Current sheet stability and accelerated reconnection of magnetic field lines

Abstract

The stability of a narrow current sheet embedded in a sheared magnetic field to short wavelength microtearing modes is examined and a mechanism for the accelerated reconnection of magnetic field lines during the fast crash of sawteeth in tokamaks is presented. The stability of the sheet is shown to depend crucially on the sign of Il/ k′∥, where Il is the line current in the sheet and k∥ ≡dk∥(x)/dx is the rate of change of the parallel wave number in the direction of the inhomogeneity, as well as on three important characteristic lengths: Δr, the distance between the center of the current sheet and the resonant surface of the mode; the width σs of the sheet; and the resonant layer width wr of the mode. When Il/k′∥ is positive, the current sheet can be unstable only if Δr < 1/2 max(σs,wr); otherwise, the sheet is stable. Conversely, when Il/k∥ is negative the sheet can be unstable only if Δr > 1/2 max(σs,wr); otherwise, the sheet is stable. Expressions for the unstable spectrum and the growth rate of the unstable modes as a function of the relative magnitude of Δr, σs, and wr are presented. For narrow current sheets with σs<wr, the growth rate of the tearing mode actually increases as the plasma resistivity decreases. The stability analysis is applied to the current sheet generated around the X point of a q=1 mode during the fast crash of the central temperature during sawteeth in tokamaks. The sheet is unstable if σs is less than a critical width σsc/a ≡‖Is/I0‖ ×(4/qa), where Is/I0 is the ratio of the current Is in the sheet to the total current I0, a is the minor radius, and qa is the safety factor at the edge of the plasma. For typical TFTR tokamak [Plasma Physics and Controlled Nuclear Fusion Research, Kyoto, 1986 (IAEA, Vienna, 1986), Vol. 1, p. 421] parameters, σsc is a few ion gyroradii. The sheet generated by the q=1 mode in a hot tokamak like TFTR or JET is much narrower than σsc. As a consequence, the sheet is unstable to a broad spectrum of short wavelength perturbations with m, n>100, where m and n are the poloidal and toroidal mode numbers, respectively. These modes grow rapidly with a growth time of the order of 1 μsec, much less than the sawtooth crash time of 50–100 μsec. This short wavelength turbulence will likely broaden the current sheet to the marginally stable width σsc and thereby increase the rate of reconnection of magnetic field lines by the q=1 mode, resulting in a fast crash of the central temperature.