Optimal growth of small perturbations in thin gaseous disks

A thin gaseous disk with a nearly Keplerian rotation profile and free boundaries in the external gravitational field of a point gravitating object does not generate any growing perturbation eigenmodes. In spite of this, a significant transient growth of linear perturbations measured by the evolution of their total acoustic energy is possible in such a disk. This is shown within the framework of the simplest model of an inviscid polytropic thin disk with a finite radial extent in which small adiabatic perturbations that are a linear combination of neutral eigenmodes with a corotation radius beyond the outer flow boundary are considered.