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We investigate the effect of a four-dimensional Fourier transform on theformulation of the Navier-Stokes equation in Fourier space and the way theenergy is transferred between Fourier components. Since time in a sampled highintensity turbulence must be considered a stochastic variable in the energyexchange between scales, we refer to these dynamic triad interactions as modalinteractions, rather than the commonly referred to triad interactions in theclassical 3-dimensional analysis. The inclusion of time as a parameter broadensthe phase match condition from the classical one, $\Delta \vec{k} \cdot \vec{r}= \left [ \vec{k} - (\vec{k}_1 + \vec{k}_2 ) \right ] \cdot \vec{r}$, to themore general formulation that also includes temporal frequencies: $\Delta\vec{k} \cdot \vec{r} - \Delta \omega t = \left [ \vec{k} - (\vec{k}_1 +\vec{k}_2 ) \right ] \cdot \vec{r} - \left [ \omega - \left (\omega_1 +\omega_2 \right ) \right ] t$. This renders possible the occurrence of`delayed' and `advanced' interactions. The observation that mismatches in thewavevector triadic interactions may be compensated by a corresponding mismatchin the frequencies supports the empirically deduced delayed interactionsreported in [Josserand \textit{et al.}, \textit{J. Stat. Phys.} (2017)]. Theseresults explain the occurrence and inherent time development of the so-calledRichardson cascade and also how finite temporal overlap of wave components canresult in significant non-local interactions and consequently non-equilibriumturbulence, e.g., fractal grid generated turbulence. The consequences ofincluding time as a parameter in practical experiments or simulations in termsof limited resolution, domain size etc. are treated in the companion paper(Part 2) of the present work.

Dynamic Triad Interactions and Evolving Turbulence -- Part 1: 4D Modal Interactions