  We propose a framework that elucidates the input-output characteristics of
flows with complex dynamics arising from nonlinear interactions between
different time scales. More specifically, we consider a periodically
time-varying base flow, and perform a frequency-domain analysis of periodic
perturbations about this base flow; the response of these perturbations is
governed by the harmonic resolvent, which is a linear operator similar to the
harmonic transfer function introduced by Wereley (1991). This approach makes it
possible to explicitly capture the triadic interactions that are responsible
for the energy transfer between different time scales in the flow. For
instance, perturbations at frequency $\alpha$ are coupled with perturbations at
frequency $\omega$ through the base flow at frequency $\omega-\alpha$. We draw
a connection with resolvent analsyis, which is a special case of the harmonic
resolvent when evaluated about a steady base flow. We show that the left and
right singular vectors of the harmonic resolvent are the optimal response and
forcing modes, which can be understood as full spatio-temporal signals that
reveal space-time amplification characteristics of the flow. We illustrate the
method on examples, including a three-dimensional system of ordinary
differential equations and the flow over an airfoil at near-stall angle of
attack.

Fluid Dynamics