The nonlinear evolution of the long-wavelength external m=1 (kink) mode of a general, straight screw pinch is investigated analytically by means of bifurcation theory. The investigation covers both the ordinary screw pinch ordering of parameters, Bz approximately Bphi , as well as the tokamak limit, Bz>>Bphi . Conducting wall effects are included in both cases. In the case of the ordinary screw pinch, general expressions, valid up to second order in the axial wavenumber, are derived for the coefficients D1 and D3 in the mode evolution equation eta n+D1 eta +D3 eta 3=0. Generally, the nonlinear stability near a linear stability boundary is found to depend on the current profile, but not on the axial field profile. Furthermore, peaked current profiles are found to be more stable than broad profiles, but for complete nonlinear stability a conducting wall is in general required. In the tokamak case, a general expression for D3, covering all previously calculated values of this coefficient (for particular profiles), is derived. Near the first marginal point qa=1/b2 (b is the wall radius normalized to the plasma radius), the shape of the current profile is found to be of decisive importance for the nonlinear evolution of the m=1 mode. In general, sufficiently peaked profiles lead to nonlinear stability, whereas broad profiles lead to explosive growth of the mode. Near the second marginal point qa=1, D3 is shown to have a singularity either of order (1-qa)-1, or of order (1-qa)-2, depending on whether Ja not=0 or Ja=0, respectively, where Ja is the current density at the plasma edge.
An analytical solution of the linearized Vlasov equation for nearly rigid displacements