Landau damping of disturbances in nearly inviscid inflectional shear flows

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 $ω$ 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 ${\rm Im}\,ω<0$ (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.