics of the particles, or, correspondingly, to the bailout embedding. Consider the following stochastic discrete-time dynamics Ϫ␥ x ϩ ϪT͑x ϩ ͒ϭe T͉ ͑x ϩ ϪT͑x ͒͒ϩ␰ , ͑15͒ n 2 n 1 “ xn n 1 n n in which, as in Sec. III, x represents the particle coordinates and T(x) the ﬂuid parcel evolution. New here is the noise ␰ term n , with statistics

␰ ϭ ␰ ␰ ϭ␧ Ϫ Ϫ2␥͒␦ ͑ ͒ ͗ n͘ 0, ͗ n m͘ ͑1 e mnI. 16 This term forces the particle away from the fluid trajectory at every step of the dynamics. However, the actual magnitude of the fluctuations induced in x will be modulated by the properties of “T—the flow gradients—along the particle trajectory. For practical reasons we shall use the convention of varying the noise intensity in correspondence with ␥ in order to obtain comparable fluctuations at different values of this parameter. FIG. 1. The standard map for kϭ7 has a chaotic sea covering almost the We consider the standard map as modeling the basic entire torus, except for a tiny period-two KAM island near position 0,2/3. flow, and its noisy bailout embedding to represent a particu- 1000 random initial conditions were chosen and iterated for 20000 steps, late impurity subject to both fluid drag and noise forces. We then the next 1000 iterations are shown. ͑a͒ Original map, ͑b͒ ␥ϭ1.4, ͑c͒ are interested in the asymptotic behavior of an ensemble of ␥ϭ1.3, ͑d͒ ␥ϭ1.2. such particles which, by the ergodicity of the fluctuations, should be well represented by the histogram of visits that a To illustrate this, Fig. 1 shows the phenomenon in a single particle pays over time to each bin of the space—the situation in which the nonlinearity parameter has been set to full phase space for the basic flow, but a projection of the full the value kϭ7. This corresponds to a very chaotic regime of phase space for the particles. the standard map, characterized by the existence of minute Figure 2 displays a sequence of these histograms in a KAM islands within a sea of chaos that covers almost all the scaled color code for the same nonlinearity parameter, k ϭ available phase space. Figure 1͑a͒ is a close-up image—to 7, as in Fig. 1, corresponding to the extremely chaotic make it easily visible—of the largest of these islands, and the regime of the standard map considered in Sec. III. The se- dots there represent the successive positions of a set of 1000 quence of images corresponds to decreasing bailout param- ␥ fluid parcels spread initially at random over the unit cell, eter . The images make evident the fine filamentary struc- evolving according to the standard map. Since none of these ture developed by the asymptotically invariant distributions parcels were initially located inside the island, this is seen as due to the combined effects of noise and the ability of par- a white region never visited by the parcels. In contrast, in ticles to separate from the basic flow. Remarkably, however, ␥ Figs. 1͑b͒–1͑d͒, the dots are the successive positions, after a these structures appear even when the values are outside number of equilibration iterations, of particles initially the range required to produce a spontaneous detachment of placed as the parcels were in Fig. 1͑a͒, but allowing a very the particle trajectories without noise. ␦ ϭ Ϫ The filamentation here arises from the existence of av- small initial discrepancy 0 x1 T(x0) between the two dynamics. Now, although having started initially outside the enues in the phase space that lead to the small KAM islands island, some of the particles settle inside, in a process that on which the particles prefer to stay. A more detailed analysis becomes increasingly marked as the bailout parameter ␥ de- shows that on these avenues the average value of the squared creases. separation between particle and fluid trajectories is relatively small. Roughly parallel to these avenues, on the other hand, there are strips of the phase space that the particles avoid. IV. NOISY DYNAMICS There, the separation between particle and fluid trajectories By virtue of volume preservation, the invariant measure is on average much larger. Filamentation is thus due to the of the fluid-parcel dynamics is either uniform, if the system tendency of the particles to avoid neighboring regions. is ergodic, or else disintegrates into a foliation of KAM tori The same mechanism may also lead to patchiness, not and ergodic regions. In any case, the addition of a small necessarily filamentary. For flows with weaker chaos, for ex- amount of white noise, which may be considered to represent ample, for which relatively large KAM islands coexist with the effects of small scale turbulence, thermal fluctuations, comparably sized regions of chaos, a situation such as that at

Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp Chaos, Vol. 12, No. 2, 2002 Small particles in ﬂuid ﬂows 493

FIG. 2. ͑Color͒ Histograms with intensity encoding as the square root of invariant probability. Parameters are standard-map nonlinearity parameter kϭ7, noise parameter ␧ϭ10Ϫ8, and bailout parameter ͑a͒ ␥ϭ1.6, ͑b͒ 1.55, ͑c͒ 1.5, ͑d͒ 1.4, ͑e͒ 1.3, and ͑f͒ 1.2. The striped color scale runs from pink at high densities to blue at low densities. kϭ1.5, shown in Fig. 3, is typical. In the histograms, there are relatively small avoided regions that separate large patches of greater concentrations of particles around the nonchaotic islands. This picture is also testimonial to a property that distinguishes the present mechanism for producing inhomogeneity from others mentioned earlier.4,5 While in those cases the particles group around the unstable manifolds of the homoclinic intersections of the basic flows, here the impurity dynamics instead tend to avoid the invariant mani- FIG. 3. ͑Color͒ Histograms ͑left-hand side͒ and temperature plots ͑right- folds. This is because following these manifolds would mean hand side͒ for standard map nonlinearity parameter kϭ1.5, noise parameter eventually hitting regions in which the velocity gradient ei- ␧ϭ10Ϫ8, and bailout parameter ͑a͒, ͑b͒, ␥ϭ0.7; ͑c͒, ͑d͒, ␥ϭ0.65; ͑e͒, ͑f͒, genvalues are closer to one, which locally amplifies the ef- ␥ϭ0.6; ͑g͒, ͑h͒, ␥ϭ0.55; and ͑i͒, ͑j͒, ␥ϭ0.5. The striped histogram color scale runs from yellow at high densities to cyan at low densities, while the fect of fluctuations on the dynamics of the particles. temperature color scale runs from red ͑high͒ to blue ͑low͒. ͑or, equivalently, of the bailout parameter͒ beyond the range V. DETACHMENT AND AVOIDANCE for which this phenomenon occurs in deterministic systems We have seen earlier that inhomogeneities in the distri- when a relatively small amount of noise is added to the par- bution of particles may arise at values of the Stokes number ticle dynamics, and hence, more generally, to the bailout dy-

Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp 494 Chaos, Vol. 12, No. 2, 2002 Cartwright, Magnasco, and Piro

namics. We show in the following that there are two stages to ͗␦ 2͘ ϱ j 2 Ϫ j␥ ␶͑ ͒ϭ ϭ ͩ ͉ Ϫk ͪ ͑ ͒ the modulation of the invariant density in the small-noise x 2 ͚ e ͟ T T (x) . 23 ͗␰ ͘ ϭ ϭ “ limit as the bailout parameter is decreased. At first the bailout j 0 k 0 is everywhere stable, as the bailout parameter is not small Thus ␶(x) defines a sort of temperature for the fluctuations ␦. enough to occasion that the particle trajectories detach from As long as the ␦ are infinitesimally small, they do not— the fluid ones, but fluctutations around this stable embedding and cannot—affect the x dynamics, which has collapsed onto may be restored toward the stable manifold at different rates, the classical trajectories; thus they do not influence the in- and thus acquire different expectation values. These fluctua- variant density P(x) either, and hence P(x) is asymptotic to tions leave a mark on the invariant density through a mecha- the Lebesgue measure. For infinitesimally small ͗␰2͘,as␥ is nism similar to spatially modulated temperature,26,27 namely, made smaller, the sum acquires more and more terms be- the dynamics prefer to escape the hot regions. This is bal- cause the prefactor eϪ j␥ decays more and more slowly. For anced in a nontrivial fashion by mixing in the dynamics to any value of ␥, the products of the gradients grow or shrink create interesting scars in the invariant density. As the bailout roughly as the exponential of the Lyapunov exponent times parameter is lowered, the noise prefactor can diverge, the j. Thus, when ␥ equals the local Lyapunov exponent at x, the embedding loses stability at some points, and detachment series defining ␶(x) stops being absolutely convergent at x ensues. and may blow up. As ␥ is lowered further, more and more In order to analyze this critical transition, we proceed points x have local Lyapunov exponents greater than ␥, and with the map approximation with the conviction that gener- so ␶(x) formally diverges at more and more points x. ␶ ϭϱ ͗␦ 2͘ ͗␰2͘ alizing this analysis to the continuous-time limit is straight- Where (x) it means that n is finite even if forward. We can separate the two-step recurrence, Eq. ͑15͒, is infinitesimally small. Thus the embedding trajectories have into two one-step recurrences, detached from the actual trajectories, and the above-given approximations break down. Detachment is the process that x ϩ ϭT͑x ͒ϩ␦ , ͑17͒ n 1 n n we first envisioned as being characteristic of bailout Ϫ␥ 16 ␦ ϩ ϭe T͉ ␦ ϩ␰ . ͑18͒ embeddings. However, by employing noise in the embed- n 1 “ xn n n ding, and carefully controlling its use, we can see the process The second equation is affine, being linear in the ␦ plus a that occurs before detachment. If ␶(x) is finite and smaller homogeneously added noise process, so it could be solved than 1/͗␰2͘, then we have a regime in which the ␦ ’s behave ␦ analytically for if we knew what the x were in the past. as a noise term added to the classical trajectories: xnϩ1 ␦ ϭ ϩ␦ ␦ 2 ϭ ␰2 ␶ Under the assumption that the are infinitesimally small, we T(xn) n with ͗ ͘ ͗ ͘ (x). ϭ n get the classical orbits xnϩ1 T(xn), and we can write down We have lost the whiteness of the noise process, since ␦ ␦ ␦ explicitly the solution for the , nϩ1 and n are not any longer statistically independent. Ϫ␥ Ϫ␥ However, this is secondary to the fact that the noise process ␦ ϩ ϭ␰ ϩe T͉ ϫ͑␰ Ϫ ϩe T͉ n 1 n “ xn n 1 “ xnϪ1 amplitude, being modulated as a function of position, will ϫ ␰ ϩ Ϫ␥ ͉ ␰ ϩ ͒͒͒ ͑ ͒ immediately lead to inhomogeneity in the dynamics: hot re- ͑ nϪ2 e T x ͑ nϪ3 , 19 “ nϪ2 ¯ gions will be avoided while cold regions will preserve the or, after unwrapping, dynamics. All of this is in a context in which the embedding Ϫ␥ Ϫ2␥ is essentially stable throughout. Thus this process is not de- ␦ ϩ ϭ␰ ϩe T͉ ␰ Ϫ ϩe T͉ T͉ ␰ Ϫ n 1 n “ xn n 1 “ xn“ xnϪ1 n 2 tachment per se, but rather avoidance. ϩ Ϫ3␥ ͉ ͉ ͉ ␰ ϩ ͑ ͒ We can illustrate this best in the context of the standard e T x T x T x nϪ3 , 20 “ n“ nϪ1“ nϪ2 ¯ map acting as before as the base flow. Figure 3 shows side by which may be written more compactly as side the visit histogram—the invariant measure—͑left-hand ͒ n j side together with the corresponding space-dependent tem- ͑ ͒ ␦ ϭ ͩ ␰ Ϫ j␥ ͉ ͪ ͑ ͒ perature right-hand side for a decreasing sequence of the nϩ1 ͚ nϪ je ͟ T x Ϫ . 21 jϭ0 kϭ0 “ n k bailout parameter ␥ and fixed values of the standard-map nonlinearity and noise parameters. While, for ␥ larger than ␰ Then, given that the are uncorrelated, the expectation value 0.55, the temperature is a well-defined function of the space ␦ 2 of is given as the sum of the squares of the terms, or coordinates, it shows signs of divergence—the red regions, ͗␦ 2͘ ϱ j 2 which become larger as ␥ decreases—for ␥ smaller than ϭ ͚ ͩ eϪ j␥ ͟ T͉ ͪ , ͑22͒ 0.55. On the other hand, however, the invariant measure dis- ͗␰2͘ “ xnϪk jϭ0 kϭ0 plays features related to the structure of ␶ on both sides of ͗ ͘ ␰ this transition, i.e., even before detachment occurs. where the • are averages over the process. Clearly, as ␥→ϱ, this expression tends to 1. In the regime in which ␥ӷ0 and ͗␰2͘Ӷ1, the ͗␦ 2 ͘ nϩ1 VI. DISCUSSION AND CONCLUSIONS Ϸ͗␰2͘Ӷ1 and hence the trajectories collapse upon the clas- ϭ ϩ␦ Ϸ sical orbits: xnϩ1 T(xn) n T(xn). Under these circum- It has previously been demonstrated that small neutrally stances, the embedding is always stable, and there is no de- buoyant particles immersed in a fluid flow, and therefore tachment. In this regime we can compute explicitly the subject to drag forces, may follow trajectories that spontane- above-given expression, Eq. ͑22͒, which depends only on the ously separate from those of the fluid parcels in some regions current value of the position: of the flow. Specifically, this occurs when the strain is very

Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp Chaos, Vol. 12, No. 2, 2002 Small particles in fluid flows 495 strong compared to the Stokes number of the particles. Given knowledges the financial support of the Spanish CSIC, Plan that, in general, the smaller the particles, the greater the Nacional del Espacio Contract No. ESP98-1347. M.O.M. ac- strain necessary for this phenomenon to manifest itself, the knowledges the support of the Meyer Foundation. O.P. ac- conditions for separation to occur may not be fulfilled in knowledges the Spanish Ministerio de Ciencia y Tecnologia, some fluid flows of physical interest. However, we have Proyecto CONOCE, Contract No. BFM2000-1108. shown here that the addition of noise to the forces acting on the particles can extend the action of this phenomenon beyond the range of Stokes numbers for which separation is possible in a deterministic system, allowing the generation of inhomogeneities in the asymptotic distributions of these par- 1 E. R. Abraham, Nature ͑London͒ 391, 577 ͑1998͒. ticles even in cases where the flow is a highly efficient mixer. 2 S. A. Levin and L. A. Segal, Nature ͑London͒ 259, 659 ͑1976͒. There is a large variety of examples in which noise 3 Spatial Patterns in Plankton Communities, edited by H. J. Steele ͑Plenum, ought naturally to be added to the dynamical equations of New York, 1978͒. 4 ͑ ͒ particulate impurities. Thermal or concentration fluctuations, C. Jung, T. Te´l, and E. Ziemniak, Chaos 3,555 1993 . 5 Z. Toroczkai et al., Phys. Rev. Lett. 80, 500 ͑1998͒. for instance, should be considered in a range of small-scale 6 Z. Neufeld, C. Lo´pez, and P. H. Haynes, Phys. Rev. Lett. 82, 2606 ͑1999͒. laboratory experiments. The effect of small-scale turbulence 7 A. Babiano, J. H. E. Cartwright, O. Piro, and A. Provenzale, in Coherent on drifters in oceanographic applications might be another Structures in Complex Systems, Lecture Notes in Physics, Vol. 567, edited relevant example. In the case of plankton dynamics, in which by D. Reguera, L. Bonilla, and M. Rubi ͑Springer, New York, 2001͒,pp. the emergence of inhomogeneous distributions is an issue, 114–126. 8 A. Babiano, J. H. E. Cartwright, O. Piro, and A. Provenzale, Phys. Rev. the autonomous swimming abilities of individuals might be Lett. 84, 5764 ͑2000͒. viewed as an internal source of noise. But could the phenom- 9 M. Feingold, L. P. Kadanoff, and O. Piro, J. Stat. Phys. 50,529͑1988͒. enon described here be the basis of the plankton distribution 10 O. Piro and M. Feingold, Phys. Rev. Lett. 61, 1799 ͑1988͒. patchiness? If each individual plankton is naively considered 11 J. H. E. Cartwright, M. Feingold, and O. Piro, Physica D 76,22͑1994͒. 12 as an impurity particle, the answer is obviously no: their size J. H. E. Cartwright, M. Feingold, and O. Piro, J. Fluid Mech. 316,259 ͑ ͒ is far too small for these effects to be appreciable. But if 1996 . 13 J. H. E. Cartwright, M. Feingold, and O. Piro, Phys. Rev. Lett. 75, 3669 there are grounds to consider plankton in large-scale colonies ͑1995͒. moving more or less rigidly in the ocean, the consequences 14 M. Abel, A. Celani, D. Vergni, and A. Vulpiani, Phys. Rev. E 64, 046307 of the phenomenon described here need to be taken seriously. ͑2001͒. We conclude with an epistemological note: the dynamics 15 C. Lo´pez et al., Chaos 11,397͑2001͒. 16 of neutrally buoyant particles in flows has suggested to us a J. H. E. Cartwright, M. O. Magnasco, and O. Piro, Phys. Rev. E 65, 045203͑R͒͑2002͒. generalization to maps that helps to solve some problems in 17 M. R. Maxey and J. J. Riley, Phys. Fluids 26, 883 ͑1983͒. the domain of Hamiltonian dynamics. This generalization in 18 E. E. Michaelides, J. Fluids Eng. 119, 233 ͑1997͒. turn pays us back by suggesting a way in which inhomoge- 19 G. I. Taylor, Proc. R. Soc. London, Ser. A 120, 260 ͑1928͒. neous distributions may arise in fluid-dynamical problems. 20 T. R. Auton, J. C. R. Hunt, and M. Prud’homme, J. Fluid Mech. 197,241 ͑ ͒ We believe that this is a remarkable instance of mutual sup- 1988 . 21 J. Boussinesq, C. R. Acad. Sci. Paris 100, 935 ͑1885͒. port in the interdisciplinary marriage between the two fields. 22 A. B. Basset, Philos. Trans. R. Soc. London 179,43͑1888͒. 23 H. Faxeń, Ann. Phys. ͑Paris͒ 4,89͑1922͒. ACKNOWLEDGMENTS 24 A. Okubo, Deep-Sea Res. 17, 445 ͑1970͒. 25 J. B. Weiss, Physica D 48, 273 ͑1991͒. We should like to thank Leo Kadanoff, Rubeń Pasman- 26 M. Bu¨ttiker, Z. Phys. B: Condens. Matter 68,161͑1987͒. ter, and Marcelo Viana for useful discussions. J.H.E.C. ac- 27 R. Landauer, J. Stat. Phys. 53,233͑1988͒.

Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp