The local instability of steady astrophysical flows with non circular streamlines with application to differentially rotating disks with free eccentricity

We carry out a general study of the stability of astrophysical flows that appear steady in a uniformly rotating frame. Such a flow might correspond to a stellar pulsation mode or an accretion disk with a free global distortion giving it finite eccentricity. We consider perturbations arbitrarily localized in the neighbourhood of unperturbed fluid streamlines. When conditions do not vary around them, perturbations take the form of oscillatory inertial or gravity modes. However, when conditions do vary so that a circulating fluid element is subject to periodic variations, parametric instability may occur. For nearly circular streamlines, the dense spectra associated with inertial or gravity modes ensure that resonance conditions can always be satisfied when twice the period of circulation round a streamline falls within. We apply our formalism to a differentially rotating disk for which the streamlines are Keplerian ellipses, with free eccentricity up to 0.7, which do not precess in an inertial frame. We show that for small $e,$ the instability involves parametric excitation of two modes with azimuthal mode number differing by unity in magnitude which have a period of twice the period of variation as viewed from a circulating unperturbed fluid element. Instability persists over a widening range of wave numbers with increasing growth rates for larger eccentricities. The nonlinear outcome is studied in a follow up paper which indicates development of small scale subsonic turbulence