Symmetric and nonsymmetric Holmboe instabilities in an inviscid flow

Abstract

Using linear stability analysis we studied the effect of displacing a thin density interface with respect to the center of the shear layer on the stability of an inviscid, stably stratified, parallel flow. When no interface displacement is present and the flow is unbounded, pure Holmboe instabilities exist at all bulk Richardson numbers and are the most unstable instabilities for values of the bulk Richardson number greater than 0.046. When the interface displacement is nonzero the two modes of a Holmboe instability split into a stronger and a weaker mode. As the height of the vertical domain decreases the roles of the two modes switch with the originally weaker mode becoming the stronger mode and vice versa. The importance of including the height of the vertical domain in the stability analysis was illustrated by comparing theoretical results with the field data of Yoshida et al. [Yoshida, Ohtani, Nishida, and Linden, in Physical Processes in Lakes and Oceans, edited by J. Imberger (American Geophysical Union, Washington, DC, 1998), pp. 389–400]. The assumption that the instabilities are initially two-dimensional is examined. When the flow is unbounded, both symmetric and nonsymmetric Holmboe instabilities are initially two-dimensional. When boundaries are included, the two-dimensional assumption is valid except when the total vertical domain is small in which case three-dimensional primary instabilities are possible.