Force-free thin flux tubes: Basic equations and stability

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Abstract
The thin flux tube approximation is considered for a straight, symmetrical, force-free, rigidly rotating flux tube. The derived set of equations describes tube, body sausage, and Alfvén wave modes and is valid for any values of β. The linear waves and instabilities of force-free flux tubes are considered. The comparison of approximate and exact solutions for an untwisted, nonrotating flux tube is performed. It is shown that the approximate and exact dispersion equations coincides, except the 20% discrepancy of sausage frequencies. An effective cross section is proposed to introduce the removal of this discrepancy. It makes the derived approximation correct for the force-free thin flux tube dynamics, except the detailed structure of radial eigenfunction. The dispersion of Alfvén torsional waves in a force-free tubes appears. The valve effect of one directional propagation of waves in rotating twisted tube is revealed. The current and rotational sausage instabilities of a force-free, thin flux tube are considered.