Inertial particle approximation to solutions of the shallow water equations on the rotating spherical Earth

The present study assesses the relationship between solutions of the mechanical problem of particle motion on the surface of a rotating sphere subject only to the gravitation force (called inertial particle motion) and the fluid dynamical problem there, described by the shallow water equations (SWE). Trajectories of fluid parcels advected by a time-dependent velocity field subject to the SWE on the sphere are computed numerically and compared to analytical formulae of inertial particle trajectories. In addition, the free surface height of an ensemble of non-interacting particles is estimated within the classical mechanics framework and compared to computed height of the SWE. The comparison between solutions of the two systems shows very good qualitative as well as quantitative agreement for times of several inertial periods in the following basic low-energy cases: inertial particle oscillations in mid-latitudes (corresponding to inertial waves in fluid dynamics) and inertial motion near the equator. Moreover, for realistic values of the reduced gravity (gH of 1 to 100 m² s^-2) and for time interval of 1-2 d the periods of the trajectories of fluid parcels nearly coincide with those of inertial particles. These results are obtained for a wide range of initial velocity fields and they imply that, at least for time intervals considered, the Coriolis force dominates the motion even after the pressure gradient forces become sufficiently large to affect the motion. They also highlight the fact that fluid parcels of non-linear inertialwaves are subject to the same westward drift as inertial particles and provide an explanation for existence of so called 'inertial peak' in the internal oceanic wave spectrum.