Structure Functions and Dispersion Laws in Two-Dimensional Turbulence

Abstract

The classical phenomenological relations between dispersion laws,second-order structure functions and energy spectra are reexaminedfrom a more quantitative standpoint. It is shown that when a nonlocalenergy spectrum (steeper than k^{-3}) is substituted into the relationgiving structure functions or dispersion laws, an infrared divergenceoccurs so that the structure functions or dispersion laws at inertial-rangeseparation are not dominated by contributions from the inertial rangespectrum; they saturate and become independent of spectral steepness.It follows that the spectral steepness of real flows in the enstrophyinertial range must be extremely difficult to estimate from correlationor dispersion measurements alone. This might explain why the existenceof steep spectra, speculated on the basis of numerical modeling,has not been confirmed by real flow measurements.