Twisted magnetic tubes with field aligned flow

I. Linear twist and uniform longitudinal field^⋆

¹ Departament de FísicaUniversitat de les Illes Balears, 07122 Palma de Mallorca, Spain
e-mail: toni.diaz@uib.es, ramon.oliver@uib.es, dfsjlb0@uib.es
² Centre for Plasma Astrophysics, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
e-mail: roberto.soler@wis.kuleuven.be

Received: 13 December 2010
Accepted: 17 June 2011

Abstract

Aims. We study the equilibrium and stability of twisted magnetic flux tubes with mass flows along the field lines. Then, we focus on the stability and oscillatory modes of magnetic tubes with uniform twist B₀ = B₀(r/p   e_ϕ + e_z) in a zero-β plasma, surrounded by a uniform, purely longitudinal field.

Methods. First we investigate the possible equilibriums, and then consider the linearised MHD equations and obtain a system of two first-order differential equations. These are solved numerically, while analytical approximations involving confluent hypergeometric functions are found in the thin tube limit. Finally, new appropriate boundary conditions are deduced and the outer solution considered (with the apparition of cut-off frequencies). We use this to derive a dispersion relation, from which the frequencies of the normal modes can be obtained.

Results. Regarding the equilibrium, the only value of the flow that satisfies the equations for this magnetic field configuration is a super-Alfvénic one. Then, we consider the normal modes of this configuration. The thin-tube approximation proves accurate for typical values, and it is used to prove that the equilibrium is unstable, unless the pitch is large. The stability criteria for twisted tubes are significantly lowered.

Conclusions. The twisted tube is subject to the kink instability unless the pitch is very high, since the Lundquist criterion is significantly lowered. This is caused by the requirement of having a magnetic Mach number greater than 1, so the magnetic pressure balances the magnetic tension and fluid inertia. This type of instability might be observed in some solar atmospheric structures, like surges.