  The resolvent formulation of the Navier$\text{--}$Stokes equations gives a
means for the characterization and prediction of features of turbulent
flows$\text{---}$such as statistics, structures and their nonlinear
interactions$\text{---}$using the singular value decomposition of the resolvent
operator based on the appropriate turbulent mean, following the framework
developed by McKeon & Sharma (2010). This work will describe a methodology for
approximating leading resolvent (i.e., pseudospectral) modes for shear-driven
turbulent flows using prescribed analytic functions. We will demonstrate that
these functions, which arise from the consideration of wavepacket
pseudoeigenmodes of simplified linear operators (Trefethen 2005), in particular
give an accurate approximation of the class of nominally wall-detached modes
that are centered about the critical layer. Focusing in particular on modeling
wall-normal vorticity modes, we present a series of simplifications to the
governing equations that result in scalar differential operators that are
amenable to such analysis. We demonstrate that the leading wall-normal
vorticity response mode for the full Navier$\text{--}$Stokes equations may be
accurately approximated by considering a second order scalar operator, equipped
with a non-standard inner product. The variation in mode shape as a function of
wavenumber and Reynolds number may be captured by evolving a low dimensional
differential equation in parameter space. This characterization provides a
theoretical framework for understanding the origin of observed structures, and
allows for rapid estimation of dominant resolvent mode characteristics without
the need for operator discretization or large numerical computations. We relate
our findings to classical lift-up and Orr amplification mechanisms in
shear-driven flows.
