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. Se veral authors have introduced model nonlinear dynamic systems with a small number of degrees of freedom which can illustrate the generic characteristi cs of such oscillations. In these models, one focuses on physically "releva nt'' degrees of freedom, without attempting to simulate all the myriad deta ils of the fundamentally nonlinear tokamak phenomena. Such degrees of freed om often involve the plasma macroscopic quantities such as pressure or dens ity and also some measure of the plasma turbulence, which is thought to con trol transport. In addition, "coherent'' modes may be involved in the dynam ics of relaxation, as well as radial electric fields, sheared flows, etc. I n the present work, an extension of an earlier sawtooth model (which involv ed only two degrees of freedom) due to the authors is presented. The dynami cal consequences of a pressure-driven "coherent'' mode, which interacts wit h the turbulence in a specific manner, are investigated. Varying only the t wo parameters related to the coherent mode, the bifurcation properties of t he system have been studied. These turn out to be remarkably rich and varie d and qualitatively similar to the behavior found experimentally in actual tokamaks. The dynamic model presented involves only continuous nonlineariti es 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 appro priate regions of parameter space. The results suggest that linear stabilit y 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 equi libria. [S1070-664X(99)00506-6].