# Physical interpretation of an intermittency definition

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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 Wick 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 ...

Is your flatness criterion related to what people call kurtosis in stats? The formula looks a bit similar.

This post imported from StackExchange Physics at 2014-03-09 16:26 (UCT), posted by SE-user twistor59
@twistor59, yeah the first definition looks similar to what is in my book and maybe the "peakedness" interpretation would make most sence in the context of intermittency (?), not sure ...

This post imported from StackExchange Physics at 2014-03-09 16:26 (UCT), posted by SE-user Dilaton
High kurtosis means a strong peak plus long tails. The long tails mean that the extreme values can't be neglected (like they could for, say, a Gaussian distribution). So in the fluid case, where they're looking at velocity gradients, this would mean that the velocity gradient was mostly low (the high peak), but there are a significant number of isolated scattered pockets where there is a very high velocity gradient (giving the long tails).

This post imported from StackExchange Physics at 2014-03-09 16:26 (UCT), posted by SE-user twistor59
@twistor59 that quite fits the phenomenology of intermittency as far as I understand it.

This post imported from StackExchange Physics at 2014-03-09 16:26 (UCT), posted by SE-user Dilaton

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