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.Comment: Accepted for publication in Astronomy and Astrophysic