Spatial versus temporal instabilities in a parametrically forced stratified mixing layer

Abstract

A spatial instability of parametrically excited stratified mixing layer flows is considered together with the related temporal instability problem. A relatively simple iteration procedure yielding solutions of both temporal and spatial problems is proposed. Using this procedure a parametric analysis of the temporal and spatial Kelvin–Helmholtz and Holmboe instabilities is performed and characteristic features of the instabilities are compared. Both inviscid and viscous models are considered. The parametric dependence on the mixing layer thickness and on the Richardson and Reynolds numbers is studied. It is shown that in the framework of this study the Gaster transformation is valid for the Kelvin–Helmholtz instability, but cannot be applied to the Holmboe one. The neutral stability curves are calculated for the viscous flow case. It is found that the transition between Kelvin–Helmholtz and Holmboe instabilities is continuous in the spatial case and in the temporal case occurs via the codimension-two bifurcation at which a complex pair of the leading eigenvalues merges into a multiple real eigenvalue. It is also found that for the same governing parameters the spatial upstream and downstream Holmboe waves have different amplification rates and different absolute phase velocities, with larger difference observed at larger Richardson numbers. It is shown that at large Richardson and small Reynolds numbers the primary temporal and spatial instabilities set in as a three-dimensional oblique Holmboe wave.