Locking of multiple resonant mode structures in the reversed-field pinch

Abstract

Locking of a rotating mode by applying a resonant magnetic perturbation having the same helicity has been observed on various devices. Experiments have been carried out on the Madison Symmetric Torus reversed-field pinch ͑RFP͒ ͓Dexter et al., Fusion Technol. 19, 131 ͑1991͔͒ which show that an externally applied magnetic perturbation can cause locking of the dominant magnetic modes ͑poloidal mode number mϭ1, toroidal mode numbers nϭ5 -10) when the perturbation is resonant with them. A perturbation which is not resonant (mϭ0 or 2͒ produces no such effect. Thus, resonant torques may lock a stochastic magnetic structure arising from several modes, as likely exists in the RFP, as well as a distinct island as exists in tokamaks, although the details of the interaction are likely to be different.

Observation of tearing mode deceleration and locking due to eddy currents induced in a conducting shell

“…In this work, we need only be concerned with error fields that have the same poloidal and toroidal mode numbers (m,n) as the single large mode, since only such resonant error fields can affect the mode's rotation. 8,17,18 As discussed further below, this large mode has (m,n)ϭ (1,5) or ͑1,6͒, depending on the magnetic equilibrium. The error field at the poloidal gap has a fairly broad m and n spectrum, including ͑1,5͒ and ͑1,6͒ components.…”

“…8,18 It has been demonstrated in MST that a sufficiently large mϭ1 error field applied at the poloidal gap can decelerate and lock the mϭ1 modes. 17 Given the large amplitude to which the QSH dominant mode grows, even a relatively small error might induce locking or, at least, affect the shape of the braking curve. There is a finite error field at the poloidal gap in all the plasmas studied here.…”

“…We have also ruled out other causes of braking and locking, such as the partially corrected magnetic error field, which play a role in some MST plasmas. 16,17 Concurrent with the deceleration and locking of the single tearing mode in these MST plasmas, there is also a deceleration of the bulk plasma. This mode and plasma deceleration occurs without any significant change to the global equilibrium, in terms of, e.g., the plasma current and electron density.…”

Momentum Transport from Nonlinear Mode Coupling of Magnetic Fluctuations

“…Therefore, for a mode with given k, the fluctuation-driven mean Lorentz force density at its resonant surface is F k ͗ j k 3 b k ͘. This force is responsible for the well-studied phenomenon of mode locking in which a tearing mode becomes locked to an external magnetic field structure [7][8][9][10][11][12][13]. If the magnetic structure is a static field error, then the mode becomes stationary in the lab frame.…”

The sine-Gordon equation in reversed-field pinch experiments

“…Theoretical 4 -11 and experimental 2,12-21 work involving mode locking, magnetic field errors, and rotational control through resonant magnetic perturbations has been performed on several RFP and tokamak devices; among the RFP experiments are the MST, 2,14,21,22 the reversed field experiment ͑RFX͒, 19 and the toroidal pinch experiment-RX ͑TPE-RX͒. [15][16][17]23 In MST, the dynamo modes, which are responsible for generating the reversed magnetic field, continually exhibit a sawtooth behavior as a function of time 2 as shown in Fig.…”