Lognormal intermittency model (${\zeta }_p=p/3-\mu p(p-3)/18$)

If $\log ({ϵ}_r/ϵ)$ is Gaussian with variance ${\sigma }_r^2$, normalization gives $⟨{ϵ}_r^q⟩/{ϵ}^q=\exp [q(q-1){\sigma }_r^2/2]$. Taking $\exp ({\sigma }_r^2)=B(\ell /r{)}^{\mu }$ predicts ${\zeta }_p=p/3-\mu p(p-3)/18$. Its eventual negative high-order exponents show that this extrapolation cannot hold for arbitrarily large orders with finite uniform velocity moments.