Damped perturbations in inviscid shear flows: Non-resonant Landau damping in stable inflectional flows

Abstract

We analyze the dynamics of small two-dimensional disturbances in stable plane-parallel inviscid shear flows under linear theory. Using a velocity profile Vx=U(y) with an inflection point but stable according to Fjørtoft's theorem, we illustrate that the continuum spectrum of van Kampen modes, possessing real phase velocities c=ω/k, aggregates into Landau damping solutions or “quasi-modes,” which exhibit exponential decay. It was found that the real part of the complex phase velocity cL(k) of these solutions may lie outside the allowable range for van Kampen modes, suggesting a non-resonant damping mechanism for these quasi-modes. This conclusion was reached by solving the eigenvalue problem and observing the evolution of initial perturbations, calculated by directly solving the evolutionary equation for vorticity as well as by decomposing the initial disturbance into van Kampen modes. Landau damping of the total vorticity across the channel emerges as an intermediate stage before transitioning to power-law damping.