Time scales of turbulent relative dispersion

Tracers in a turbulent flow separate according to the celebrated $t^{3/2}$ Richardson-Obukhov law, which is usually explained by a scale-dependent effective diffusivity. Here, supported by state-of-the-art numerics, we revisit this argument. The Lagrangian correlation time of velocity differences increases too quickly for validating this approach, but acceleration differences decorrelate on dissipative time scales. Phenomenological arguments are used to relate the behavior of separations to that of a “local energy dissipation,” defined as the average ratio between the cube of the longitudinal velocity difference and the distance between the two tracers. This quantity is shown to stabilize on short time scales and this results in an asymptotic diffusion $\propto$$t^{1/2}$ of velocity differences. The time of convergence to this regime is shown to be that of deviations from Batchelor's initial ballistic regime, given by a scale-dependent energy dissipation time rather than the usual turnover time. It is finally demonstrated that the fluid flow intermittency should not affect this long-time behavior of the relative motion.

Figure 1

Time evolution of the mean-square separation for $R_{\lambda }=730$ and various initial separations. The dashed line represents the behavior (2). The solid line is a fit to the Richardson regime (4) with $g=0.52$ and $C=1.6$. Inset: $t_0$ as a function of $r_0$ in dissipative-scale units. The solid line is an Eulerian average, the circles are Lagrangian measurements, and the dashed line is the turnover time $\tau \left(r_0\right)$.

Figure 2

Lagrangian time autocorrelation of the acceleration difference $\delta \mathit{a}$ for various $r_0$ and $R_{\lambda }=730$. Inset: For the same separations $r_0$, variance of the acceleration difference amplitude conditioned on the longitudinal velocity difference $\delta u^{\parallel }$ as a function of the local dissipation rate ${\left[\delta u^{\parallel }\right]}^3/r_0$.

Figure 3

Time evolution of the averaged squared modulus of the velocity difference for $R_{\lambda }=730$ and different $r_0$ (same symbols as in Fig. 1). The short-time prediction (7) is shown as a dashed line. The diffusive behavior ${\langle \left|\delta \mathit{u}\right|}^2{\rangle }_{r_0}\simeq S_2\left(r_0\right)+2.3\epsilon t$ is represented as a dashed-dotted line. Inset: Time evolution of ${\langle {\left[\delta u^{\parallel }\right]}^3/\delta x\rangle }_{r_0}$; the dashed line corresponds to the value $6\epsilon$.

Figure 4

Fourth- and sixth-order moments of the displacement $\left|\delta \mathit{x}\right(t)-\delta \mathit{x}(0\left)\right|$ for $R_{\lambda }=730$. Following ideas from Ref. [23], they are represented as a function of the second-order moment. The two gray dashed lines show scale-invariant behaviors of the form ${\langle |\delta \mathit{x}\left(t\right)-\delta \mathit{x}\left(0\right)|}^p{\rangle \propto \langle |\delta \mathit{x}\left(t\right)-\delta \mathit{x}\left(0\right)|}^2{\rangle }^{p/2}$.