Note on the Kelvin–Helmholtz instability of stratified fluids

Abstract

In this paper the conditions under which the Kelvin–Helmholtz instability in stratified fluids is absolute or convective are investigated. It is first shown that at transition from convective to absolute instability two double roots of the dispersion relation coalesce. Based on this property an analytical condition for absolute instability is derived. It is found that the instability is absolute for almost all values of the flow velocity and stratification. Convective instability is found only for a narrow range of flow velocities over the instability threshold when the density ratio parameter, r=ρ1/(ρ1+ρ2), lies in the range 1/3<r<1/2. The different behavior near the instability threshold can be related to the signs of the group velocities of the two waves which coalesce to create the instability: For r<1/3, the group velocities of the two waves have opposite signs, and the resulting instability is absolute, whereas for 1/3<r<1/2, the two waves have group velocities with the same sign, and the instability is convective. This result is also shown to be reflected in the form of the amplitude equation at the instability threshold, which is the linearly unstable Klein–Gordon equation.