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^{1}, Armin Kaleck

^{1}, Michalis Psimopoulos

^{1}and Günter Hasselberg

^{1}

### Abstract

It is shown that certain specific properties are requested from the current density profile in order to arrive at a coherent theoretical explanation of the sawtooth phenomenon in tokamaks: (1) the shear *s*̂ must be small at the *q*=1 surface (hence the current density profile has a plateau or shoulder in the neighborhood of *q*=1); (2) *J* ’ varies abruptly not far away from *q*=1 (in particular, the outer edge of the plateau is close to *q*=1). Indeed, when these conditions are satisfied, the contribution to the jump of the derivative of the magnetohydrodynamic (MHD) solution ([Ψ’]_{ r 1 }) from those terms that are proportional to ε^{2} (ε is the inverse aspect ratio) is not negligible. It is, moreover, opposite in sign to the usual contribution ([Ψ^{(0)’ }]_{ r 1 }∝*s*̂_{1}) so that the inverse of the logarithmic jump 1/Δ’∝Ψ_{ r 1 }/[Ψ’]_{ r 1 } can now take on either positive values (*s*̂_{1}>*s*̂_{1,c }), or negative ones (*s*̂_{1}<*s*̂_{1,c }), or can ‘‘diverge’’ (1/Δ’→±∞ for *s*̂_{1}→*s*̂_{1,c }).

The growth rate reaches very large values when 1/Δ’→−∞ (i.e., [Ψ’]_{ r 1 }→0^{−}) and its rapid acceleration observed shortly before the crash can now be explained: The ontology of the sawtooth rise and crash requires indeed that *s*̂_{1} always be smaller than *s*̂_{1,c } (hence, 1/Δ’<0 opposite to conventional wisdom!) and approach this value from below at the onset of the collapse. Current density [*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* *1* *9* *8* *6* (IAEA, Vienna, 1987), Vol. 1, p. 263] as well as density [*C* *o* *n* *t* *r* *o* *l* *l* *e* *d* *F* *u* *s* *i* *o* *n* *a* *n* *d* *P* *l* *a* *s* *m* *a* *P* *h* *y* *s* *i* *c* *s* *1* *9* *8* *9* (EPS, Geneva, 1989), Vol. 13 B, II‐565] plateaus and shoulders have been observed in TEXTOR discharges. This and other comparisons with experiment, including the acceleration rate of the perturbation, are discussed. Analysis of the particle trajectories in the perturbed magnetic structure yields an amplitude criterion for the onset of collapse that agrees well with observations on TEXTOR. The ontology of the crash phase requires an ambipolar electric field to build up inside the *q*=1 surface so that only suprathermal electrons are lost. This enables the calculation of ratios of the density, current density, and electron temperature crashes; these are found to agree with the measured ratios. Theory finally suggests that the sawtooth limitation of the central temperature is not unavoidable.

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