Classification of magnetic instabilities

Abstract

Linearised hydromagnetic stability problems can often be formulated as eigenvalue problems with solutions proportional to exp[-iωt] where iω = p + is is the eigenvalue and t is time. For a hydromagnetic system in the geometry of an infinite cylindrical annulus, we have revealed the presence of double eigenvalues at various locations in the parameter (Λ, n)-space. Here, Λ 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 tracking a particular eigenvalue around a closed path in parameter space does not necessarily return the original eigenvalue. This phenomena was examined by Jones (1987), in the context of Poiseuille flow. Jones showed that such changes are due to the presence of double (and multiple) eigenvalue points lying within the closed path. Thus, care must be taken when following any eigenvalue in parameter space since the final result can be path dependent. In the hydromagnetic problem, we find that the most unstable mode (i.e. the mode we are most interested in) often behaves in this manner. If great care is not taken when using the methods (such as inverse iteration) that follow a single eigenvalue and the effects of double eigenvalues accounted for, it is possible to mistakenly overestimate critical parameter values. Another consequence of this phenomenon is that classifying magnetic instabilities when Λ ∼ O(1) as either ideal or resistive is not possible. This distinction only makes sense in the perfectly conducting limit Λ → ∞.

Key Words:

• Magnetic instabilities
• double eigenvalues.