Asymptotic theory of the nonlinearly saturated m=1 mode in tokamaks with q<1

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Abstract
In this paper the necessary and sufficient conditions required for the existence of a nonlinearly saturated m=1 tearing mode in tokamaks with q01(r) are solved exactly, in closed form for an arbitrary mean profile ψ0(r) in leading order. This is shown to result in a ‘‘no disturbance’’ theorem: the m=1 perturbation must be confined to within the radius ri such that q(ri)=1. The bifurcation relation for the nondimensional perturbation amplitude is then constructed by solving the nonlinear inner critical layer equations using an ordered iterative technique. For monotonically increasing q profiles, the equation has a solution if and only if the toroidal current density of the unperturbed equilibrium has a maximum within ri and the parameter d log q(r)/[d log η(r)] [where η(r) is the resistivity profile consistent with the q profile of the unperturbed equilibrium] is sufficiently small at ri. The considerations are extended to nonmonotonic profiles as well. When the conditions are met, a nonlinearly saturated m=1 tearing mode is shown to exist with a novel island structure, quite different from those obtained from the usual Δ’ analysis, which is shown to be inappropriate to the present problem. The relevance of the results of the present theory to sawtooth phenomena reported in JET [Plasma Physics and Controlled Nuclear Fusion Research (IAEA, Vienna, 1989), Vol. 1, p. 377] and other tokamaks is briefly discussed. The solution constitutes an analytically solved test case for numerical simulation codes to leading orders in a/R and the shear parameter d log q/d log η.