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For axisymmetric models for coronal loops the relationship between thebifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and thepoints of linear ideal MHD instability is investigated imposing line-tiedboundary conditions. Using a well-studied example based on the Gold-Hoyleequilibrium, it is demonstrated that if the equilibrium sequence is calculatedusing the Grad-Shafranov equation, the instability corresponds to the secondbifurcation point and not the first bifurcation point because the equilibriumboundary conditions allow for modes which are excluded from the linear idealstability analysis. This is shown by calculating the bifurcating equilibriumbranches and comparing the spatial structure of the solutions close to thebifurcation point with the spatial structure of the unstable mode. If theequilibrium sequence is calculated using Euler potentials the first bifurcationpoint of the Grad-Shafranov case is not found, and the first bifurcation pointof the Euler potential description coincides with the ideal instabilitythreshold. An explanation of this results in terms of linear bifurcation theoryis given and the implications for the use of MHD equilibrium bifurcations toexplain eruptive phenomena is briefly discussed.

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