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*m*=1 mode in tokamaks with

*q*

_{0}<1

^{1}and F. A. Haas

^{1}

### Abstract

In this paper the necessary and sufficient conditions required for the existence of a nonlinearly saturated *m*=1 tearing mode in tokamaks with *q* _{0}<1 are considered in cylindrical tokamak ordering using the asymptotic techniques developed by one of the authors in an earlier paper [Phys. Fluids **2** **4**, 1716 (1981)]. The outer equations for the helical perturbation amplitude ψ_{1}(*r*) are solved exactly, in closed form for an arbitrary mean profile ψ_{0}(*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 *r* _{ i } such that *q*(*r* _{ i })=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 *r* _{ i } 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 *r* _{ i }. 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 [*P* *l* *a* *s* *m* *a* *P* *h* *y* *s* *i* *c* *s* *a* *n* *d* *C* *o* *n* *t* *r* *o* *l* *l* *e* *d* *N* *u* *c* *l* *e* *a* *r* *F* *u* *s* *i* *o* *n* *R* *e* *s* *e* *a* *r* *c* *h* (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.

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