Energy principles for self-gravitating barotropic flows - II. The stability of Maclaurin discs

Abstract

We analyze stability conditions of "Maclaurin flows" (self-gravitating, barotropic, two dimensional, stationary streams moving in closed loops around a point) by minimizing their energy, subject to fixing all the constants of the motion including mass and circulations.Necessary and sufficient conditions of stability are obtained when gyroscopic terms in the perturbed Lagrangian are zero. To illustrate and check the properties of this new energy principle, we have calculated the stability limits of an ordinary Maclaurin disk whose dynamical stability limits are known. Perturbations are in the plane of the disk. We find all necessary and sufficient conditions of stability for single mode symmetrical or antisymmetrical perturbations. The limits of stability are identical with those given by a dynamical analysis. Regarding mixed types of perturbations the maximally constrained energy principle give for some the necessary and sufficient condition of stability, for others only sufficient conditions of stability. The application of the new energy principle to Maclaurin disks shows the method to be as powerful as the method of dynamical perturbations.

Noether currents for Eulerian variational principles in non-barotropic magnetohydrodynamics and topological conservations laws

“…1988; Katz et al. 1993; Yahalom, Katz & Inagaki 1994; Vladimirov, Moffatt & Ilin 1996, 1997, 1999; Yahalom 2011). The theory can also be used to study the evolution of waves with respect to a given MHD configuration, such approach was used by Webb et al.…”

Simplified variational principles for non-barotropic magnetohydrodynamics

“…I anticipate applications of this study both to linear and non-linear stability analysis of known barotropic magnetohydrodynamic configurations [20,21,22]. I suspect that for achieving this we will need to add additional constants of motion constraints to the action as was done by [23,24] see also [25,26,27]. As for designing efficient numerical schemes for integrating the equations of fluid dynamics and magnetohydrodynamics one may follow the approach described in [28,29,30,31].…”

Variational principles for topological barotropic fluid dynamics

“…We have also given a formula for the flow helicity in terms of our variational variables and have shown that our variable do not entail trivial helicity and that not only global helicity is conserved but also helicity per unit of vortex flux. The variational principles can clearly be used to obtain new flows using both numerical and analytical techniques and to study their stability as was done with other variational methods in Yahalom (2011), Ophir et al (2012), Yahalom (2013), Arnold (1965a), Arnold (1965b), Katz et al (1993) and Yahalom et al (1994). The reduction in the number of dynamical variables should facilitate such calculations.…”