Theory of ideal and resistive m=1 modes in tokamaks

Abstract

A review of the present theoretical understanding of the linear stability of internal m=1 modes is presented and its connection to phenomena observed in toroidal magnetic confinement experiments, i.e. 'sawtooth' and 'fishbone' oscillations, is discussed. Particular attention is devoted to the analysis of non-magnetohydrodynamic (non-MHD) effects, such as those due to finite diamagnetic and electron drift frequencies and to finite ion Larmor radius, and to the special role played by energetic particles (whose response is primarily kinetic and which can stabilize low frequency modes and destabilize new, higher frequency, branches). Some recent developments in the non-linear theory of these modes are also discussed. This review complements that of Kuvshinov and Savrukhin (Sov. J. Plasma Phys. 16 (1990) 353)

Diamagnetic thresholds for sawtooth cycling in tokamak plasmas

“…13. It can be observed that the transition between "kink cycling" and "sawtooth cycling" corresponds fairly well with the stability boundary in the diagram presented by Migliuolo, 13 provided that the case with small x Ãe =x Ãi is retained.…”

“…[15][16][17] When diamagnetic flows are included in the description of the internal kink, the growth rate of the resistive instability is further reduced and the mode can become completely stable. 8,13 It is also noted that the ideal MHD time scale required to recover the short crash times observed in the experiment appears to be at odds with the slow growth rate (in the resistive time scale) of the precursor oscillations that precede the crash. Several authors have proposed models that involve an acceleration of the reconnection rate during the late stage of the sawtooth ramp.…”

“…13 Fig. 7 shows such a diagram for a sequence of instability obtained with our simulations (c g is the growth rate of the resistive kink mode, while …”

“…[6][7][8][9][10][11][12][13][14] It is noted that in ideal magnetohydrodynamics (MHD), with no dissipation mechanism available to enable magnetic reconnection, an unstable kink mode results in a three dimensional equilibrium with a m=n ¼ 1=1 helicity component (no kink cycles). In resistive MHD, kink cycles can be recovered, but the duration of kink crashes, s crash $ g À1=2 , is several orders of magnitude too slow with respect to the sawtooth crashes observed in large tokamaks.…”

Mode structure of disruption precursors in TFTR enhanced reversed shear discharges

“…However, ideal MHD theory is not yet completely successful. An example is the understanding of the sawtooth behavior [8,9,10]. The MHD activity predicted near the central core is sensitive t o t h e q v alue at the axis and the magnetic shear q 0 =q at the surface where q = 1 so that the validity of ideal MHD theory near the core still requires further experimental and theoretical investigations.…”