  In this work we examine the turbulence maintained in a Restricted Nonlinear
(RNL) model of plane Couette flow. This model is a computationally efficient
approximation of the second order statistical state dynamics (SSD) obtained by
partitioning the flow into a streamwise averaged mean flow and perturbations
about that mean, a closure referred to herein as the RNL$_\infty$ model. The
RNL model investigated here employs a single member of the infinite ensemble
that comprises the covariance of the RNL$_\infty$ dynamics. The RNL system has
previously been shown to support self-sustaining turbulence with a mean flow
and structural features that are consistent with DNS. This paper demonstrates
that the RNL system's self-sustaining turbulent state is supported by a small
number of streamwise varying modes, which form the natural support for the
self-sustaining process maintaining RNL turbulence. Remarkably, truncation of
the RNL system's support to a single streamwise varying mode can suffice to
sustain the turbulent state. The close correspondence between RNL simulations
and DNS that has been previously observed along with the results presented here
suggest that the fundamental mechanisms underlying wall-turbulence can be
analyzed using these highly simplified RNL systems.
