On the application of asymptotic analysis to the dynamical theory of the pole tide

Summary. This article examines the effects of boundaries on the pole tide in an ocean of constant depth. The cyclically continuous global ocean solutions to Laplace's tidal equations with the pole tide forcing are used as the particular solutions to the problem. The approach here is to find approximate asymptotic solutions to the homogeneous tidal equations which can be added to the global particular solutions so that the normal component of velocity will vanish at the boundary. At the very long period of the pole tide, the unforced motions are assumed to be non-divergent, and so only the homogeneous vorticity equation must be solved.
The first case considered is a zonal ocean bounded by parallels of latitude equidistant from the equator. Asymptotic solutions are found in order to satisfy the zonal boundary condition, and this gives rise to a narrow zonal boundary current. The contribution of these solutions is exponentially small compared to the forced global pole tide except in the immediate vicinity of the northern and southern boundaries.
Next, the effect of meridional boundaries is considered. When a linear form of bottom friction is assumed, two approximate homogeneous solutions are found to construct a general solution that satisfies the meridional boundary conditions. One solution decays exponentially in longitude and gives rise to a western boundary current, while the other solution is independent of longitude. The meridional boundary conditions are used to match the homogeneous and particular solutions, and so the solution for the interior of the ocean satisfies the eastern boundary condition. The resulting solution for the pole tide has a western boundary current term, while over the majority of the ocean domain the solution has a term varying with the wavelength of the forcing (the global solution) and a zonal motion term used to satisfy the eastern boundary condition. Comparisons are made with the wind-driven ocean circulation problem.