Global Aspects of Elliptical Instability in Tidally Distorted Accretion Disks

Abstract

Tidally distorted accretion disks in binary star systems are subject to a local hydrodynamic instability which excites $m=1$ internal waves. This instability is three dimensional and approximately incompressible. We study the global aspects of this local instability using equations derived under the shearing sheet approximation, where the effects of the azimuthal variation along distorted orbital trajectories are included in source terms which oscillate with local orbital phase. Linear analyses show that the excitation of the instability is essentially local, i.e. insensitive to radial boundary conditions. The region of rapid growth feeds waves into the region of slow or negligible growth, allowing the instability to become global. The global growth rate depends the maximum local growth rate, the size of the rapid growth region, and the local group velocity. We present an empirical expression for the global growth rate. We note that the local nature of the instability allows the excitation of waves with $m\ne 1$ when the local growth rate is large.