Relaxation Instability in Tokamaks

Abstract

Several tokamak experiments have exhibited an unstable behavior having the form of relaxation oscillations in which the soft-x-ray emission has a saw-tooth time dependence 1 ' 2 ' 3 . In the central region the results indicate a slowly rising temperature followed by a rapid fall. The whole process occurs repeatedly with a period of the order of a millisecond.The following general explanation of this behavior has gained some acceptance. The inner temperature rises due to Ohmic heating. The resulting increase in conductivity leads to an increase in the current density on axis and as a consequence the safety factor q falls below unity. The plasma then undergoes an instability which transports the energy which has been produced by Ohmic heating out to larger radii. In some way the plasma relaxes back to an axisymmetric state having q > 1 and the whole process is then repeated.The question arises as to precisely how this phenomenon occurs. A model has been suggested by Kadomtsev, 4 the basic elements of which are as follows, When the value of q falls below unity an m = 1 instability occurs. As a result the plasma surfaces are displaced to one side and resistivity allows a magnetic island to form on the opposite side of the plasma around the q = 1 surface. This island grows and displaces the original set of magnetic surfaces which then decay away. The resulting value of q is greater than unity but the concentration of the current toward the magnetic axis leads to a lowering of q until instability reap-12 A. Van Deursen, A. Van Lumig, and J. Reuss, Int. J. Mass. Spectrom. Ion Phys 0 18, 129 (1975).pears and the whole cycle is repeated.Our purpose here is to describe the results of a three-dimensional, nonlinear calculation which reproduces relaxation instabilities of the type observed experimentally and which demonstrates the basic features of Kadomtsev's model. The configuration studied is cylindrical and, in order to achieve acceptable computation times, a higher value of /3 has been used than is obtained in tokamak experiments.The equations solved are the time-dependent hydromagnetic equations including resistivity, viscosity, Ohmic heating, and an energy loss. The resistivity is taken to be proportional to T e " 3/2 . The viscosity used is small and is taken to be constant. The Ohm's law is E + vXB = r?(T)"j, and the resulting Ohmic heating is included together with an energy-loss term in the energy equationThe form of the energy-loss term is arbitrary but it represents an attempt to describe the more rapid energy-loss rate in the outer region.The calculations were carried out on a rectangular grid using a generalized form of the Lax-Wendroff method. Relaxation oscillations were first demonstrated on a 10x10x7 grid. The calculations described here were carried out on a more refined 14x14x10 grid and showed the A three-dimensional, nonlinear, numerical simulation is described in which saw-tooth oscillations occur as a result of a hydromagnetic relaxation instability. The time evolution of the magnetic topology demons...

Self-organized Te redistribution during driven reconnection processes in high-temperature plasmas

“…This is consistent with the flattening process of the pressure profile (current profile) in the full reconnection model. These processes are illustrated in frames [8][9][10][11]. As the heat is removed from the core, the closed field line topology is established and recovers the poloidal symmetry as shown in frame 12.…”

“…In the full reconnection model [7] proposed by Kadomtsev and supported by simulation [8] by Sykes, as the plasma current density in the core region increases (q(0) drops below unity), the m/n=1/1 mode becomes unstable and a pressure driven instability flattens the central pressure as well as the plasma current through an "X-point" reconnection zone along the magnetic pitch of the q~1 surface on the reconnection time scale defined by earlier references [7,8]. On the other hand, in the quasi-interchange model [9] proposed by Wesson, the core plasma having a flat q (q~1) profile inside the inversion radius becomes unstable due to a slight change of magnetic pitch angle instead of a pressure driven instability.…”

Comparison Study of 2D Images of Temperature Fluctuations during Sawtooth Oscillation with Theoretical Models

“…The first measurements with the ECEI system on the reconnection process in TEXTOR have been recently published [8]. This Letter focuses on a direct comparison with three leading physical models developed for the sawtooth crash phenomenon.In the full reconnection model [9,10], the plasma current density in the core region increases [q 0 drops below unity], and the m=n 1=1 internal kink mode becomes unstable due to a pressure driven instability. Island formation starts due to an influx of the cooler part of the plasma PRL 96,…”

“…In the full reconnection model [9,10], the plasma current density in the core region increases [q 0 drops below unity], and the m=n 1=1 internal kink mode becomes unstable due to a pressure driven instability. Island formation starts due to an influx of the cooler part of the plasma PRL 96,…”

New Insights to the Sawtooth Oscillation (“m/n = 1/1 mode”) in Hot Plasmas based on High Resolution 2-D Images of Te Fluctuations

“…In order to identify the control mechanism(s), it is thus imperative to have a better understanding of the underlying physics of the sawtooth oscillation. The observed 2-D ECE images are directly compared with the predicted 2-D pattern of the prominent theoretical models for the sawtooth oscillation: the full reconnection [3,4], quasi-interchange [5], and ballooning mode [6,7] models. The time evolution of the growth/decay of the island/hot spot before the crash and the heat flow pattern after the crash resembles that of the full reconnection model; however, the reconnection process is not helically symmetric and the crash time is not even close to the resistive time scale.…”