Flow instabilities of magnetic flux tubes III. Toroidal flux tubes

Context. The stability properties of toroidal magnetic flux tubes are relevant for the storage and emergence of magnetic fields in the convective envelope of cool stars. In addition to buoyancyand magnetic tension-driven instabilities, flux tubes are also susceptible to an instability induced by the hydrodynamic drag force. Aims. Following our investigation of the basic instability mechanism in the case of straight flux tubes, we now investigate the stability properties of magnetic flux rings. The focus lies on the influence of the specific shape and equilibrium condition on the thresholds of the friction-induced instability and on their relevance for emerging magnetic flux in solar-like stars. Methods. We substitute the hydrodynamic drag force with Stokes law of friction to investigate the linear stability properties of toroidal flux tubes in mechanical equilibrium. Analytical instability criteria are derived for axial symmetric perturbations and for flux rings in the equatorial plane by analysing the sequence of principal minors of the coefficient matrices of dispersion polynomials. The general case of non-equatorial flux rings is investigated numerically by considering flux tubes in the solar overshoot region. Results. The friction-induced instability occurs when an eigenmode reverses its direction of propagation due to advection, typically from the retrograde to the prograde direction. This reversal requires a certain relative velocity difference between plasma inside the flux tube and the environment. Since for flux tubes in mechanical equilibrium the relative velocity difference is determined by the equilibrium condition, the instability criterion depends on the location and field strength of the flux ring. The friction-induced instability sets in at lower field strengths than buoyancyand tension-driven instabilities. Its threshold is independent of the strength of friction, but the growth rates depend on the strength of the frictional coupling between flux tube and environment. Conclusions. The friction-induced instability lowers the critical magnetic field strength beyond which flux tubes are subject to growing perturbations. Since its threshold does not depend explicitely on the friction parameter, this mechanism also applies in case of the quadratic velocity dependence of the hydrodynamic drag force. Whereas buoyancyand tension-driven instabilities depend on the magnetic field strength alone, the dependence of hydrodynamic drag on the tube diameter gives rise to an additional dependence of growth times on the magnetic flux.