“Locally homogeneous turbulence”: Is it an inconsistent framework?

In his first 1941 paper Kolmogorov assumed that the velocity has increments that are homogeneous and independent of the velocity at a suitable reference point. This assumption of local homogeneity is consistent with the nonlinear dynamics only in an asymptotic sense when the reference point is far away. This inconsistency is illustrated numerically using the Burgers equation. Kolmogorov’s derivation of the four-fifths law for the third-order structure function and its anisotropic generalization are actually valid only for homogeneous turbulence, but a local version due to Duchon and Robert still holds. A Kolmogorov-Landau approach is proposed to handle the effect of fluctuations in the large-scale velocity on small-scale statistical properties; it is only a mild extension of the 1941 theory and does not incorporate intermittency effects.