Landau damping of disturbances in nearly inviscid inflectional shear flows

TL;DR

The paper investigates Landau damping in nearly inviscid inflectional shear flows, showing that even infinitesimal viscosity converts Landau quasi-modes into true eigenmodes with oscillatory resonance in the inner region. It develops a rigorous analytical framework using matched asymptotics and contour integrals to derive first-order viscous corrections, dispersion relations, and approximate eigenfunctions, complemented by a high-precision Legendre-basis numerical approach that validates the theory and tracks the EF evolution as $\mu\to0$. The findings reveal a distinctive intermediate-time dominance of the Landau mode followed by final decay governed by slower continuum modes, and they establish quantitative predictions for the resonance width, period, and amplitude growth. The work draws deep parallels with collisionless-to-weakly-collisional transitions in plasmas and stellar systems, while highlighting unique features of shear-flow spectra and posing open questions about quasi-mode survival across velocity profiles and Reynolds numbers.

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 $ω$ 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.

Figures (6)

• Figure 1: Stable spectrum of eigenmodes of nearly inviscid free mixing layer $U=\tanh y$ for $k=1.05$, $\mu\equiv (k\,{\rm Re})^{-1} =10^{-4}$, obtained numerically for $N_{\rm max}=300, 1000$ and 5000. The single blue point above the triple point represents the Landau mode, which remains essentially unchanged across all $N_{\rm max}$ values (points overlaid). In purely inviscid flow the spectrum consists of van Kampen modes that cover the interval $-1\le c \le 1$ on the real $c$-axis (not marked in the figure), which generate the Landau quasi-mode.
• Figure 2: The ratio $\Delta\sigma(k,\mu)/\mu\equiv [\sigma(k,\mu)-\sigma_{\rm L}(k)]/\mu$, where $\sigma_{\rm L}(k)\equiv \gamma_{\rm L}(k)/k$ and $\gamma_{\rm L}(k)$ is Landau damping rate, as a function of wavenumber $k$, computed using Eq.(\ref{['eq:Delta_sigma']}) with contour deformation into the lower half-plane. The red plus marks the analytical result at $k=1$; blue circles show inviscid limit values given in Table \ref{['tab1']}.
• Figure 3: Modulus of expansion coefficients $|\bar{A}_n|$, $n \geq 1$, for the Landau quasi-mode with $k=1.05$ at four $\mu$ values. $\bar{A}_0 = 1$ for all $\mu$. $|\bar{A}_n|$ peaks at $n_* \approx (\sigma/\mu)^{1/2}$ for very low $\mu$.
• Figure 4: Least-damped mode EFs for varying $\mu$ and $k$ values, comparing numerical solutions (solid) with analytical approximations (dashed) and nonresonant solutions (dotted). Left: EF deformation for $k=1.05$ as $\mu$ decreases (solid: numerical solutions from (\ref{['eq:A2zeta']}), dashed: analytical approximation (\ref{['eq:zeta_inner']})). Right: Resonance range width dependence on $\sigma$ for three $k$ values, supporting our analytical result (\ref{['eq:hw']}). EFs are plotted using symlog scaling (logarithmic with linear scaling near zero, indicated by gray shading).
• Figure 5: Evolution of total vorticity across the flow, ${\cal N}(t)$, shown in symlog format. Parameters: $k=1.05$, $\mu=10^{-4}$, $\gamma\equiv k\,\sigma \approx 0.0335$, $N_{\rm max}=1000$. The oscillation period $T_{\rm t}$ at $t\gtrsim 350$ is equal to 5.98.