Effect of dynamical traps on chaotic transport in a meandering jet flow
aa r X i v : . [ n li n . C D ] D ec Effect of dynamical traps on chaotic transport in ameandering jet flow
M.Yu. Uleysky, M.V. Budyansky, and S.V. PrantsNovember 12, 2018
Abstract
We continue our study of chaotic mixingand transport of passive particles in a simplemodel of a meandering jet flow [Prants, et al,Chaos , 033117 (2006)]. In the present pa-per we study and explain phenomenologicallya connection between dynamical, topological,and statistical properties of chaotic mixingand transport in the model flow in terms ofdynamical traps, singular zones in the phasespace where particles may spend arbitrarylong but finite time [Zaslavsky, Phys. D , 292 (2002)]. The transport of passiveparticles is described in terms of lengths anddurations of zonal flights which are events be-tween two successive changes of sign of zonalvelocity. Some peculiarities of the respec-tive probability density functions for shortflights are proven to be caused by the so-called rotational-islands traps connected withthe boundaries of resonant islands (includ-ing the vortex cores) filled with the parti-cles moving in the same frame and the saddletraps connected with periodic saddle trajec-tories. Whereas, the statistics of long flights can be explained by the influence of the so-called ballistic-islands traps filled with theparticles moving from a frame to frame. Chaotic advection of water masses withtheir physical and biochemical charac-teristics in quasi two-dimensional geo-physical flows in the ocean and atmo-sphere can be studied within the frame-work of Hamiltonian dynamics. In a re-cent paper [1] we have studied chaoticmixing and transport of passive parti-cles in a simple kinematic model of ameandering jet flow motivated by theproblem of lateral mixing in the west-ern boundary currents in the ocean.We found all the possible bifurcationsof advection equations, described thestructure of the phase space (which isthe physical space for advected par-ticles), and computed some statisti-cal characteristics of chaotic transport.In the present paper we establish a henomenological connection betweendynamical, topological, and statisti-cal properties of chaotic transport andmixing in the same flow. Specific sin-gular zones in the phase space whereparticles may spend arbitrary long butfinite time (dynamical traps in termi-nology by Zaslavsky [2]), are respon-sible for anomalous statistical proper-ties. The dynamical traps, connectedwith rotational islands and saddle tra-jectories, are responsible, mainly, foranomalous mixing, whereas those ones,connected with ballistic islands — foranomalous transport. These dynamicaltraps may have strong impact on trans-port and mixing in real geophysical jetflows. Methods of the theory of dynamical sys-tems are actively used to describe advectionof water (air) masses and their properties inthe ocean and atmosphere [1, 14, 13, 15, 16,26, 3, 4, 5, 6, 7, 17, 18, 19, 20, 21, 32]. Thegeophysical jet currents, like the Gulf Stream,the Kuroshio, the Antarctic circumpolar cur-rent, and others in the ocean and the polarnight Antarctic jet in the atmosphere, are ro-bust structures whose form typically changesin space and time in a meander-like way. Ifadvected particles rapidly adjust their ownvelocity to that of background flow and donot affect the flow properties, then the par-ticles are called passive (scalars, tracers, orLagrangian particles) and their equation ofmotion is very simple d r dt = v ( r , t ) (1) where r = ( x, y, z ) and v = ( u, v, w ) are theposition vector and the particle velocity ata point ( x, y, z ) . If the corresponding Eu-lerian velocity field is supposed to be regu-lar, the vector Eq. (1) in nontrivial cases is aset of three nonlinear deterministic differen-tial equations whose phase space is a physicalspace of advected particles. It is well knownfrom dynamical systems theory that solutionsof this kind of equations can be chaotic inthe sense of exponential sensitivity to smallvariations in initial conditions and/or con-trol parameters. As to advection equations,it was Arnold [8] who firstly suggested chaosin the field lines (and, therefore, in trajecto-ries) for a special class of three-dimensionalstationary flows (so-called ABC flows), andthis suggestion has been confirmed numer-ically by H´enon [9]. In the approximationof incompressible planar flows, the velocitycomponents can be expressed in terms ofa stream function [10]: u = − ∂ Ψ /∂y and v = ∂ Ψ /∂x . The equations of motion (1) ina two-dimensional incompressible flow havenow a Hamiltonian form with the streamfunc-tion Ψ playing the role of a Hamiltonian andthe coordinates ( x, y ) of a particle playing therole of canonically conjugated variables.Advection of passive particles has beenshown to be chaotic in a number of theoreti-cal [4, 5, 6, 7, 1, 14, 13, 15, 16, 17, 26, 32]and laboratory [18, 19, 20, 21] models ofgeophysical jet currents. In our recent pa-per [1] (Paper I) we have studied mixing,transport, and chaotic advection in a sim-ple kinematic model of a meandering two-dimensional jet flow with a Bickley zonal ve-locity profile u ∼ sech y being motivated by2he problem of lateral mixing of water masses(together with salinity, heat, nutrients, pollu-tants, and other passive scalars) in the west-ern boundary currents in the ocean. We de-rived advection equations in the frame mov-ing with the phase velocity of a running waveimposed on the Bickley jet, found the station-ary points, conditions of their stability, andall the possible bifurcations of these equationswhich were shown to be autonomous in theco-moving frame. Under a periodic perturba-tion of the wave amplitude, the phase planeof the chosen model flow has been shownto consist of a central eastward jet, periph-eral westward currents, and chains of north-ern and southern circulations (vortex cores)immersed in a chaotic sea which, in turn,contains islands of regular motion. Statis-tical properties of chaotic transport of ad-vected particles have been characterized interms of particle’s zonal flights (any eventbetween two successive changes of the signof the particle’s zonal velocity u ). Probabil-ity density functions (PDFs) of durations andlengths of flights, computed with a numberof very long chaotic trajectories, have beenfound to be complicated functions with localmaxima and fragments with exponential andpower-law decays.The aim of this paper is to study a phe-nomenological connection between dynami-cal, topological, and statistical properties ofchaotic mixing and transport in the mean-dering jet flow considered in Paper I and toexplain transport properties by phenomenonof the so-called dynamical traps. Followingto Zaslavsky [2], the dynamical trap is a do-main in the phase space of a Hamiltonian sys- tem where a particle (or, its trajectory) canspend arbitrary long finite time, performingalmost regular motion, despite the fact thatthe full trajectory is chaotic in any appropri-ate sense. In fact, it is the definition of aquasi-trap. Absolute traps, where particlescould spend an infinite time, are possible inHamiltonian systems only with a zero mea-sure set. The dynamical traps are due toa stickiness of trajectories to some singulardomains in the phase space, largely, to theboundaries of resonant islands, saddle trajec-tories, and cantori. There are no classifica-tion and description of the dynamical traps.Zaslavsky described two types of dynamicaltraps in Hamiltonian systems: hierarchical-islands traps around chains of resonant is-lands [27, 28, 12] and stochastic-layer trapswhich are stochastic jets inside a stochasticsea where trajectories can spend a very longtime [29, 2, 12]. It is expected that classi-fication and description of the most typicaldynamical traps would help us to constructkinetic equations which will be able to de-scribe transport properties of chaotic systemsincluding anomalous ones [2, 11, 12].This paper is organized as follows. Westart with advection equations derived in Pa-per I in the frame of reference moving withthe phase velocity of a meander whose am-plitude changes in time periodically. Wecompute in Sec. II PDFs for lengths x f anddurations T f of zonal flights for a numberof chaotic trajectories and show analyticallythat all the flights start and finish only insidea strip confined by two curves whose formis defined by the condition u = 0 . Someprominent peaks in statistics of short flights3 | x f | < π ) are proved to be caused by sticki-ness of trajectories to the boundaries of rota-tional resonant islands filled with regular par-ticles rotating in the same frame (Sec. III).We call this type of dynamical traps as arotational-island trap (RIT). In Sec. IV westudy dynamical traps connected with peri-odic saddle trajectories, emerged from sad-dle points of the unperturbed system (4) un-der the perturbation (5), and prove numeri-cally that the saddle traps (STs) contributeto the statistics of short flights as well. An-other type of islands, ballistic islands (filledwith regular particles moving from frame toframe), is proved to contribute to the statis-tics of long flights ( | x f | ≫ π ) in Sec. V. Boththe ballistic-islands trap (BITs) and RITs be-long to the class of hierarchical-island trapsby the Zaslavsky’s classification. We take the following specific stream functionas a kinematic model of a meandering jet flowin the laboratory frame of reference: Ψ ′ ( x ′ , y ′ , t ′ ) == − Ψ ′ tanh y ′ − a cos k ( x ′ − ct ′ ) λ p k a sin k ( x ′ − ct ′ ) ! , (2)where the width of the jet is λ . Meanderingis provided by a running wave with the am- plitude a , the wave-number k , and the phasevelocity c . The normalized streamfunction inthe frame moving with the phase velocity is Ψ = − tanh (cid:18) y − A cos xL √ A sin x (cid:19) + Cy, (3)where x = k ( x ′ − ct ′ ) and y = ky ′ are newscaled coordinates, and A = ak , L = λk ,and C = c/ Ψ ′ k are the control parame-ters. Equations, governing advection of pas-sive particles (1) in the co-moving frame, arethe following: ˙ x = 1 L √ A sin x cosh θ − C, ˙ y = − A sin x (1 + A − Ay cos x ) L (cid:0) A sin x (cid:1) / cosh θ ,θ = y − A cos xL √ A sin x , (4)where dot denotes differentiation with re-spect to dimensionless time t = Ψ ′ k t ′ .In Paper I (for more details see [17]) wehave found and analyzed all the stationarypoints, their stability, and the bifurcationsof the equations of motion (4). Being moti-vated by the problem of mixing and trans-port of water masses and their propertiesin oceanic western boundary currents likethe Gulf Stream and the Kuroshio, we chosethe phase portrait shown in Fig. 1a amongall the possible flow regimes. Passive par-ticles can move along stationary (in the co-moving frame) streamlines in a different man-ner. They can move to the east in the jet( J ) and to the west in northern and southern(with respect to the jet) peripheral currents4 P ). There are also particles rotating in thenorthern and southern circulation cells (C)in a periodic way. The northern separatrixconnects the saddle points at x ( n ) s = 2 πn and y ( n ) s = L Arcosh p /LC + A and the south-ern one connects the saddle points at x ( s ) s =(2 n + 1) π and y ( s ) s = − L Arcosh p /LC − A ,where n = 0 , ± , . . . .As a perturbation, we took in Paper I thesimple periodic modulation of the meander’samplitude A = A + ε cos( ωt + φ ) . (5)Under the perturbation, there arise reso-nances between the perturbation frequency ω and the frequencies f of the particle’s rota-tion in the circulations C . A frequency map f ( x , y ) , computed in Paper I (see Fig. 2 inthat paper), shows the values of f for parti-cles with initial positions ( x , y ) in the un-perturbed flow. With a given value of theperturbation frequency and fixed values ofthe other control parameters, the vortex coresin the circulations survive, stochastic layersappear along the unperturbed separatrix, andthe central jet J is a barrier to transport ofparticles across the jet. In Paper I we fixedthe scaled values of the parameters of the un-perturbed flow, the jet’s width L = 0 . ,the meander’s amplitude A = 0 . andits phase velocity C = 0 . , that are inthe range of the realistic values for the GulfStream [22, 23], and took the initial phaseto be φ = π/ . The perturbation frequency ω = 0 . chosen in Paper I is close to thevalues of the rotation frequency f of the par-ticles circulating in the inner core of the re-gions C (see Fig. 2 in Paper I). In Fig. 1 b we show the Poincar´e section (for a large numberof trajectories) of the meandering jet whoseamplitude is modulated with the frequency w = 0 . and the strength ε = 0 . .The vortex cores survive under this pertur-bation, the stochastic layers appear along theunperturbed separatrix, and a central jet J isa barrier to transport of particles across thejet.The equations of motion (4) with the per-turbation (5) are symmetric under the follow-ing transformations: (1) t → t , x → π + x , y → − y and (2) t → − t , x → − x , y → y . Itimplies that the meridional transport (north-south and south-north) is symmetric but thezonal transport (west-east and east-west) issymmetric under a time reversal. Due tothese symmetries motion can be consideredon the cylinder with ≤ x ≤ π and y ≥ .The part of the phase space with πn ≤ x ≤ π ( n + 1) , n = 0 , ± , . . . , is called a frame.It is convenient to characterize chaotic mix-ing and transport in terms of zonal flights. Azonal flight is a motion of a particle betweentwo successive changes of signs of its zonalvelocity, i. e. the motion between two succes-sive events ˙ x = u = 0 . Particles (and cor-responding trajectories) in chaotic jet flowscan be classified in terms of the lengths offlights x f as follows. The trajectories with | x f | < π correspond to the particles movingin the same frame or in neighbor frames. Inthe global stochastic layer there are particlesmoving chaotically forever in the same framebut they are of a zero measure. Among theparticles with inter-frame motion, there areregular and chaotic ballistic ones. Regularballistic trajectories can be defined as those5hich cannot have two flights with | x f | > π in succession. They correspond to particlesmoving in regular regions of the phase spacepersisting under the perturbation, (eastwardmotion in the jet and western motion inthe peripheral current) and those moving inthe stochastic layer (trajectories belonging toballistic islands). Typical chaotic trajecto-ries have complicated distributions over thelengths and durations of flights.In the laboratory frame of reference, allthe fluid particles move to the east togetherwith the jet flow and a flight is a motion be-tween two successive events when the parti-cle’s zonal velocity U is equal to the mean-der’s phase velocity c . If U < c , the corre-sponding particle is left behind the meander(it is a western flight in the co-moving frame),if
U > c , it passes the meander (an easternflight in the co-moving frame). Short flightswith | x f | < π (motion in the same spatialframe in the co-moving frame of reference)correspond to the motion in the laboratoryframe when two successive events U = c oc-cur on the space interval less than the me-ander’s spatial period π/k . Ballistic flightsbetween the spatial frames in the co-movingframe with | x f | > π correspond to the mo-tion in the laboratory frame when the parti-cles move through more than one meander’screst between two successive events U = c . As in Paper I, we will characterize statisti-cal properties of chaotic transport by proba-bility density functions (PDFs) of lengths offlights P ( x f ) and durations of flights P ( T f ) for a number of very long chaotic trajec-tory. Both regular and chaotic particles maychange many times the sign of their zonal ve-locity ˙ x = u . From the condition ˙ x = 0 inEq. (4), it is easy to find the equations forthe curves which are loci of turning points Y ± ( x, A ) = ± L p A sin x ×× Arsech q LC p A sin x + A cos x. (6)We consider the northern curve, i. e., Eq. (6)with the positive sign. Taking into accountthat the perturbation has the form (5), werealize that all the northern turning pointsare inside a strip confined by two curves ofthe form (6) with A = A ± ε . Let us analyzethe derivative over the varying parameter A ∂Y∂A = cos x ++ ACL sin x D (cid:18) √ D − √ − D (cid:19) , (7)where D = LC √ A sin x . If the deriva-tive at a fixed value of x does not change itssign on the interval A − ε ≤ A ≤ A + ε , then Y varies from Y ( x, A − ε ) to Y ( x, A + ε ) ,and for each value of y we have a single valueof the perturbation parameter A . However,there may exist such values of x for whichthe equation ∂Y /∂A = 0 has a solution onthe interval mentioned above. In this caseone may have more than one values of A fora single value of y . Thus, the width of thestrip, containing turning points, is defined by6he values of Y at the extremum points andat the end points of the interval of the val-ues of A . In Fig. 1 c we show the turningpoints of a single chaotic trajectory on thecylinder ≤ x ≤ π confined between twocorresponding curves.In the numerical simulation throughout thepaper we use the Runge-Kutta integrationscheme of the fourth order with the constanttime step ∆ t . . To study chaotic trans-port we have carried out numerical exper-iments with tracers initially placed in thestochastic layer. It was found that statisti-cal properties of chaotic transport practicallydo not depend on the number of tracers pro-vided that the corresponding trajectories aresufficiently long ( t ≃ ). The PDFs for thelengths x f and durations T f of flights for fivetracers with the computation time t = 5 · for each tracer are shown in Figs. 2 a andb, respectively, both for the eastward (e) andwestward (w) motion. Both P ( x f ) and P ( T f ) are complicated functions with local extremadecaying in a different manner for differentranges of x f and T f . The main aim of ourstudy of chaotic transport is to figure outthe basic peculiarities of the statistics andattribute them to specific zones in the phasespace, namely, to dynamical traps stronglyinfluencing the transport. It is well known, that in nonlinear Hamilto-nian systems a complicated structure of thephase space with islands, stochastic layers,and chains of islands, immersed in a stochas- tic sea, arises under a perturbation due toa variety of nonlinear resonances and theiroverlapping [24]. The motion is quasiperi-odic and stable in the islands. The bound-aries of the islands are absolute barriers totransport: particles can not go through themneither from inside nor from outside. Invari-ant curves of the unperturbed system (seeFig. 1 a) are destroyed under the perturba-tion (5) (see Fig. 1 b). As the perturbationstrength ε increases, a closed invariant curvewith frequency f is destroyed at some criticalvalue of ε . If the f /ω is a rational number,the corresponding curve is replaces by an is-land chain, while the curves with irrationalfrequencies are replaced by cantori (for a re-view see [25]). There are uncountably manycantori forming a complicated hierarchy. Nu-merical experiments with a variety of Hamil-tonian systems with different number of de-grees of freedom provide an evidence for thepresence of strong partial barriers to trans-port around the island’s boundaries (for re-view, see [12]) which manifest themselves onPoincar´e sections as domains with increaseddensity of points.In Paper I we have found that with chosenvalues of the control parameters there existin each frame a vortex core (which is an is-land of the primary resonance ω = f ) im-mersed into a stochastic sea, where there aresix islands of a secondary resonance emergedfrom three islands of the primary resonance f = 2 ω (see Fig. 3 in Paper I). Chains ofsmaller size islands are present around thevortex core and the secondary-resonance is-lands. Particles belonging to all of these is-lands (including the vortex core) rotate in the7ame frame performing short flights with thelengths | x f | < π . So we will call them ro-tational islands and distinguish from the so-called ballistic islands to be considered below.Stickiness of particles to boundaries of therotational islands has been demonstrated inPaper I. It means that real fluid particlescan be trapped for a long time in a singu-lar zone nearby the borders of the rotationalislands which we will call rotational-islandstraps (RITs). To illustrate the effect of theRITs we demonstrate in Figs. 3 and 4 thePoincar´e sections of a chaotic trajectory inthe frame ≤ x ≤ π sticking to the vortexcore and to the secondary-resonance islands,respectively. The contour of the vortex coreis shown in Fig. 4 by the thick line. Thesmall points are tracks of the particle’s po-sition at the moments of time t n = 2 πn/ω (where n = 1 , , . . . ) and the thin curvesare fragments of the corresponding trajec-tory on the phase plane. Increased densityof points indicates the presence of dynamicaltraps near the boundaries of the rotationalislands. Contribution of the vortex-core RIT(Fig. 3) to chaotic transport is expected tobe much more significant than the one of theRITs of the other islands (Fig. 4).It is reasonable to suppose that RITs con-tribute to the statistics of short flights. Byshort flights we mean the flights with thelength shorter than π . In Fig. 5 we show thepart of the full PDF P ( T f ) (Fig. 2 b) for theeastward (e) and westward (w) short flightsseparately. There are a comparatively smallnumber of the eastward flights with T f < .Let us note the prominent peak of the cor-responding PDF at T f ≃ followed by an exponential decay. As to the westward shortflights, there are two small local peaks around T f ≃ and .To estimate the contribution of the vortex-core RIT to the statistics of short flights, wecompute and compare the statistics of the du-rations of flights T f for two trajectories: aregular quasiperiodic one with the initial po-sition close to the inner border of the vortexcore (Fig. 6 a) and a chaotic one with the ini-tial position close to the vortex-core borderfrom the outside (Fig. 6 b). Each full rota-tion of a particle in a frame consists of twoflights, eastward and westward, with differentvalues of T f because of the zonal asymmetryof the flow. The statistics for the chaotic tra-jectory, sticking to the vortex core (Fig. 6 b),may be considered as a distribution of the du-rations of flights in the vortex-core RIT. Theminimal flight duration in this RIT is T f ≃ (the flights with smaller values of T f are rareand they occur outside the trap). Positionsof the local maxima of the PDF for the stick-ing trajectory in Fig. 6 b correlate approxi-mately with the corresponding local maximaof the PDF for the regular trajectory insidethe core in Fig. 6 a. The similar correlationshave been found (but not shown here) be-tween the local maxima of the PDFs for thelengths of flights P ( x f ) for the interior reg-ular and sticking chaotic trajectories. Thesecorrelations and positions of the peaks provenumerically that short flights with | x f | < π and . T f . may be caused by theeffect of vortex-core RIT. We conclude fromFig. 6 b that the vortex-core RIT contributesto the statistics of the short flights in therange . T f . for the eastward flights8ith the prominent peak at T f ≃ and inthe range . T f . for the westwardflights with small peaks at T f ≃ and .The effect of the RIT of the secondary-resonance islands is illustrated in Fig. 4.To find the characteristic times of this RITwe compute two trajectories: a regularquasiperiodic one with the initial position in-side one of these islands and a chaotic onewith the initial position close to the outerborder of the island. The respective PDFs P ( T f ) , shown in Figs. 7 a and b, demonstratestrong correlations between the correspond-ing peaks at T f ≃ , , and . Computed(but not shown here) PDFs P ( x f ) for thesetrajectories confirm the effect of the islandsRIT on the statistics of short flights. As a result of the periodic perturbation (5),the saddle points of the unperturbed system(4) at x ( n ) s = 2 πn , y ( n ) s = L Arcosh p /LC + A and at x ( s ) s = (2 n + 1) π , y ( s ) s = − L Arcosh p /LC − A ( n = 0 , ± , . . . ) be-come periodic saddle trajectories. These hy-perbolic trajectories have their own stableand unstable manifolds and play a role ofspecific dynamical traps which we call saddletraps (ST). In this section we demonstratethat the STs influence strongly on chaoticmixing and transport of passive particles andcontribute, mainly, in the short-time statis-tics of flights.Tracers with initial positions close to astable manifold of a saddle trajectory aretrapped for a while performing a large num- ber of revolutions along it. To illustrate theeffect of the STs we show in Fig. 8 a andb fragments of two chaotic trajectories stick-ing to the saddle trajectory and performingabout 20 full revolutions before escaping tothe east (Fig. 8 a) and to the west (Fig. 8 b).We have managed to detect and locate thecorresponding periodic unstable saddle tra-jectory which is situated in Figs. 8 a andb in the domain where a few fragments ofthe chaotic trajectory imposed on each other.Because of the flow asymmetry, the dura-tion of eastern flights of a particle along thesaddle trajectory T e ≃ . is shorter thanthe duration of western flights T w ≃ . .The black points are the tracks of the par-ticle’s positions on the flow plane at the mo-ments of time t n = 2 πn/ω ≃ . n (where n = 1 , , . . . ). They belong to smooth curveswhich are fragments of the stable and unsta-ble manifolds of the saddle trajectory at thechosen initial phase φ = π/ .To estimate the contribution of the STs tothe statistics of short flights shown in Fig. 5,we compute and plot in Fig. 8 c the number ofthe eastward ( N e ) and westward ( N w ) shortflights with a given duration T f for those twochaotic trajectories sticking to the saddle tra-jectory arising from the saddle point with theposition x s = 0 , y s ≃ . . Each full ro-tation of the particles consists of an eastwardflight with the duration T e ≃ . and anwestward flight with the duration T w ≃ . .The flights with T e ≃ . contribute tothe main peak in Fig. 5 and the flights with T w ≃ . to “the wesward” plateau in thatfigure.The mechanism of operation of the STs9an be described as follows. Each saddletrajectory γ ( t ) possesses time-dependent sta-ble W s ( γ ( t )) and unstable W u ( γ ( t )) materialmanifolds composed of a continuous sets ofpoints through which pass at time t trajec-tories of fluid particles that are asymptoticto γ ( t ) as t → ∞ and t → −∞ , respectively.Under a periodic perturbation, the stable andunstable manifolds oscillate with the periodof the perturbation. It was firstly proved byPoincar´e that W s and W u may intersect eachother transversally at an infinite number ofhomoclinic points through which pass doublyasymptotic trajectories. To give an image ofa fragment of the stable manifold of the peri-odic saddle trajectory, we distribute homoge-neously . · particles in the rectangular [ − . ≤ x ≤ .
45; 2 ≤ y ≤ . and computethe time the particles need to escape the rect-angular. The color in Fig. 9 modulates thetime T when particles with given initial posi-tions ( x , y ) reach the western line at x = − or the eastern line at x = 1 . The particleswith initial positions marked by the black andwhite colors move close to the stable manifoldof the saddle trajectory and spend a maximaltime near it before escaping. The black andwhite diagonal curve in Fig. 9 is an image ofa fragment of the corresponding stable man-ifold. The particles with initial positions tothe north from the curve escape to the westalong the unstable manifold of the saddle tra-jectory whereas those with initial positions tothe south from the curve escape to the eastalong its another unstable manifold.We have found that particles quit the STalong the unstable manifolds in accordancewith specific laws. We distribute a large number of particles along the segment with x = 0 and y = [2 .
02; 2 . , crossing thestable manifold W s , and compute the time T particles with given initial latitude po-sitions y need to quit the ST. More pre-cisely, T ( y ) is a time moment when a par-ticle with the initial position y reaches thelines with x = − or x = 1 . The “experi-mental” points in Fig. 10 a fit the law T e =( − . ± . − (31 . ± . y s − y ) for the particles which quit the trap movingto the east and the law T w = ( − . ± . − (28 . ± . y − y s ) for those parti-cles which move to the west when quitting thetrap, where y s = 2 . is a crossingpoint of W s with the segment of initial posi-tions.The ST attracts particles and force themto rotate in its zone of influence performingshort flights, the number of which n dependson particle’s initial positions y . The n ( y ) is a steplike function (see Fig. 10 b) withthe lengths of the steps decreasing in a ge-ometric progression in the direction to thesingular point, l j = l q − j , where l j is thelength of the j -th step and q ≃ . for thewestern exits and q ≃ . for the easternones. The seeming deviation from this lawin the range y = [2 . . (see a smallwestern segment between two larger ones inFig. 10 b) is explained by crossing the ini-tial line y = [2 .
02; 2 . by the curve of zerozonal velocity u . To have the correct law forthe western exits, it is necessary to add thetwo segments of that cut step. The asymme-try of the functions T ( y ) and n ( y ) is causedby the asymmetry of the flow.10 Ballistic-islands traps
Besides the rotational islands with parti-cles moving around the corresponding ellip-tic points in the same frame, we have foundin Paper I ballistic islands situated both inthe stochastic layer and in the peripheralcurrents. Regular ballistic modes [30] corre-spond to stable quasiperiodic inter-frame mo-tion of particles. Only the ballistic islands inthe stochastic layer are important for chaotictransport. Mapping positions of the regularballistic trajectories at the moments of time t n = 2 πn/ω ( n = 1 , , . . . ) onto the firstframe, we obtain chains of ballistic islandsboth in the northern and southern stochasticlayers, i. e., between the borders of the north-ern (southern) peripheral currents and of thecorresponding vortex cores. A chain withthree large ballistic islands is situated in thosestochastic layers. The particles, belonging tothese islands, move to the west, and theirmean zonal velocity can be easily calculatedto be h u f i = − π/ T = − ω/ ≃ − . .There are also chains of smaller-size ballisticislands along the very border with the periph-eral currents.We have demonstrated in Paper I a sticki-ness of chaotic trajectories to the borders ofthose three large ballistic islands (see Figs. 6and 7 in Paper I). The Poincar´e sectionwith fragments of two chaotic trajectoriesin the northern stochastic layer is shown inFig. 11 a. One particle performs a long flightsticking to the very border with the regularwestward current, and another one moves tothe west sticking to the very boundaries ofthree large ballistic islands. A magnification of a fragment of the border and tracks of asticking trajectory around a smaller-size bal-listic island are demonstrated in Fig. 11 b.Fig. 11 c demonstrates the effective size of thetrap of the large ballistic islands with tracksof a sticking trajectory around them.It is reasonable to suppose that the ballistic-islands traps (BIT) contribute,largely, to the statistics of long flightswith | x f | ≫ π . All the ballistic particles,moving both to the west and to the east,can finish a flight and make a turn onlyin the strip shown in Fig. 1 c. The lociof the corresponding turning points havea complicated fractal-like structure. Weconsider further only long westward flights,taking place in the northern stochastic layer,because it is much wider than the stochasticlayer between the regular central jet andthe southern parts of the vortex cores whereeastward flights take place.To distinguish between contributions ofthe traps of different ballistic islands (and,maybe, other zones in the phase space) to thestatistics of long flights, we compute for fivelong chaotic trajectories (up to t = 5 · ) thedistribution of a number of westward flightswith T f ≥ over the mean zonal veloci-ties h u f i = x f /T f of the particles perform-ing such flights. The distribution in Fig. 12has a prominent peak centered at the meanzonal velocity h u f i ≃ − . which corre-sponds to a large number of long flights ofthose particles (and their trajectories) whichstick to the very boundaries of the large bal-listic islands (see Fig. 11 a) moving with themean velocity h u f i ≃ − . . The flat leftwing of the distribution N ( h u f i ) corresponds11o the traps of smaller-size ballistic islandsnearby the border with the peripheral cur-rent. There are different families of these is-lands (see one of them in Fig. 11 b) with theirown values of the mean zonal velocity whichare in the range − . . h u f i . − . .Stickiness to the boundaries of the border is-lands is weaker because they are smaller thanthe large islands and their contribution tothe statistics of long flights is comparativelysmall.The right wing of the distribution N ( h u f i ) with − . . h u f i . − . deserves fur-ther investigation. The value h u f i ≃ − . is a minimal value of the zonal velocity forlong westward flights possible in the north-ern stochastic layer. Increasing the mini-mal duration of a flight from T f = 10 to T f = (2 ÷ · , we have found splitting ofthe broad distribution with − . . h u f i . − . into a number of small distinct peaks.Comparing trajectories with the values of h u f i corresponding to these peaks, we havefound that all they move around the largeballistic islands. The particles with smallervalues of h u f i used to penetrate further tothe south from the islands more frequentlythan those with larger values of h u f i whichprefer to spend more time in the northernpart of the dynamical trap connected withthose islands. Thus, we attribute the rightwing of the distribution N ( h u f i ) to an effectof the trap situated around the large ballisticislands.To estimate the contribution of differentBITs to the statistics of long westward flightsin Fig. 2 we have computed the PDFs P ( x f ) and P ( T f ) for particles performing westward flights with x f ≥ and T f ≥ andwith the mean zonal velocity h u f i to be cho-sen in three different ranges shown in Fig. 12: − . . h u f i . − . (particles stick-ing to the border islands) − . < h u f i . − . (particles sticking to the very bound-ary of three large islands), and − . < h u f i . − . (the trap of the three largeislands). All the PDFs P ( x f ) decay ex-ponentially but with different values of theexponents equal to ν ≃ − . and ν ≃− . ÷− . for the traps of border andthe large ballistic islands, respectively. Thetail of the PDF P ( x f ) for westward flights,shown in Fig. 2, decays exponentially with ν ≃ − . . Thus, the contribution of thelarge island’s BIT to the statistics of longwestward flights is dominant. As to tem-poral PDFs P ( T f ) for westward long flights,they are neither exponential nor power-lawlike with strong oscillations at the very tails.The slope for the border BITs is again smallerthan for the large ballistic islands trap. A meandering jet is a fundamental structurein oceanic and atmospheric flows. We de-scribed statistical properties of chaotic mix-ing and transport of passive particles in akinematic model of a meandering jet flow interms of dynamical traps in the phase (phys-ical) space. The boundaries of rotational is-lands (including the vortex cores) in circula-tion zones are dynamical traps (RITs) con-tributing, mainly, to the statistics of shortflights with | x f | < π . Characteristic times12nd spatial scales of the RITs have beenshown to correlate with the PDFs for thelengths x f and durations T f of short flights.The stable manifolds of periodic saddle tra-jectories play a role of saddle traps (STs) withthe specific values of the lengths and dura-tions of short flights of the particles stickingto the saddle trajectories. The boundaries ofballistic islands in the stochastic layers (in-cluding those situated along the border withthe peripheral current) are dynamical traps(BITs) contributing, mainly, to the statisticsof very long flights with | x f | ≫ π .Dynamical traps are robust structures inthe phase space of dynamical systems in thesense that they present at practically all val-ues of the corresponding control parameters.We never know exact values of the param-eters in real flows, especially, in geophysicalones. We don’t know exactly the structureof the corresponding phase space, however,we know that typical features, like islands ofregular motion, vortices, and jets, exist inreal flows (see their images in some labora-tory flows in Ref. [31]). In this paper wechose specific values of the control parame-ters for which specific PDFs have been com-puted and explained by the effect of those dy-namical traps that exist under the chosen pa-rameters. We have carried out computer ex-periments with different values of the controlparameters and found that the phase spacestructure has been changed, of course, withchanging the values of the parameters, butthe corresponding RITs, STs, and BITs withspecific temporal and spatial characteristicshave been found to contribute to the corre-sponding statistics. After finishing our work, we were ac-quainted with Ref. [32] where meridionalchaotic transport, associated with a similarkinematic model of a meandering jet, hasbeen studied by the method of lobe dynam-ics [33]. In difference from our study of zonalchaotic transport, a geometric structure ofcross jet transport has been considered inRef. [32] where values of the control parame-ters have been chosen to be sufficiently largeto break up the central jet as a barrier totransport of particles across the jet. Themechanisms for particles to cross the jet havebeen described in terms of lobe dynamics andthe mean time to cross the jet for particlesentering the jet and the mean residence timefor particles in the jet have been estimatedin Ref. [32]. We have studied a more real-istic situation (at least, for surface oceanicjet currents) when the jet is an absolute bar-rier to cross jet transport and we explainedstatistical properties of transport in terms ofdynamical traps of saddle trajectories, rota-tional and ballistic islands. The method oflobe dynamics is hardly applicable for studyzonal chaotic transport since it is practicallyimpossible to trace out lobe evolution for alarge number of frames. Acknowledgments
The work was supported by the RussianFoundation for Basic Research (Grant no. 06-05-96032), by the Program “MathematicalMethods in Nonlinear Dynamics” of the Rus-sian Academy of Sciences, and by the Pro-gram for Basic Research of the Far Eastern13ivision of the Russian Academy of Sciences.
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Chaotic Transport inDynamical System (Springer-Verlag,New York, 1992). -3-2-1 0 1 2 3 0 1 2 3 4 5 6 7 8 9 y xa)JC P JCP CC -3-2-1 0 1 2 3 0 1 2 3 4 5 6 7 8 9 y xb) y xc) Figure 1: (a) Stationary streamfunction ofa meandering jet in the co-moving frame(3). The flow is divided into three differentregimes: circulations ( C ), jet ( J ), and pe-ripheral currents ( P ). (b) Poincar´e sectionof the perturbed meandering jet in the co-moving frame. The parameters of the steadyflow are: the jet’s width L = 0 . , the me-ander’s amplitude A = 0 . and its phasevelocity C = 0 . . The perturbation am-plitude and frequency are: ε = 0 . and ω = 0 . . (c) Turning points of a singlechaotic trajectory on the cylinder ≤ x ≤ π are in a strip confined by two curves (6) with A = A ± ε .15 l n P x f a)w e -12-8-4 0 4 2 0 2 4 l og P w l og P e log T f b)w e Figure 2: Probability density functions of(a) lengths x f and (b) durations T f of thewestward (w) and eastward (e) flights. ThePDFs P w ( T f ) and P e ( T f ) are normalized tothe number of westward ( . · ) and east-ward ( · ) flights, respectively. Statisticsfor five tracers with the computation time t = 5 · for each one. y x Figure 3: The vortex-core trap. Poincar´esection of a chaotic trajectory in the frame ≤ x ≤ π with a fragment of a trajectory. y x Figure 4: The secondary resonance islandstrap. A fragment of a chaotic trajectorysticking to the islands is shown. P w P e T f w e Figure 5: The PDFs for the eastward (e) andwestward (w) flights with the length shorterthan π . The PDFs P w ( T f ) and P e ( T f ) are normalized to the number of westward( . · ) and eastward ( . · ) flights,respectively. Statistics for five tracers withthe computation time t = 5 · for each one.16 P w P e T f a)w e P w P e T f b)w e Figure 6: The vortex-core trap PDFs of du-rations T f of the eastward (e) and westward(w) flights. (a) Regular quasiperiodic trajec-tory with the duration t = 2 · inside thevortex core close to its boundary. Both thePDFs are normalized to the number · ofcorresponding flights. (b) Chaotic trajectorywith the duration t = 2 · sticking to theboundary of the vortex core from the outside.Both the PDFs are normalized to the number · of corresponding flights. P w P e T f a)w e P w P e T f b)w e Figure 7: The secondary-resonance islandstrap. The PDFs of durations T f of the east-ward (e) and westward (w) flights. (a) Regu-lar quasiperiodic trajectory inside the islandswith the duration t = 5 · . Both the PDFsare normalized to the number . · of corre-sponding flights. (b) Chaotic trajectory stick-ing to the island’s boundary from the outsidewith the duration t = 5 · . P w ( T f ) and P e ( T f ) are normalized to the number of west-ward ( . · ) and eastward ( · ) flights,respectively.17 y x a) y x b) N w N e T f c)w e Figure 8: The saddle trap. Fragments of twochaotic trajectories sticking to the periodicsaddle trajectory one of which escapes to theeast (a) and another one to the west (b). (c)The number of the eastward ( N e ) and west-ward ( N w ) short flights with duration T f forthose two trajectories. Statistics with twotrajectories with the duration t = 10 andthe total number of western N w = 55 andeastern N e = 51 flights. y −0.4 −0.2 0 0.2 0.4 x w e T Figure 9: The saddle-trap map. Color modu-lates the time T which . · particles withgiven initial positions ( x , y ) need to reachthe lines at x = − or x = 1 escaping tothe west (w) and to the east (e), respectively.The black and white diagonal curve is an im-age of a fragment of the stable manifold of thesaddle trajectory. The cross is a position of aparticle on that trajectory at the initial timemoment. The integration time is t = 500 .18 T y a)e w n y b)e w Figure 10: (a) Time T a particle with an ini-tial latitude position y needs to quit the sad-dle trap. (b) The number of short flights n such a particle performs before quitting thesaddle trap. The ranges of y from which par-ticles quit the trap moving to the west andeast are denoted by “w” and “e”, respectively. y a) b) xc) Figure 11: (a) Poincar´e section of the north-ern stochastic layer where stickiness to thevery border with the regular westward cur-rent and to three large ballistic islands areshown. Increased density of points along theborder with the peripheral current is causedby the traps of the border ballistic islands oneof which is shown in (b). (c) The trap of thelarge ballistic islands.19 -88 -84 -80 -76 N f × -3 × Figure 12: The distribution of a number oflong westward flights with T f ≥ over theirmean zonal velocities h u f i . The sharp peakcorresponds to the trap connected with thevery boundaries of the large ballistic islands,the left wing — to a number of traps of fam-ilies of the border ballistic islands, and theright wing — to the trap situated around thelarge ballistic islands. Statistics for five trac-ers with the total number of long westwardflights N f = 5 · and the computation time t = 5 ·8