On the deficiency of structure functions as inertial range diagnostics

Abstract

In the limit of infinite Reynolds number, Re, Kolmogorovs two-thirds and five-thirds laws are formally equivalent. However, for the sorts of Reynolds numbers encountered in terrestrial experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. We explain that this is because the second-order structure function is a poor diagnostic, mixing information about energy and enstrophy, and about small and large scales. It is shown that, as a result, the form of the structure function in the inertial range is not a simple power law, but rather a combination of two powers, of the form ((Δv)2)(r) = a + br2 + cr2/3, where the coefficients a, b and c are functions of Re. It is only as Re → ∞ that a pure two-thirds law is obtained. Similar problems arise for higher-order structure functions, which also display combined power laws.