  Normal mode oscillations in Saturn excite density and bending waves in the C
Ring, providing a valuable window into the planet's interior. Saturn's
fundamental modes (f modes) excite the majority of the observed waves, while
gravito-inertial modes (rotationally modified g modes) associated with stable
stratification in the deep interior provide a compelling explanation for
additional density waves with low azimuthal wavenumbers m. However, multiplets
of density waves with nearly degenerate frequencies, including an m=3 triplet,
still lack a definitive explanation. We investigate the effects of rapid and
differential rotation on Saturn's oscillations, calculating normal modes for
independently constrained interior models. We use a non-perturbative treatment
of rotation that captures the full effects of the Coriolis and centrifugal
forces, and consequently the mixing of sectoral f modes with g modes
characterized by very different spherical harmonic degrees. Realistic profiles
for differential rotation associated with Saturn's zonal winds can enhance
these mode interactions, producing detectable oscillations with frequencies
separated by less than 1%. Our calculations demonstrate that a three-mode
interaction involving an f mode and two g modes can feasibly explain the finely
split m=3 triplet, although the fine-tuning required to produce such an
interaction generally worsens agreement with seismological constraints provided
by m=2 density waves. Our calculations additionally demonstrate that sectoral f
mode frequencies are measurably sensitive to differential rotation in Saturn's
convective envelope. Finally, we find that including realistic equatorial
antisymmetry in Saturn's differential rotation profile couples modes with even
and odd equatorial parity, producing oscillations that could in principle
excite both density and bending waves simultaneously.
