Localness of energy cascade in hydrodynamic turbulence, I. Smooth coarse-graining

Gregory L. Eyink, Hussein Aluie

Published 2009-09-13Version 1

We introduce a novel approach to scale-decomposition of the fluid kinetic energy (or other quadratic integrals) into band-pass contributions from a series of length-scales. Our decomposition is based on a multiscale generalization of the ``Germano identity'' for smooth, graded filter kernels. We employ this method to derive a budget equation that describes the transfers of turbulent kinetic energy both in space and in scale. It is shown that the inter-scale energy transfer is dominated by local triadic interactions, assuming only the scaling properties expected in a turbulent inertial-range. We derive rigorous upper bounds on the contributions of non-local triads, extending the work of Eyink (2005) for low-pass filtering. We also propose a physical explanation of the differing exponents for our rigorous upper bounds and for the scaling predictions of Kraichnan (1966,1971). The faster decay predicted by Kraichnan is argued to be the consequence of additional cancellations in the signed contributions to transfer from non-local triads, after averaging over space. This picture is supported by data from a $512^3$ pseudospectral simulation of Navier-Stokes turbulence with phase-shift dealiasing.