  This paper extends the resolvent formalism for wall turbulence proposed by
McKeon and Sharma(2010) to account for the effect of streamwise-constant
riblets. Under the resolvent formulation, the Navier-Stokes equations are
interpreted as a forcing-response system: the nonlinear convective term is
interpreted as a feedback forcing on the remaining linear terms, which
generates a velocity and pressure response. A gain-based decomposition of the
linear forcing-response transfer function --- the resolvent operator --- yields
highly amplified velocity and pressure modes, which can be considered key
building blocks of the turbulent flow field. Previous work has shown that these
high-gain modes provide substantial insight into turbulence statistics,
structure, and control of smooth-walled flows. To introduce the effect of
riblets within this framework, a linear spatially-varying body force is added
to the governing equations. In other words, volume penalization is used to
approximate the surface features. Predictions for spanwise-periodic and
streamwise-constant riblets show that specific high-gain modes identified from
the modified governing equations reproduce observations made in prior direct
numerical simulations with limited computation. The deterioration in
performance with increasing riblet size is predicted well and so is the
emergence of spanwise rollers resembling Kelvin-Helmholtz vortices. This new
modeling framework is also used to pursue limited riblet shape optimization.
