  The quadratic convection term in the incompressible Navier-Stokes equations
is considered as a non-linear forcing to the linear operator, and it is studied
in the Fourier domain through the analysis of interactions between triadically
compatible wavenumber-frequency triplets. Interaction coefficients are proposed
to quantify the contribution to the forcing by each pair of triplets and are
computed using data from direct numerical simulations of a turbulent channel at
$Re_{\tau} \approx 550$. The coefficients show the importance of interactions
involving streamwise large scales. The regions of non-linear interactions
permitted under the quasi-linear (QL) and generalized quasi-linear (GQL)
assumptions are shown to be significant contributors to the forcing and
increasing the number of GQL-large scales is shown to monotonically increase
the total forcing captured, providing a possible reason for the success of QL
and GQL simulations. The tools presented are expected to be useful for
improving modeling of the nonlinearity, especially in QL, GQL, and resolvent
analyses, and understanding the amplitude modulation mechanism relating
large-scale fluctuations to the modulation of near-wall structures.
