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 ...

  • $\begingroup$ Is your flatness criterion related to what people call kurtosis in stats? The formula looks a bit similar. $\endgroup$
    – twistor59
    Apr 2, 2013 at 11:25
  • $\begingroup$ @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 ... $\endgroup$
    – Dilaton
    Apr 2, 2013 at 12:19
  • $\begingroup$ 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). $\endgroup$
    – twistor59
    Apr 2, 2013 at 12:30
  • $\begingroup$ @twistor59 that quite fits the phenomenology of intermittency as far as I understand it. $\endgroup$
    – Dilaton
    Apr 2, 2013 at 13:02


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