The nonlinear saturation of a resistive modified tearing mode in a toroidal plasma is calculated by Simon saturation theory for a weakly linearly unstable (over-stable) regime. The linearly unstable mode is a (1, 1) mode, where the first 1 indicates n, the toroidal mode number about an axisymmetric equilibrium and the second 1 indicates the multiple of the non-zero frequency at which the n = 1 mode oscillates when linearly marginally stable. The saturation amplitude is found to be proportional to (Δ – Δc)½ where Δ, a measure of the driving energy of the instability, is proportional to the difference between the logarithmic derivatives of the radial (i.e. perpendicular to the magnetic surface) perturbation magnetic field in the simplest case, and Δc measures toroidal stabilization due to average magnetic line curvature. To obtain the Simon saturation condition, one must go to third order in the small parameter (Δ –Δc)½. Starting with (1, 1), (1, – 1),( – 1, – 1), and( –1, 1 ) modes in first order in (Δ – Δc)½, the modes (0, 0), (0, 2), (2, 2), (2, 0) are obtained in second order, and these are driven with first-order modes to a third-order (1,1) mode which yields a saturation condition. In the quasi-linear approximation, only the near-zero frequency modes (0, 0) and (2, 0) are considered in the second order. In the present paper, only the axisymmetric perturbation (0,0) is used in second order. This gives a relation between nonlinear saturation amplitude and frequency shift, but does not determine either uniquely because of the undetermined parameter γ in the perturbation solutions. This parameter is determined exactly by the requirement of finiteness of the solutions when the (2, 0) non-axisymmetric near-zero frequency perturbation is taken into account in second order. However, magnetic island width at saturation can still be estimated taking only the (0,0) second order mode because this width depends so insensitively on γ, namely as γ–¼. Under this restriction to (0, 0) in second order, the frequency shift is found to be negative. Its absolute value is proportional to Δ – Δc, while the magnetic island width is proportional to (Δ – Δc)¼ and has a scale determined by t, the thickness of the resistive layer, proportional to η⅓, the cube root of the electrical resistivity.
Matched Pair of AOTFs with Net Zero Frequency-shift
