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Inertial particle approximation to solutions of the Shallow Water Equations on the rotating spherical Earth
Nathan Paldor, Andrey Sigalov
Hebrew University of Jerusalem
nathan.paldor@huji.ac.il(Abstract received 05/03/2005 for session B)
ABSTRACT
The work estimates qualitative and quantitative relationships between solutions of two classical problems associated with the horizontal dynamics on the surface of the rotating spherical Earth. The first problem, where explicit expressions exist, is the mechanical problem of particle motion subject only to the gravitation force (called Inertial particle motion) and the second problem is the fluid dynamical problem, described by the Shallow Water Equations (SWE), where the relevant results can only be obtained by numerical integration of the nonlinear partial differential equations. Trajectories of fluid parcels advected by a time-dependent velocity field subject to the SWE on the sphere are computed numerically and compared to inertial particle trajectories. In addition the density (i.e. 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 zero gravity (i.e. g=0) case is considered a test case for the reliability of our numerical method for solving the SWE and we find that trajectories of fluid parcels generated by the SWE coincide with trajectories of inertial particles in this, g=0, limit. For g>0 agreement between corresponding solutions is guaranteed by continuity only for small gravity and for short times. Nevertheless, comparison between solutions of two systems shows very good qualitative as well as quantitative agreement for times of several inertial periods in the following elementary low-energy cases: inertial particle oscillations in mid-latitudes (corresponding to inertial waves in fluid dynamics) and divergent motion near the equator. Moreover, for realistic values of the reduced gravity and height (gH of 1 to 100 (m/s)^2) and for time interval of 1--2 days the periods of the trajectories of fluid parcels coincide with those of inertial particles. Our numerical calculations also show that the \beta-effect on inertial waves is very similar to its effect on particle motion: in both cases it causes a westward drift of fluid parcels or particles, respectively.
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2005 LAPCOD Meeting, Lerici, Italy, June 13-17, 2005