# Variance and Intermittency in turbulence

In the turbulent transport of a scalar field, $\Phi$, decomposed into mean and fluctuating components, $\Phi=\left<\Phi\right>+\phi^\prime$, the scalar variance is defined as $\left<{\phi^\prime}^2\right>$.

Today I got a bit reamed out in an academic meeting for using the variance to synonymously describe the intermittence of the flow. I realize the difference between variance and intermittence, but in this context are the two connected? If so, can somebody show this mathematically? If not, can somebody explain how they are clearly separated?

• I'm not a specialist, but the way I see this, intermittency describes switching between different modes of behavior. If possible, I would define separate mean and variance for each of the modes. – gigacyan Oct 22 '13 at 6:43

In a scale invariant flow, the $n$'th order correlation function will take a scaling form, $$\Delta\Psi(n,r) = \langle \left[\Phi(x+r,t)-\Phi(x,t)\right]^n\rangle = \langle \left[\phi'(x+r,t)-\phi'(x,t)\right]^n\rangle \sim r^{\zeta_n} \, .$$ If the statistics of $\Phi$ are gaussian, all the correlation functions are determined by the second one, $\Delta\Psi(2,r)$, and we have $$\zeta_n = \frac{\zeta_2}{2}n \, .$$ All the scaling exponents are linearly related. Note that Kolmogorov theory uses this assumption and the fact that $\zeta_3 = 1$ to get $\zeta_n = n/3$.
In general though, the statistics of $\Phi$ are not Gaussian and this does not hold. Intermittency is the phyisical process that leads deviations of $\zeta_n$ from $n\zeta_2/2$. It is called "intermittency" because it can be traced down to the tale of the probability distribution of gradients of $\Psi$, i.e. to large but rare fluctuations which do not appear in a Gaussian distribution. These happen intermittently because they are so rare.