Locality of the Turbulent Energy Cascade

Question

Locality appears to be an omnipresent feature of physics, including within turbulent fluid dynamics.
In 1941 Kolmogorov postulated that the turbulent energy cascade is driven by scale-local interactions, where the eddies at a given scale solely interact with eddies of nearby scales. Hence they transfer kinetic energy in a waterfall-like "cascade" from the large integral scale all the way down to the dissipative ones, wherein the energy is converted into heat.
As far as I understand, much numerical investigation has been done on this, and this scale-local nature of interactions has largely been confirmed to be true. See e.g. (DOI) 10.1103/PhysRevFluids.3.084601 or 10.1063/1.3266883 (arXiv).
However, I cannot find much on the scale-locality of correlations. It seems that the only correlations that are measured in turbulence research is the correlations between eddies in real-space, i.e. in terms of their physical seperation distance. Why has nobody investigated the correlations across scale? Does the scale-locality of interactions imply that also the correlations between eddies are scale-local?

tpg2114 · Accepted Answer

The study of interscale interactions from a scale perspective is investigated using what has come to be called triadic interaction analysis. The idea is to convert the equations to their Fourier representation and then look at which wave numbers interact with each other and by how much. A good representative paper from early in the development of the method is Interscale dynamics and local isotropy in high Reynolds number turbulence within triadic interactions.
The studies reveal that there are local, non-local, and distant interactions between scales. Under high Reynolds number, homogeneous conditions these interactions are dominated by local behavior as Kolmogorov hypothesized. However, under more realistic conditions there are other effects that become important, such as non-local interactions where small scales transfer energy to large ones in an inverse cascade (also know as back-scatter). It's also much more complex for compressible turbulence, where the interactions happen amongst 4 wave-number pairs instead of 3. These are all active areas of investigation.

Locality of the Turbulent Energy Cascade

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