Ideal kink instabilities in line-tied coronal loops - Growth rates and geometrical properties

A detailed analysis of the ideal kink instability in line-tied cylindrically symmetric coronal loops is presented. Using a rapidly converging Fourier series expansion technique, the growth rate, as well as the eigenfunction, of ideal m = 1 kink modes is calculated for two topologically distinct models of force-free static MHD equilibria: one in which all the magnetic field lines are connected to the photosphere and one presenting a polarity inversion surface. The growth rates depend crucially on the loop length. Loops of the former type are found to be more unstable, and possess higher growth rates, than loops of the latter type, which are unstable to sausage-tearing modes and may also be unstable to m = 1 resistive kink modes. Applications of these results to the structure of coronal loops are presented.