A nonlinear dynamic model of relaxation oscillations in tokamaks

A nonlinear dynamic model of relaxation oscillations in tokamaks

A nonlinear dynamic model of relaxation oscillations in tokamaks 150 150 Mathew

A nonlinear dynamic model of relaxation oscillations in tokamaks

Tokamaks exhibit several types of relaxation oscillations such as sawteeth, fishbones and Edge Localized Modes (ELMs) under appropriate conditions. Several authors have introduced model nonlinear dynamic systems with a small number of degrees of freedom which can illustrate the generic characteristics of such oscillations. In these models, one focuses on physically ‘‘relevant’’ degrees of freedom, without attempting to simulate all the myriad details of the fundamentally nonlinear tokamak phenomena. Such degrees of freedom often involve the plasma macroscopic quantities such as pressure or density and also some measure of the plasma turbulence, which is thought to control transport. In addition, ‘‘coherent’’ modes may be involved in the dynamics of relaxation, as well as radial electric fields, sheared flows, etc. In the present work, an extension of an earlier sawtooth model (which involved only two degrees of freedom) due to the authors is presented. The dynamical consequences of a pressure-driven ‘‘coherent’’ mode, which interacts with the turbulence in a specific manner, are investigated. Varying only the two parameters related to the coherent mode, the bifurcation properties of the system have been studied. These turn out to be remarkably rich and varied and qualitatively similar to the behavior found experimentally in actual tokamaks. The dynamic model presented involves only continuous nonlinearities and is the simplest known to the authors that can yield features such as sawteeth, ‘‘compound sawteeth’’ with partial crashes, ‘‘monster’’ sawteeth, metastability, intermittency, chaos, periodic and ‘‘grassy’’ ELMing in appropriate regions of parameter space. The results suggest that linear stability analysis of systems, while useful in elucidating instability drives, can be misleading in understanding the dynamics of nonlinear systems over time scales much longer than linear growth times and states far from stable equilibria.

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01/06/1999