On the interchange instability


  • B. Rogers,

  • B. U. Ö Sonnerup


We reexamine the standard method for analyzing the stability of planetary magnetospheres against interchange motions of magnetic flux tubes and the plasma contained in them. This method consists of finding the change in potential energy ε = ∫ [p/(γ - 1) + ρϕ] dV of two tubes of equal flux which exchange position, leaving the energy of the surrounding plasma and field unchanged. As an illustration, we study a cylindrically symmetric plasma with an embedded purely toroidal magnetic field. Our results for this geometry and for the nonrotating case are as follows. (1) The standard method, which takes into account changes in internal energy, p/(γ - 1), and gravitational energy, ρϕ, but not magnetic energy has been claimed to be valid for vanishing plasma pressure. We show it to be correct up to and including terms of order β, β being the ratio of plasma to magnetic pressure. (2) Toroidal equilibria having β of order unity or larger can be analyzed by interchanging tubes of unequal flux such that to first order in the displacement the surroundings are left unperturbed and by taking into account changes in the magnetic energy, B²/2µ0, of the two tubes. (3) Such equilibria can also be analyzed by considering the displacement of a single flux tube, taking into account resulting changes in the energy of the tube as well as the environment. For rotating equilibria we find that the standard method of replacing ρϕ by ρ(ϕ - r²Ω²/2) in ε only gives a sufficient condition for stability. However, for the cylindrical equilibrium an exact marginal condition can be obtained from energy arguments, provided Coriolis effects are taken into account.