The relationship between q profiles, transport, and sawteeth in tokamaks

Abstract

The purpose of this paper is to investigate the possible relationships between transport properties such as thermal diffusivity and resistivity on the one hand, and the magnetic properties such as q profile and toroidal flux change on the other, in tokamaks that exist under macroscopically quasistationary conditions. It is experimentally well established that when the sources are held constant over times long compared with energy and particle confinement times, tokamak discharges can exist in a quasistationary state with or without periodic sawteeth superposed on the basic equilibrium. The principal aim of this study is to provide qualitative physical insight into the nature of such states. For simplicity, single-fluid equations are assumed and the analysis is restricted to a cylinder model. The complete set of conservation equations comprising continuity, pressure balance, Ohm’s law, and energy balance are used along with appropriate sources. The conductive energy loss is assumed to occur due to an anomalous thermal diffusivity. Following earlier time-dependent studies of tokamak transport phenomenology due to the authors [Haas and Thyagaraja, Plasma Phys. Controlled Fusion 28, 1093 (1986)], the friction force in the generalized Ohm’s law is assumed to be described by an effective resistivity tensor with its principal axes in the poloidal and toroidal directions. The toroidal resistivity is taken to be of order Spitzer while the poloidal component ηθ is assumed to be anomalously large compared with ηz. A range of phenomena involving Ohmic discharges, auxiliary heated discharges, and noninductively driven currents is investigated. Attention is drawn to the joint implications of the conservation laws and the general forms of the constitutive relations for the structure of the profiles and conditions for the existence of equilibria. In particular, it is shown that βp>1 is achievable in macroscopically steady conditions, only if a sufficiently strong particle source is present in addition to an energy source. It also follows from the analysis that low βp discharges require an anomalous thermal force type term in Ohm’s law, in addition to the anomalous poloidal resistivity. Recent experimental results on sawtoothing discharges are used to extend the theoretical considerations from strictly steady discharges to those involving sawteeth. A simple nonlinear dynamical model of sawtoothing is constructed and used to illustrate some of the features of sawtooth oscillations. The paper is intended to complement fuller numerical studies. It may help in understanding transport and large-scale relaxation processes like the sawtooth in tokamaks by providing a set of theoretical relations that can be subjected to a direct experimental test. It is a feature of the results that they are independent of the details of the turbulent dynamics, which are ultimately thought to be responsible both for the form and scaling of the constitutive properties considered.