Mixing properties in the advection of passive tracers via recurrences and extreme value theory

In this paper we characterize the mixing properties in the advection of passive tracers by exploiting the extreme value theory for dynamical systems. With respect to classical techniques directly related to the Poincaré recurrences analysis, our method provides reliable estimations of the characteristic mixing times and distinguishes between barriers and unstable fixed points. The method is based on a check of convergence for extreme value laws on finite datasets. We define the mixing times in terms of the shortest time intervals such that extremes converge to the asymptotic (known) parameters of the generalized extreme value distribution. Our technique is suitable for applications in the analysis of other systems where mixing time scales need to be determined and limited datasets are available.

Figure 1

Poincaré sections for the stream function in Eq. (16). The parameters are $\alpha =1,\epsilon =0.63$ (left), and $\alpha =1,\epsilon =0.8$ (right).

Figure 2

$\kappa$ for the observable $g_3$, $\beta =3$ for an ensemble of 2000 points. The theoretical expected value is $\kappa =-1/6$ (yellow). See text for a description.

Figure 3

Left: $\kappa$ for the observable $g_1$. Right: $\kappa$ for the observable $g_3$, $\beta =3$. Ensemble of 2000 points. The theoretical expected value are $\kappa =0$ for $g_1$ and $\kappa =-1/6$ for $g_3$. See text for a description.

Figure 4

$\kappa$ for the observable $g_1$ at different bin lengths $m$ for three different ensemble of points. The theoretical expected values are $\kappa =0$ (black line). See text for a description.