The Cassini Ultraviolet Imaging Spectrograph observed numerous, opaque, icicle-shaped features in stellar occulations by Saturn’s F ring (Esposito et al. 2008; Meinke et al. 2012). Gravitational instabilities that rapidly form transient km-scale opaque clusters of ring particles have been offered as the most likely explanation for these features, but a theoretical model of the formation process has been lacking up to now. The dynamical problem is complicated by the strong gravitational perturbations that the two nearby shepherding satellites, Pandora and Prometheus, exert on the F ring. In particular, the shepherding satellites excite eccentric streamlines of ring particles that evolve into high-density satellite wakes that have been described both analytically (Stewart 1991) and by high-resolution N-body simulations (Lewis and Stewart 2005, 2011). Here I describe a new theoretical model that can predict local gravitational instabilities inside nonlinear satellite wakes like those seen in the F ring. Starting from a Hamiltonian variational principle for gravitational instabilities in planetary rings (Stewart 2019 EPSC-DPS), the analytical formulation of a satellite wake reported by Stewart (1991) is used as the initial state of the system. By taking advantage of the separation in scales between the ~ 1000 km wavelength of the satellite wake and the km-scale of the local instability, the variational principle can be approximated asymptotically to yield an approximate solution for transient instabilities inside the satellite wake structure of the F ring. This process is also likely to occur on the edges of the Keeler Gap in Saturn’s A ring where satellite wakes are excited by the gap moon, Daphnis.