Nonlinear Generation of One-armed Spirals in Self-gravitating Disks

Single-armed spiral instabilities may play an important role in self-gravitating disks. However, past studies have indicated that disks which are less massive than an embedded central star are in general linearly stable to m = 1 perturbations. In this paper, we outline how the nonlinear interaction of two growing modes with azimuthal mode numbers m = n and m = n + 1 can excite the development of a one-armed disturbance which grows exponentially at a rate equal to the sum of the growth rates of the two linearly unstable modes. Within the context of a second order, semianalytic analysis, we discuss the special case of a unique nonwinding, exponentially growing solution, and then we consider further solutions based on physically relevant initial conditions. We find that the nonwinding solution describes the general dynamics of the mode at later times. We then compare the results of our nonlinear analysis with hydrodynamic simulations, illustrating that the process we have considered is indeed an important effect.