Relative dispersion in turbulence

In the previous sections we considered flows with a smooth spatial structure in which the relative dispersion of fluid trajectories is exponential in time and can be characterized by a single timescale, the inverse of the Lyapunov exponent. This is also valid for two-dimensional turbulent flows that have a smooth velocity field in the small-scale enstrophy cascade range (Bennett, 1984). A similar behavior occurs in any dimension at scales below the Kolmogorov scale (the so-called Batchelor or viscous-convective range, see below). In the inertial range of fully developed three-dimensional turbulence, however, the velocity field has a broad range of timescales and they all contribute to the relative dispersion of particle trajectories and affect the transport properties of the flow. 

The relative motion of particle pairs in three-dimensional homogeneous turbulence was first described by L.F. Richardson based on experimental data on atmospheric dispersion. Well before the theoretical results of Kolmogorov on turbulent flows he suggested that the average distance between pairs of particles grows according to a power law of the form (Richardson, 1926) 

Using the result of Kolmogorov for the longitudinal velocity structure function 

Note that the Richardson law indicates a faster than diffusive separation and can also be interpreted as a scale dependent diffusion coefficient D = Ce1 v2 — r 114/3. A classic example of such length-scale dependent diffusivity is illustrated in the diagram by Okubo (1971), Fig 2.22, that is based on data from various experiments and observations on horizontal dispersion in oceanic flows. These data can be approximately described by a power law dependence of the form Deff(l) Z1-15. The deviation from the Richardson law can be seen as a consequence of the more complex structure of oceanic currents. 

Clearly, the Richardson dispersion law is qualitatively different from the exponential separation of particles in chaotic advection. The finite-size Lyapunov exponent (FSLE) is a generalization of the Lyapunov exponent introduced to characterize flows with multiple length 

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