Nonlinear MHD Simulations Research Articles

Resistivity effect in the vicinity of a coronal magnetic null point

Introduction: We aim to examine how magnetic resistivity impacts the movement of magnetoacoustic waves near a magnetic null-point in the solar corona.Method: The resistive, nonlinear MHD simulations are solved by the PLUTO code in 2.5D for different amount of the resistivity.Results and Discussion: Propagation of magnetoacoustic waves in the vicinity of a magnetic null point has the potential to create current sheets with high current density excitation and plasmoid generation. During the entire duration of the simulation, it is discovered that plasma density became significant due to the plasmoid and also current density is high for high resistivity. It is depicted that high resistivity also leads to bigger plasmoids or magnetic islands in comparison to small resistivity.

The Effect of Resistivity on the Periodicity of Oscillatory Reconnection

The oscillatory reconnection mechanism is investigated for a parameter study of eight orders of magnitude of resistivity, with a particular interest in the evolution of the oscillating current density at the null point and its associated periodicity. The resistive, nonlinear MHD simulations are solved in 2.5D for different levels of resistivity. Three methods (wavelet analysis, Fourier transform, and ANOVA) are used to investigate the effect of resistivity versus resultant period. It is found that there is an independence between the level of background resistivity and the period of the oscillatory reconnection mechanism. Conversely, it is found that resistivity has a significant effect on the maximum amplitude of the current density and the nature of its decay rate, as well as the magnitude of ohmic heating at the null.

Plasmoid-mediated reconnection during nonlinear peeling–ballooning edge-localized modes

Plasmoid-mediated reconnection is investigated for nonlinear peeling–ballooning edge-localized modes (P-B ELMs). The formation of current sheets and the transition to 3D current sheet instability is demonstrated through fully nonlinear resistive MHD simulations of P-B ELMs in DIII-D discharges. Large-scale axisymmetric current sheets, as well as small-scale poloidally extended current sheets, are formed as the coherent P-B ELM filaments nonlinearly evolve. It is observed that, at high Lundquist numbers, these current sheets break during a reconnection burst, i.e. a secondary exponential growth of intermediate modes followed by relaxation due to the suppression of P-B drive.

Transition in particle transport under resonant magnetic perturbations in a tokamak

Nonlinear 3D MHD simulations and validations reveal that the hybrid particle-MHD transport is a key process for driving the pump-out in the presence of Resonant Magnetic Perturbations (RMPs) in the KSTAR tokamak. Particle transport and the resulting density pump-out by RMPs are shown to be composed of not only the classical flow convection near magnetic islands due to polarization but also the neoclassical ion diffusion across perturbed magnetic surfaces. The latter is known as the Neoclassical Toroidal Viscosity (NTV) and is integrated into nonlinear MHD simulations here for the first time, revealing that the two-stage pump-outs observed in KSTAR experiments are reproduced only with such integrated nonlinear MHD and transport evolution. Near-resonant responses, which have received less attention than the resonant response, play distinct roles in the pump-out along with the island formation. In addition, this modeling is used to investigate the pump-outs in double-null-like plasmas and numerically capture the effect of the double-null shape on the pump-outs, which may explain the difficulty of Edge Localized Mode (ELM) suppression access in double-like plasmas. This reveals new aspects of the impact toroidal geometry and mode coupling have on 3D physics and reveals the importance of near-resonant components in suppressing ELMs.

MHD simulations of formation, sustainment and loss of quiescent H-mode in the all-tungsten ASDEX Upgrade

Periodic edge localized modes (ELMs) are the non-linear consequences of pressure-gradient-driven ballooning modes and current-driven peeling modes becoming unstable in the pedestal region of high confinement fusion plasmas. In future tokamaks like ITER, large ELMs are foreseen to severely affect the lifetime of wall components as they transiently deposit large amounts of heat onto a narrow region at the divertor targets. Several strategies exist for avoidance, suppression, or mitigation of these instabilities, such as the naturally ELM-free quiescent H-mode (QH-mode). In the present article, an ASDEX Upgrade (AUG) equilibrium that features a QH-mode is investigated through non-linear extended magneto-hydrodynamic simulations covering the dynamics over tens of milliseconds. The equilibrium is close to the ideal peeling limit and non-linearly develops saturated modes at the edge of the plasma. A dominant toroidal mode number of n = 1 is found, for which the characteristic features of the edge harmonic oscillation are recovered. The saturated modes contribute to heat and particle transport preventing pedestal build-up to the ELM triggering threshold. The non-linear dynamics of the mode, in particular its interaction with the evolution of the edge safety factor, are studied, and suggest a possible new saturation mechanism for the QH-mode. The simulations show good qualitative and quantitative agreement with experiments in AUG. In particular, the processes leading to the termination of QH-mode above a density threshold are studied, which results in the transition into an ELM regime. In the vicinity of this threshold, limit cycle oscillations are observed.

Viscosity impact on 3D non-linear MHD simulations of RFP fusion plasmas

Several studies pointed out the joint role of resistivity η and viscosity ν in determining the dynamics and the emergence of helical regimes of reversed-field pinch (RFP) plasmas. In this framework, the self-consistent time evolution of the η and ν coefficients still lacks of a fully satisfying modeling, being constrained by many approximations. In this work, the hypothesis of a flat viscosity profile is relaxed: A viscosity profile inspired by the Braginskii perpendicular viscosity is implemented in the code. This choice is motivated by the fact that the magnetohydrodynamics field instabilities relevant for the RFP configuration dynamics (resistive-kink/tearing modes) are active in the direction perpendicular to the magnetic field. Such a non-monotonous profile causes a localized damping of plasma flow in the regions, where the viscosity is stronger, close to the plasma edge. This results in the reduction of the flow shear, in turn allowing the enhancement of edge magnetic field modes amplitude. The impact on the magnetic topology and on connection length to the wall is also analyzed.

MHD modeling of shattered pellet injection in JET

Nonlinear 3D MHD simulations of shattered-pellet injection (SPI) in JET show prototypical SPI-driven disruptions using the M3D-C1 and NIMROD extended-MHD codes. Initially, radiation-driven thermal quenches are accelerated by MHD activity as the pellet crosses rational surfaces, leading to a radiation spike, global stochasticization of the magnetic field, and a complete thermal quench. Eventually, current quenches, preceded by a current spike are seen as the Ohmic heating becomes equal to the radiative cooling. The results are qualitatively similar for both a single monolithic pellet, pencil-beam model, and a realistic shatter to represent the SPI plume. A scan in viscosity from 500 to 2000 m2 s−1 for MHD simulations finds that reducing viscosity increases MHD activity and decreases thermal quench time slightly. A realistic cloud of fragments modeling shows that mixed-D–Ne pellet travels deeper into the plasma core before the thermal quench. At the slow pellet speeds, the pellet is found to be moving slowly enough inward that even the 5% neon in the mixed pellet is enough to effectively radiate the thermal energy available. Radiation toroidal peaking is predicted to be at levels consistent with experimental observations and reduced as the pellet travels deeper into the plasma. These simulations lay the ground work for more-sophisticated validative and predictive modeling of SPI in JET using both M3D-C1 and NIMROD.

On the origin of the plasma current spike during a tokamak disruption and its relation with magnetic stochasticity

A JOREK 3D non-linear MHD simulation of a disruption triggered by an argon massive gas injection in JET, which quantitatively reproduces the plasma current () spike (Nardon et al 2021 Plasma Phys. Control. Fusion 63 115006), is analyzed in order to investigate the origin of the spike and its relation with magnetic stochasticity. The spike is associated to a current density (j φ ) profile relaxation which appears to result from Shear Alfvén Wave (SAW) propagation along stochastic field lines, as proposed by Boozer (2019 Plasma Phys. Control. Fusion 61 024002; 2020 Phys. Plasmas 27 102305), possibly complemented by a macroscopic E×B flow structure. Using axisymmetric JOREK simulations involving a mean field Ohm’s law, we verify that the level of hyper-resistivity associated to SAWs is consistent with the prediction made in (Boozer 2019 Plasma Phys. Control. Fusion 61 024002; Boozer 2020 Phys. Plasmas 27 102305), which connects the spike with the level of stochasticity. The relaxation comprises two main phases, the first one corresponding to a fast (0.1 ms) and almost complete j φ flattening in the q < 2 region, while the second one is longer (0.5 ms) and corresponds to a more gradual, global and incomplete j φ flattening. During the first phase, strong E×B flows develop that play a key role in mixing impurities into the core.

Role of edge-localized neoclassical tearing modes in quiescent H-mode plasmas in the DIII-D tokamak

The edge-harmonic oscillations (EHOs) in standard quiescent H-mode (QH-mode) plasmas in DIII-D are consistent with edge-localized neoclassical tearing modes (NTMs) based on nonlinear two-fluid MHD simulations. Using kinetic equilibria constrained by edge profile measurements, the MHD simulations show that the n = 1 NTM and its harmonics can be destabilized at the pedestal top of QH-mode plasma by the edge bootstrap current. The simulations further show that the unstable NTMs can saturate either at small (&amp;lt;2% ψN) or large (&amp;gt;4% ψN) island width depending on the magnitude of the edge bootstrap current, where ψN is the normalized radius in poloidal flux. The onset of the EHO also results in a prompt decrease in the pedestal width and height, consistent with simulation results for the onset of the NTM at the top of the QH-mode pedestal. This suggests that the avoidance of edge-localized modes (ELMs) in QH-mode can be attributed to the enhanced local transport induced by the NTM that is sufficient to prevent the expansion of the pedestal to an unstable width, analogous to the mechanism explored for ELM suppression by resonant magnetic perturbations. Nonlinear MHD simulations scanning the E × B frequency and the ratio of parallel and perpendicular thermal diffusivity (χǁ/χ⊥) at the pedestal top show that edge-localized NTMs are destabilized for conditions of high E × B frequency, high pedestal temperature, and low pedestal density, qualitatively consistent with experimental conditions required for observing the EHO.

Dynamics of ultralow-q plasmas in the RFX-mod device

The results are presented of an experimental activity performed in the RFX-mod device aimed at characterizing plasma dynamics in the so-called Ultralow-q (ULq) magnetic configuration, which corresponds to edge safety factor values below 1. The role of the edge safety factor in determining plasma dynamics is studied. In particular, a characterization of MHD activity is performed. The results of dedicated non-linear 3D visco-resistive MHD simulations are in good quantitative agreement with the experimental observations. In particular, the predicted tendency for ULq plasmas to be characterized by magnetic spectra dominated by a single mode (either a kink or a double resonant internal mode) is confirmed by experiment. Magnetic reconnection plays a relevant role in determining the dynamics of the magnetic topology. Both almost quiescent and largely fluctuating plasmas are observed with a strong sensitivity on the edge safety factor. The main MHD properties of the ULq are compared to those of RFP and tokamak discharges, also produced in the RFX-mod device. MHD modes exhibit toroidal rotation at a frequency depending on mode amplitude. Differently from what encountered in RFP plasmas at comparable current levels, no wall locking is detected.

A generalised formulation of G-continuous Bezier elements applied to non-linear MHD simulations

The international tokamak ITER is progressing towards assembly completion and first-plasma operation, which will be a physics and engineering challenge for the fusion community. In the preparation for ITER experimental scenarios, non-linear MHD simulations are playing an essential role to actively understand and predict the behaviour and stability of tokamak plasmas in future fusion power plant. The development of MHD codes like JOREK is a key aspect of this research effort, and provides invaluable insight into the plasma stability and the control of global and localised plasma events, like Edge-Localised-Mode and disruptions. In this paper, we present an operational implementation of a new, generalised formulation of Bezier finite-elements applied to the JOREK code, a significant advancement from the previously G1-continuous bi-cubic Bezier elements. This new mathematical method enables any polynomial order of Bezier elements, with a guarantee of G-continuity at the level of (n−1)/2, for any odd n, where n is the order of the Bezier polynomials. The generalised method is defined, and a rigorous mathematical proof is provided for the G-continuity requirement. Key details on the code implementation are mentioned, together with a suite of tests to demonstrate the mathematical reliability of the finite-element method, as well as the practical usability for typical non-linear tokamak MHD simulations. A demonstration for a state-of-the-art simulation of an Edge-Localised-Mode instability in the future ITER tokamak, with realistic grid geometry, finalises the study.

Nonlinear MHD simulation of core plasma collapse events in Wendelstein 7-X

Three-dimensional nonlinear MHD simulations study the core collapse events observed in a stellarator experiment, Wendelstein 7-X. In the low magnetic shear configuration like the Wendelstein 7-X, the rotational transform profile is very sensitive to the toroidal current density. The 3D equilibrium with localized toroidal current density is studied. If the toroidal current density follows locally in the middle of the minor radius, the rotational transform is also changed locally. Sometimes, the magnetic topology changes due to appearing the magnetic island. A full three-dimensional nonlinear MHD code studies the nonlinear behaviors of the MHD instability. It was found that the following sequence. At first, the high-n ballooning-type mode structure appears in the plasma core, and then the mode linearly grows. The high-n ballooning modes nonlinearly couple and saturate. The mode structure changes to the low-n mode. The magnetic field structure becomes strongly stochastic into the plasma core due to the nonlinear coupling in that phase. Finally, the plasma pressure diffuses along the stochastic field lines, and then the core plasma pressure drops. This is a crucial result to interpret the core collapse event by strong nonlinear coupling.

Comparing spontaneous and pellet-triggered ELMs via non-linear extended MHD simulations

Injecting frozen deuterium pellets into an ELMy H-mode plasma is a well established scheme for triggering edge localized modes (ELMs) before they naturally occur. This paper presents non-linear simulations of spontaneous type-I ELMs and pellet-triggered ELMs in ASDEX Upgrade performed with the extended MHD code JOREK. A thorough comparison of the non-linear dynamics of these events is provided. In particular, pellet-triggered ELMs are simulated by injecting deuterium pellets into different time points during the pedestal build-up described in A Cathey et al (2020 Nuclear Fusion 60 124007). Realistic ExB and diamagnetic background plasma flows as well as the time dependent bootstrap current evolution are included during the build-up to accurately capture the balance between stabilising and destabilising terms for the edge instabilities. Dependencies on the pellet size and injection times are studied. The spatio-temporal structures of the modes and the resulting divertor heat fluxes are compared in detail between spontaneous and triggered ELMs. We observe that the premature excitation of ELMs by means of pellet injection is caused by a helical perturbation described by a toroidal mode number of n = 1. In accordance with experimental observations, the pellet-triggered ELMs show reduced thermal energy losses and a narrower divertor wetted area with respect to spontaneous ELMs. The peak divertor energy fluence is seen to decrease when ELMs are triggered by pellets injected earlier during the pedestal build-up.

Transition from no-ELM response to pellet ELM triggering during pedestal build-up—insights from extended MHD simulations

Pellet edge localized mode (ELM) triggering is a well-established scheme for decreasing the time between two successive ELM crashes below its natural value. Reliable ELM pacing has been demonstrated experimentally in several devices, increasing the ELM frequency considerably. However, it was also shown that the frequency cannot be increased arbitrarily due to a so-called lag-time. During this time, after a preceding natural or triggered ELM crash, neither a natural ELM crash occurs nor is it possible to trigger an ELM crash by pellet injection. For this article, pellet ELM triggering simulations are advanced beyond previous studies in two ways. Firstly, realistic E × B and diamagnetic background flows are included. And secondly, the pellet is injected at different stages of the pedestal build-up. This allows us to recover the lag time for the first time in simulations and investigate it in detail. A series of nonlinear extended MHD simulations is performed to investigate the plasma dynamics resulting from an injection at different time points during the pedestal build-up. The experimentally observed lag-time is qualitatively reproduced. In particular, a sharp transition is observed between the regime where no ELMs can be triggered and the regime where pellet injection causes an ELM crash. Via variations of pellet parameters and injection time, the two regimes are studied and compared in detail, revealing pronounced differences in the nonlinear dynamics. The toroidal mode spectrum is significantly broader when an ELM crash is triggered, enhancing the stochasticity and therefore also the losses of thermal energy along magnetic field lines. In the heat fluxes to the divertor targets, pronounced toroidal asymmetries are observed. In the case of high injection velocities leading to deep penetration, the excitation of core modes like the 2/1 neoclassical tearing mode is also observed.

Pedestal collapse by resonant magnetic perturbations

Pedestal collapse (i.e., the complete loss of the edge transport barrier (ETB)) in DIII-D H-mode plasmas occurs when resonant magnetic perturbations (RMPs) penetrate the steep gradient region at the plasma edge. Normally, RMP driven magnetic islands can occur at the top and bottom of the H-mode pedestal and these islands generate conditions consistent with edge-localized-mode (ELM) suppression and density pump-out, respectively, based on nonlinear two-fluid MHD simulations. In contrast, MHD simulations show that the steep pressure gradient region between the top and bottom of the DIII-D pedestal is generally immune to resonant field penetration due to large local E × B and diamagnetic flows. By this fortuitous circumstance, the edge-transport-barrier and H-mode confinement can be maintained while achieving ELM suppression. However, pedestal collapse can occur in DIII-D when the screening flows are inadequate to prevent field penetration in the steep gradient region of the pedestal. Non-linear two-fluid MHD simulations support the role of resonant field penetration in pedestal collapse for DIII-D H-mode plasmas with weak edge E × B and diamagnetic screening flows. ITER will likely have weaker edge screening flows than present experiments due to its much larger size, making it more susceptible to resonant field penetration in the steep gradient region of the pedestal. Analysis of model ITER equilibria demonstrates that resonant field penetration in the steep pressure gradient region is possible for RMP levels of the order required for ELM suppression. The effect of such penetration on the ITER pedestal will depend sensitively on the resulting degree of island overlap.

Radiation asymmetry and MHD destabilization during the thermal quench after impurity shattered pellet injection

The radiation response and the MHD destabilization during the thermal quench after a mixed species shattered pellet injection with impurity species neon and argon are investigated via 3D non-linear MHD simulation using the JOREK code. Both the n = 0 global current profile contraction and the local helical cooling at each rational surface caused by the pellet fragments are found to be responsible for MHD destabilization after the injection. Significant current driven mode growth is observed as the fragments cross low order rational surfaces, resulting in rapidly inward propagating stochastic magnetic field, ultimately causing the core temperature collapse. The thermal quench (TQ) is triggered as the fragments arrive on the q = 1 or q = 2 surface depending on the exact q profile and thus mode structure. When injecting from a single toroidal location, strong radiation asymmetry is found before and during the TQ as a result of the unrelaxed impurity density profile along the field line and asymmetric outward heat flux. Such asymmetry gradually relaxes over the course of the TQ, and is entirely eliminated by the end of it. Simulation results indicate that the aforementioned asymmetric radiation behavior could be significantly mitigated by injection from toroidally opposite locations, provided that the time delay between the two injectors is shorter than 1 ms. It is also found that the MHD response are sensitive to the relative timing and injection configuration in these multiple injection cases.

Wide Operational Windows of Edge-Localized Mode Suppression by Resonant Magnetic Perturbations in the DIII-D Tokamak.

Edge-localized mode (ELM) suppression by resonant magnetic perturbations (RMPs) generally occurs over very narrow ranges of the plasma current (or magnetic safety factor q_{95}) in the DIII-D tokamak. However, wide q_{95} ranges of ELM suppression are needed for the safety and operational flexibility of ITER and future reactors. In DIII-D ITER similar shape plasmas with n=3 RMPs, the range of q_{95} for ELM suppression is found to increase with decreasing electron density. Nonlinear two-fluid MHD simulations reproduce the observed q_{95} windows of ELM suppression and the dependence on plasma density, based on the conditions for resonant field penetration at the top of the pedestal. When the RMP amplitude is close to the threshold for resonant field penetration, only narrow isolated magnetic islands form near the top of the pedestal, leading to narrow q_{95} windows of ELM suppression. However, as the threshold for field penetration decreases with decreasing density, resonant field penetration can take place over a wider range of q_{95}. For sufficiently low density (penetration threshold) multiple magnetic islands form near the top of the pedestal giving rise to continuous q_{95} windows of ELM suppression. The model predicts that wide q_{95} windows of ELM suppression can be achieved at substantially higher pedestal pressure in DIII-D by shifting to higher toroidal mode number (n=4) RMPs.

Towards validated MHD modeling of edge harmonic oscillation in DIII-D QH-mode discharges

The extended-MHD NIMROD code (Sovinec C.R. and King J.R. 2010 J. Comput. Phys. 229 5803) is used to simulate the dynamics of an edge harmonic oscillation (EHO) in quiescent H-mode (QH-mode) DIII-D (Luxon J.L. 2002 Nucl. Fusion 42 614) discharge 163 518. EHOs observed in non-linear MHD simulations have n = 1 and n = 2 as dominant modes akin the DIII-D experiment. Kinetic equilibrium reconstructions during the time of the fully-developed EHO include the effect of the MHD profile relaxation and are found below the stability boundary. This paper discusses methods to include additional instability drives to the experimental equilibria in order to trigger EHO formation. The experimental equilibrium for the DIII-D discharge 163 518 is modified to include two levels of instability drive by increasing the experimental pressure gradient. In order to do a more direct comparison of the simulation results with the experiment, a synthetic BES diagnostic is used to compute cross-correlation and cross-power spectral densities associated with the simulated density perturbations. It is shown that the amplitude of the experimental density perturbations is between the computed density perturbation amplitude for the two levels of instability drive. The synthetic cross-power spectral density shows a transition from a double to a single peak in frequency when the BES analysis shifts from near the LCFS towards the steep gradient region of the pedestal. This observation is similar to the experiment, but the first peak frequency for the weak instability drive is found below the experimental frequencies, and the second peak for the strong instability drive is found above the experimental peak frequencies. However, these peak frequencies are in agreement with the local flow estimate and a MHD turbulence bursty behavior in the simulations with the strong instability drive.

Empirical scaling of the n = 2 error field penetration threshold in tokamaks

This paper presents a multi-machine, multi-parameter scaling law for the n = 2 core resonant error field threshold that leads to field penetration, locked modes, and disruptions. Here, n is the toroidal harmonic of the non-axisymmetric error field (EF). While density scalings have been reported by individual tokamaks in the past, this work performs a regression across a comprehensive range of densities, toroidal fields, and pressures accessible across three devices using a common metric to quantify the EF in each device. The metric used is the amount of overlap between an EF and the spectrum that drives the largest linear ideal MHD resonance, known as the "dominant mode overlap". This metric, which takes into account both the external field and plasma response, is scaled against experimental parameters known to be important for the inner layer physics. These scalings validate non-linear MHD simulation scalings, which are used to elucidate the dominant inner layer physics. Both experiments and simulations show that core penetration thresholds for EFs with toroidal mode number n = 2 are of the same order as the n = 1 thresholds that are considered most dangerous on current devices. Both n = 1 and n = 2 thresholds scale to values within the ITER design tolerances, but data from additional devices with a range of sizes are needed in order to increase confidence in quantitative extrapolations of n = 2 thresholds to ITER.

The role of edge resonant magnetic perturbations in edge-localized-mode suppression and density pump-out in low-collisionality DIII-D plasmas

Two-fluid nonlinear MHD simulations using the TM1 code demonstrate that the formation of magnetic islands at the top and bottom of the H-mode pedestal, together with the strong screening of resonant fields in the gradient region of the pedestal, can account for ELM suppression and density pump-out by n = 2 resonant magnetic perturbations (RMPs) in low-collisionality DIII-D ITER similar shape (ISS) plasmas. Using experimentally relevant transport coefficients, neoclassical resistivity, electron collisionality, and RMP amplitudes, nonlinear MHD simulations reproduce the observed level of density reduction (density pump-out) in DIII-D due to the formation of narrow magnetic islands in the resistive foot of pedestal. For large amplitude RMPs (Br/Bt > 1×10−4) simulations predict resonant field penetration and resulting in pressure reduction at the top of the pedestal, consistent with experimental observations at the onset of ELM suppression. The predicted reduction in the height and width of the pedestal by magnetic island enhanced collisional transport provides a quantitative mechanism for the stabilization of the peeling-ballooning mode (PBM). Importantly, these simulations predict strong screening of resonant fields in the steep gradient region of the pedestal due to strong E×B rotation and diamagnetic flows. However, if the plasma resistivity is made artificially larger (∼ 10X) than neoclassical, the simulations predict magnetic stochasticity throughout the plasma edge and the collapse of the pedestal due to the reduction in the penetration threshold with increasing resistivity. A scaling relation for resonant field penetration at the pedestal top, using several hundred nonlinear simulations, reproduces the density and E×B rotation dependence of the ELM suppression threshold observed in DIII-D. Simulations using ITER model equilibria suggest that the penetration threshold at the top of the ITER pedestal will be lower than in DIII-D due to the anticipated lower perpendicular flow velocities in ITER, resulting in the weaker screening of resonant fields.