A physical mechanism is proposed for the glacial-interglacial variations in the rate of vertical mixing and in sea ice cover in the Southern Ocean. Such variations were postulated without an explanation by Toggweiler <1999> and Stephens and Keeling <2000>, who used them to explain the glacial-interglacial CO2 variations. In the physical mechanism explored here, initially given by Gildor and Tziperman <2001b>, changes in the stratification of the Southern Ocean due to the cooling of North Atlantic Deep Water (NADW) during glacial maxima reduce the rate of vertical mixing of the surface water with the deep water. The changed temperature of the NADW arriving to the Southern Ocean and the reduced vertical mixing there also increase the Southern Ocean sea ice cover during glacial maxima. These vertical mixing and sea ice cover changes are shown to be a natural result of the sea ice switch mechanism of the glacial cycles. A box model of the climate system is used to demonstrate the above physical mechanism and its effect on the atmospheric CO2. Because of the uncertainties in the exact dependence of the vertical mixing on vertical stratification, it is impossible to quantify the exact separate contribution of reduced vertical mixing and larger sea ice cover to the glacial CO2 variations. The CO2 variations are not essential to the existence of the glacial cycles in the sea ice switch mechanism, yet they are shown to amplify the glacial-interglacial variability, consistent with the view of the role of CO2 deduced from proxy observations.