Alfvén instability of steady state flux tubes. II. Upflows in stratified atmospheres

Context. MHD instabilities play an important role in the dynamics and energetics of the solar atmosphere.

Aims: An open vertical magnetic flux tube is permeated by an upflow in a stratified atmosphere with variable temperature. The stability of the tube is investigated with respect to small-amplitude torsional perturbations generated at the footpoint by random convective motions.

Methods: A steady state equilibrium incorporating the effects of a vertical body force, heating, and losses is derived analytically. The governing equations for torsional motions are integrated with a fourth-order Runge-Kutta method and matched with the analytical solutions in the upper regions to obtain a numerical dispersion relation. The dependence of the eigenmode frequencies on different parameters is analysed. Unstable modes are found for a range of Alfvén and flow speeds in the photosphere, as well as expansion factors of the flux tubes. Both supersonic and subsonic flows are considered.

Results: It is shown that torsional perturbations are exponentially amplified in time if a section of the tube exists where the upflowing plasma decelerates and the tube expands. The flow speeds required for the instability are sub-Alfvénic.

Conclusions: The instability may be important for understanding the abundance of Alfvén waves seen in recent observations and the associated heating in magnetic regions of the solar atmosphere.