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

We introduce a novel approach to scale decomposition of the fluid kinetic energy (or other quadratic integrals) into band-pass contributions from it 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 interscale 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 nonlocal triads, extending the work of Eyink [Physica D 207, 91 (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 [Phys. Fluids 9, 1728 (1966): J Fluid Mech 47, 525 (1971)] The faster decay predicted by Kraichnan is argued to be the consequence of additional cancellations in the signed contributions to transfer from nonlocal 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. (C) 2009 American Institute of Physics. [doi:10.1063/1.3266883]