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We carry out a general study of the stability of astrophysical flows thatappear steady in a uniformly rotating frame. Such a flow might correspond to astellar pulsation mode or an accretion disk with a free global distortiongiving it finite eccentricity. We consider perturbations arbitrarily localizedin the neighbourhood of unperturbed fluid streamlines.When conditions do notvary around them, perturbations take the form of oscillatory inertial orgravity modes. However, when conditions do vary so that a circulating fluidelement is subject to periodic variations, parametric instability may occur.For nearly circular streamlines, the dense spectra associated with inertial orgravity modes ensure that resonance conditions can always be satisfied whentwice the period of circulation round a streamline falls within. We apply ourformalism to a differentially rotating disk for which the streamlines areKeplerian ellipses, with free eccentricity up to 0.7, which do not precess inan inertial frame. We show that for small $e,$ the instability involvesparametric excitation of two modes with azimuthal mode number differing byunity in magnitude which have a period of twice the period of variation asviewed from a circulating unperturbed fluid element. Instability persists overa widening range of wave numbers with increasing growth rates for largereccentricities. The nonlinear outcome is studied in a follow up paper whichindicates development of small scale subsonic turbulence.Comment: Accepted for publication in Astronomy and Astrophysic

astronomy and astrophysics

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