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Quasi-mode evolution in a stochastic magnetic field
Abstract We present a multi-scale model of quasi-mode evolution in a stochastic magnetic field. The similarity between a quasi-mode and a ballooning mode enables us to address the challenges arising from the disparate geometries in the theories of ballooning modes in the presence of resonant magnetic perturbations. We obtain useful insights into our understanding of ballooning mode dynamics in a stochastic background. To maintain quasi-neutrality at all scales, the beat between the quasi-mode and the stochastic magnetic field drives microturbulence, which drives the turbulent background that promotes mixing and damps the quasi-mode. As a result of the broad mode structure of the quasi-mode, the turbulent viscosity and the turbulent diffusivity produced by the microturbulence are larger than those in our related study on resistive interchange modes. The stochastic magnetic field can also enhance the effective plasma inertia and reduce the effective drive, thereby slowing the mode growth. A nontrivial correlation between the microturbulence and the magnetic perturbations is shown to develop. This could account for the reduction in the Jensen–Shannon complexity of pedestal turbulence in the Resonant Magnetic Perturbation Edge-Localized Mode suppression phase observed in recent experiments. Directions for future experimental and theoretical studies are suggested.
Global gyrokinetic study of shaping effects on electromagnetic modes at NSTX aspect ratio with ad hoc parallel magnetic perturbation effects
Plasma shaping may have a stronger effect on global turbulence in tight-aspect-ratio tokamaks than in conventional-aspect-ratio tokamaks due to the higher toroidicity and more acute poloidal asymmetry in the magnetic field. In addition, previous local gyrokinetic studies have shown that it is necessary to include parallel magnetic field perturbations in order to accurately compute growth rates of electromagnetic modes in tight-aspect-ratio tokamaks. In this work, the effects of elongation and triangularity on global, ion-scale, linear electromagnetic modes are studied at National Spherical Torus Experiment (NSTX) aspect ratio and high plasma β using the global gyrokinetic particle-in-cell code XGC. The effects of compressional magnetic perturbations are approximated via a well-known modification to the particle drifts that was developed for flux-tube simulations [Joiner et al., Phys. Plasmas 17, 072104 (2010)], without proof of its validity in a global simulation, with the gyrokinetic codes GENE and GEM being used for local verification and global cross-verification. Magnetic equilibria are re-constructed for each distinct plasma profile that is used. Coulomb collision effects are not considered. Within the limitations imposed by the present study, it is found that linear growth rates of electromagnetic modes (collisionless microtearing modes and kinetic ballooning modes) are significantly reduced in a high-elongation and high-triangularity NSTX-like geometry compared to a circular NSTX-like geometry. For example, growth rates of kinetic ballooning modes at high- β are reduced to the level of that of collisionless trapped electron modes.
Impact of pedestal density gradient and collisionality on ELM dynamics
BOUT++ turbulence simulations are conducted to capture the underlying physics of small ELM characteristics achieved by increasing separatrix density via controlling strike points from vertical to horizontal divertor plates for three EAST discharges. BOUT++ linear simulations show that the most unstable modes change from high-n ideal ballooning modes to intermediate-n peeling–ballooning modes and eventually to peeling–ballooning stable plasmas in the pedestal. Nonlinear simulations show that the fluctuation is saturated at a high level for the lowest separatrix density. The ELM size decreases with increasing separatrix density, until the fraction of this energy lost during the ELM crash becomes less than 1% of the pedestal stored energy, leading to small ELMs. Simulations indicate that small ELMs can be triggered either by the marginally peeling–ballooning instability near the peak pressure gradient position inside the pedestal or by a local instability in the pedestal foot with a larger separatrix density gradient. The pedestal collisionality scan for type-I ELMs with steep pedestal density gradient shows that both linear growth rate and ELM size decrease with increasing collisionality. On the contrary, the pedestal collisionality and pedestal density width scan with a weak pedestal density gradient indicate small ELMs can either be triggered by a high-n ballooning mode or by a low-n peeling mode in a low collisionality region 0.04–0.1. We report the simulations indicate the weaker the linear unstable modes near marginal stability with small linear growth rate, the lower nonlinearly saturated fluctuation intensity and the smaller turbulence spreading from the linear unstable zone to stable zone in the nonlinear saturation phase, leading to small ELMs.
The effect of triangularity on predicted pedestals for the CFETR
In this work, the dominant unstable magnetohydrodynamic mode and its stable region in the pedestal of the China Fusion Engineering Test Reactor are studied by numerical modeling with the peeling-ballooning theory over a wide range of triangularities (δ) and collisionalities (υ*). A new accessible stable region is found at δ<−0.1 for large βN,ped and υ*. This new stable region expands toward lower βN,ped and υ* with decreasing δ and is totally covered by the peeling unstable region with δ=−0.5 because of the increasing trapped particle fraction (ft,eff). The sensitivity of this new stable region to the kinetic ballooning mode constraint and elongation is studied. For negative and low δ<0.1, the boundary of the first stable region is determined from the ballooning mode. For δ>0.1, the peeling mode becomes dominant as the boundary approaches low s with low υ*, while the ballooning mode is still dominant at the boundary with high υ*. When δ increases beyond 0.46, the first stable region is expanded, and access to the second stable region of the ballooning mode opens up. The entire newly connected region of the first and second stable regions keeps expanding with further increases in δ until δ=0.6. Beyond this value, the ballooning mode becomes more unstable with increasing δ, while the peeling mode is approximately unchanged with increasing δ in this range. The change in the dominant mode and the stable region with increasing δ can be explained by the trade-off between the stabilization effect from the deeper poloidal magnetic well and destabilization due to the enlarged drive term.
Tokamak ITG-KBM transition benchmarking with the mixed variables/pullback transformation electromagnetic gyrokinetic scheme
Electromagnetic gyrokinetic simulation of high temperature plasma is required to predict confinement in magnetic fusion devices and has posed challenges for existing codes. In this paper, we demonstrate successful global gyrokinetic simulation of the ion temperature gradient-driven mode-kinetic ballooning mode transition in a toroidal fusion plasma test case using the mixed variables/pullback transformation (MV/PT) scheme with the particle-in-cell codes XGC and ORB5, and compare to results from a conventional continuum code from the literature. Furthermore, the MV/PT scheme combines explicit time integration with mitigation of the well-known electromagnetic gyrokinetic “cancelation problem.” We calculate eigenmodes in the electrostatic and parallel vector potentials, and find good agreement in growth rate, real frequency, and the normalized plasma pressure of mode transition.
Development of a data-driven neural network model for electron thermal transport in NSTX
A data-driven electron thermal transport neural network (ETT-NN) model, trained on TRANSP interpretative analysis results of National Spherical Torus Experiment (NSTX), was developed to enable faster and more accurate ETT computation for spherical tokamaks (STs). The model incorporates both convolutional NNs and recurrent NNs, allowing it to simultaneously account for the spatial and temporal non-localities and multi-scale features of turbulent transport, which have been considered only in a limited manner in conventional models. The model was validated through interpretative analysis and predictive simulations using Tokamak Reactor Integrated Automated Suite for Simulation and Computation, demonstrating relatively high accuracy. Additionally, parameter scans were performed on test discharges known to exhibit specific turbulent modes, such as microtearing mode, trapped electron mode, kinetic ballooning mode, and electron temperature gradient mode. The scanning results revealed that the ETT-NN model exhibits the same trends as those observed in conventional gyrokinetic simulations or theories, while also capturing the global nature of turbulent transport, indicating that the data-driven model accurately reflects the underlying physical characteristics. Furthermore, due to the dimensionless nature of the model, we can feasibly expand its applicability by incorporating data from other devices and uncovering the characteristics of ETT in STs in the future.
MHD stability analysis against pressure and current-driven modes in the SMall Aspect Ratio Tokamak
Linear magnetohydrodynamic (MHD) simulations for the SMall Aspect Ratio Tokamak (SMART) have been carried out for the first time, for both positive (PT) and negative triangularity (NT) shaped plasmas using the MARS-F code. The MHD stability of projected SMART plasmas against internal kinks, infernal modes and edge peeling-ballooning modes have been analyzed for a wide range of realistic equilibria. A stabilization of internal kinks and infernal modes is observed when increasing the safety factor profile and reducing plasma beta. PT shaped plasmas are more stable against both internal kinks and infernal modes than their counterpart NT shaped plasmas. Toroidal flows have little impact on the MHD stability of the internal kinks, but they have a strong stabilizing effect on infernal modes, which can be further mitigated in NT shaped plasmas. The MHD stability of peeling-ballooning modes is reduced in NT shaped plasmas, as observed in conventional tokamaks.
Inter-ELM pedestal evolution and the role of edge fluctuations in the C-Mod and DIII-D tokamaks
Edge localized modes (ELMs) lead to a cyclical behaviour of the H-mode pedestal density, temperature and pressure. Substantial evidence exists associating the ELM crash with violation of ideal MHD stability, via current driven kink/peeling modes and/or pressure gradient driven ballooning modes. Recovery from ELM crashes and buildup to the next ELM crash is an active and critical area of edge physics research for ITER projections. In one set of theories, the pedestal pressure profile in H-mode discharges is predicted to be limited by micro-instabilities. One such micro-instability is the kinetic ballooning mode (KBM), which is hypothesized to provide a ‘soft’ limit that regulates the edge transport, thereby restricting the local pressure gradient. In this picture, e.g. in the EPED model, the pedestal width expands slowly until the ideal MHD stability limit is reached. Experiments were recently performed on the C-Mod and DIII-D devices to search for instabilities correlated with the pedestal evolution between ELMs. The results show correlations between the onset of quasi-coherent fluctuations between type I ELMs and a critical temperature gradient. Linear gyrokinetic calculations have shown that these fluctuations have characteristics similar to those expected for KBMs. These results provide additional data toward validation of the EPED model, increasing confidence in ITER projections.
Density limit in peeling-limited pedestals at and above the Greenwald value in DIII-D high poloidal beta plasmas
High pressure, peeling limited pedestals with pedestal normalized beta β N,ped >2 and pedestal top density n ped at or above the Greenwald density nG have been achieved in DIII-D high poloidal beta plasmas, with high global normalized beta β N >3 and energy confinement H 98 ~1.2-1.7. Higher β N allows higher pedestal density above the Greenwald value and higher pedestal pressure, even with a low injected torque. MHD modeling confirms that the experimental profiles lie near the peeling-mode unstable boundary with high normalized pressure gradient βMHD and high edge current density. Experimental analysis and stability calculations indicate that the high poloidal beta with strong Shafranov shift, high βMHD and weak/negative magnetic shear improves the pedestal stability by decoupling the peeling and ballooning modes and stabilizing the ballooning modes, thus facilitating access to the second stable region of peeling-ballooning mode. The access to the second stable peeling-ballooning stability region opens Super-H-like channels without extremely strong shaping or strong torque injection. The high-pressure peeling pedestal allows the pedestal density to go beyond the Greenwald limit with strong ExB shear maintained: pedestal pressure increases with pedestal density even when n ped >n G , until reaching the ideal MHD instability boundary, where giant ELMs occur. The giant ELMs are dominated by a strong n=1 component and cause a large reduction of the edge pressure, but a negligible change of the core pressure, consistent with kink/peeling-mode induced instability. The pedestal recovers from the collapse and typically sustains a high baseline density, around the Greenwald limit, during the whole discharge duration. Experiments also found that internal transport barriers and n ped ~0.9nG, peeling limited pedestals could be simultaneously achieved in high β N plasmas, while an internal feedback between ITB strength and pedestal performance is found.
Understanding the negative triangularity ELM trigger and ELM free state on DIII-D with ECE-imaging
Here, the Electron Cyclotron Emission Imaging (ECEI) diagnostic was used to observe a finite-n interchange mode structure in the edge of negative triangularity shaped plasmas on DIII-D. At a small negative triangularity (δ u = -0.2), the plasma is in the H-mode with ELMs that are triggered by a low-n interchange mode. At a larger negative triangularity (δ u = -0.4) and low NBI power (2 MW), a dithering oscillation is observed that is triggered by a low-n interchange mode, whereas at higher NBI power (>2 MW), the edge reverts to L-mode and the low-n interchange mode is present continuously. In all cases, the edge pressure gradient is clamped when the interchange mode is present. It is concluded that the low-n interchange mode prevents the plasma from transitioning to H-mode at a large negative triangularity. This agrees with linear BOUT++ simulations which suggest that the interchange-type MHD can be a resistive ballooning mode whereby resistivity can significantly increase the finite-n ballooning mode growth rate. The absence of H-mode at large negative triangularity can, thus, be explained by the excitation of low-n pressure driven resistive ballooning modes in the plasma edge.
Ballooning instability of the earth's plasma sheet region in the presence of parallel flow
Stability of the plasma sheet and plasma sheet boundary layer against the ballooning mode instability is investigated. The equilibrium state of a two-dimensional plasma sheet configuration with parallel sheared flow is modeled. This equilibrium is shown to be ballooning unstable when delta-W is not positive definite, where delta-W is the potential energy. The eigenmode structure of the ballooning mode is found by imposing the boundary conditions that the waves are totally reflected from the ionosphere, and that no waves are coming in from infinity. The eigenmode structure of the unstable balloning modes is highly oscillatory, extending beyond about 100 R(E). The ballooning modes are thus a possible candidate for explaining the MHD waves and other dynamical events observed in the magnetotail by ISEE 3 and other spacecraft.
Critical role of current-driven instabilities for ELMs in NSTX
The impact of different extended-magnetohydrodynamic (MHD) contributions on the stability thresholds of peeling-ballooning modes in ELMing and ELM-free plasmas in the spherical tokamak NSTX is investigated with the initial value code M3D-C1. We show that ELMing discharges in NSTX are limited by resistive current-driven peeling modes, whereas non-ELMing wide-pedestal H-mode discharges are located near the ideal pressure-driven ballooning threshold. It is demonstrated that extended-MHD can lead to more reliable edge stability predictions than existing ideal-MHD models. Resistive peeling-ballooning modes are found to exist well before the ideal stability threshold is met, and kink-peeling modes exhibit considerable sensitivity to plasma resistivity. Other effects not considered in ideal-MHD models affect PB modes in NSTX in a weaker way. Gyroviscous stress appears stabilizing such that the stability boundary lies closer to the experimental point. Equilibrium rotation can suppress ideal core modes and thus isolate edge modes. These results are important for the development of a predictive pedestal model for low-aspect ratio tokamaks.
Nonlinear second order electromagnetic gyrokinetic theory for a tokamak plasma
The steep plasma pressure gradient that forms at the edge of the high confinement, H-mode regime of tokamak operation provides free energy to drive electromagnetic micro-instabilities that are widely believed to influence the transport processes in this so-called pedestal region. This high pressure gradient also provides a high current density (bootstrap current), known to influence ballooning mode stability and to be important for driving kink modes in the ideal magneto-hydrodynamic plasma model (so-called peeling-ballooning modes). Furthermore, efficient, steady state future tokamak power plants must operate with a large bootstrap current in the core and especially concerning spherical tokamaks, confinement will be influenced by electromagnetic turbulence. To accommodate these important situations, conventional electromagnetic gyrokinetic theory is extended to incorporate neoclassical effects in the equilibrium drives, allowing $B_{\vartheta} ~ B_0$ (B0 is the confining magnetic field, and $B_{\vartheta}$ is its poloidal component). This provides a global gyrokinetic model that self-consistently captures the consequences of large bootstrap current fractions on the equilibrium distribution functions.
Pedestal stability analysis on MAST in preparation for MAST-U
In preparation for the upcoming MAST-U campaign, pedestal stability of spherical tokamaks is revisited by investigating standard H-mode discharges on MAST. As a step beyond previous studies, both ion and electron profiles are used for obtaining equilibria and a diverse set of pedestals is evaluated. Stability analysis with the ELITE and CGYRO codes shows that MAST pedestals are constrained by kinetic ballooning modes and medium toroidal mode number peeling-ballooning modes, with most unstable modes ranging from n = 25 to n = 45. In discharges with a steep q profile at the edge a larger number of poloidal harmonics is excited for each toroidal mode. In conclusion, a comparison with discharges on DIII-D with matched shape and similar non-dimensional parameters indicates that the increased shear at lower aspect ratio stabilizes low n peeling modes.
Performance prediction applying different reduced turbulence models to the SMART tokamak
The SMall Aspect Ratio Tokamak (SMART) is currently being commissioned at the University of Seville and will be able to compare the performance of positive and negative triangularity plasmas at low aspect ratio. Predictive simulations have been performed for different machine scenarios and heating schemes using the TRANSP code. The objectives of these simulations are to predict the parameters expected in positive triangularity plasmas, to guide diagnostic development, and to validate transport models. Several reduced turbulence models have been used to predict electron and ion temperatures for the operational phase 2. All models provide similar results from approximately mid-radius to the separatrix but important discrepancies are found in the core region. These positive triangularity results are compared with experiments from a similar size machine like GLOBUS-M2. The multi-mode model (MMM) shows the best agreement. Simulations with different boundary conditions have been performed and no strong differences have been observed between them. The impact of neutral beam injection (NBI) on the predicted profiles has also been addressed. Rotation reduces turbulence levels so higher temperatures are achieved when included in the simulations. Studying the different contributions to the thermal diffusivities, it is observed that electron temperature gradient (ETG) turbulence dominates at the plasma core while micro-tearing modes (MTM) dominate at the edge in the electron channel. In the ion channel, the neoclassical contribution is dominant at the core and at the very edge while the Weiland component, which includes ion temperature gradient mode (ITG), trapped electron mode (TEM), kinetic ballooning mode (KBM), peeling mode (PM) and collisionless and collision dominated magnetohydrodynamic (MHD) modes governs the mid-radius region. For phase 3, two plasmas with different electron densities have been studied. The case with lower density matches well a specific discharge of GLOBUS-M2. The higher density plasma shows high performance with β N ≈ 3.8.
Anomalous transport in the H-mode pedestal of Alcator C-Mod discharges
Anomalous transport in the H-mode pedestal region of five Alcator C-Mod discharges, representing a collisionality scan is analyzed. The understanding of anomalous transport in the pedestal region is important for the development of a comprehensive model for the H-mode pedestal slope. In this research, a possible role of the drift resistive inertial ballooning modes (Rafiq et al 2010 Phys. Plasmas 17 082511) in the edge of Alcator C-Mod discharges is analyzed. The stability analysis, carried out using the TRANSP code, indicates that the DRIBM modes are strongly unstable in Alcator C-Mod discharges with large electron collisionality. An improved interpretive analysis of H-mode pedestal experimental data is carried out utilizing the additive flux minimization technique (Pankin et al 2013 Phys. Plasmas 20 102501) together with the guiding-center neoclassical kinetic XGC0 code. The neoclassical and neutral physics are simulated in the XGC0 code and the anomalous fluxes are computed using the additive flux minimization technique. The anomalous fluxes are reconstructed and compared with each other for the collisionality scan Alcator C-Mod discharges. It is found that the electron thermal anomalous diffusivities at the pedestal top increase with the electron collisionality. This dependence can also point to the drift resistive inertial ballooning modes as the modes that drive the anomalous transport in the plasma edge of highly collisional discharges.
Exploring the transition from continuous turbulence fluctuations to bursting ELMs in high SOL density regimes
BOUT++ turbulence simulations of the DIII-D reveal that the density profile between the separatrix and pedestal plays a crucial role in the dynamics of edge localized modes (ELMs) and edge plasma turbulent transport. Nonlinear simulations demonstrate that small ELMs in the DIII-D hybrid scenario under high SOL density conditions are predominantly driven by local ballooning modes near the separatrix, stabilizing global instabilities while enhancing localized pressure fluctuations. A key control parameters for ELM dynamics is the separatrix-to-pedestal density ratio, n e,sep /n e,ped . A high ratio indicates a shallow gradient, favoring small ELMs, while a lower ratio signals a steep gradient, which increases the likelihood of large ELMs. Comprehensive parameter scans, including n e,sep /n e,ped , density gradient profiles near the separatrix, and resistivity, reveal the critical role of these parameters in shaping transitions between turbulence-driven transport and ELM bursting. The scans demonstrate that in high SOL density regimes, small ELMs can result from either global resistive MHD instabilities or local ballooning modes near the separatrix, depending on the steepness of the separatrix density gradient. These findings also highlight the transition from continuous turbulence to small ELMs. The post-crash peak in pressure fluctuations, δP rms serves as a critical metric for identifying transition from continuous turbulence fluctuations to ELM bursting. Larger δP rms values correlate with ELM bursts driven by local or global instabilities, whereas smaller values indicate turbulence-dominated transport. Drift-Alfvén and resistive ballooning turbulence enhance the entrainment of fluctuations from the pedestal to the SOL, contributing to the complex interplay of dynamics in this regime. These findings emphasize the importance of separatrix density shaping and pedestal gradient control for optimizing ELM behavior in ITER and future fusion devices.