A shallow-water theory for annular sections of Keplerian disks

¹ Astronomy Unit, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK e-mail: mumurhan@ccsf.edu
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

Received: 3 February 2008
Accepted: 21 July 2008

Abstract

Context. We present a scaling argument that we develop into a shallow water theory of non-axisymmetric disturbances in annular sections of thin Keplerian disks.

Aims. We develop a theoretical approach to understand physically the relationship between two-dimensional vortex dynamics that is known and their three-dimensional counterparts in Keplerian disks.

Methods. Using asymptotic scaling arguments, varicose disturbances of a Keplerian disk are considered on radial and vertical scales consistent with the height of the disk while the azimuthal scales are the full 2π angular extent of the disk. For simplicity perturbations are assumed to be homentropic according to a polytropic equation of state. The timescales considered are long compared to the local disk rotation time.

Results. The scalings relate to dynamics that are radially geostrophic and vertically hydrostatic. A potential vorticity quantity emerges and is shown to be conserved in a Lagrangian sense. Uniform potential vorticity linear solutions are explored and the theory is shown to contain an incarnation of the strato-rotational instability under channel flow conditions. Linearized solutions of a single defect on an infinite domain are developed and shown to support a propagating Rossby edgewave. Linear non-uniform potential vorticity solutions are also developed and shown to be similar in some respects to the dynamics of strictly two-dimensional inviscid flows. The relationship of the scalings and some of the resulting dynamics are considered with respect to other approximations employed in the literature. Based on the framework of this theory, arguments based on geophysical notions are presented to support the assertion that the strato-rotational instability is in a generic class of barotropic/baroclinic potential vorticity instabilities. Extensions of this formalism are also proposed.

Conclusions. The shallow water formulation achieved by the asymptotic theory developed here opens a new approach in studying disk dynamics.