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The excitation of density and bending waves in Saturn's C ring by planetaryoscillation modes presents a unique opportunity to learn about gas giantinteriors and rotation. However, theoretical complications related to Saturn'srapid and differential rotation pose a barrier to the full utilization of ringwave detections. We calculate oscillation modes using a complete,non-perturbative treatment of differential rotation modelled after Saturn'szonal winds in self-consistently computed, polytropic equilibria. We find thatprevious, approximate treatments of the effects of differential rotation inSaturn overestimate shifts in the frequencies of fundamental modes (f-modes)thought to be responsible for the majority of the waves detected in the C ring,due to an omitted modification of the equilibrium shape and structure of theplanet by differential rotation. The bias introduced by these frequencyoverestimates is small, but significant relative to the uncertainties affordedby Cassini data. We additionally consider the non-perturbative effects ofSaturn-like differential rotation on the rotational mixing of f-modes andinternal gravity modes (g-modes), which is relevant to detections of multipledensity waves with very closely split pattern speeds. We find that higher orderrotational effects can produce orders-of-magnitude enhancements in the surfacegravitational perturbations of g-modes dominated by large spherical harmonicdegrees $\ell$, regardless of frequency separation from the sectoral f-mode.Despite this enhancement, we find that the observed fine-splitting of densitywaves is unlikely to involve g-modes dominated by $\ell\gtrsim 10$. Thisrestriction may aid in the inference of possible internal structures forSaturn.

Impacts of zonal winds on planetary oscillations and Saturn ring seismology