Physical interpretation of an intermittency definition
A random function $v(t)$ is said to be intermittent at small scales of its "Flatness" $F$, given as
$$ F(\Omega) = \frac{\langle (v_{\Omega}^{>}(t))^4\rangle}{\langle v_{\Omega}^{>}(t))^2\rangle} = \frac{\langle v_{\Omega}^{>}(t)v_{\Omega}^{>}(t))v_{\Omega}^{>}(t))v_{\Omega}^{>}(t))\rangle}{\langle v_{\Omega}^{>}(t)v_{\Omega}^{>}(t)\rangle \langle v_{\Omega}^{>}(t)v_{\Omega}^{>}(t)\rangle} $$
diverges as the high pass filter velocity $\Omega \rightarrow \infty$.
$v(t)$ which can for example be the velocity in the fluid, is decomposed into its Fourier components
$$ v(t) = \int_{R^3}d\omega \, e^{i\omega t}\hat{v}_{\omega} $$
and $v_{\Omega}^{>}(t)$ is the high frequency part of for example the velocity in the fluid
$$ v_{\Omega}^{>}(t) = \int_{|\omega| > \Omega} d\omega \, e^{i\omega t}\hat{v}_{\omega} $$
I am struggling hard to understant the physical meaning of this definition and nead some help in this:
First of all, why is $F$ called "flatness", it should be a measure of flatness of what? What does it mean for $F$ to diverge in the high frequency (UV) limit? Looking at the expression for $F$, I got the impression that it could mean that higher order correlations in time ("n-point functions") start to dominate in case of intermittency and more "local" 2-point interactions, which are responsible to maintain a scale invariant inertial subrange (?), become negligible such that the turbulent system starts to deviate from a Kolmogorov inertial subrange for example.
In addition, other measures of intermittency can be defined involving higher order correlations in length $l$, such as the so called hyper-flattness defined as $F_6 (l) = S_6 (l)/(S_2(l)^3)$ etc... Does this mean that one could more generally say that for a turbulent system that shows intermittency, the <PERSON> theorem can not be applied to calculate higher order n-point functions from 2-point functions?
I am finally interested in understanding intermittency from a quantum field theory point of view, which is unfortunatelly not the point of view of the book I have taken these definitions from ... 453