Scaling exponents and exit-time statistics

The actual values of the scaling exponents can be directly extracted from the statistics of the passive scalar, which is not spoiled by large-scale objects. In Figure 2.8 we plot the exponents $\zeta ^{\theta }_p$ as obtained by looking at the local slopes of the structure functions $S_{p}^{\theta}(r)$, in comparison with the exponents predicted by the Lagrangian exit-time statistics, according to

$$\langle \exp[-\alpha p T_L(r)] \rangle \sim r^{\zeta_p}$$ (2.38)

Figure 2.8: The scaling exponents of the passive scalar $\zeta ^{\theta }_p$ ($+$). We also show the exponents obtained from the exit times statistics ($\odot$) according to $\langle \exp[-\alpha p T_L(r)] \rangle \sim r^{\zeta_p^{\theta}}$ with average over about $2 \times 10^{5}$ couple of Lagrangian particles. The error-bars are estimated by the r.m.s. fluctuation of the local slope.

The exit-time statistics has been obtained with the following procedure:

• At time $t=0$ $2 \times 10^{5}$ couples of Lagrangian particles are located at random position in the periodicity box $(0 < x < L, 0 < y < L)$ of the velocity field, with initial separation $d0$ between the two particles of each couple.
• A set of thresholds $\{ r_i \}$ within the inertial range are fixed in geometric progression $r_i = r_0 \lambda^i$.
• The trajectories of Lagrangian particles are numerically computed according to $\dot{\mbox{\boldmath$x$}} = {\mbox{\boldmath$u$}}({\mbox{\boldmath$x$}},t)$ where the velocity field ${\mbox{\boldmath$u$}}$ has been obtained by parallel integration of Navier-Stokes equation.
• During the integration the distance between the two particles of each couple is measured, and the time $T_i$ when it reaches the threshold $r_i$ is stored.
• When the distance of a couple reaches the integral scale $L$ at time $T_L$, the exit-times $T_L(r_i)$ are computed as
  

  $$T_L(r_i) = T_L - T_i \;.$$ (2.39)

  
  

• The distance of the couple is then renormalized to $d0$ and the procedure is repeated.

Exit-times $T_L(r_i)$ provide an excellent tool for estimating the scaling exponent of the field. Since the thresholds can be chosen exactly within the inertial range, the scaling of Lagrangian structure function $\langle \exp[-\alpha p T_L(r)] \rangle \sim r^{\zeta_p}$ is not spoiled by contamination of the viscous range, and the excellent scaling allows for a precise measurement of the scaling exponents, which are in good agreement with those directly observed from the structure function of the passive field. Moreover, while the finite-time Lyapunov exponents are achievable only in numerical simulations, the measure of exit-times can be performed also in experiments by means of couple of Lagrangian tracers.

From the exit-time statistics it is also possible to recover the right tail of the Cramér function $S(\gamma )$ as the inverse Legendre transform [41] of the scaling exponents $\zeta_p$:

$$\zeta_p = \min_{\gamma} \left\{ p,[p \alpha + S(\gamma)]/\gamma \right\} \;.$$ (2.40)

which perfectly matches the Cramér function directly measured from the statistics of finite-time Lyapunov exponents (see Figure 2.9).

Figure 2.9: The Cramér function $S(\gamma )$ computed from finite time Lyapunov exponents (symbols) and exit-time statistics (line).