Nonlinear Stability of an Ideal Plasma

  • V. A. Gordin
  • V. I. Petviashvili
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)


The functional series for the first integrals of helicity are used for the derivation of the sufficient stability conditions of ideal plasma in the sense of Lyapunov. The Euler equations for the Lyapunov functional imply the Grad-Shafranov equation with pressure profile determined by the toroidal field. The calculations were carried out both for cylindrical and toroidal configurations. In the considered example the magnetic field is mostly created by currents in plasma. The safety factor is less than one and decreases in the direction of the plasma boundary. The toroidal field behaves similarly: decreasing but not turning to zero. The pressure and density of helicity and their gradients, however, turn to zero at the boundary.


Euler Equation Jacobi Equation Plasma Boundary Toroidal Field Functional Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.M. Lyapunov. Collection of Works, v. 2. Gostekhizdat, Leningrad, 1956 (in Russian).Google Scholar
  2. 2.
    Y.I. Arnold. Mathematical Methods of Classical Mechanics. Nauka, Moscow, 1974 (in Russian).Google Scholar
  3. 3.
    L.A. Diky. Hydrodynamical Stability and Atmospheric Dynamics. Hydrometeoizdat, Leningrad, 1976 (in Russian).Google Scholar
  4. 4.
    V.A. Gordin. Mathematical Problems of the Hydrodynamical Weather Forecast, v.1,2. Hydrometeoizdat, Leningrad, 1987 (in Russian).Google Scholar
  5. 5.
    B.B. Kadomtsev. In: Reviews of Plasma Physics, v.2. Consultants Bureau, 1966.Google Scholar
  6. 6.
    G. Bateman. MHD Instabilities. The MIT Press, 1978.Google Scholar
  7. 7.
    V.A. Gordin, V.I. Petviashvili. Sov. Phys.-JETP. Lett., 1987, 45, 267.ADSGoogle Scholar
  8. 8.
    V.A. Gordin, V.I. Petviashvili. Sov. Phys.-JETP, 1989, 97, 1711.Google Scholar
  9. 9.
    Y. Murakami, Z. Yoshida, N.Ynoue. Nuclear Fusion, 1988, 28, 449.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • V. A. Gordin
    • 1
  • V. I. Petviashvili
    • 1
  1. 1.Kurchatov Institute of Nuclear EnergyMoscowUSSR

Personalised recommendations