The kink instability in infinite cylindrical flux tubes: Eigenvalues for power-law twist profiles

Abstract

We develop simple accurate methods of calculating ideal MHD instability eigenvalues λ for infinitely-long cylindrical tubes, with twist functionT(r)=B _θ/rB _z . A complete theoretical treatment is presented for force-free magnetic equilibria with arbitraryT(r), and detailed semi-analytic results for the kink instability are given for the particular case of a power-law twistT(r)=r ^ν, where the index ν is non-negative. Our results show that the most rapidly growing and energetic instabilities occur in the Gold-Hoyle ν=0 field, with the instability progressively weakening with increasing ν. However, the maximum force eigenvalue is always small, so that even in the Gold-Hoyle case (where λ=O(10⁻²) in dimensionless units) only a small proportion of the available magnetic energy can be released in the linear phase. Our results also confirm that the linear pinch (ν=∞) is remarkably weak (λ=O(10⁻³)) yet relatively resistant to line-tying! It is shown that the weakness of the force eigenvalue implies that the influence of uniform gas pressure on stability is negligible. Implications for the energy-release mechanism in solar flares are discussed.