= Intermittency
{disambiguate=turbulence}
<Intermittency (turbulence)> is the uneven concentration of turbulent fluctuations or dissipation into relatively intense events separated by quieter regions or intervals. It often appears statistically as increasingly non-Gaussian velocity or magnetic-field increments at smaller separations. A turbulent signal can therefore have a moderate average variance while rare events dominate its higher moments.<Intermittency in the solar wind turbulence through probability distribution functions of fluctuations by Luca Sorriso-Valvo and others>[Abstract: scale-dependent non-Gaussian fluctuation distributions and mixtures of variances]{ref}
One useful diagnostic is the increment flatness
$$
F(r)=\frac{\langle(\delta u_L)^4\rangle}{\langle(\delta u_L)^2\rangle^2}
=\frac{S_4(r)}{S_2(r)^2},
$$
using the same <structure function (turbulence)> component and convention in both moments. A centered Gaussian has $F=3$. Increasing $F$ towards small scales indicates increasingly strong relative tails, although measurement noise and nonstationary mixtures also need checking.
A toy distribution isolates the rare-event effect. Let $X$ vanish with probability $1-p$ and take values $+\sigma/\sqrt p$ and $-\sigma/\sqrt p$ with probability $p/2$ each. Then
$$
\mathbb E X^2=\sigma^2,\qquad \mathbb E X^4=\frac{\sigma^4}{p},\qquad F=\frac1p.
$$
At $p=0.05$, the variance is still $\sigma^2$ but the flatness is $20$. This is an illustrative marginal distribution, not a construction of a complete turbulent field.
Spacecraft measurements provide a concrete application. Analysis of Helios 2 magnetic and velocity records compared increments at different lags and heliocentric distances, finding scale-dependent flatness in both fast and slow <solar wind>. Mapping time lag to spatial separation requires an advection assumption. Flatness characterizes the fluctuations; by itself it does not establish the sign or magnitude of an <energy cascade>.<Radial evolution of solar wind intermittency in the inner heliosphere by Roberto Bruno and others>[Abstract, flatness analysis and Figure 1, Helios 2 intervals between 0.3 and 1 AU]{ref}
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