Intermittency

The previous results which has been presented with intuitive but not-rigorous arguments, are formally valid only in a mean field sense, i.e. assuming a constant stretching rate $\lambda$. This is not the general case.

While in the limit of infinite time the stretching rate is the same for almost all trajectories in an ergodic region, and is given by the Lyapunov exponent, the stretching rates at a finite time $t$ are given by the finite time Lyapunov exponents $\gamma$, which are defined as

$$\gamma = \lim_{\delta_0 \to 0} {1 \over t} \ln {\delta(t) \over \delta_0}$$ (2.17)

Because of their local character they can assume different values depending on the initial positions of the trajectories along which they are measured. For large $t$ their distribution approaches the asymptotic form

$$P(\gamma,t) \sim t^{1/2} \exp[-S(\gamma) t]$$ (2.18)

The Cramér function $S(\gamma )$ (also called entropy function) is concave, positive, with a quadratic minimum in $\lambda$ (the maximum Lyapunov exponent) $S(\lambda)=0$, and its shape far from the minimum depends on the details of the velocity statistics [38,39]. The quadratic minimum of $S(\gamma )$ correspond to a Gaussian behavior for the core of the probability distribution of the stretching rate $\gamma$, which can be predicted in the general case thanks to the Central Limit Theorem. In the limit $t \to \infty$ the distribution became a $\delta$-distribution with support for $\gamma = \lambda$.

Local fluctuations of the stretching rates are the origin of the intermittent behavior of the passive scalar statistic. Indeed, in order to obtain the correct evaluation of the structure functions of passive scalar fluctuations, it is necessary to average over the distribution of finite-time Lyapunov exponents:

$$S^{\theta}_p (r) \equiv \langle (\delta_r \theta)^p \rangle ... ...]/\gamma} \sim \left({r \over L} \right)^{\zeta^{\omega}_p}\;.$$ (2.19)

The scaling exponents are evaluated from Eq. (2.20) by a steepest descent argument as:

$$\zeta^{\theta}_p = \min_{\gamma} \left\{ p,[p \alpha + S(\gamma)]/\gamma \right\}\;.$$ (2.20)

Intermittency manifests itself in the nonlinear dependence of the exponents $\zeta ^{\theta }_p$ on the order $p$.

In the Gaussian approximation for $P(\gamma)$, which holds near its core, the Cramér function has the quadratic expression:

$$S(\gamma) = {(\gamma - \lambda)^2 \over 2 \Delta}$$ (2.21)

and it is possible to obtain an explicit expression for the scaling exponents $\zeta^{\omega}_p$:

$$\zeta^{\theta}_p = \min \left\{p, {1 \over \Delta} \left( \sqrt{\lambda^2 +2 p \alpha \Delta} -\lambda \right) \right\}$$ (2.22)