Time-dependent runaway electron simulations: Ampere–Faraday equations implemented in CQL3D

Abstract

The runaway electron (RE) distributions driven by a large toroidal electric field induced by the drop in the temperature profile due to disruption or pellets are comprehensively simulated by the 3D Fokker–Planck (FP) solver CQL3D (Harvey and McCoy 1992 Proc. of IAEA TCM), recently coupled to the Ampere–Faraday (AF) equations. The evolution of the toroidal current in a plasma occurs on a resistive time scale, τres = 4πa2/(c2η), which is typically of the order of seconds in present tokamaks. Here, a and η are respectively plasma radius or radial extent of a current density perturbation, and Ohmic resistivity. From the Faraday EM equation, the toroidal electric field is proportional to the time derivative of the poloidal magnetic field, which, from the Ampere equation, is proportional to the toroidal current. Thus, the toroidal electric field rapidly increases due to an abrupt temperature drop decrease in conductivity, to prevent change in the toroidal current faster than τres. This is a example of Lenz’s law. For example, in simulations with KPRAD (Whyte et al 2003 J. Nucl. Mater. 313–6 1239) of neon pellet injection into a DIII-D shot, Te drops from 2 keV to 10 eV in 0.1 ms and Zeff increases 1–4, giving that the electric field increases 3500× to 0.8 V cm−1. As described in Harvey et al (2000 PoP 7 4590), this places much of the tail electron distribution beyond the Dreicer runaway velocity, giving so-called ‘hot-tail runaways’ which for a time are the dominant source of runaways, more so than the knockon source. In this prior calculation, performed for a single flux surface, the toroidal current density is held constant, on the basis that τres is large. Most of the initial current can be converted to runaway current, which is then dangerous, particularly for ITER. A more comprehensive A–F model recently implemented in CQL3D, taking into account the time-development of the full-plasma-width toroidal electric field on time-scales of order τres applies an iterative technique for the toroidal field previously developed for a different application (Kupfer et al 1996 PoP 3 3644), maintaining the implicit-in-time evolution of CQL3D. The degree of runaway current formation is reduced in AF augmented CQL3D, but the basic mechanism of ‘hot-tail runaways’ remains a dominant contribution to the REs at early times after the Te drop in these simulations. On the other hand, a NIMROD (Sovinec et al 2004 J. Comput. Phys. 195 355) simulation of shattered-pellet shutdown of DIII-D plasma (Kim 2018 APS/DPP Meeting), gives a slower thermal quench; when the plasma profiles and electric field are coupled one-way to CQL3D, the ‘hot-tail’ REs are much less, and growth of RE is dominated by the knockon process.