Effects of the earth's curvature on the dynamics of isolated objects. PartI: The disk

A disk over the frictionless surface of the earth shows an interaction betw een the center of mass and internal motions. At low energies, the former is an "inertial oscillation" superimposed to a uniform zonal drift c and the latter is a rotation with variable vertical angular velocity omega (as meas ured by a terrestrial observer). The dynamics is understood best in a stereographic frame following the secu lar drift. The center of mass has a circular but not uniform motion; its me ridional displacement induces the variations of the orbital and internal ro tation rates. On the other hand, the temporal mean of the Coriolis forces d ue to both rotations produces the secular drift. In spherical terrestrial coordinates geometric distortion complicates the d escription. For instance, the zonal velocity of the center of mass U is not equal to the average zonal component of the particle velocities (u), as a result of the earth's curvature. The drift c and the temporal means (U) ove r bar and ((u)) over bar are all three different. In addition, omega differ s from the local vertical angular velocity sigma (as measured by an observe r following the disk). The classical "beta plane" approximation predicts co rrectly the value of c but makes order-one errors in everything else (e.g., it makes (U) over bar = ((u)) over bar = c and omega = sigma). The results of this paper set up the basis to study curvature effects on an isolated vortex. This, more difficult, problem is discussed in Parr II.