CLASSIFICATION OF MAGNETIC INSTABILITIES

Linearised hydromagnetic stability problems can often be formulated as eigenvalue problems with solutions proportional to exp[-i omega t] wh ere - i omega = p + is is the eigenvalue and t is time. For a hydromag netic system in the geometry of an infinite cylindrical annulus, we ha ve revealed the presence of double eigenvalues at various locations in the parameter (Lambda, n)-space. Here, Lambda is the Elsasser number, a non-dimensional inverse measure of the magnetic diffusivity, and n is the axial wavenumber of the field and flow. We have found that trac king a particular eigenvalue around a closed path in parameter space d oes not necessarily return the original eigenvalue. This phenomena was examined by Jones (1987), in the context of Poiseuille flow. Jones sh owed that such changes are due to the presence of double (and multiple ) eigenvalue points lying within the closed path. Thus, care must be t aken when following any eigenvalue in parameter space since the final result can be path dependent. In the hydromagnetic problem, we find th at the most unstable mode (i.e. the mode we are most interested in) of ten behaves in this manner. If great care is not taken when using the methods (such as inverse iteration) that follow a single eigenvalue an d the effects of double eigenvalues accounted for, it is possible to m istakenly overestimate critical parameter values. Another consequence of this phenomenon is that classifying magnetic instabilities when Lam bda similar to O(1) as either ideal or resistive is not possible. This distinction only makes sense in the perfectly conducting limit Lambda --> infinity.