Holmboe modes revisited

A scaling analysis is presented better identifying the conditions in which the Boussinesq approximation may be used to study shear disturbances like that of Holmboe modes. The classic Holmboe normal mode instability is then reanalyzed by including baroclinic effects whose introduction alters the onset of Holmboe's traveling-wave instability depending on the direction of the propagating modes. Since the introduction of baroclinicity is tantamount to relaxing the Boussinesq assumption, it means that in the presence of shear there is now a vertical variation of the horizontal momentum flux that alters the phase speed and structure of the classic Holmboe modes; the physical source of their broken right-left propagatory symmetry is associated with this physical effect. Furthermore, the regions of parameter space in which Holmboe's classic analysis predicts there to be nonpropagating double instabilities now supports propagating Holmboe modes when baroclinic effects are included. We also find that a globally constant shear profile behaves as a stabilizing agent, in contradiction to the destabilizing role that shear normally plays in the classic Kelvin-Helmholtz problem of a shear-density interface. The general relationship between the normal modes of this type of system to that of the continuous spectrum is also noted. We also find that the baroclinic effects explored here probably do not manifest in terrestrial oceanographic and laboratory conditions, although they may do so in atmospheres.