  Streamwise and quasi-streamwise elongated structures have been shown to play
a significant role in turbulent shear flows. We model the mean behavior of
fully turbulent plane Couette flow using a streamwise constant projection of
the Navier Stokes equations. This results in a two-dimensional, three velocity
component ($2D/3C$) model. We first use a steady state version of the model to
demonstrate that its nonlinear coupling provides the mathematical mechanism
that shapes the turbulent velocity profile. Simulations of the $2D/3C$ model
under small amplitude Gaussian forcing of the cross-stream components are
compared to DNS data. The results indicate that a streamwise constant
projection of the Navier Stokes equations captures salient features of fully
turbulent plane Couette flow at low Reynolds numbers. A system theoretic
approach is used to demonstrate the presence of large input-output
amplification through the forced $2D/3C$ model. It is this amplification
coupled with the appropriate nonlinearity that enables the $2D/3C$ model to
generate turbulent behaviour under the small amplitude forcing employed in this
study.
