Essence of inviscid shear instability: a point view of vortex dynamics

Abstract

The essence of shear instability is fully revealed both mathematically and physically. A general sufficient and necessary stable criterion is obtained analytically within linear context. It is the analogue of Kelvin-Arnol'd theorem, i.e., the stable flow minimizes the kinetic energy associated with vorticity. Then the mechanism of shear instability is explored by combining the mechanisms of both Kelvin-Helmholtz instability (K-H instability) and resonance of waves. It requires both concentrated vortex and resonant waves for the instability. The waves, which have same phase speed with the concentrated vortex, have interactions with the vortex to trigger the instability. We call this mechanism as "concentrated vortex instability". The physical explanation of shear instability is also sketched. Finally, some useful criteria are derived from the theorem. These results would intrigue future works to investigate the other hydrodynamic instabilities.