Nonlinear frequency chirping of the energetic-particle-driven geodesic acoustic mode (EGAM) is investigated using a hybrid simulation code for energetic particles interacting with a magnetohydrodynamic fluid. It is demonstrated in the simulation result that both frequency chirping up and chirping down take place in the nonlinear evolution of the EGAM. It is found that two hole-clump pairs are formed in the energetic particle distribution function in two-dimensional velocity space of pitch angle variable and energy. One pair is formed in the phase space region that destabilizes the instability, while the other is formed in the stabilizing region. The transit frequency of the hole (clump) in the destabilizing region chirps up (down), while in the stabilizing region the hole (clump) chirps down (up). The transit frequencies of particles in the holes and clumps are in good agreement with the chirping EGAM frequency indicating that the particles are kept resonant with the EGAM during the nonlinear frequency chirping. Continuous energy transfer takes place from the destabilizing phase space region to the stabilizing region during the spontaneous frequency chirping of the wave.

Strongly magnetized plasmas are rich in spatial and temporal scales, making a computational approach useful for studying these systems. The most accurate model of a magnetized plasma is based on a kinetic equation that describes the evolution of the distribution function for each species in six-dimensional phase space. High dimensionality renders this approach impractical for computations for long time scales. Fluid models are an approximation to the kinetic model. The reduced dimensionality allows a wider range of spatial and/or temporal scales to be explored. Computational modeling requires understanding the ordering and closure approximations, the fundamental waves supported by the equations, and the numerical properties of the discretization scheme. Several ordering and closure schemes are reviewed and discussed, as are their normal modes, and algorithms that can be applied to obtain a numerical solution.

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Optimal strategies for disruption mitigation benefit from the understanding of details both spatially and temporally. Beyond the assessment of the efficacy of a particular proposed Disruption Mitigation System (DMS), ITER's longevity will require accounting of both mitigated and unmitigated disruptions. Accurate models and validated simulations that detail multiple ITER scenarios with mitigated and unmitigated disruptions are essential for accurate estimates of load damage. The primary candidate for ITER's DMS is Shattered Pellet Injection (SPI); its efficacy must be evaluated within the next several years. To perform critical time dependent 3-D nonlinear simulations, we have developed a particle based SPI model in the NIMROD code coupled to its modified single fluid equations with impurity and radiation [Izzo, Nucl. Fusion 46(5), 541 (2006)]. SPI validation simulations of the thermal quench and comparisons to DIII-D impurity scan experiments [Shiraki et al., Phys. Plasmas 23(6), 062516 (2016)] are presented. We also present an initial ITER Q = 10 pure neon SPI simulation and compare it with the DIII-D SPI simulations. NIMROD SPI simulations demonstrate that the ablating fragment drives strong parallel flows that transport the impurities and governs the thermal quench. Analysis of SPI simulations shows that the mixed deuterium/neon SPI results in a more benign thermal quench due to the enhanced transport caused by the additional deuterium. These results suggest that an optimal pellet mixture exists for the SPI system.

We present a comparison study of 3-D pressureless resistive MHD (rMHD) and 3-D presureless two-fluid MHD models of the Helicity Injected Torus with Steady Inductive helicity injection (HIT-SI). HIT-SI is a current drive experiment that uses two geometrically asymmetric helicity injectors to generate and sustain toroidal plasmas. The comparable size of the collisionless ion skin depth d i to the resistive skin depth predicates the importance of the Hall term for HIT-SI. The simulations are run with NIMROD, an initial-value, 3-D extended MHD code. The modeled plasma density and temperature are assumed uniform and constant. The helicity injectors are modeled as oscillating normal magnetic and parallel electric field boundary conditions. The simulations use parameters that closely match those of the experiment. The simulation output is compared to the formation time, plasma current, and internal and surface magnetic fields. Results of the study indicate 2fl-MHD shows quantitative agreement with the experiment while rMHD only captures the qualitative features. The validity of each model is assessed based on how accurately it reproduces the global quantities as well as the temporal and spatial dependence of the measured magnetic fields. 2fl-MHD produces the current amplification I tor I inj and formation time s f demonstrated by HIT-SI with similar internal magnetic fields. rMHD underestimates I tor I inj and exhibits much a longer s f . Biorthogonal decomposition (BD), a powerful mathematical tool for reducing large data sets, is employed to quantify how well the simulations reproduce the measured surface magnetic fields without resorting to a probe-by-probe comparison. BD shows that 2fl-MHD captures the dominant surface magnetic structures and the temporal behavior of these features better than rMHD. V C 2013 AIP Publishing LLC. [http://dx.

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, τ res  =  4πa 2/(c 2 η), 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 τ res. 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, T e drops from 2 keV to 10 eV in 0.1 ms and Z eff 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 τ res 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 τ res 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 T e 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.

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