Forcing of slow density waves in the C ring by Saturn's quasi-toroidal normal modes

We calculate Saturn's quasi-toroidal normal modes of azimuthal order three for several candidate models for the interior stratification. We propose that these oscillations are responsible for exciting a class of slowly propagating spiral density waves observed in the C ring, possessing the same azimuthal symmetry and pattern speeds, that cluster near the rotation rate of the planet. Both prograde and retrograde propagation relative to the planetary rotation are observed. We find that we can construct interior models for which the pattern speeds of the normal modes align with those of a majority of the slow density waves. Excitation of the retrograde density waves re-quires retrograde toroidal modes with matching pattern speeds. We find that in order to support such modes, the interior model must include a layer of positive static stability in the molecular hydrogen envelope. The model that performs best places the peak of the static stability at a fractional radius of 0.85, near the radius of the semi -conductive layer where Saturn's zonal jets are inferred to decay. The properties of the toroidal modes are less sensitive to the stratification in the deep interior, hence our results neither confirm nor contradict the presence of a dilute core. When differential rotation is included in the form of zonal winds based on analysis of Cassini gravity and magnetic field data, we find that several toroidal modes become inertially unstable, raising the possibility that this instability is the principal mechanism for their excitation. However, while we are able to construct interior models that align the pattern speeds of a few unstable retrograde modes with those of prominent density waves in the ring, the same models fail with respect to the unstable prograde modes. Consequently, the significance of the instability mechanism remains an open question. Our results suggest that the interaction of Saturn's quasi-toroidal modes with the C ring provides an important source of new, complementary constraints on the planet's internal stratification and that further study of these modes is warranted.