  The perspective of statistical state dynamics (SSD) has recently been applied
to the study of mechanisms underlying turbulence in various physical systems.
An example implementation of SSD is the second order closure referred to as
stochastic structural stability theory (S3T), which has provided insight into
the dynamics of wall turbulence and specifically the emergence and maintenance
of the roll/streak structure. This closure eliminates nonlinear interactions
among the perturbations has been removed, restricting nonlinearity in the
dynamics to that of the mean equation and the interaction between the mean and
perturbation covariance. Here, this quasi-linear restriction of the dynamics is
used to study the structure and dynamics of turbulence in plane Poiseuille flow
at moderately high Reynolds numbers in a closely related dynamical system,
referred to as the restricted nonlinear (RNL) system. RNL simulations reveal
that the essential features of wall-turbulence dynamics are retained.
Remarkably, the RNL system spontaneously limits the support of its turbulence
to a small set of streamwise Fourier components giving rise to a naturally
minimal representation of its turbulence dynamics. Although greatly simplified,
this RNL turbulence exhibits natural-looking structures and statistics.
Surprisingly, even when further truncation of the perturbation support to a
single streamwise component is imposed the RNL system continues to produce
self-sustaining turbulent structure and dynamics. RNL turbulence at the
Reynolds numbers studied is dominated by the roll/streak structure in the
buffer layer and similar very-large-scale structure (VLSM) in the outer layer.
Diagnostics of the structure, spectrum and energetics of RNL and DNS turbulence
are used to demonstrate that the roll/streak dynamics supporting the turbulence
in the buffer and logarithmic layer is essentially similar in RNL and DNS.
