Kelvin-Helmholtz instability of kink waves in photospheric twisted flux tubes

¹ Faculty of Physics, Sofia University, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
e-mail: izh@phys.uni-sofia.bg
² Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria
e-mail: teimuraz.zaqarashvili@oeaw.ac.at
³ Abastumani Astrophysical Observatory at Ilia State University, 2 University Street, 0160 Tbilisi, Georgia

Received: 30 April 2012
Accepted: 11 September 2012

Abstract

Aims. We investigate conditions under which kink magnetohydrodynamic waves propagating along photospheric uniformly twisted flux tubes with axial mass flows become unstable as a consequence of the Kelvin-Helmholtz instability.

Methods. We employed the dispersion relations of kink waves derived from the linearised magnetohydrodynamic equations. We assumed real wave numbers and complex angular wave frequencies, namely complex wave phase velocities. The dispersion relations were solved numerically at fixed input parameters and several mass flow velocities.

Results. We show that the stability of the waves depends upon four parameters, the density contrast between the flux tube and its environment, the ratio of the background magnetic fields in the two media, the twist of the magnetic field lines inside the tube, and the value of the Alfvén-Mach number (the ratio of the jet velocity to Alfvén speed inside the flux tube). We assume that the azimuthal component of the magnetic field in the tube is proportional to the distance from the tube axis and that the tube is only weakly twisted (i.e., the ratio of the azimuthal and axial components of the magnetic field is low). At certain densities and magnetic field twists, an instability of the Kelvin-Helmholtz type of kink (m = 1) mode can arise if the Alfvén-Mach number exceeds a critical value. In particular, for an isolated twisted magnetic flux tube (magnetically free environment) at a density contrast (the ratio of the mass density of the environment to that of the tube itself) equal to 2 and a magnetic field twist (defined as the ratio of azimuthal magnetic field component at the inner surface of the tube to the background magnetic field strength) equal to 0.4, the threshold Alfvén-Mach number has a magnitude of 1.250075, which means that for an Alfvén speed inside the tube of 10 km s^-1 the jet velocity should be higher than 12.5 km s^-1 to ensure the onset of the Kelvin-Helmholtz instability of the kink (m = 1) mode. Speeds of that order can be detected in photospheric tubes.

Conclusions. The observed mass flows may trigger the Kelvin-Helmholtz instability of the kink (m = 1) mode in weakly twisted photospheric magnetic flux tubes at critical Alfvén-Mach numbers lower that those in untwisted tubes if the magnetic field twist lies in the range 0.36−0.4 and the flow speed exceeds a critical value. A weak external magnetic field (with a ratio to the magnetic field inside the tube in the range 0.1−0.5) slightly increases that critical value.