Millennium timescale Dansgaard-Oeschger oscillations of glacial climate, evidence for the occurrence of which was first provided on the basis of oxygen isotopic data from a Greenland ice-core 25 years ago, have recently been shown to arise naturally in a fully coupled climate model description of the interactions between the overturning circulation of the oceans, the atmosphere, and sea ice under near maximum glacial conditions. The fast transitions from cold stadial to warm interstadial conditions in a typical D-O oscillation are characterized by the opening of a 'super polynya' in the Irminger Sea basin south of Denmark Strait that arises due to the onset of intense vertical mixing below the stadial sea ice lid. Further analyses have suggested on qualitative grounds that the salinification of the surface mixed layer that occurs leading up to the onset of water column instability may be the mechanism that allows the instability to occur by sharply reducing the stabilizing salinity gradient across the pycnocline at the base of the mixed layer. Through detailed linear stability analysis of the water column beneath the region where the polynya first forms, the authors seek to understand whether the onset of intense sub-sea ice mixing is simply explicable and to characterize the operative mechanism. These analyses demonstrate that the transition is explicable in terms of the onset of a linear instability but that the evolving salinity structure deeper in the water column below the pycnocline is demonstrably more important than that across the pycnocline itself. This deeper source of instability is associated with the salt fingering mechanism rather than that associated with thermally driven convection.