Magneto-acoustic waves in magnetic twisted flux tubes

At present,many works about MHD wave diagnostics in magnetic flux tubes are based on some pioneering works not considering the contributions of magnetic twist.Other works considered the effect on MHD waves,but the dispersion relationship they presented only gave the wave modes of m=0,1,2...The kink mode of m=-1 was absent.Therefore,in this work we present a complete dispersion relationship that includes both magnetic twist and the wave mode of m=-1.Analogous to the m=+1 wave mode,the mode of m=-1 also exhibits the mode change at finite kr;,from body to surface mode.The phase speeds of this mode are usually less than those of m=+1 mode.The harmonic curves of m=±1 modes in dispersion relationship diagrams are approximately symmetric in respect to a characteristic velocity,e.g.the tube velocity in flux tubes.Based on the present dispersion relationship,we revisit the issue of spiral wave patterns in sunspots and find that the magnetic twist has no great influence on their morphology in the frame of linear perturbation analysis.