Lagrangian Tracers on a Surface Flow: The Role of Time Correlations

Abstract

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...

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