The ocean is the largest reservoir of carbon in the climate system. Atmospheric $CO_2$ is efficiently transferred to the deep ocean by a process called the biological carbon pump: photosynthetic fixation of $CO_2$ at the sea surface and remineralization of sinking organic carbon at depths are main causes for the vertical contrast of carbon in the ocean. The sequestered carbon to the deep ocean returns to the sea surface by ocean circulation. Part of the upwelled $CO_2$ leaks into the atmosphere through air-sea gas exchange. It has been suggested that the air-sea partitioning of carbon has varied in concert with the glacial-interglacial climate variations, due partly to changes in ocean circulation. In this review paper, we briefly summarize key concepts of the oceanic carbon pump. We also discuss the response of the air-sea carbon partitioning to change in ocean circulation in the context of the glacial-interglacial climate change.