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

## TL;DR

This paper investigates how parasitic modes influence nonlinear closure in fluid flows by analyzing the feedback loop of the resolvent operator, revealing their role in nonlinear forcing and flow fluctuation structures.

## Contribution

It introduces a method to identify the influence of parasitic modes within the resolvent framework, enhancing understanding of nonlinear flow interactions beyond dominant linear mechanisms.

## Key findings

- Parasitic modes contribute significantly to nonlinear forcing.
- Resolvent modes align with DMD modes at fundamental frequencies.
- Discrepancies occur at higher harmonics due to lack of low-rank structure.

## Abstract

We use the feedback loop of McKeon \& Sharma (2010), where the nonlinear term in the Navier-Stokes equations is treated as an intrinsic forcing of the linear resolvent operator, to educe the structure of fluctuations in the range of scales (wavenumbers) where linear mechanisms are not active. In this region, the absence of dominant linear mechanisms is reflected in the lack of low-rank characteristics of the resolvent and in the disagreement between the structure of resolvent modes and actual flow features. To demonstrate the procedure, we choose low Reynolds number cylinder flow and the Couette equilibrium solution EQ1, which are representative of very low-rank flows dominated by one linear mechanism. The former is evolving in time, allowing us to compare resolvent modes with Dynamic Mode Decomposition (DMD) modes at the first and second harmonics of the shedding frequency. There is a match between the modes at the first harmonic but not at the second harmonic where there is no separation of the resolvent operator's singular values. We compute the self-interaction of the resolvent mode at the shedding frequency and illustrate its similarity to the nonlinear forcing of the second harmonic. When it is run through the resolvent operator, the `forced' resolvent mode shows better agreement with the DMD mode. A similar phenomenon is observed for the fundamental streamwise wavenumber of the EQ1 solution and its second harmonic. The importance of parasitic modes, labeled as such since they are driven by the amplified frequencies, is their contribution to the nonlinear forcing of the main amplification mechanisms as shown for the shedding mode, which has subtle discrepancies with its DMD counterpart.