Kelvin-Helmholtz instability in coronal mass ejecta in the lower corona

Abstract

We model an imaged Kelvin-Helmholtz (KH) instability on a coronal mass ejecta (CME) in the lower corona by investigating conditions under which kink ($m = 1$) and $m = -3$ magnetohydrodynamic (MHD) modes in an uniformly twisted flux tube moving along its axis become unstable. We employ the dispersion relations of MHD modes derived from the linearised magnetohydrodynamic equations. We assume real wave numbers and complex angular wave frequencies, namely complex wave phase velocities. The dispersion relations are solved numerically at fixed input parameters (taken from observational data) and various mass flow velocities. It is shown that the stability of the modes 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 Alfv\'en Mach number (the ratio of the tube velocity to Alfv\'en speed inside the flux tube). For a twisted magnetic flux tube at a density contrast of $0.88$, background magnetic field ratio of $1.58$, and a normalised magnetic field twist of $0.2$, the critical speed for the kink ($m = -3$) mode (where $m$ is the azimuthal mode number) is $678$ km\,s$^{-1}$ just as it is observed. The growth rate for this harmonic at KH wavelength of $18.5$ Mm and ejecta width of $4.1$ Mm is equal to $0.037$ s$^{-1}$, in agreement with observations. KH instability of the $m = -3$ mode may also explain why the KH vortices are seen only at the one side of arising CME. The good agreement between observational and computational data shows that the imaged KH instability on CME can be explained in terms of emerging KH instability of the $m = -3$ MHD mode in twisted magnetic flux tube moving along its axis.