Presentation #301.08 in the session Ring Systems: Planetary Rosetta Stones.
In an unperturbed planetary ring, local gravitational instabilities are observed to produce transient elongated clumps of ring particles that are called self-gravity wakes (SGW). These SGWs generally have a finite pitch angle relative to the azimuthal direction, leading to gravitational torques that can significantly enhance the local angular momentum flux in the rings. In resonantly perturbed regions of planetary rings, spiral density waves (DW) are excited that can lead to strong periodic variations in the local surface density and mean flows. These periodic DW variations can lead to larger scale SGWs with pitch angles that vary with the phase of the DW, as has been observed in high-resolution Cassini images of nonlinear DWs. But do these SGWs produce a significant angular momentum flux that can alter the propagation of the DW? To address this question, a variational principle has been derived that includes both (1) a local model of the periodic variations caused by the DW, (2) the dynamics of SGWs, and (3) their mutual interactions. The advantage of the variational principle is that conservation laws are readily obtained that describe how angular momentum is exchanged between the DW and the SGWs. Our model borrows much of its theoretical formalism from the extensive theory of wave-mean-flow interactions in geophysical fluid dynamics. A novel addition to the theory includes the treatment of self-gravity due to SGWs with arbitrary pitch angle using a local plane-wave approximation described by Cook and Franklin (1964). As already pointed out by Maxwell (1859), the azimuthal component of self-gravity can trigger local instabilities when purely axisymmetric instabilities are inhibited. This is the basic reason that SGWs with large pitch angles are observed near the crests of strong DWs where the large radial pressure gradient inhibits axisymmetric instabilities.