We investigate the effect of a four-dimensional Fourier transform on the formulation of the Navier-Stokes equation in Fourier space and the way the energy is transferred between Fourier components. Since time in a sampled high-intensity turbulence must be considered a stochastic variable in the energy exchange between scales, we refer to these dynamic triad interactions as modal interactions, rather than the commonly referred to triad interactions in the classical three-dimensional analysis. The inclusion of time as a parameter broadens the phase match condition from the classical one, $\mathrm{\Delta }\mathit{k}·\mathit{r}=[\mathit{k}-({\mathit{k}}_1+{\mathit{k}}_2)]·\mathit{r}$, to the more general formulation that also includes temporal frequencies: $\mathrm{\Delta }\mathit{k}·\mathit{r}-\mathrm{\Delta }\omega t=[\mathit{k}-({\mathit{k}}_1+{\mathit{k}}_2)]·\mathit{r}-[\omega -({\omega }_1+{\omega }_2)]t$. This renders possible the occurrence of “delayed” and “advanced” interactions. The observation that mismatches in the wavevector triadic interactions may be compensated by a corresponding mismatch in the frequencies supports the empirically deduced delayed interactions reported in Josserand et al. [J. Stat. Phys. 167, 596 (2017)]. These results explain the occurrence and inherent time development of the so-called Richardson cascade and also how finite temporal overlap of wave components can result in significant nonlocal interactions and consequently nonequilibrium turbulence, e.g., fractal grid generated turbulence. The consequences of including time as a parameter in practical experiments or simulations in terms of limited resolution, domain size, etc., are treated in the companion paper (Part II) of the present work.

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• Classical mechanics
• Shear layer turbulence
• Turbulence
• Turbulent mixing
• Wakes & jets