Hydrodynamic instabilities in the solar tachocline

University of Leeds, School Of Mathematics, Department of Applied Mathematics, University of Leeds, LS2 9JT Leeds, UK e-mail: fayyaz@maths.leeds.ac.uk

Received: 24 April 2008
Accepted: 27 June 2008

Abstract

Aims. We consider the stability of simple hydrodynamic models of the solar tachocline. This study addresses only the non-magnetic instabilities. In our model there is a strong radial shear, which because of the rotation is coupled to a latitudinal temperature gradient. We also assume there is a strong stable stratification, appropriate for the slow tachocline.

Methods. These instabilities are calculated by finding the eigenvalues and eigenfunctions using a matrix-based collocation method. We also find analytic expressions for the key instabilities in the relevant asymptotic limits of strong stable stratification and small Prandtl number.

Results. We find two distinct types of instability, axisymmetric modes of the Goldreich-Schubert type, and three-dimensional Eady-type modes. Both types of mode are affected by thermal diffusion. We find that the axisymmetric modes considered by Knobloch & Spruit ([CITE], A&A, 113, 261) are likely to dominate in the slow tachocline. The non-axisymmetric baroclinic instability of Eady-type may be important in the layers closer to the base of the convection zone.