Structure Functions and Dispersion Laws in Two-Dimensional Turbulence

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  • 1 Laboratoire de Météorologie Dynamique, Ecole Normale Supéerieure, Paris Cedex 05
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Abstract

The classical phenomenological relations between dispersion laws, second-order structure functions and energy spectra are reexamined from a more quantitative standpoint. It is shown that when a nonlocal energy spectrum (steeper than k−3) is substituted into the relation giving structure functions or dispersion laws, an infrared divergence occurs so that the structure functions or dispersion laws at inertial-range separation are not dominated by contributions from the inertial range spectrum; they saturate and become independent of spectral steepness. It follows that the spectral steepness of real flows in the enstrophy inertial range must be extremely difficult to estimate from correlation or dispersion measurements alone. This might explain why the existence of steep spectra, speculated on the basis of numerical modeling, has not been confirmed by real flow measurements.

Abstract

The classical phenomenological relations between dispersion laws, second-order structure functions and energy spectra are reexamined from a more quantitative standpoint. It is shown that when a nonlocal energy spectrum (steeper than k−3) is substituted into the relation giving structure functions or dispersion laws, an infrared divergence occurs so that the structure functions or dispersion laws at inertial-range separation are not dominated by contributions from the inertial range spectrum; they saturate and become independent of spectral steepness. It follows that the spectral steepness of real flows in the enstrophy inertial range must be extremely difficult to estimate from correlation or dispersion measurements alone. This might explain why the existence of steep spectra, speculated on the basis of numerical modeling, has not been confirmed by real flow measurements.

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