The free surface of a fluid-filled dish assumes a parabolic shape when spun about its vertical axis. If filled with a material that hardens while spinning, the resulting parabolic dish provides a laboratory analog for free motion near the Earth's surface. This fact was pointed out by Satoshi Sakai and used to demonstrate inertial oscillations as part of the "Weather in a Tank" teaching module produced by MIT. These demonstrations use the fact that, for small curvatures and small-amplitude motion, a rotating parabolic dish can be approximated by an f-plane. However, the restriction to f-plane-like dynamics obscures the physics of the inertial oscillations. Further, allowing the possibility of large curvatures and amplitudes leads to significantly richer behavior.
Exact solutions for free motion on a parabolic dish are presented and their behavior examined in both the laboratory and co-rotating frames. These solutions have a different mathematical structure than those previously presented for free motion on a rotating planet because the intrinsic curvature of a paraboloid is opposite that of a sphere. The behavior of the exact solutions is compared to those obtained through the f-plane approximate. In particular, the exact solutions precess around the rotation axis, leading to the equivalent of a β-induced drift in the co-rotating frame.