Smooth-filamental transition

The smooth-filamental transition predicted for $\alpha=\lambda$ in the mean field case, can be recast taking into account the intermittent behavior. According to Eq. (2.20) the exponent of the $p$-th structure function is the minimum between $p$, which corresponds to the smooth behavior, and $\min_{\gamma} \left\{[p \alpha + S(\gamma)]/\gamma \right\}$ that describes the rough, filamental field.

In the Gaussian approximation the crossover between the linear and non-linear behavior for the scaling exponent happens for:

$$p^*={2 \over \Delta} (\alpha -\lambda)$$ (2.23)

where the curve $1/\Delta ( \sqrt{\lambda^2 +2 p \alpha \Delta} -\lambda )$ intersects the line $p$. Below the smooth-filamental transition, when $\alpha < \lambda$, the intersection is for $p^* <0$, thus all the positive moments of the distribution of passive scalar fluctuation follow the non-linear behavior. This means that the filamental part of the field is dominant. On the contrary, when $\alpha > \lambda$ there is a crossover between the smooth scaling exponent for the structure functions of order $p \le p^*$ and the non-linear behavior for $p > p^*$. In this case only the higher moments of the passive scalar statistics are able to detect the filamental structures, while the lower moments are dominated by the smooth behavior of the field.

It is worthwhile to underline that $\zeta ^{\theta }_p$ in the non-smooth case is determined by the value of $S(\gamma^*_p)$ where $\gamma^*_p$ is the finite-time Lyapunov exponent which minimizes $[p \alpha + S(\gamma)]/\gamma$. Since $\gamma^*_p$ is a growing function of $p$ (in the quadratic approximation $\gamma^*_p = \sqrt{\lambda^2 +2 p \alpha \Delta}$ ) higher moments of passive scalar fluctuations are determined by the tail of the distribution of the stretching rates, where the parabolic approximation for $S(\gamma )$ does not hold in general. Thus the behavior of high order structure functions is extremely dependent on the detail of the velocity field. Nevertheless the presence of a smooth-filamental transition is determined by the asymptotic behavior of the exponents for $p \to 0$ where the Gaussian approximation holds thanks to the Central Limit Theorem, thus the smooth-filamental transition is always present for $\alpha=\lambda$, while for $\alpha > \lambda$ the exact value of the critical moment $p^*$ can deviate from Eq. (2.23).

We notice that relations obtained in the Gaussian approximation are exact for the peculiar case of the Kraichnan-model, that is a smooth velocity field $\delta$-correlated in time, in which the Cramér function $S(\gamma )$ is exactly parabolic.