σ-stability analysis of toroidal magnetic fields in stars

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Abstract
This study uses σ-stability analysis to explore the hydromagnetic instabilities of a star with an axisymmetric toroidal magnetic field. Necessary and sufficient conditions for σ-stability are derived and are used to obtain the maximum growth rate (MGR) of instability. The concepts determining the point of MGR and the σ-stability line are introduced as rigorous tools to investigate the instabilities. The σ-stability analysis of a star with a weak toroidal magnetic field is considered in detail. It is shown that the hydromagnetic stability of a star with a weak toroidal magnetic field depends solely on the shape of the impressed toroidal magnetic field. Relevant hydromagnetic instabilities are in almost all cases due to an unstable stratification of the toroidal magnetic field with respect to the colatitude. The MGR depends on the strength of the impressed magnetic field, but important properties of the hydromagnetic instabilities, such as the extent of the instability region, the position of the point of MGR, and the shape of the σ-stability line, depend only on the topology of the magnetic field. The results of the σ-stability analysis of a star with a weak magnetic field are used to investigate the instabilities caused by admissible weak toroidal magnetic fields H Ω = Kρβ (r sin θ)2β − 1 (with β a parameter) in a polytrope n = 3. These toroidal magnetic fields are always most unstable under non-axisymmetric perturbations with |m| = 1 (for β < 2) or under axisymmetric perturbations (β ≧ 2). The growth rates of the most unstable perturbations lead to very short e-folding times when applied to real stars with weak magnetic fields. Instabilities with growth rates comparable to the MGR are primarily internal phenomena, but interesting instabilities with e-folding times still relatively short compared to the nuclear time scale may occur further out.

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