Rapid and extensive growth of sea ice cover was suggested to play a major role in the sea ice switch mechanism for the glacial cycles as well as on shorter millennial scales <Gildor and Tziperman, 2000>. This mechanism also predicts a hysteresis between sea ice and land ice, such that land ice grows when sea ice cover is small and withdraws when sea ice cover is more extensive. The switch-like sea ice growth and the hysteresis were previously demonstrated using a simple, highly idealized box model. In this work we demonstrate a switch-like sea ice behavior as well as the sea ice--land ice hysteresis using a coupled climate model that is continuous in the latitudinal dimension. It is shown that the switch-like sea ice growth occurs when the initial meridional atmospheric temperature gradient is not too strong. It is also shown that the meridional extent to which sea ice grows in a switch-like manner is not affected by the intensity of the thermohaline circulation, which does, however, influence the climate cooling that is needed to trigger such rapid sea ice growth.