Landau damping of disturbances in nearly inviscid inflectional shear flows

TL;DR

This paper studies how small viscosity affects the damping of disturbances in nearly inviscid shear flows with inflection points, revealing the transformation of continuous modes into true eigenmodes with preserved vorticity structure.

Contribution

It demonstrates how tiny viscosity converts singular continuous modes into regular eigenmodes, providing insights into the transition from inviscid to viscous flow behavior.

Findings

Exponential decay of perturbations in inviscid flows resembles Landau damping.

Small viscosity transforms singular modes into true eigenmodes.

Vorticity structure is preserved in the viscous eigenmode.

Abstract

We investigate the structure of damped two-dimensional perturbations in unstable plane-parallel shear flows with an inflection point. In inviscid flows within the stable wavenumber region $k$, no regular eigenmodes exist -- the frequency spectrum $\omega$ consists of a continuous set of singular van Kampen modes with real frequencies. Nevertheless, initial perturbations of the total vorticity integrated across the flow decay exponentially, resembling the behavior of an eigenmode with complex eigenfrequency ${\rm Im}\,\omega<0$ (Landau damping). However, the vorticity itself does not decay but becomes increasingly corrugated across the flow. We demonstrate that accounting for arbitrarily small viscosity transforms this exponentially decaying perturbation into a true eigenmode in which the vorticity preserves its spatial form. We numerically trace the transformation of the vorticity structure of this mode and its disappearance as viscosity approaches zero. We discuss similarities and differences between the behavior of damped perturbations in the transition from inviscid to nearly inviscid flows in hydrodynamics and their behavior in plasma and homogeneous stellar systems during the analogous transition from collisionless to very weakly collisional systems.