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