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major western boundary currents in the ocean are meandering jets separating water masses with different physical and biogeochemical characteristics . the prominent examples are the gulf stream in the atlantic ocean and the kuroshio in the pacific ocean . these and similar `` heat engines '' define the climate in large regions of the planet . similar jets in the stratosphere play important role in transport and distribution of chemical substances . from the hydrodynamic point of view , they may be considered as jet flows with running waves of different wave lengths and phase velocities imposed . the simplest kinematic model of such a flow is a two - dimensional jet of an ideal fluid with a given velocity profile that is perturbed by an amplitude - modulated wave traveling from the west to the east . the problem of transport and mixing of passive scalars in meandering jets has been considered by many authors in the context of atmospheric and oceanic physics @xcite . the typical phase portrait ( fig .  [ fig1 ] ) consists of two chains of circulations with a zigzag - like jet between them and resembles the phase portrait of a particle in the field of two running waves . in the frame moving with the velocity of one of the waves , the problem is topologically equivalent to the motion of a periodically perturbed nonlinear physical pendulum that is known to demonstrate chaotic oscillations @xcite . different aspects of chaotic mixing of passive particles in meandering jets in the atmosphere and the ocean have been studied in the papers @xcite .    in the paper we focus on topological and statistical aspects of the chaotic transport and mixing in a specific kinematic model of an eastward meandering jet which has been introduced in refs . @xcite some years ago . we are motivated by the desire to get a more deep insight into the evolution of material lines in the flow and to establish a connection between dynamical , topological and statistical characteristics of the flow . the equations of motion of passive particles advected by a planar incompressible flow is known to have a hamiltonian form @xmath0 with the streamfunction @xmath1 playing the role of a hamiltonian and the coordinates @xmath2 of a particle being canonically conjugated variables . thus , nonstationary two - dynamical advection is equivalent to a hamiltonian system with one and half degrees of freedom whose phase space coincides with its configuration space . this property is very useful in visualizing geometric invariant sets in real and numerical experiments . to be specific we consider a two - dimensional bickley jet with the velocity profile @xmath3 whose argument is modulated by a zonal running wave @xcite . the streamfunction in the fixed frame reference is the following : @xmath4 where @xmath5 , @xmath6 and @xmath7 are amplitude , wavenumber and phase velocity respectively , @xmath8 is a measure of the jet s width . after introducing the following notations : @xmath9 and @xmath10 we get the advection equations ( [ 1 ] ) in the frame moving with the phase velocity @xmath7 : @xmath11    ) in the frame moving with the meander s phase velocity @xmath7 . ] the respective streamfunction @xmath12 has three normalized control parameters : @xmath13 , @xmath14 and @xmath15 are the jet s width , meander s amplitude and its phase velocity . the scaling chosen results in translational invariance of the phase portrait along the @xmath16-axis with the period @xmath17 .    the detailed analysis of stationary points and bifurcations of eqs . ( [ 11 ] ) has been done in ref . stationary points may exist only under the condition @xmath18 . there are four stationary points , two of them are always stable and the other ones are stable under the condition @xmath19 . if the additional condition @xmath20 is fulfilled , there are additional four unstable saddle points . resuming one gets :    1 . @xmath21 , there are no stationary points . 2 . @xmath22 and @xmath23 , there are two centers and two saddles with two separatrices connecting the saddles . there are two separatrices each of which passes through its own saddle . the jet between the separatrices is westward . @xmath24 and @xmath25 , the jet is eastward with the same stationary points as in the case 2 . @xmath26 , there are eight stationary points and two separatrices . there are two separatrices , each of which connects two saddle points . the jet is westward . @xmath27 and @xmath25 , the jet is eastward with the same stationary points as in the case 4 . therefore , from the point of view of existence and stability of stationary states , there are three possibilities : 1 ) there are no stationary points ; 2 ) there are four stationary points , two centers and two saddles ; 3 ) there are eight stationary points , four centers and four saddles . a bifurcation between the first and second regimes consists in arising two pairs `` saddle - center '' . a bifurcation between the second and third regimes consists in arising two saddles and a center between them instead of one saddle ( a fork - type bifurcation ) . there is one more bifurcation that does not change the number and stability of stationary points but changes the topology of the flow . the values of the streamfunction on the separatrices are equal on modulo but of opposite signs . there is a critical value of the phase velocity @xmath28 under which the separatrices coincide and the respective streamfunction is equal to zero . if @xmath29 , a free flow between the separatrices is westward , whereas with @xmath30 it is eastward . it is difficult to find @xmath28 analytically but it may be shown @xcite that @xmath31 , if @xmath32 otherwise @xmath33 . the respective phase portraits are typical with hamiltonian systems with running waves in shear flows @xcite . in dependence on the values of the phase velocity @xmath15 one can get different topologies : a homoclinic connection , a heteroclinic connection and a separatrix reconnection . being motivated by eastward jet currents in the ocean and atmosphere , we deal in this paper with the case  3 ( see the phase portrait in fig . streamlines with the streamfunction ( [ 12 ] ) , that is time - independent in the moving frame , are shown in fig . 1 . the plot demonstrates three different regions of the flow : a central eastward jet ( j ) , north and south circulations ( c ) and peripheral westward currents ( p ) . the centers of the circulations are at critical lines to be defined by the condition @xmath34 , @xmath35 , and they are divided by two separatrices connecting saddle points . no exchange between the north and south circulations is possible in the unperturbed system . even the simplest periodic perturbation of the meander s amplitude , @xmath36 , results in spitting stable and unstable manifolds of the saddle points . there arise stochastic layers with chaotic mixing and transport of water masses fluxes of which depend on the width of a layer , a number of overlapping resonances and other factors . as an example , we plot in fig . 2 the poincar section of the perturbed flow . the control parameters throughout the paper are chosen as follows : @xmath37 , @xmath38 , @xmath39 , @xmath40 , @xmath41 and @xmath42 . due to the zonal and meridian symmetries it is enough to consider mixing in the northern part of the first eastern frame only @xmath43 . in fig . 3a we plot the respective poincar section . the vortex core , that survives under the perturbation , is immersed into a stochastic sea where one can see 6 small islands belonging to the same resonance . particles , belonging to these islands , rotate around the elliptic point of the vortex core . zoom of the section nearby two of the islands is shown in fig . the feature we want to pay attention to is a stickiness to the boundaries of the vortex core and of the islands that is visualized by increasing the density of points near the respective boundaries . it is a typical phenomenon with hamiltonian systems @xcite which influences essentially the transport of passive particles . chaotic trajectories for @xmath44 steps . the parameters are : @xmath40 and @xmath45 . ( b ) for short eastward flights , @xmath46 , the pdf decays exponentially . ( c ) the tails of both the pdfs decay as a power law with the slope @xmath47 obtained with westward ( squares ) long flights @xmath48 . ]    without perturbation , the transport properties are very simple : particles either rotate in circulations c or move eastward in the jet j and westward in the peripheral currents p. under a perturbation , the motion in the stochastic layers become extremely sensitive to small variations in initial conditions , and one is forced to use an statistical approach to describe transport . a commonly used statistical measure of transport is the variance @xmath49 , where the averaging is supposed to be done over an ensemble of particles . long time behavior of @xmath50 and of the probability density function of particle s displacement @xmath51 may be anomalous with typical chaotic hamiltonian systems @xcite . when @xmath52 and @xmath51 is a gaussian distribution , the advection is known to be normal with a well - defined diffusion coefficient @xmath53 . when @xmath54 , with @xmath55 , @xmath56 is either zero or infinite @xcite , and one gets either subdiffusion ( @xmath57 ) or super - diffusion ( @xmath58 ) . if the trajectories are dominated by sticking regions nearby boundaries of islands , where particles spend a long time , subdiffusion results . superdiffusion occurs when particles in the jet travel long distances between sticking events . the respective length and time pdfs are expected to be non gaussian . we will call `` a flight '' any event between two successive changes of signs of the particle s zonal velocity . in this terminology a sticking consists of a number of flights with approximately equal flight times . the poincar section of the flow with the perturbation strength @xmath40 and frequency @xmath45 is shown in fig . the flight pdfs are computed with @xmath59 particles ( initially placed in the first east frame inside a stochastic layer ) up to the time @xmath60 . the pdf of the lengths of flights is shown in fig . [ fig4]a for both the directions . the asymmetry between the eastward ( `` positive '' ) and westward ( `` negative '' ) flights is evident . both the pdfs can be roughly split into three distinctive regions . the very short flights with small values of @xmath61 ( @xmath62 ) are supposed to be dominated by sticking to the boundaries of the vortex core and oscillatory islands . the pdf for eastward flights with the lengths in the range @xmath46 decays exponentially ( fig . [ fig4]b ) . the tails of both the pdfs are close to a power - law decay @xmath63 . in fig . [ fig4]c we estimate the exponent for long westward flights , @xmath48 , with corresponding error by least - square fitting of the straight line to the log - log plot of the data to be @xmath47 .         plane , ( b ) dependence of the zonal position on time . stickiness and flight events are evident with the green chaotic trajectory.,scaledwidth=80.0% ]    besides resonant islands with particles moving around the elliptic point in the same frame ( fig . 3 ) , we have found so - called ballistic islands which were situated both in the chaotic sea and in the peripheral jets ( fig . ballistic modes @xcite correspond to the stable periodic motion of particles from one frame to another . particles , belonging to a ballistic island , move with a constant zonal velocity from one frame to another . when mapping their positions at the moments @xmath64 @xmath65 onto the first frame , we see a chain of islands which are visible in fig . 5a . in fig . 5b a zoom of the ballistic island is shown . stickiness to the island s border is evident . in fig . 6 we demonstrate two ballistic trajectories corresponding to the ballistic island shown in fig . 5b . if a particle at @xmath66 is placed inside the island , it travels to the west in a regular way ( the upper blue trajectory in fig . 6a ) . its zonal position @xmath16 grows linearly with time ( the lower blue trajectory in fig . 6b ) . the lower green trajectory in fig . 6a corresponds to a particle placed initially nearby the border of the same ballistic island from outside . intermittent flight and sticking events are evident in fig . 6b ( the upper green curve in that figure ) . of particles that reach the lines @xmath67 after @xmath68 turns around the elliptic points . @xmath69 is a time particles need to reach the lines @xmath67 . indices @xmath70 and @xmath71 mean particles moving in the eastward and westward directions , respectively.,scaledwidth=80.0% ]     are marked by the respective letters . ] poincar sections provide good impression about the structure of the phase space but not about geometry of mixing . in this section we consider the evolution of a material line consisting of a large number of particles distributed initially on a straight line that transverses the stochastic layer at @xmath72 . a typical stochastic layer consists of an infinite number of unstable periodic and chaotic orbits with islands of regular motion to be imbeded . all the unstable invariant sets are known to possess stable and unstable manifolds . when time progresses particle s trajectories nearby a stable manifold of an invariant set tend to approach the set whereas the trajectories close to an unstable manifold go away from the set . because of such a very complicated heteroclinic structure , we expect a diversity of particle s trajectories . some of them are trapped forever in the first eastern frame @xmath43 rotating around the elliptic point along heteroclinic orbits . other ones quit the frame through the lines @xmath72 or @xmath73 , and then either are trapped there or move to the neighbor frames ( including the first one ) , and so on to infinity .    to get a more deep insight into the geometry of chaotic mixing we follow the methodology of our works @xcite and compute the time @xmath69 , particles spend in the neighbor circulation zones @xmath74 before reaching the critical lines @xmath75 , and the number of times @xmath68 they wind crossing the lines @xmath76 . in the upper panel in fig . 7 the functions @xmath77 and @xmath78 are shown . the upper parts of each function ( with @xmath79 and @xmath80 ) represent the results for the particles with initial positive zonal velocities which they have simply due to their locations on the material line at @xmath72 . these particles enter the eastern frame and may change the direction of their motion many times before leaving the frame through the lines @xmath72 or @xmath73 . the time moments of those events we fix for all the particles with @xmath81 . the lower `` negative '' parts of the functions @xmath77 and @xmath78 represent the results for the particles with initial negative zonal velocities ( @xmath82 ) which move initially to the first western frame ( @xmath83 ) . in fact , @xmath84 and @xmath85 are the time moments when a particle with the initial position @xmath86 quits the eastern or western frames , respectively . both the functions consist of a number of smooth u - like segments intermittent with poorly resolved ones . border points of each u - like segments separate particles belonging to stable and unstable manifolds of the heteroclinic structure . the corresponding initial @xmath87-positions is a set ( of zero measure ) of particles to be trapped forever in the respective frame . a fractal - like structure of chaotic advection in both the frames is shown in the upper panel in fig . 7 , and its fragments for the first levels are shown in the middle panel for the eastern and the western fragments separately . particles with even values of @xmath88 quit one the frames through the border @xmath72 , those with odd @xmath88  through the border @xmath73 for the eastern frame and @xmath89 for the western one . let us consider in detail the fractal - like structure in the eastern frame keeping in mind that the results are similar with any other frame . the @xmath90-dependence is a complicated hierarchy of sequences of segments of the material line . following to the authors of the paper @xcite , we call as an epistrophe a sequence of segments of the @xmath91-level , converging to the ends of a segment of a sequence of the @xmath88-th level , whose length decrease in accordance with a law . at @xmath92 we see in fig . 7 an epistrophe with segment s length a , b , c , d and so on decreasing as @xmath93 with @xmath94 . letters a and b in fig . 7 denote the first segments of the epistrophes at the level @xmath95 , whereas d and c the first segments of the epistrophes at the level @xmath96 . the respective laws for all those epistrophes are not exponential .    in fig . 8 we demonstrate fragments of the evolution of the material line in the first eastern frame at the moments indicated in the figure . letters on the line mark the corresponding segments of the @xmath90 and @xmath84 functions in fig . 7 . as an example , let us explain formation of the epistrophe abcd at the level @xmath92 . with the period of perturbation @xmath97 , a portion from the north end of the material line leaves the frame through its eastern border . look at the segments a and b at @xmath98 and @xmath99 . they quit the first frame as a fold through the period @xmath100 . the other segments  c , and d ( not shown in fig . 8) do the same job . the epistrophe s segments at the odd levels ( @xmath101 ) quit the frame with the period of perturbation @xmath102 one by one being folded ( c and d segments ) . the folds of the segments of the @xmath103-level are exterior with respects to the folds of the segments of the @xmath104-level . the following empirical law is valid : @xmath105 , where @xmath106 is a time when the first segments of the epistrophes at the level @xmath103 ( a with @xmath92 ) reach the line @xmath73 , and @xmath107 the respective time for the first segments of the epistrophes at the level @xmath108 ( c and d segments with @xmath96 ) .    segments of the epistrophes of the even levels ( @xmath109 ) leave the frame with the period @xmath102 as well but through the border @xmath72 moving to the west . we show the evolution of some of them at the moments @xmath110 and @xmath111 in fig . thus , the material line evolves by stretching and folding , and folds quit the frame in both directions with the period of perturbation . we have treated the problem of mixing and transport of passive particles in a kinematic model of a meandering oceanic jet current from the point of view of dynamical system s theory . a careful simulation of the hamiltonian equations of advection has shown a complicated character of mixing under a time - 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we model lagrangian lateral mixing and transport of passive scalars in meandering oceanic jet currents by two - dimensional advection equations with a kinematic stream function with a time - dependent amplitude of a meander imposed . 
 the advection in such a model is known to be chaotic in a wide range of the meander s characteristics . 
 we study chaotic transport in a stochastic layer and show that it is anomalous . 
 the geometry of mixing is examined and shown to be fractal - like . 
 the scattering characteristics ( trapping time of advected particles and the number of their rotations around elliptical points ) are found to have a hierarchical fractal structure as functions of initial particle s positions . 
 a correspondence between the evolution of material lines in the flow and elements of the fractal is established . 
 chaotic advection , meandering jet , fractals 47.52.+j,47.53.+n , 92.10.ty