A&A 485, 351-361 (2008)

DOI: 10.1051/0004-6361:200809564

## Flow instabilities of magnetic flux tubes

##### III. Toroidal flux tubes

**V. Holzwarth**

Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany

e-mail: holzwarth@mps.mpg.de

Received 12 February 2008 / Accepted 11 April 2008

** Abstract ***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 buoyancy- and 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
buoyancy- and 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 buoyancy- and 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.

**Key words:**magnetic fields

**--**magnetohydrodynamics (MHD)

**--**instabilities

**--**Sun: magnetic fields

**--**stars: magnetic fields

**©**

*ESO 2008*