  We study the input-output response of a streamwise constant projection of the
Navier-Stokes equations for plane Couette flow, the so-called 2D/3C model.
Study of a streamwise constant model is motivated by numerical and experimental
observations that suggest the prevalence and importance of streamwise and
quasi-streamwise elongated structures. Periodic spanwise/wall-normal (z-y)
plane stream functions are used as input to develop a forced 2D/3C streamwise
velocity field that is qualitatively similar to a fully turbulent spatial field
of DNS data. The input-output response associated with the 2D/3C nonlinear
coupling is used to estimate the energy optimal spanwise wavelength over a
range of Reynolds numbers. The results of the input-output analysis agree with
previous studies of the linearized Navier-Stokes equations. The optimal energy
corresponds to minimal nonlinear coupling. On the other hand, the nature of the
forced 2D/3C streamwise velocity field provides evidence that the nonlinear
coupling in the 2D/3C model is responsible for creating the well known
characteristic "S" shaped turbulent velocity profile. This indicates that there
is an important tradeoff between energy amplification, which is primarily
linear and the seemingly nonlinear momentum transfer mechanism that produces a
turbulent-like mean profile.
