Localness of energy cascade in hydrodynamic turbulence, II. Sharp spectral filter

We investigate the scale-locality of subgrid-scale (SGS) energy flux and interband energy transfers defined by the sharp spectral filter. We show by rigorous bounds, physical arguments and numerical simulations that the spectral SGS flux is dominated by local triadic interactions in an extended turbulent inertial-range. Inter-band energy transfers are also shown to be dominated by local triads if the spectral bands have constant width on a logarithmic scale. We disprove in particular an alternative picture of “local transfer by nonlocal triads,” with the advecting wavenumber mode at the energy peak. Although such triads have the largest transfer rates of all individual wavenumber triads, we show rigorously that, due to their restricted number, they make an asymptotically negligible contribution to energy flux and log-banded energy transfers at high wavenumbers in the inertial-range. We show that it is only the aggregate effect of a geometrically increasing number of local wavenumber triads which can sustain an energy cascade to small scales. Furthermore, non-local triads are argued to contribute even less to the space-average energy flux than is implied by our rigorous bounds, because of additional cancellations from scale-decorrelation effects. We can thus recover the -4/3 scaling of nonlocal contributions to spectral energy flux predicted by Kraichnan’s ALHDIA and TFM closures. We support our results with numerical data from a 5123 pseudospectral simulation of isotropic turbulence with phase-shift dealiasing. We also discuss a rigorous counterexample of Eyink (1994), which showed that non-local wavenumber triads may dominate in the sharp spectral flux (but not in the SGS energy flux for graded filters). We show that this mathematical counterexample fails to satisfy reasonable physical requirements for a turbulent velocity field, which are employed in our proof of scale-locality. We conclude that the sharp spectral filter has a firm theoretical basis for use in large-eddy simulation (LES) modeling of turbulent flows.