There are three fundamental features which characterize large glacial millennial (Dansgaard-Oeschger) oscillations: (1) the climatic transitions were abrupt and large; (2) the lengths of both interstadials and stadials and the period of Dansgaard-Oeschger oscillations were not uniform; and (3) there were no large millennial oscillations during an early stage of a glacial period and a peak glacial period. In this modeling study we offer a consistent explanation for these three features by employing an Earth system Model of Intermediate Complexity. We demonstrate that a moderate global cooling forces the Atlantic meridional overturning circulation (MOC) into an unstable state and hence causes the flip-flop of the Atlantic MOC between a strong mode and a weak mode. The durations of both interstadials and stadials associated with these millennial oscillations are modulated by the changing background climate in qualitative agreement with the observations. In a warm climate the Atlantic MOC is strong and stable, with the deep water formed mainly by intense heat loss to the atmosphere. In a cold climate the Atlantic MOC is weak and stable, and this mode is largely maintained by the process of sea ice brine rejection. Since the Dansgaard-Oeschger oscillations result from an alternation between these two MOC states during an intermediate phase climate, we conclude that brine rejection plays a necessary role in the oscillations, confirming a hypothesis suggested in some proxy data studies.