  Turbulence in the restricted nonlinear (RNL) dynamics is analyzed and
compared with DNS of Poiseuille turbulence at $R=1650$. The structures are
obtained by POD analysis of the two components of the flow partition used in
RNL dynamics: the streamwise-mean flow and fluctuations. POD analysis of the
streamwise-mean flow indicates that the dominant POD modes, in both DNS and
RNL, are roll-streaks harmonic in the spanwise. However, we conclude that these
POD modes do not occur in isolation but rather are Fourier components of a
coherent roll-streak structure. POD analysis of the fluctuations in DNS and RNL
reveals similar complex structures consisting in part of oblique waves
collocated with the streak. The origin of these structures is identified by
their correspondence to POD modes predicted using a stochastic turbulence model
(STM). These predicted POD modes are dominated by the optimally growing
structures on the streak, which the STM predicts correctly to be of sinuous
oblique wave structure. This close correspondence between the roll-streak
structure and the associated fluctuations in DNS, RNL and the STM implies that
the self-sustaining mechanism operating in DNS is essentially the same as that
in RNL, which has been previously associated with optimal perturbation growth
on the streak.
