Counterbalanced interaction locality of developed hydrodynamic turbulence

The problem of interaction locality in k space is studied in a diagrammatic perturbation approach for the Navier-Stokes equation in quasi-Lagrangian variables. Analyzing the whole diagram series we have found an exact relation between the asymptotic behavior of the triple-correlation function of velocities that governs the energy transfer over scales and the double-correlation function giving the energy distribution. Namely, at r≪R, we obtain ${\mathit{S}}_3$(r,R,R-r)∝${\mathit{S}}_2$(R)(r/R) [${\mathit{S}}_3$(r)/${\mathit{S}}_2$(r)]∝${\mathit{R}}_2^{\mathrm{\zeta }}$-1${\mathit{r}}_2^{2\mathrm{-}\mathrm{\zeta }}$, where ${\mathrm{\zeta }}_2$ is the static exponent of double-velocity moment. This relation between two different physical quantities (in principle, measurable independently) is accessible to an experimental check. Also, this relation allows us to describe an energy exchange between distant scales in k space: For any steady spectrum carrying constant energy flux, the interactions of the given k-eddies with large (${\mathit{k}}_1$≪k) and small eddies (${\mathit{k}}_2$≫k) are shown to decrease by the same law with the distance in k space, such as (${\mathit{k}}_1$/k${)}_2^{2\mathrm{-}\mathrm{\zeta }}$ and (k/${\mathit{k}}_2$ ${)}_2^{2\mathrm{-}\mathrm{\zeta }}$. It means a balance of interactions for such a spectrum. Considering, in particular, the multifractal picture of developed turbulence, we analyze the range of exponents h of the velocity field [δv(r)∝${\mathit{r}}^{\mathit{h}}$] which provides the locality of interaction in the k space. It is shown that the condition of infrared locality of interaction (with larger ${\mathit{k}}_1$-eddies) could give only the upper restriction for the exponent. The upper limit thus found (${\mathit{h}}_{\max }$=1) coincides with the boundary exponent of singularity of energy dissipation. As far as an interaction locality in the ultraviolet limit (${\mathit{k}}_2$≫k) is concerned, we prove that any reasonable dimension function D(h) provides locality whatever small h is considered.