Hydrodynamic stability and mode coupling in Keplerian flows: local strato-rotational analysis

¹ E. Kharadze Georgian National Astrophysical Observatory, 2a Kazbegi Ave., 0160 Tbilisi, Georgia e-mail: Jean-Paul.Zahn@obspm.fr
² LUTH, Observatoire de Paris, CNRS, Université Paris-Diderot, Place Jules Janssen, 92195 Meudon, France

Received: 31 July 2007
Accepted: 17 October 2007

Abstract

Aims.We present a qualitative analysis of key (but yet unappreciated) linear phenomena in stratified hydrodynamic Keplerian flows: (i) the occurrence of a vortex mode, as a consequence of strato-rotational balance, with its transient dynamics; (ii) the generation of spiral-density waves (also called inertia-gravity or waves) by the vortex mode through linear mode coupling in shear flows.

Methods.Non-modal analysis of linearized Boussinesq equations were written in the shearing sheet approximation of accretion disk flows.

Results.It is shown that the combined action of rotation and stratification introduces a new degree of freedom, vortex mode perturbation, which is in turn linearly coupled with the spiral-density waves. These two modes are jointly able to extract energy from the background flow, and they govern the disk dynamics in the small-scale range. The transient behavior of these modes is determined by the non-normality of the Keplerian shear flow. Tightly leading vortex mode perturbations undergo substantial transient growth, then, becoming trailing, inevitably generate trailing spiral-density waves by linear mode coupling. This course of events – transient growth plus coupling – is particularly pronounced for perturbation harmonics with comparable azimuthal and vertical scales, and it renders the energy dynamics similar to the 3D unbounded plane Couette flow case.

Conclusions.Our investigation strongly suggests that the so-called bypass concept of turbulence, which has been recently developed by the hydrodynamic community for spectrally stable shear flows, can also be applied to Keplerian disks. This conjecture may be confirmed by appropriate numerical simulations that take the vertical stratification and consequent mode coupling into account in the high Reynolds number regime.