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This chapter reviews the fundamental governing equations for atmosphere and ocean dynamics. This material lays the groundwork for our later examination of coupled atmosphere–ocean variability. The equations of motion are derived first for a Cartesian coordinate system, and then for a rotating spherical coordinate system. The scales of the terms of the momentum equations are analyzed, and a number of simplifications are developed for later use (including the f-plane and β-plane coordinate systems, the hydrostatic approximation for the atmosphere and the Boussinesq approximation for the ocean, and geostrophic balance and the thermal wind).