The dynamics of the Arctic ocean are studied on a polar projection of the sphere. The density structure is idealized as a two-layer system, and a general formulation is developed which allows inclusion of latitudinal and longitudinal depth variations as well as asymmetries in the boundaries of the ocean. For simplicity, the density structure is neglected when depth variations are present. Time dependent displacements from equilibrium levels are assumed to be waves of constant zonal wave number; no radial propagation is considered. Amplitude equations are derived for these displacements, subject to the assumption that the polar basin is small enough to keep only a first approximation to the curvature of the Earth. A semi-qualitative investigation of the possible solutions is made in the case of a symmetrical basin, using the Method of Signatures, and existence criteria are found for the solutions in the presence of radial depth variations. Concentrating thereafter on planetary waves, explicit solution for such motions in the simplest case (depth constant, symmetrical boundaries) allows comparison with the results of other investigators (Longuet-Higgins, 1964 b; Goldsbrough, 1914 a) . It is found that the polar projection and first approximation to the curvature give quite good results, so that this method may be applied to polar regions in the same way as the β-plane is used in mid-latitudes. The general effects of radial bottom slopes are discussed and a simple example treated more explicitly. Some theorems of Ball (1963) on the motions of shallow rotating fluids in paraboloidal basins are found to hold for such basins in the polar plane approximation to the sphere. / Science, Faculty of / Physics and Astronomy, Department of / Graduate