A Thermo-resistive Tearing Mode Model of the Density Limit in Tokamaks

Abstract

An upper limit on the maximum achievable plasma density is observed in all tokamaks. Since the fusion power is proportional to the square of plasma density, it is vitally important to understand the physical mechanism of the phenomenon to achieve high efficiency fusion energy generation. The exact formulation of this empirical limit evolved as better measurements became available. The most successful empirical scaling law of the density limit is the Greenwald density limit, which agrees well with the measurements on different tokamaks with various cross-sectional shapes. The phenomenology and existing theories of the Greenwald density limit are briefly reviewed. Our work is inspired by some generally accepted observations of the density limit. The density limit is associated with cooling of the edge plasma and radiation as well as the onset of MHD modes. In this work, we present a thermo-resistive tearing mode formalism that predicts the density limit as the critical density of local power balance inside the magnetic island. In tokamaks, there are always some impurities cooling the plasma, especially at the edge. As the plasma density approaches the limit, the current density profile becomes more peaked, revealed by larger normalized internal inductance in the measurements. The Ohmic heating power drops and the impurity radiation power increases significantly, therefore the power balance inside the magnetic island changes from net heating to net cooling. The tearing mode then grows to a much larger size under net cooling. Large magnetic islands connect the core with the edge and lead to the confinement degradation or even a disruption. Semi-analytical calculations show quantitative agreement of the model with the empirical scaling law. Three-dimensional MHD simulations also confirm that the thermo-resistive tearing mode model explains the density limit with the correct phenomenology. Heat diffusion experiments reveal that the perpendicular heat transport inside the island is greatly reduced, which greatly strengthens the thermal effects on the tearing mode growth. A simple rigorous analytic model of the tearing mode, which is not directly related to the thermo-resistive tearing mode formalism, is presented in the appendix. Future work will focus on developing a self-consistent heat transport model coupled with the tearing mode growth model and comparing the model with experimental measurements directly.