The role of parasitic modes in nonlinear closure via the resolvent feedback loop

Resolvent modes are compared to Dynamic Mode Decomposition (DMD) modes at the first and second harmonics of the shedding frequency for cylinder flow. Sharma et al. (2016) and Towne et al. (2018) have discussed when these modes are likely to agree. While there is a match between the modes at the first harmonic, the structure predicted by resolvent analysis bears no resemblance to the DMD mode at the second harmonic where there is no separation of the resolvent operator's singular values. We use the feedback loop of McKeon and Sharma (2010), where the nonlinear term is the intrinsic forcing of the resolvent operator, to educe the structure of fluctuations where linear mechanisms are not active. The self-interaction of the resolvent mode at the shedding frequency is similar to the second harmonic's nonlinear forcing. When it is run through the resolvent operator, the `forced' resolvent mode agrees with the DMD mode. The role of parasitic modes, labeled as such since they are driven by the amplified frequencies, is important in terms of their contribution to the nonlinear forcing of the main amplification mechanisms. This is demonstrated for the shedding mode which has subtle discrepancies with its DMD counterpart.