Existence of Purely Alvén Waves in Magnetic Flux Tubes with Arbitrary Cross-Sections

We study linear torsional Alfvén waves in a magnetic flux tube with an arbitrary cross-section. assume that the equilibrium field is propagating z-direction Cartesian coordinates x,y, and z. The cross-section bounded by smooth closed curve. Both plasma are homogeneous outside this magnitude function of x y, while density product two functions: one dependent on z other y. As result, speed also equal to V0(x,y) times define as do not disturb density. show these can exist only when V0. When condition existence satisfied, polarised directions tangent level lines orthogonal field. found wave amplitude has specific form depends particular coordinate system.