Nonlinear theory of the long-wavelength external m = 1 mode in screw pinches and tokamaks

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 ∼ Bp. 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 ηtt + D1η + D3η3 = 0. Generally, the nonlinear stability near a linear stability boundary is found to depend on the current profile, but no 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. (Author)