Kelvin–Helmholtz instability in a twisting solar polar coronal hole jet observed by SDO/AIA

We investigate the conditions under which the fluting (m=2), m=3, and m=12 magnetohydrodynamic (MHD) modes in a uniformly twisted flux tube moving along its axis become unstable in order to model the Kelvin–Helmholtz (KH) instability in a twisting solar coronal hole jet near the northern pole of the Sun. We employed the dispersion relations of MHD modes derived from the linearized MHD equations. We assumed real wavenumbers and complex angular wave frequencies, namely complex wave phse velocities. The dispersion relations were solved numerically at fixed input parameters (taken from observational data) and varying degrees of torsion of the internal magnetic field. It is shown that the stability of the modes depends upon five parameters: the density contrast between the flux tube and its environment, the ratio of the external and internal axial magnetic fields, the twist of the magnetic field lines inside the tube, the ratio of transverse and axial jet’s velocities, and the value of the Alfvén Mach number (the ratio of the tube axial velocity to Alfvén speed inside the flux tube). Using a twisting jet of 2010 August 21 by SDO/AIA and other observations of coronal jets we set the parameters of our theoretical model and have obtained that in a twisted magnetic flux tube of radius of 9.8Mm, at a density contrast of 0.474 and fixed Alfvén Mach number of ≅0.76, for the three MHD modes there exist instability windows whose width crucially depends upon the internal magnetic field twist. It is found that for the considered modes an azimuthal magnetic field of 1.3–1.4G (computed at the tube boundary) makes the width of the instability windows equal to zero, that is, it suppress the KH instability onset. On the other hand, the times for developing KH instability of the m=12 MHD mode at instability wavelengths between 15 and 12Mm turn out to be in the range of 1.9–4.7min that is in agreement with the growth rates estimated from the temporal evolution of the observed unstable jet’s blobs in their initial stage.