  An analytic perturbation theory is suggested in order to find finite-size
corrections to the scaling power laws. In the frame of this theory it is shown
that the first order finite-size correction to the scaling power laws has
following form $S(r) \cong cr^{\alpha_0}[\ln(r/\eta)]^{\alpha_1}$, where $\eta$
is a finite-size scale (in particular for turbulence, it can be the Kolmogorov
dissipation scale). Using data of laboratory experiments and numerical
simulations it is shown shown that a degenerate case with $\alpha_0 =0$ can
describe turbulence statistics in the near-dissipation range $r > \eta$, where
the ordinary (power-law) scaling does not apply. For moderate Reynolds numbers
the degenerate scaling range covers almost the entire range of scales of
velocity structure functions (the log-corrections apply to finite Reynolds
number). Interplay between local and non-local regimes has been considered as a
possible hydrodynamic mechanism providing the basis for the degenerate scaling
of structure functions and extended self-similarity. These results have been
also expanded on passive scalar mixing in turbulence. Overlapping phenomenon
between local and non-local regimes and a relation between position of maximum
of the generalized energy input rate and the actual crossover scale between
these regimes are briefly discussed.
