  An analysis of the statistics of the non-linear terms in resolvent analysis
is performed in this work for turbulent Couette flow at low Reynolds number.
Data from a direct numerical simulation of a minimal flow unit, at Reynolds
number 400, is post-processed using Fourier analysis in both time and space,
leading to the covariance matrix of the velocity. From the same data, we
computed the non-linear terms of the Navier-Stokes equations (treated as
forcing in the present formulation), which allowed us to compute the covariance
matrix of the forcing for this case. The two covariances are related exactly by
the resolvent operator; based on this, we explore the recovery of the velocity
statistics from the statistics of the forcing as a function of the components
of the forcing term. This is carried out for the dominant structures in this
flow, which participate in the self-sustaining cycle of turbulence: (i)
streamwise vortices and streaks, and (ii) spanwise coherent fluctuations of
spanwise velocity. The present results show a dominance by four of the
non-linear terms for the prediction of the full statistics of streamwise
vortices and streaks; a single term is seen to be dominant for spanwise
motions. A relevant feature observed in these cases is that forcing terms have
significant coherence in space; moreover, different forcing components are also
coherent between them. This leads to constructive and destructive interferences
that greatly modify the flow response, and should thus be accounted for in
modelling work.
