Approximate particular solutions for the pole tide in a global ocean

Approximate particular solutions are presented for Laplace's tidal equations for the pole tide forcing on an ocean-covered earth. The system of equations only needs to satisfy cyclic continuity and is reduced to separate equations for the meridional velocity and the deviation of the tide from equilibrium. These two equations are characterized by a small parameter multiplying the derivatives and so approximate particular solutions are obtained by equating the undifferentiated terms with the forcing terms. With this approximation, linear friction does not introduce a phase shift in the solutions. There is a near geostrophic balance between the horizontal velocities and the small deviation of the tide from equilibrium in the momentum equations, while to the first approximation the horizontal velocity divergence in the continuity equation yields the equilibrium tide. The horizontal velocities are expressed in terms of a stream function and velocity potential, which are shown to satisfy the vorticity and divergence equations. These results show that for the long-period pole tide forcing (a velocity potential forcing in the divergence equation), both a stream function and velocity potential are necessary to describe the motion.