We show that, even in the most favorable case, the motion of a small spherical tracer suspended in a fluid of the same density may differ from the corresponding motion of an ideal passive particle. We demonstrate furthermore how its dynamics may be applied to target trajectories in Hamiltonian systems.PACS numbers: 47.52.+j, 05.45.Gg, 45.20.Jj We show with the simplest model for the force acting on a small rigid neutrally buoyant spherical tracer particle in an incompressible two-dimensional fluid flow that tracer trajectories separate from fluid trajectories in those regions where the flow has hyperbolic stagnation points. A tracer will only evolve on fluid trajectories with Lyapunov exponents bounded by the value of its reciprocal Stokes number. By making the Stokes number large enough, one can force a tracer in a flow with chaotic pathlines to settle on either the regular KAM-tori dominated regions or to selectively visit the chaotic regions with small Lyapunov exponents. These findings should be of interest for the interpretation of Lagrangian observations, for example in oceanography, and in laboratory fluid experiments that use small neutrally buoyant tracers. Moreover, since a two-dimensional incompressible flow is a particular instance of a generically chaotic Hamiltonian system, our results constitute a tool for targeting trajectories in Hamiltonian systems.Our starting point is the equation of motion for a small rigid spherical tracer in an incompressible fluid,

We discuss a series of numerical experiments on the dispersion of neutrally buoyant particles in two-dimensional turbulent flows. The topology of two-dimensional turbulence is parametrized in terms of the relative dominance of deformation or rotation; this leads to a segmentation of the turbulent field into hyperbolic and elliptic domains. We show that some of the characteristic structural domains of two-dimensional turbulent flows, namely coherent structures and circulation cells, generate particle traps and peculiar accelerations which induce several complex properties of the particle dispersion processes at intermediate times. In general, passive particles are progressively pushed from the coherent structures and tend to concentrate in highly hyperbolic regions in the proximity of the isolines of zero vorticity. For large dispersion times, the background turbulent field is a privileged domain of particle richness; there is however a permanent particle exchange between the background field and the energetic circulation cells which surround the coherent structures. At intermediate times, an anomalous dispersion regime may appear, depending upon the relative weight of the different topological domains active in two-dimensional turbulence. The use of appropriate conditional averages allows the basic topology of two-dimensional turbulence to be characterized from a Lagrangian point of view. In particular, an intermediate $t^{\frac{5}{4}}$ anomalous dispersion law is shown to be associated with the action of hyperbolic regions where deformation dominates rotation; the motion of the advected particles in strongly elliptic regions where rotation dominates over deformation is shown to be associated with a $t^{t^{\frac{5}{3}}$ dispersion law. Because neutral particles concentrate on average in hyperbolic regions, the $t^{\frac{5}{4}}$ dispersion law is quite robust and it can be observed under very general circumstances.}

In this paper we study the statistical laws of relative dispersion in two-dimensional turbulence by deriving an exact equation governing its evolution in time, then evaluating the magnitude of its various terms in numerical experiments, which allows us to check the validity of the classical dispersion laws: the equivalent to the Richardson-Obukhov t3 law in the energy cascade range, and the Kraichnan-Lin exponential law in the enstrophy cascade range. We examine theoretically and experimentally the conditions of validity of both laws. It is found that the t3 law is obtained in the energy inertial range provided the separation scale of the particles is smaller by an order of magnitude than the injection scale. When the t3 law is reached, the relative acceleration correlations are observed to have reached a statistical quasistationary stage: this would tend to justify in the energy inertial range of two-dimensional turbulence a working hypothesis formulated by Lin & Reid (1963); also, the necessity of starting from very small initial separations to get the t3 law may be explained by the time necessary for relative acceleration correlations to reach the statistical quasi-stationary regime. On the other hand, the Kraichnan-Lin exponential law is, strictly speaking, never observed; it is in fact reduced to a very short transient stage when the relative dispersion characteristic time reaches its minimum value, as predicted by Batchelor.

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