Stability in the weak variational principle of barotropic flows and implications for self-gravitating discs

Abstract

In this study, stability conditions of self-gravitating disc models are obtained. The self-gravitating disc models under study include known models such as the Maclaurin disc and the infinite, self-gravitating, rotating sheet. These models also include a new class of analytically solvable models denoted by ‘generalized Maclaurin discs’. These self-gravitating, finite discs are differentially rotating with adiabatic index γ > 2 and have the property that the derivatives of densities go smoothly to zero at the boundary. Stability conditions of the various models are obtained through the ‘weak energy principle’ introduced by Katz, Inagaki & Yahalom. It is shown that necessary and sufficient conditions of stability are obtained when we have only pair coupling in the gyroscopic terms of the perturbed Lagrangian; otherwise, the ‘weak energy principle’ gives only sufficient conditions. All perturbations considered are in the same plane as the configurations. For differentially rotating discs, we consider only radial perturbations. The limits of stability are identical with those given by a dynamical analysis when available, and with the results of the strong energy principle analysis when given. Thus, although the ‘weak energy’ method is mathematically more simple than the ‘strong energy’ method of Katz et al., since it does not involve solving second-order partial differential equations, it is by no means less effective. Additional results also derived through the ‘weak energy principle’ include stability conditions for the 2D Rayleigh flows and Toomre’s local criterion for the stability of rotating discs. Among the most interesting results is an exact extension of Toomre’s criterion to the global stability of generalized Maclaurin discs, whereby a necessary condition for local stability becomes a sufficient condition for global stability.