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Separate a smooth large-scale velocity $U_L$ from a remaining smaller-scale component $w_L$. If the large-scale variation length is much greater than $r$, their longitudinal increment has the formFor a positive integer $p$, the leading expansion of its structure function isThus even a velocity component at scales much larger than $r$ enters through its gradients and through mixed moments with the finer field. It is not generally legitimate to discard those mixed moments: large-scale strain can organize and modulate the finer motions. For example, at fourth order the expansion containsin addition to the small-scale fourth moment and the generally present odd mixed terms. Even if the components were independent and symmetric, the displayed mixed second-moment term would survive. This explicitly exhibits large-scale contributions to structure functions beyond second order.

The implication for K62 refined similarity is that an observed structure function need not measure a purely scale-local velocity fluctuation. Dividing it by $({ϵ}_{r}r{)}^{1/3}$ may still leave dependence on outer-scale gradients, correlations or finite-range cutoffs. The universal conditional coefficients ${\beta }_p$ and the identification of a measured exponent with the lognormal intermittency model therefore require additional locality and scale-separation tests. Finite-range mixed contributions can imitate or obscure anomalous scaling. This is an objection to treating the 1962 dimensional argument as an unconditional law for measured increments; it is not, by itself, a proof that an asymptotic refined similarity regime cannot exist.