# Nonlinear forcing in the resolvent analysis of exact coherent states of the Navier-Stokes equations

## TL;DR

This paper introduces a velocity-vorticity formulation and Helmholtz decomposition into resolvent analysis of Navier-Stokes equations, enhancing the low-dimensional modeling of exact coherent states with fewer degrees of freedom.

## Contribution

It presents a novel velocity-vorticity based resolvent framework with Helmholtz decomposition, improving the efficiency of representing coherent states in fluid dynamics.

## Key findings

- Reduced degrees of freedom in the model
- Enhanced accuracy in representing coherent states
- Simplified input/output formulation

## Abstract

The resolvent analysis of McKeon & Sharma (2010) recasts the Navier-Stokes equations into an input/output form in which the nonlinear term is treated as a forcing that acts upon the linear dynamics to yield a velocity response. The framework has shown promise with regards to producing low-dimensional representations of exact coherent states. Previous work has focused on a primitive variable output; here we show a velocity-vorticity formulation of the governing equations along with a Helmholtz decomposition of the nonlinear forcing term reveals a simplified input/output form in the resolvent analysis. This approach leads to an improved method for compact representations of exact coherent states for both forcing and response fields, with a significant reduction in degrees of freedom in comparison to the primitive variable approach.