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We investigate the effect of a four-dimensional Fourier transform on theformulation of Navier-Stokes equation in Fourier space and the way the energyis transferred between Fourier components (modal interactions, commonlyreferred to as triad interactions in the classical 3-dimensional analysis). Wespecifically consider the effect of a spatially and temporally finite,digitally sampled velocity record on the modal interactions and find thatFourier components may interact within a broadened frequency window as comparedto the usual integrals over infinite ranges. We also see how these finitevelocity records have a significant effect on the efficiency of the differentmodal interactions and thereby on the shape and development of velocity powerspectra. The observation that mismatches in the wavevector triadic interactionsmay be compensated by a corresponding mismatch in the frequencies supports theempirically deduced delayed interactions reported in [Josserand \textit{etal.}, \textit{J. Stat. Phys.} (2017)]. Collectively, these results explain theoccurrence and time development of the so-called Richardson cascade and alsowhy deviations from the classical Richardson cascade may occur. Finally, wequote results from companion papers that deal with measurements and computersimulations of the time development of velocity power spectra in a turbulentjet flow into which a single Fourier mode (narrow band oscillation) isinjected.

Dynamic triad interactions and evolving turbulence spectra