Linear dynamics of weakly viscous accretion disks: a disk analog of Tollmien-Schlichting waves

¹ Astronomy Unit, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK e-mail: o.umurhan@qmul.ac.uk
² Department of Geophysics and Space Sciences, Tel-Aviv University, Tel-Aviv, Israel
³ Astronomy Department, City College of San Francisco, San Francisco, CA 94112, USA
⁴ Department of Physics, Technion-Israel Institute of Technology, 32 000 Haifa, Israel e-mail: gioras@physics.technion.ac.il
⁵ Institute of Theoretical Astrophysics, University of Heidelberg, 69120 Heidelberg, Germany

Received: 19 February 2008
Accepted: 14 November 2008

Abstract

This paper discusses new perspectives and approaches to the problem of disk dynamics where, in this study, we focus on the effects of the viscous instabilities influenced by boundary effects. The Boussinesq approximation of the viscous large shearing box equations is analyzed in which the azimuthal length scale of the disturbance is much larger than the radial and vertical scales. We examine the stability of a non-axisymmetric potential vorticity mode, i.e. a PV-anomaly. in a configuration in which buoyant convection and the strato-rotational instability do not to operate. We consider a series of boundary conditions that show the PV-anomaly to be unstable both on finite and semi-infinite radial domains. We find these conditions lead to an instability that is the disk analog of Tollmien-Schlichting waves. When the viscosity is weak, evidence of the instability is most pronounced by the emergence of a vortex sheet at the critical layer located away from the boundary where the instability is generated. For some boundary conditions, a necessary criterion for the onset of instability for vertical wavelengths that are a sizable fraction of the layer's thickness and when the viscosity is low is that the appropriate Froude number of the flow be greater than one. This instability persists if more realistic boundary conditions are applied, although the criterion on the Froude number is more complicated. The unstable waves studied here share qualitative features to the instability seen in rotating Blasius boundary layers. The implications of these results are discussed. An overall new strategy for exploring and interpreting disk instability mechanisms is also suggested.