Shear instabilities in a fully compressible polytropic atmosphere

¹ Department of Mathematics, City University London, Northampton Square, London, EC1V 0HB, UK
e-mail: Veronika.Witzke.1@city.ac.uk; Lara.Silvers.1@city.ac.uk
² Aix-Marseille Université, CNRS, École Centrale Marseille, IPHE UMR 7342, 49 rue F. Joliot-Curie, 13013 Marseille, France
e-mail: Favier@irphe.univ-mrs.fr

Received: 5 November 2014
Accepted: 8 March 2015

Abstract

Shear flows have a significant impact on the dynamics in an assortment of different astrophysical objects, including accretion discs and stellar interiors. Investigating shear flow instabilities in a polytropic atmosphere provides a fundamental understanding of the motion in stellar interiors where turbulent motions, mixing processes, and magnetic field generation take place. Here, a linear stability analysis for a fully compressible fluid in a two-dimensional Cartesian geometry is carried out. Our study focuses on determining the critical Richardson number for different Mach numbers and the destabilising effects of high thermal diffusion. We find that there is a deviation in the predicted stability threshold for moderate Mach number flows, along with a significant effect on the growth rate of the linear instability for small Péclet numbers. We show that in addition to a Kelvin-Helmholtz instability, a Holmboe instability can appear, and we discuss the implication of this in stellar interiors.