Potential vorticity dynamics in the framework of disk shallow-water theory

II. Mixed barotropic-baroclinic instability

School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK School of Natural Sciences, University of California Merced, Merced, CA 95343, USA Astronomy Deplartment, City College of San Francisco, San Francisco, CA 94112, USA
e-mail: oumurhan@ucmerced.edu

Received: 10 January 2012
Accepted: 11 April 2012

Abstract

Context. We consider potential vorticity dynamics for astrophysical disks.

Aims. The aim of this work is to extend exploration of shear instabilities in cold astrophysical disks that are three-dimensional and whose mean states are baroclinic. In particular, we seek to demonstrate the potential existence of traditional baroclinic instabilities in meteorological studies. We show this in a simplified two-layer Philips disk model of a midplane symmetric disk.

Methods. We analyze the dynamical normal-mode response of thin annular disks with two strongly localized potential vorticity gradients, one in each of two disk layers. Each disk layer is of constant but differing densities. The resulting mean azimuthal velocity profile shows a variation in the vertical direction, implying that the system is baroclinic in the mean state. The stability of the system is treated in the context of disk shallow-water theory, wherein azimuthal disturbances are much longer than the corresponding radial or vertical scales. The normal-mode problem is solved numerically using two different methods.

Results. The results of a symmetric single-layer barotropic model is considered and it is found that instability persists for models in which the potential vorticity profiles are not symmetric, consistent with previous results. The instaiblity is interpreted in terms of interacting Rossby waves. For a two-layer system, in which the flow is fundamentally baroclinic, we report here that instability takes on the form of mixed barotropic-baroclinic type: instability occurs, but it qualitatively follows the pattern of instability found in the barotropic models. Instability arises because of the phase locking and interaction of the Rossby waves between the two layers. The strength of the instability weakens as the density contrast between layers increases.

Conclusions. Baroclinic instability is feasible for astrophysical disks but has the character of mixed barotropic-baroclinic type. The instability as explored in this study could be present in protoplanetary disks with weak vertical density stratification.