Eqns. (2.11-13).

It must be mentioned that marginal interactions and renormalizability are consequence of the dimensional relation implicit in Kolmogorov scaling: $[q^\smalL]=-[t]$. In general, had we set: $$\gamma_k\sim k^r\qquad H_k\sim k^s
\eqno(2.18)$$ we would have obtained: $$[q^\smalL]=\frac{r-s-2}{2}\qquad [\delta\z]=\frac{3r-s}{2}$$ $$[\lambda_{\gamma_n}]=[\lambda_{H_n}]
=[\lambda_{G_n}] =-n([t]+[q^\smalL]) 
=\frac{n}{2}(s+2-3r)
\eqno(2.19)$$ We thus see that super-renormalizability $[\lambda]<0$ and non-renormalizability $[\lambda]>0$ of the theory occur, respectively, for positive and negative $[q^\smalL]+[t]$ [@zinn-justin]. This corresponds to the two regimes of eddy decay time becoming asymptotically longer (shorter) than the eddy turn-over time, and hence the nonlinearity becoming dominant (negligible) at large scales.

Marginality of the interactions means that logarithmic divergences may arise both at large and small $k$. At small $k$, however, such divergences are not expected, due to the subtraction in the definition of $\delta\z$. The reason is sketched below (more details in a separate publication; it must be said, anyway, that this is not a surprise: Lagrangian closures [@kraichnan65] were introduced precisely to cure the infrared divergences arising in the original Eulerian theories). As it appears from Eqn. (2.17), small $k$ divergence is due to internal lines in a loop diagrams involving the field $\delta\z$. The scaling of Eqn. (2.17) is associated with large, not with small $k$ 