On the stability of incompressible MHD modes in magnetic cylinder with twisted magnetic field and flow

In this work, we studied MHD modes in a magnetically twisted flux tube with a twisted flow that is embedded in the uniform magnetic field. We consider when the azimuthal magnetic field and velocity are linear functions of radius (case i) and also more generally when they are arbitrary functions of radius (case ii). Under these assumptions, we obtain the dispersion equation in the incompressible limit. This solution can also be used to describe the MHD perturbations in plasma pinches and vortices. The dispersion equation is simplified by implementing the thin flux tube approximation. It is shown that sausage modes (m = 0) become unstable for large enough azimuthal flow speeds. Also, we obtained the unstable modes for m > 0. It is shown that the stability criterion of the m = 1 mode (for case i) is independent of the background azimuthal components of the plasma velocity and magnetic field. These criteria fully coincide with the result that was previously obtained by Syrovatskiy for a plane interface. Moreover, this result even remains valid when the azimuthal magnetic field and velocity have an arbitrary dependence on radius (case ii). A criterion for the stability of the m ≥ 2 modes is also obtained. It was found that instability of these modes is determined by both longitudinal and azimuthal flows. It is shown that if there is sufficient azimuthal background flow, then all modes with m ≥ 2 will become unstable.