The spectrum of MHD flows about X points

Abstract

A recently proposed method to calculate the spectrum of linear, incompressible, unbounded plasma flows is applied to magnetohydrodynamic flows about X points. The method transforms the two-dimensional spectral problem in physical space into a one-dimensional problem in Fourier space. The latter problem is far easier to solve. Application of this method to X-point plasma flows results in two kinds of essential spectra. One kind corresponds to stable perturbations and the other one to perturbations that become overstable whenever the square of the poloidal Alfvén Mach number becomes larger than 1. Apart from these two spectra, no other spectral values were found.

Mode coupling in two-dimensional magnetohydrodynamic flows

“…This is a very special feature, which occurs for cylindrical, incompressible plasmas. Equation (4.5) was also found for an X-point geometry, whereas (4.4) is modified because of the difference in field-line curvature (Nijboer et al 1998). …”

“…This is a dense set of eigenvalues that discretizes when a wall at a finite distance is introduced. For closed flux surfaces, the discrete modes cluster towards a cluster point, which was termed the 'classical' continuum by Nijboer et al (1998).…”

“…2. The system (A 1), (A 2) is equivalent to that for X-point geometries as described by Nijboer et al (1998).…”

“…For a compressible plasma, this means that one has to solve these two coupled equations for the magnetic flux and the poloidal Alfvén Mach number, as is done, for instance, by Goedbloed and Lifschitz (1997). For an incompressible plasma, however, the poloidal Alfvén Mach number and the density profiles become flux functions, and the Bernoulli equation reduces to a definition of a Bernoullitype of function H, which, for a cylindrical geometry, takes the form (Nijboer et al 1998)…”

“…This was shown by Hameiri and Hammer (1979), who considered a cylindrical geometry with longitudinal magnetic field and poloidal flow, and by Hellsten and Spies (1979), who considered a toroidal geometry with toroidal flow and poloidal magnetic field. Nijboer et al (1998) considered an X-point geometry. A new continuous spectrum was found for unbounded domains.…”