The dispersion functional for multidimensional equilibria

• Published in: The Physics of fluids (1958) 1985-12, Vol.28 (12), p.3546-3556
• Main Authors: Lewis, H. Ralph, Barnes, Daniel C., Schwarzmeier, James L., Seyler, Charles E.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Magnetic confinement and equilibrium; Physics; Physics of gases, plasmas and electric discharges; Physics of plasmas and electric discharges
• ISSN: 0031-9171; 2163-4998
• DOI: 10.1063/1.865309

Numerical study of the linear stability of plasmas is very difficult when one or more of the plasma species is collisionless and the equilibrium is multidimensional, that is, characterized by two or more nonignorable spatial coordinates. The problem arises, for example, in evaluating kinetic stabilizing effects on the internal tilting mode (an n=1 ballooning mode) in field‐reversed configurations. In this paper, the Laplace transform of the perturbation distribution function for a collisionless species is derived for all classes of phase‐space trajectories and used to construct the dispersion functional for multidimensional equilibria. The kinetic part of the dispersion functional is expressed in terms of the Laplace transform of autocorrelation functions with respect to a certain delay time. It is shown how to obtain the same result formally by using Liouville eigenfunctions. For the case of the Vlasov‐fluid model, the dispersion functional is transformed in a way that is particularly appropriate for computation of the kinetic stability of field‐reversed configurations to the internal tilting mode.