  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 3-dimensional analysis. The inclusion of time as a parameter broadens
the 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 the
more 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 the
wavevector triadic interactions may be compensated by a corresponding mismatch
in the frequencies supports the empirically deduced delayed interactions
reported in [Josserand \textit{et al.}, \textit{J. Stat. Phys.} (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 non-local interactions and consequently non-equilibrium
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 2) of the present work.
