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At present, many works about magneto-hydrodynamics wave diagnostics inmagnetic flux tubes are based on some pioneer works not considering thecontributions of magnetic twist. Other works considered the effect onmagneto-hydrodynamics waves, but the dispersion relationship they presentedonly gave the wave modes of m=0,1,2. The kink mode of m=-1 was absent.Therefore, in this work we would like to present a complete dispersionrelationship 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 changeat finite kr0, from body to surface mode. The phase speeds of this mode areusually less than those of m=+1 mode. The harmonic curves of m=+1/-1 modes indispersion relationship diagrams are approximately symmetric in respect to acharacteristic velocity, e.g. the tube velocity in flux tubes. Based on thepresent dispersion relationship, we revisit the issue of spiral wave patternsin sunspot and find that the magnetic twist has no great influence on theirmorphology in the frame of linear perturbation analysis.

Magneto-Acoustic Waves in Magnetic Twisted Flux Tube