On frequency shift and splitting induced by a deformation of the cross-section of a pinch

Abstract

The influence of noncircularity of the plasma cross-section on the MHD spectrum of pinches is studied by applying a standard expansion method to a simple model equation. Conditions for frequency shift and splitting of circular modes in terms of the angular dependence of the shape of the cross-section are derived. It is found that the splitting always is symmetric, implying that removal of the degeneracy to first order always decreases the stability margin. Higher-order splitting is, however, always superimposed on a simple second-order frequency shift which in principle can be stabilizing. It is also found that the expansion breaks down for specific axial wavenumbers. This breakdown is due to a jump in the degree of the degeneracy when the wavenumber passes a value at which modes coupled by the noncircularity have the same zero-order eigenfrequency. The conclusions are tested on a surface current Z-pitch. An additional conclusion emerging from this test is that a noncircularity causes a resonance coupling between m=1 and some higher mode, which pushes the Kruskal-Shafranov limit towards higher q values.