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Recent advances in GYRO allow simulations to map out the linear stability of many eigenvalues and eigenvectors of the gyrokinetic equation (as opposed to only the most unstable) at low computational cost. In this work, GYRO's new linear capabilities are applied to a pressure scan about the pedestal region of DIII-D shot 131997. MHD calculations in the infinite-n limit of the ideal ballooning mode, used in the very successful EPED model to predict pedestal height and width, demonstrate clear onset of the instability at 70% of the experimental pressure. Presented GYRO results first demonstrate that the ion temperature gradient driven mode and microtearing mode are dominant at the top of the pedestal, while an unnamed group of drift waves are found to be most unstable in the peak gradient region of the pedestal. The peak gradient modes have very extended ballooning structure, peak near the inboard midplane and have drift frequencies at or near the electron diamagnetic drift direction, even for very low wavenumbers (k θ ρ s ∼ 0.2). Connection is made to the MHD calculations by demonstrating the kinetic ballooning mode (KBM) is present but subdominant in the DIII-D pedestal, and the pressure required for onset of the KBM in the gyrokinetic limit is in near agreement with MHD predictions. Finally, comparisons and analysis of GYRO with two independent gyrokinetic codes, GEM (initial value) and HD7 (1D eigenvalue), are presented.

The E × B shear flow plays a dual role on peeling-ballooning modes and their subsequently triggered edge localized mode (ELM) crashes. On one hand, the flow shear can stabilize high-n modes and twist the mode in the poloidal direction, constraining the mode's radial extent and reducing the size of the corresponding ELM. On the other hand, the shear flow also introduces the Kelvin-Helmholtz drive, which can destabilize peeling-ballooning modes. The overall effect of equilibrium shear flow on peeling-ballooning modes and ELM crashes depends on the competition between these two effects. When the flow shear is either small or very large, it can reduce ELM size. However, for moderate values of flow shear, the destabilizing effect from the Kelvin-Helmholtz term is dominant and leads to larger ELM crashes.

A minimum set of equations based on the peeling–ballooning (P–B) model with nonideal physics effects (diamagnetic drift, E × B drift, resistivity and anomalous electron viscosity) is found to simulate pedestal collapse when using the BOUT++ simulation code, developed in part from the original fluid edge code BOUT. Linear simulations of P–B modes find good agreement in growth rate and mode structure with ELITE calculations. The influence of the E × B drift, diamagnetic drift, resistivity, anomalous electron viscosity, ion viscosity and parallel thermal diffusivity on P–B modes is being studied; we find that (1) the diamagnetic drift and E × B drift stabilize the P–B mode in a manner consistent with theoretical expectations; (2) resistivity destabilizes the P–B mode, leading to resistive P–B mode; (3) anomalous electron and parallel ion viscosities destabilize the P–B mode, leading to a viscous P–B mode; (4) perpendicular ion viscosity and parallel thermal diffusivity stabilize the P–B mode. With addition of the anomalous electron viscosity under the assumption that the anomalous kinematic electron viscosity is comparable to the anomalous electron perpendicular thermal diffusivity, or the Prandtl number is close to unity, it is found from nonlinear simulations using a realistic high Lundquist number that the pedestal collapse is limited to the edge region and the ELM size is about 5–10% of the pedestal stored energy. This is consistent with many observations of large ELMs. The estimated island size is consistent with the size of fast pedestal pressure collapse. In the stable α-zones of ideal P–B modes, nonlinear simulations of viscous ballooning modes or current-diffusive ballooning mode (CDBM) for ITER H-mode scenarios are presented.

We develop and test a model, EPED1.6, for the H-mode pedestal height and width based upon two fundamental and calculable constraints: (1) onset of non-local peeling–ballooning modes at low to intermediate mode number, (2) onset of nearly local kinetic ballooning modes at high mode number. Calculation of these two constraints allows a unique, predictive determination of both pedestal height and width. The present version of the model is first principles, in that no parameters are fit to observations, and includes important non-ideal effects. Extensive successful comparisons with existing experiments on multiple tokamaks, including experiments where predictions were made prior to the experiment, are presented, and predictions for ITER are discussed.

The BOUT++ code is used to simulate edge localized modes (ELMs) in a shifted circle equilibrium. Reduced ideal MHD simulations are first benchmarked against the linear ideal MHD code ELITE, showing good agreement. Diamagnetic drift effects are included finding the expected suppression of high toroidal mode-number modes. Nonlinear simulations are performed, making the assumption that the anomalous kinematic electron viscosity is comparable to the anomalous electron thermal diffusivity. This allows simulations with realistically high Lundquist numbers (S = 108), finding ELM sizes of 5–10% of the pedestal stored thermal energy. Scans show a strong dependence of the ELM size on resistivity at low Lundquist numbers, with higher resistivity leading to more violent eruptions. At high Lundquist numbers relevant to high-performance discharges, ELM size is independent of resistivity as hyper-resistivity becomes the dominant dissipative effect.

We present edge kinetic ion transport simulations of tokamak plasmas in magnetic divertor geometry using the fully nonlinear (full-f) continuum code TEMPEST. Besides neoclassical transport, a term for divergence of anomalous kinetic radial flux is added to mock up the effect of turbulent transport. To study the relative roles of neoclassical and anomalous transport, TEMPEST simulations were carried out for plasma transport and flow dynamics in a single-null tokamak geometry, including the pedestal region that extends across the separatrix into the scrape-off layer and private flux region. A series of TEMPEST simulations were conducted to investigate the transition of midplane pedestal heat flux and flow from the neoclassical to the turbulent limit and the transition of divertor heat flux and flow from the kinetic to the fluid regime via an anomalous transport scan and a density scan. The TEMPEST simulation results demonstrate that turbulent transport (as modelled by large diffusion) plays a similar role to collisional decorrelation of particle orbits and that the large turbulent transport (large diffusion) leads to an apparent Maxwellianization of the particle distribution. We also show the transition of parallel heat flux and flow at the entrance to the divertor plates from the fluid to the kinetic regime. For an absorbing divertor plate boundary condition, a non-half-Maxwellian is found due to the balance between upstream radial anomalous transport and energetic ion endloss.

The injected power required to induce a transition from L-mode to H-mode plasmas is found to depend strongly on the injected neutral beam torque and consequent plasma toroidal rotation. Edge turbulence and flows, measured near the outboard midplane of the plasma (0.85 < r/a < 1.0) on DIII-D with the high-sensitivity 2D beam emission spectroscopy (BES) system, likewise vary with rotation and suggest a causative connection. The L–H power threshold in plasmas with the ion ∇B drift directed away from the X-point decreases from 4–6 MW with co-current beam injection, to 2–3 MW near zero net injected torque and to B drift directed towards the X-point exhibit a qualitatively similar though less pronounced power threshold dependence on rotation. 2D edge turbulence measurements with BES show an increasing poloidal flow shear as the L–H transition is approached in all conditions. As toroidal rotation is varied from co-current to balanced in L-mode plasmas, the edge turbulence changes from a uni-modal character to a bi-modal structure, with the appearance of a low-frequency (f = 10–50 kHz) mode propagating in the electron diamagnetic direction, similar to what is observed as the ion ∇B drift is directed towards the X-point in co-rotating plasmas. At low rotation, the poloidal turbulence flow near the edge reverses prior to the L–H transition, generating a significant poloidal flow shear that exceeds the measured turbulence decorrelation rate. This increased poloidal turbulence velocity shear appears to facilitate the L–H transition. No such reversal is observed in high rotation plasmas. The high-frequency poloidal turbulence velocity spectrum exhibits a transition from a geodesic acoustic mode zonal flow to a higher-power, lower frequency zero-mean-frequency zonal flow as rotation varies from co-current to balanced during a torque scan at constant injected neutral beam power, perhaps also facilitating the L–H transition. This reduced power threshold at lower toroidal rotation may benefit inherently low-rotation plasmas such as ITER.

We present edge gyrokinetic simulations of tokamak plasmas using the fully non-linear (full-f) continuum code TEMPEST. A non-linear Boltzmann model is used for the electrons. The electric field is obtained by solving the 2D gyrokinetic Poisson equation. We demonstrate the following. (1) High harmonic resonances (n > 2) significantly enhance geodesic-acoustic mode (GAM) damping at high q (tokamak safety factor), and are necessary to explain the damping observed in our TEMPEST q-scans and consistent with the experimental measurements of the scaling of the GAM amplitude with edge q 95 in the absence of obvious evidence that there is a strong q-dependence of the turbulent drive and damping of the GAM. (2) The kinetic GAM exists in the edge for steep density and temperature gradients in the form of outgoing waves, its radial scale is set by the ion temperature profile, and ion temperature inhomogeneity is necessary for GAM radial propagation. (3) The development of the neoclassical electric field evolves through different phases of relaxation, including GAMs, their radial propagation and their long-time collisional decay. (4) Natural consequences of orbits in the pedestal and scrape-off layer region in divertor geometry are substantial non-Maxwellian ion distributions and parallel flow characteristics qualitatively like those observed in experiments.

Experimental and simulation results on filamentary structures observed in the Mega-Amp Spherical Tokamak (MAST) are presented and discussed. Fast camera data have been used to determine the mode number, toroidal and radial sizes and velocities of the filaments observed in L-mode, inter-edge localized mode (ELM) periods and ELMs which are summarized. Automated methods are applied to the analysis of L-mode image data in order to measure dependence with plasma parameters. This indicates that the mode number of L-mode edge turbulence increases with density and decreases with q95, while filament width has the opposite dependence.Simulations of L-mode discharges using the 3D, 2-fluid BOUT code produce similar sizes and radial velocities to the observations, and indicate that the source of these filaments is within a region ~2 cm from the plasma edge in a spontaneously formed E × B shear layer. Ion temperature fluctuations in these filaments are found to be approximately double the magnitude of electron temperature fluctuations, probably due to fast parallel electron heat transport.

It turns out that gyrokinetic theory can be geometrically formulated as a special case of a geometrically generalized Vlasov-Maxwell system. It is proposed that the phase space of the space-time is a seven-dimensional fiber bundle P over the four-dimensional space-time M, and that a Poincare-Cartan-Einstein 1-form γ on the seven-dimensional phase space determines a particle's worldline in the phase space. Through Liouville 6-form Ω and fiber integral, the 1-form γ also uniquely defines a geometrically generalized Vlasov-Maxwell system as a field theory for the collective electromagnetic field. The geometric gyrokinetic theory is then developed as a special case of the geometrically generalized Vlasov-Maxwell system. In its most general form, gyrokinetic theory is about a symmetry, called gyrosymmetry, for magnetized plasmas, and the 1-form γ again uniquely defines the gyrosymmetry. The objective is to decouple the gyrophase dynamics from the rest of the particle dynamics by finding the gyrosymmetry in γ. ...

Nonlinear three-dimensional electromagnetic simulations are employed to study the dynamics of edge localized modes (ELMs) driven by intermediate wavelength peeling-ballooning modes. It is found that the early behavior of the modes is similar to expectations from linear, ideal peeling-ballooning mode theory, with the modes growing linearly at a fraction of the Alfven frequency. In the nonlinear phase, the modes grow explosively, forming a number of extended filaments which propagate rapidly from the outer closed flux region into the open flux region toward the outboard wall. Similarities to nonlinear ballooning theory as well as additional complexities are observed. Comparison to observations reveals a number of similarities. Implications of the simulations and proposals for the dynamics of the full ELM crash are discussed.

Direct comparisons between numerical simulations and the measured plasma fluctuations and transport are presented by performing nonlinear two-fluid simulations with the BOUT code (Xu X Q and Cohen R H 1998 Contrib. Plasma Phys. 38 158). BOUT models boundary-plasma turbulence in a realistic divertor geometry using modified Braginskii equations for plasma vorticity, density (ni), electron and ion temperature (Te, Ti) and parallel momenta. The BOUT code solves for the plasma fluid equations in a 3D toroidal segment, including the region somewhat inside the separatrix and extending into the scrape-off layer; the private flux region is also included. In this paper, the physics of resistive X-point turbulence and its relation to flow shear generation is discussed. We present comparisons between the boundary plasma turbulence observed in the BOUT code and experiments on DIII-D (Luxon J L et al 1986 Int. Conf. on Plasma Physics and Controlled Nuclear Fusion (Vienna: IAEA) p 159), the National Spherical Torus Experiment (Peng Y-K M 2000 Phys. Plasmas 7 1681), and C-Mod (Hutchinson I H et al 1994 Phys. Plasmas 1 1511). In an L-mode discharge in the DIII-D tokamak, both BOUT simulations and beam emission spectroscopy show a similar flow pattern and blob size across the last closed flux surface. In an L-mode discharge, both BOUT simulations and gas puff imaging show similar filament structures along the field line and similar frequency spectrum at the outboard midplane. In simulations of the quasi-coherent mode in the EDA regime of C-Mod, the particle flux measured from BOUT simulation is consistent with Langmuir probe measurements on C-Mod at the midplane near the separatrix. The qualitative comparisons thus indicate that BOUT contains much of the relevant physics for boundary plasma turbulence in the experimentally relevant X-point divertor geometry of present-day tokamaks and spherical tori.

Impacts of electron inertia with an electron skin depth (ESD) longer than the realistic value used in early numerical studies on non-ideal ballooning modes (NIBMs) are numerically investigated by a linearized 3-field reduced MHD model. In this paper, 4 different ESDs,are used for an resistivity dependence study of the growth rate of NIBMs, where de s the real ESD and d*e = 10 corresponds to an order of ESD used in a numerical study on collisionless ballooning mode (CBM) reported in [Kleva and Guzdar Phys. Plasmas 6, 116 (1999)]. In the case with the real ESD d*e = de, a transition from resistive ballooning mode (RBM) to CBM occurs in the edge relevant resistivity regime, while the electron inertia effect is overestimated and the growth rate is almost independent of resistivity in the cases with d∗e =√10de and 10de. These results indicate that the real ESD is one of key factors for the edge stability and turbulence analysis.

The boundary plasma turbulence code BOUT models tokamak boundary-plasma turbulence in a realistic divertor geometry using modified Braginskii equations for plasma vorticity, density (ni), electron and ion temperature (T{sub e}; T{sub i}) and parallel momenta. The BOUT code solves for the plasma fluid equations in a three dimensional (3D) toroidal segment (or a toroidal wedge), including the region somewhat inside the separatrix and extending into the scrape-off layer; the private flux region is also included. In this paper, a description is given of the sophisticated physical models, innovative numerical algorithms, and modern software design used to simulate edge-plasmas in magnetic fusion energy devices. The BOUT code's unique capabilities and functionality are exemplified via simulations of the impact of plasma density on tokamak edge turbulence and blob dynamics.