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

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, $\Delta \bm{k} \cdot \bm{r} = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r}$, to the more general formulation that also includes temporal frequencies: $\Delta \bm{k} \cdot \bm{r} - \Delta \omega t = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r} - \left [ \omega - \left (\omega_1 + \omega_2 \right ) \right ] t$. 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.