On the Relationship Between MHD Equilibrium Bifurcations and Linear Stability Theory for Models of Line-tied Coronal Loops

We present the results of an investigation of the relationship between the predictions made by linear MHD stability theory for instabilities of straight flux tube models of line-tied coronal loops and the bifurcations of the underlying MHD equilibria. It is found that contrary to the naive assumption that there should be a one-to-one correspondence between instability points and bifurcation points, such an one-to-one correspondence does not always exist. Using the Gold-Hoyle equilibrium as an illustrative example it is shown that additional axisymmetric bifurcations exist if the equilibrium bifurcations are calculated with Grad-Shafranov theory. In this case the onset of the m = 0-instability (sausage mode) corresponds to the second and not the first bifurcation point of the equilibrium sequence. No additional bifurcations have been found when Euler potentials are used, and in this case the first bifurcation coincides with the onset condition for the m = 0-instability. A possible reason for this difference are the different boundary conditions.