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Finite time correlations of the velocity in a surface flow are found to be important for the formation of clusters of Lagrangian tracers. The degree of clustering characterized by the Lyapunov spectrum of the flow is numerically shown to be in qualitative agreement with the predictions for the whitein-time compressible Kraichnan flow, but to deviate quantitatively. For intermediate values of compressibility the clustering is surprisingly weakened by time correlations.PACS numbers: 47.27.Ak, Inhomogeneous distribution of particles advected in a turbulent flow is a generic consequence of compressibility. This can be obtained in two situations. The first possibility is that the advecting flow is compressible itself and that the particles follow the streamlines [1]. The other possibility is that the particles do not follow the streamlines because of inertia [2,3,4,5,6,7,8] or lift [9], and that the effective velocity field is compressible. Such situations are relevant for the formation of clouds [10] or for the advection of bubbles in turbulent flows, e.g. for breaking waves on the ocean surface [11,12]. We focus here on the first possibility.While the dominant tendency of incompressible flows is to separate particle trajectories, a compressible component is responsible for particle trapping in contracting regions for long times. The Eulerian compressibility of a flow is measured by the dimensionless ratioIt takes values between 0 (incompressible flow) and 1 (potential flow). While there can be no clustering without compressible effects, the compressibility ratio C is insufficient to determine the final distribution completely [13]: the behavior depends also on the spatial roughness, the dimensionality, and, as we will demonstrate here, on the time correlations in the flow.A convenient characterization of the final distribution uses the Lagrangian Lyapunov spectrum. Dynamical systems theory shows that the asymptotic clusters are smooth along the unstable directions of positive Lyapunov exponents and fractal along the stable directions. In d dimensions, the sum of Lyapunov exponents where K is the maximal integer such thatObviously, for an incompressible flow K = d and also D L = d: the particle distribution fills the entire volume. As compressibility increases, the sum of Lyapunov exponents will become negative and K will drop below d.If the largest Lyapunov exponent λ 1 becomes negative, the final cluster will not have any smooth directions anymore and the particles will cluster in a point-like fractal. One hence distinguishes a regime of strong compressibility λ 1 < 0 with D L = 0, and one of weak compressibility λ 1 > 0.There are no general results on the spectrum of Lagrangian Lyapunov exponents in turbulent flows. For the case of incompressible, isotropic, three dimensional turbulence the numerical observation is that λ 2 ∼ 1/4λ 1 [15]. In the limit of a compressible Kraichnan flow, which is a synthetic, white-in-time, and Gaussian distributed, the spectrum is given byis an inverse time proportional to the Lag...

We introduce a method by which stochastic processes are mapped onto complex networks. As examples, we construct the networks for such time series as those for free-jet and low-temperature helium turbulence, the German stock market index (the DAX), and the white noise. The networks are further studied by contrasting their geometrical properties, such as the mean-length, diameter, clustering, average number of connection per node. By comparing the network properties of the investigated original time series with those for the shuffled and surrogate series, we are able to quantify the effect of the long-range correlations and the fatness of the probability distribution functions of the series on the constructed networks. Most importantly, we demonstrate that the time series can be reconstructed with high precisions by a simple random walk on their corresponding networks.

Inverse energy cascade regime of two dimensional turbulence is investigated by means of high resolution numerical simulations. Numerical computations of conditional averages of transverse pressure gradient increments are found to be compatible with a recently proposed self-consistent Gaussian model. An analogous low order closure model for the longitudinal pressure gradient is proposed and its validity is numerically examined. In this case numerical evidence for the presence of higher order terms in the closure is found. The fundamental role of conditional statistics between longitudinal and transverse components is highlighted. PACS number(s) : 47.27.Ak, The existence of two simultaneous inertial ranges in twodimensional turbulence, as a consequence of coupled energy and enstrophy conservation, is one of the most important phenomena in statistical fluid mechanics [1]. At variance with 3D-turbulence, the energy injected into the system at scale ℓ f flows toward the large scales, while the enstrophy cascades down on the small scales. Because of the inverse energy cascade, the Navier-Stokes equations,which rule the evolution of an incompressible (∂ i u i = 0) velocity field, cannot reach a steady-state unless an energy sink at large scales is added. Alternatively one can consider an ensemble of solutions of (1) with a fixed energy value below the condensation level [2], i.e. with an integral scale L(t) (growing in time as t 3/2 ) still much smaller than the system size. Because of the scaling of the characteristic times, the small scales (inertial range) in the system ℓ f ≪ r ≪ L can be considered in a stationary state. One of the most challenging problems is to understand the statistics of velocity fluctuations ∆u(r) = u(x + r) − u(x) [3]. In homogeneous and isotropic turbulence it amounts to study the joint probability density function (PDF) P (U, V, r) of longitudinal U and transverse V velocity differences where ∆u = Ux + Vŷ andx = r r . Recently experimental [4] and numerical [5] investigations in two dimensions have shown that the probability distribution of the pure longitudinal P (U, r) and transversal P (V, r) velocity differences at inertial scales display a close-toGaussian statistics with undetectable intermittency corrections to structure function exponents. Although the establishment of normal scaling in all inverse cascades seem to be generic [6] nevertheless the Gaussianity of the statistics in inverse cascade of the forced two dimensional turbulence is remained to be understood. From (1), a set of equations for generic mixed structure functions, i.e. S n,m (r) ≡ U n V m = A n,m r ξn,m have been obtained [7,8]. In [8] those equations are elaborated from the joint PDF equation. Unfortunately, the PDF equation is not closed, resembling the well known closure problem in turbulence. In the inverse energy cascade regime, dissipative contributions can be neglected so the remaining unclosed terms are the longitudinal and transversal pressure gradients increments. Recently Yakhot [8,9] suggested a selfcon...