On the Relationship between Equilibrium Bifurcations and Ideal MHD Instabilities for Line-Tied Coronal Loops

For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated imposing line-tied boundary conditions. Using a well-studied example based on the Gold-Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the Grad-Shafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials the first bifurcation point of the Grad-Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.Comment: 22 pages, 6 figures, accepted by Solar Physic

On the application of numerical continuation methods to two- and three-dimensional solar and astrophysical problems

In this thesis, applications of a numerical continuation method to two- and three-dimensional bifurcation problems are presented. The 2D problems are motivated by solar applications. In particular, it is shown that the bifurcation properties of a previously studied model for magnetic arcades depend strongly on the pressure function used in the model. The bifurcation properties of a straight flux model for coronal loops are investigated and compared with the results of linear ideal MHD stability analysis. It is shown that for line-tied boundary conditions, the method for the calculation of the equilibrium sequence determines whether the first or the second bifurcation point coincides with the linear stability threshold. Also, in this thesis, the 3D version of the continuation code is applied for the first time. The problems treated with the 3D code are therefore chosen with the intention to demonstrate the general capabilities of the code and to see where its limitations are. Whereas the code performs as expected for relatively simple albeit nonlinear bifurcation problems, a clear need for further development is shown by more involved problems