The local instability of steady astrophysical flows with non circular streamlines with application to differentially rotating disks with free eccentricity
Astronomy Unit, Queen Mary, University of London, Mile End Rd, London E1 4NS, UK e-mail: jcbp@maths.qmw.ac.uk
Received:
3
September
2004
Accepted:
22
November
2004
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 
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.
Key words: accretion, accretion disks / instabilities / hydrodynamics / celestial mechanics / planets and satellites: formation / stars: oscillations
© ESO, 2005