A reduced description of shear flows consistent with the Reynolds number
scaling of lower-branch exact coherent states in plane Couette flow [J. Wang et
al., Phys. Rev. Lett. 98, 204501 (2007)] is constructed. Exact time-independent
nonlinear solutions of the reduced equations corresponding to both lower and
upper branch states are found for Waleffe flow [F. Waleffe, Phys. Fluids 9,
883--900 (1997)]. The lower branch solution is characterized by fluctuations
that vary slowly along the critical layer while the upper branch solutions
display a bimodal structure and are more strongly focused on the critical
layer. The reduced model provides a rational framework for investigations of
subcritical spatiotemporal patterns in parallel shear flows.
