Nonlinear Magnetohydrodynamic Simulations Research Articles

Exact Nonlinear Decomposition of Ideal-MHD Waves Using Eigenenergies

In this paper, we introduce a new method for exact decomposition of propagating, nonlinear magnetohydrodynamic (MHD) disturbances into their component eigenenergies associated with the familiar slow, Alfvén, and fast wave eigenmodes, and the entropy and field-divergence pseudoeigenmodes. First, the mathematical formalism is introduced, where it is illustrated how the ideal-MHD eigensystem can be used to construct a decomposition of the time variation of the total energy density into contributions from the eigenmodes. The decomposition method is then demonstrated by applying it to the output of three separate nonlinear MHD simulations. The analysis of the simulations confirms that the component wave modes of a composite wavefield are uniquely identified by the method. The slow, Alfvén, and fast energy densities are shown to evolve in exactly the way expected from comparison with known linear solutions and nonlinear properties, including processes such as mode conversion. Along the way, some potential pitfalls for the numerical implementation of the decomposition method are identified and discussed. We conclude that the exact, nonlinear decomposition method introduced is a powerful and promising tool for understanding the nature of the decomposition of MHD waves as well as analyzing and interpreting the output of dynamic MHD simulations.

Non-linear dynamics of the double tearing mode

3D non-linear magnetohydrodynamics simulations of a double tearing mode with the JOREK code are presented in the context of trying to better understand the benign termination of runaway electron beams observed in some experiments. It is shown that the non-linear behavior qualitatively depends on the resistivity η via its effect on how fast secondary, non-linearly destabilized, tearing modes grow relative to the primary mode. Within a certain range of η, a violent and global relaxation is observed, consistent with the “Kadomtsev-predicted” reconnection region extending from almost the very center up to the edge of the plasma.

On the frequency bifurcations of the MHD startup modes in NSTX

The observed bifurcations of the low frequency (<50 kHz) and low toroidal periodicity (n < 5) magnetohydrodynamic (MHD) activity often present in the initial part of the National Spherical Tokamak Experiment (NSTX) discharges can be explained by the evolution of the radial profile of the safety factor (q=rBϕ/RBθ) crossing multiple rational surfaces in the core. Important performance limiting instability mechanisms in the NSTX spherical tokamak are often linked to low frequency and low-n MHD activity. They are quite common in long-pulse NSTX plasmas. They can be present at the beginning of the plasma current flat-top, at the end of the discharge or during the whole duration, and they have been observed to deleteriously impact performance over a wide range of q95. An interesting feature observed in some NSTX discharges is the presence of a bifurcation in the frequency of the low n modes, as low as n = 1, that have frequencies comparable to the plasma core rotation divided by n. Equilibrium reconstructions constrained by magnetic diagnostics data and motional stark effect pitch angle radial profiles suggest that the observed bifurcations are linked to a fast evolving minimum value of q. 3D non-linear resistive MHD simulations show that these modes are ideal and exist as non-resonant before the correspondent rational surface enters the plasma.

Experiments and non-linear MHD simulations of hot vertical displacement events in ASDEX-Upgrade

Hot vertical displacement events (VDEs) are one of the worst case scenarios for high-current tokamaks as they are associated with large heat loads and electro-magnetic forces. Non-linear magneto-hydrodynamic (MHD) simulations of the thermal and current quench can help to understand their dynamics and consequences. In order to make predictions for future devices, the validation of codes against present machines is crucial. Dedicated experiments were performed in ASDEX Upgrade to provide a basis for simulations with the non-linear extended MHD code JOREK. 2D as well as non-axisymmetric simulations at realistic parameters can reproduce quantities like the edge safety factor q 95 at the thermal quench onset, the halo current magnitude and the level of vertical forces, while the exact width of the halo current area requires more sophisticated boundary conditions and is left for future work. Small horizontal forces are observed during the hot VDEs in the experiment as well as in the simulations.

Nonlinear magnetohydrodynamic modeling of current-drive-induced sawtooth-like crashes in the W7-X stellarator

Sawtooth-like core electron temperature crashes have been observed in W7-X experiments with electron cyclotron current drive. We present nonlinear single-fluid magnetohydrodynamic simulations of this phenomenon using the newly developed stellarator modeling capability of the M3D-C1 code. The near-axis current drive gives rise to two ι = 1 resonances in the equilibrium rotational transform profile so that two consecutive (1, 1) internal kink modes are seen in the simulations. A small-amplitude crash at the inner resonance occurs first, which may correspond to the sawtooth precursors observed in the experiments. A bigger crash at the outer resonance then flattens the core temperature profile, which shows semi-quantitative agreements with experimental measurements on certain metrics such as the crash amplitude and the inversion radius of the temperature change. These results illustrate a likely mechanism of the current-drive-induced sawtooth-like crashes in W7-X and, to some extent, validate the stellarator modeling capability of M3D-C1.

Acoustic response in the transition region to transverse oscillations in a solar coronal loop

Context. Magnetohydrodynamic (MHD) waves play an important role in the dynamics and heating of the solar corona. Transverse (Alfvénic) oscillations of loops commonly occur in response to solar eruptions and are mostly studied in isolation. However, acoustic coupling has been shown to be readily observable in the form of propagating intensity variations at the loop footpoints. Aims. We extend the modelling of wave coupling between a transverse loop oscillation and slow magnetoacoustic waves in a structured loop to include a lower atmosphere. Methods. We achieve this with combined analytical modelling and fully non-linear MHD simulations. Results. Transverse loop oscillations result in the excitation of propagating slow waves from the top of the transition region and the lower boundary. The rate of excitation for the upward propagating waves at the lower boundary is smaller than for waves at the top of the transition region due to the reduced local sound speed. Additionally, slow waves are found to propagate downwards from the transition region, which reflect at the lower boundary and interfere with the upward propagating waves. Resonances are present in the normal mode analysis but these do not appear in the simulations. Due to the presence of the transition region, additional longitudinal harmonics lead to a narrower slow wave profile. The slow wave field is anti-symmetric in the direction of wave polarisation, which highlights the importance that the loop orientation has on the observability of these waves. The ponderomotive effect must be accounted for when interpreting intensity oscillations. Evidence is found for an additional short-period oscillation, which is likely a hybrid mode.

Kinematic viscosity estimates in reversed-field pinch fusion plasmas

This paper concerns the kinematic viscosity in reversed-field pinch fusion plasmas, including both the study of numerical magneto-hydrodynamics (MHD) simulations and the analysis of RFX-mod experimental data.In the first part, we study the role of non-uniform time-constant radial viscosity profiles in 3D non-linear visco-resistive MHD simulations. The new profiles induce a moderate damp (for the velocity field) and a correspondent enhancement (for the magnetic field) of the spectral components resonating in the regions where the viscosity is higher.In the second part, we evaluate the kinematic viscosity coefficient on a wide database of RFX-mod shots according to the transport theories of Braginskii (considering parallel, perpendicular and gyro viscosity coefficients), considering the action on viscosity of ITG modes (ion temperature gradient) and according to the transport theory of Finn. We then exploit the comparison with the visco-resistive MHD simulations (where the visco-resistive dissipation rules the MHD activity) to show that the classical Braginskii perpendicular viscosity produces the best agreement between simulations and data, followed by the Braginskii gyro-viscosity.

Filamentary plasma eruptions and the heating and acceleration of electrons

We present test-particle simulations of electrons during a nonlinear magnetohydrodynamic (MHD) simulation of a type-I edge localized mode to explore the effect of an eruptive plasma filament on the kinetic level. The electrons are moderately heated and accelerated during the filamentary eruption on a fast timescale of the order of 0.5 ms. A clearly non-thermal tail is formed in the distribution of the kinetic energy that is of power-law shape and reaches 90 keV for some particles. The acceleration is exclusively observed in the direction parallel to the magnetic field, i.e., with a clear preference in countercurrent direction, and we show that the parallel electric field is the cause of the observed acceleration. Most particles that escape from the system leave at one distinct strike-line in the outer divertor leg at some time during their energization. The escaping high-energy electrons in the tail of the energy distribution are not affected by collisions; thus, they show characteristics of runaway electrons. The mean square displacement indicates that transport in energy space clearly is superdiffusive, and interpreting the acceleration process as a random walk, we find that the distributions of energy-increments exhibit exponential tails, and transport in energy space is equally important of convective (systematic) and diffusive (stochastic) nature. By analyzing the MHD simulations per se, it turns out that the histograms of the parallel electric field in the edge region exhibit power-law shapes, and this clearly non-Gaussian statistics is ultimately one of the reasons for the moderately anomalous phenomena of particle transport that we find in energy space.

Growing neoclassical tearing modes seeded via transient-induced-multimode interactions

Nonlinear extended magnetohydrodynamics simulations demonstrating seeding of neoclassical tearing modes (NTMs) via magnetohydrodynamics-transient-induced multimode interactions are presented. Simulations of NTMs are enabled by two recent NIMROD code developments: the implementation of heuristic neoclassical stresses and the application of transient magnetic perturbations (MPs) at the boundary. NTMs are driven unstable by the inherently pressure driven kinetic bootstrap current, which arises due to collisional viscosity between passing and trapped electrons. These simulations use heuristic closures that model the neoclassical electron and ion stresses. NTM growth requires a seed island, which is generated by a transiently applied MP in simulations. The capability is demonstrated using kinetic-based reconstructions with flow of a DIII-D ITER Baseline Scenario discharge (La Haye et al., in Proceedings IAEA FEC, 2020). The applied MP seeds a 2/1 NTM that grows in two phases: a slow growth phase followed by a faster robust growth phase. Additionally, an evolving sequence of higher order core modes are excited at first. Power transfer analysis shows that nonlinear interactions between the core modes and the 2/1 helps drive the initial slow growth. Once the induced 2/1 magnetic island reaches a critical width, the NTM transitions to faster robust growth, which is well described by the nonlinear modified Rutherford equation. This work highlights the role of nonlinear mode coupling in seeding NTMs.

Non-resonant global mode in LHD partial collapse with net toroidal current

A transition from an interchange mode to a non-resonant mode is found in the nonlinear magnetohydrodynamic simulation for the partial collapse in a large helical device (LHD) plasma with a net toroidal current. This transition can occur when the magnetic shear is weak and the rotational transform is close to unity in the core region. In this transition, the mode number of the dominant Fourier component is reduced. As a result of the nonlinear evolution, the (m, n) = (1, 1) component can be dominant, where m and n are the poloidal and the toroidal mode numbers, respectively. This transition is considered to be a candidate to explain the observation in the LHD experiments with the net toroidal current that show partial collapses are caused by the (1, 1) mode.

Collisional-radiative non-equilibrium impurity treatment for JOREK simulations

A collisional-radiative non-equilibrium impurity treatment for JOREK 3D nonlinear magneto-hydrodynamic (MHD) simulations has been developed. The impurities are represented by marker-particles flowing along the fluid velocity field lines, while ionizing and recombining independently according to ADAS data and local fluid density and temperature. The non-equilibrium impurity contributions are then projected back to the fluid field for self-consistent time evolution. A 2D test case is used to compare the new non-equilibrium impurity model against previous coronal equilibrium (CE) impurity treatment, as well as to compare the non-equilibrium impurity behavior between the single and the two temperature model. Further, we conduct benchmark with previously published coronal non-equilibrium results by other 3D nonlinear MHD codes such as M3D-C1 and NIMROD. The new non-equilibrium treatment is shown to successfully capture the early phase cooling by weakly ionized impurities which the CE model missed. The benchmarks with M3D-C1 and NIMROD show general agreement in both the integrated quantities and the 2D profile evolution, despite the difference in the atomic model used. The above comparison and benchmark cases demonstrate the capability of the non-equilibrium impurity model for JOREK, paving the way for more sophisticated 3D non-linear massive material injection simulations which have important applications in disruption mitigation studies.

Acoustic response to transverse oscillations in a solar coronal loop

Context. Magnetohydrodynamic (MHD) waves play an important role in the dynamics and heating of the solar corona. Their investigation also reveals information about the local conditions. Transverse (Alfvénic) oscillations of loops commonly occur in response to solar eruptions. It has been shown that these oscillations elicit an acoustic response through wave coupling at the footpoint and the pondermotive force. Aims. We extend the modelling of wave coupling between a transverse loop oscillation and slow magnetoacoustic waves through line-tied footpoint boundary conditions by considering the effect of transverse loop structuring and non-linearity. Methods. We combine analytical wave modelling with fully non-linear MHD simulations to study the wave field of propagating slow waves in a two-dimensional slab loop (arcade) model. Results. We demonstrate that transverse loop oscillations generate propagating slow waves from the footpoints with the same periodicity but shorter wavelength determined by the local sound speed. The degree of wave coupling is proportional to the square root of the plasma-β. The slow wave field is anti-symmetric in the direction of transverse wave polarisation. We show through synthetic diagnostics that this has important consequences for their observability in terms of the orientation of the loop with respect to the observer. We also show that for the interpretation of intensity oscillations associated with typical loop oscillations the ponderomotive response also needs to be taken into account. The modelling presented here allows for the successful identification of the slow waves and pondermotive response in a previous observational study.

Thermal quench and current profile relaxation dynamics in massive-material-injection-triggered tokamak disruptions

3D non-linear magnetohydrodynamic simulations of a disruption triggered by a massive injection of argon gas in JET are performed with the JOREK code. The key role of the thermal drive of the m = 2, n = 1 tearing mode (i.e. the drive from helical cooling inside the island) in the disruption process is highlighted by varying the amplitude and position of the argon source across simulations, and also during a simulation. In cases where this drive persists in spite of the development of magnetic stochasticity, which is favoured by moving the argon source in an ad hoc way from the plasma edge into the 2/1 island at some point in the simulation, a relaxation in the region (roughly) takes place. This relaxation generates a plasma current spike comparable to the experimental one. Simulations are compared in detail to measurements via synthetic diagnostics, validating the model to some degree.

Approach to nonlinear magnetohydrodynamic simulations in stellarator geometry

The capability to model the nonlinear magnetohydrodynamic (MHD) evolution of stellarator plasmas is developed by extending the M3D-C 1 code to allow non-axisymmetric domain geometry. We introduce a set of logical coordinates, in which the computational domain is axisymmetric, to utilize the existing finite-element framework of M3D-C 1. A C 1 coordinate mapping connects the logical domain to the non-axisymmetric physical domain, where we use the M3D-C 1 extended MHD models essentially without modifications. We present several numerical verifications on the implementation of this approach, including simulations of the heating, destabilization, and equilibration of a stellarator plasma with strongly anisotropic thermal conductivity, and of the relaxation of stellarator equilibria to integrable and non-integrable magnetic field configurations in realistic geometries.

Nonlinear MHD simulations of external kinks in quasi-axisymmetric stellarators using an axisymmetric external rotational transform approximation

Reduced magnetohydrodynamic (MHD) equations are used to study the nonlinear dynamics of external kinks in a quasi-axisymmetric (QA) stellarator with varying fractions of external rotational transform. The large bootstrap currents associated with high beta plasmas may make QA configurations susceptible to low n external modes, limiting their operational space. The violence of the nonlinear dynamics, and, in particular, when these modes lead to a disruption, is not yet understood. In this paper, the nonlinear phase of external kinks in an unstable QA configuration with an edge safety factor below two is simulated. An axisymmetric approximation of this stellarator is constructed in the nonlinear MHD code, JOREK, capturing the influence of the external rotational transform. The use of this approximation for the considered stellarator is validated by comparing the linear dynamics against the linear viscoresistive MHD code, CASTOR3D. The nonlinear dynamics of this stellarator approximation are compared with an equivalent tokamak to understand the influence of a relatively small external rotational transform. While the external rotational transform does have a stabilizing influence on the MHD activity, it remains violent. To explore the first order influence of a larger external rotational transform, this equilibrium parameter is artificially increased for the considered stellarator, reducing the effective plasma current. The violence of the kink instability is quantified, and shown to reduce with the increasing external rotational transform. At the same time, the external kink triggers internal modes that exacerbate the loss in confinement during the nonlinear phase, such that it remains large over much of the parameter space. It is only with a significant fraction of external rotational transform that these subsequent modes are stabilized.

Magnetohydrodynamic simulations of runaway electron beam termination in JET

A runaway electron (RE) fluid model is used to perform non-linear magnetohydrodynamic simulations of a relativistic electron beam termination event in JET. The case considered is that of a post-disruption low density cold plasma in the runaway plateau phase, wherein high-Z impurities have been largely flushed out via deuterium second injection (Shot:95135). Details of the experiment are found in separate publications. Our studies reveal that a combination of low plasma density and a hollow current profile which is confirmed by experimental studies causes fast growth of a double-tearing mode, which in turn leads to stochastization of the magnetic field and a prompt loss of REs. The phenomenology of events leading to the crash and the timescales of the dynamics are in excellent agreement with the experiment. Simulations also indicate significant toroidal variation in RE deposition but without localized hotspots. The strong stochastization setting in first from the edge leads to a poloidally broad deposition footprint that partly explains the benign nature of the termination event. This work further supports the potential possibility to engineer a benign RE beam termination scenario via deuterium second injection in ITER, as proposed by Reux et al ‘Runaway electron beam suppression using impurity flushing and large magnetohydrodynamic instabilities’ (submitted to Physical Review Letters).

Non-linear extended MHD simulations of type-I edge localised mode cycles in ASDEX Upgrade and their underlying triggering mechanism

A triggering mechanism responsible for the explosive onset of edge localised modes (ELMs) in fusion plasmas is identified by performing, for the first time, non-linear magnetohydrodynamic simulations of repetitive type-I ELMs. Briefly prior to the ELM crash, destabilising and stabilising terms are affected at different timescales by an increasingly ergodic magnetic field caused by non-linear interactions between the axisymmetric background plasma and growing non-axisymmetric perturbations. The separation of timescales prompts the explosive, i.e. faster than exponential, growth of an ELM crash which lasts . The duration and size of the simulated ELM crashes compare qualitatively well with type-I ELMs in ASDEX Upgrade. As expected for type-I ELMs, a direct proportionality between the heating power in the simulations and the ELM repetition frequency is obtained. The simulations presented here are a major step forward towards predictive modelling of ELMs and of the assessment of mitigation techniques in ITER and other future tokamaks.

Fast plasma dilution in ITER with pure deuterium shattered pellet injection

JOREK 3D non-linear magnetohydrodynamic (MHD) simulations of pure deuterium shattered pellet injection in ITER are presented. Considering a 15 MA L-mode plasma with a thermal energy content of 36 MJ from the non-activated phase of ITER operation, it is shown that such a scheme could allow diluting the plasma by more than a factor 10 without immediately triggering large MHD activity, provided the background impurity density is low enough. This appears as a promising strategy to reduce the risk of hot tail runaway electron (RE) generation and possibly to avoid RE beams altogether in ITER, motivating further studies in this direction.

Extended full-MHD simulation of non-linear instabilities in tokamak plasmas

Non-linear magnetohydrodynamic (MHD) simulations play an essential role in active research and understanding of tokamak plasmas for the realization of a fusion power plant. The development of MHD codes such as JOREK is a key aspect of this research effort. In this paper, we present an operational version of the full-MHD model implemented in JOREK, a significant advancement from the reduced-MHD model used for previous studies, where assumptions were made on the perpendicular dynamics and the toroidal magnetic field. The final model is presented in detail, and benchmarks are performed using both linear and non-linear simulations, including comparisons between the new full-MHD model of JOREK and the previously extensively studied reduced-MHD model, as well as results from the linear full-MHD code CASTOR3D. For the cases presented, this new JOREK full-MHD model is numerically and physically reliable, even without the use of numerical stabilization methods. Non-linear modeling results of typical tokamak instabilities are presented, including disruption and edge-localized-mode physics, most relevant to current open issues concerning future tokamaks such as ITER and DEMO.

Parametric decay instabilities near the second-harmonic upper hybrid resonance in fusion plasmas

Parametric decay instabilities (PDIs) lead to the generation of strong frequency-shifted radiation when powerful X-mode polarized microwaves, injected for electron cyclotron resonance heating (ECRH) of fusion plasmas, cross a region of non-monotonic electron density near the second-harmonic upper hybrid resonance (UHR). For the standard second-harmonic X-mode ECRH scenarios used at the ASDEX Upgrade tokamak, the second-harmonic UHR occurs near the plasma edge, meaning that PDIs occur in connection with phenomena leading to non-monotonic edge electron density profiles, e.g. blobs, edge-localized modes (ELMs), and inter-ELM modes. We present the first study of strong signals near half the ECRH frequency in fusion-relevant plasmas and demonstrate their PDI-like features, such as an ECRH power threshold required for their occurrence, through analog modulations of the ECRH power. The signals near half the ECRH frequency are found to be far more prevalent than expected based on previous theories, and we hence present a qualitative explanation of their origin, in terms of cross-polarized microwaves generated through non-linear interactions of the waves involved in the PDIs, which captures the basic features of the experimental observations. We additionally present examples of the PDI-related microwave signals just below the ECRH frequency which indicate the location of the microwave source. Furthermore, based on a non-linear magnetohydrodynamics simulation of an ELM crash in ASDEX Upgrade, performed using the JOREK code, the duration of the PDI-like microwave spikes observed in connection with ELMs is shown to match the transit time of an ELM filament though an ECRH beam. Finally, the PDI power threshold expected theoretically, calculated using the electron density and temperature profiles from the JOREK simulation, is shown to match the experimentally observed values.