The Dynamics of Heavy Gaseous Disks

Abstract

This paper seeks to develop a fuller understanding of the behavior of thin, polytropic, self-gravitating disks. Our approach consists of performing a detailed analysis of a representative model disk that is known to be susceptible to a single, rapidly growing, two-armed spiral mode. Numerical simulations of the evolution of this disk are presented; these new simulations have both an improved radial dynamic range and more realistic boundary conditions. This numerical work indicates that the primary unstable two-armed spiral mode is robustly endemic to the disk itself and is not a pathological product of artificially reflecting disk edges. The numerical simulations are then used to motivate (1) a linear modal analysis of both the representative disk and other related disks. In describing the linear analysis, we include a discussion of the relative merits of matrix and direct integration methods for solving the linear problem. The numerical simulations are also used to motivate (2) second-order and third-order perturbative analyses of the hydrodynamic governing equations, including the derivation of a higher order analogue to the energy integral constraint. We show that the development of mass and angular momentum transport through the disk is very well explained by a second-order nonlinear process involving self-interaction of the dominant (m = 2) linear mode. We then demonstrate that the saturation phenomena seen in earlier studies of this (and other) disks is the result of a third-order nonlinear effect. This definitive explanation of saturation provides a key for understanding the quasi-steady behavior that is often seen in self-gravitating disks.