The effects of plasma shaping on the H-mode pedestal structure are investigated. High fidelity kinetic simulations of the neoclassical pedestal dynamics are combined with the magnetohydrodynamic (MHD) stability conditions for triggering edge localized mode (ELM) instabilities that limit the pedestal width and height in H-mode plasmas. The neoclassical kinetic XGC0 code [Chang et al., Phys. Plasmas 11, 2649 (2004)] is used in carrying out a scan over plasma elongation and triangularity. As plasma profiles evolve, the MHD stability limits of these profiles are analyzed with the ideal MHD ELITE code [Snyder et al., Phys. Plasmas 9, 2037 (2002)]. Simulations with the XGC0 code, which includes coupled ion-electron dynamics, yield predictions for both ion and electron pedestal profiles. The differences in the predicted H-mode pedestal width and height for the DIII-D discharges with different elongation and triangularities are discussed. For the discharges with higher elongation, it is found that the gradients of the plasma profiles in the H-mode pedestal reach semi-steady states. In these simulations, the pedestal slowly continues to evolve to higher pedestal pressures and bootstrap currents until the peeling-ballooning stability conditions are satisfied. The discharges with lower elongation do not reach the semi-steady state, and ELM crashes are triggered at earlier times. The plasma elongation is found to have a stronger stabilizing effect than the plasma triangularity. For the discharges with lower elongation and lower triangularity, the ELM frequency is large, and the H-mode pedestal evolves rapidly. It is found that the temperature of neutrals in the scrape-off-layer (SOL) region can affect the dynamics of the H-mode pedestal buildup. However, the final pedestal profiles are nearly independent of the neutral temperature. The elongation and triangularity affect the pedestal widths of plasma density and electron temperature profiles differently. This provides a new mechanism of controlling the pedestal bootstrap current and the pedestal stability.
I. INTRODUCTION
Confinement and fusion performance of tokamak discharges depend strongly on the height of the pedestal, which is a steep gradient region at the edge of the plasma.1 It has been observed that large heat pulses to the divertor and plasma facing wall are produced by edge localized modes (ELMs), which are instabilities that periodically remove the pedestal.2,3 These heat pulses, which accelerate the erosion of the divertor, may require suspension of operation until the divertor can be replaced. Once formed, the pedestal continues to grow until the steep pressure gradient and current density trigger a magnetohydrodynamic (MHD) instability that signals the onset of an ELM crash.
The pedestal forms and then collapses periodically because of ELMs that occur when the pedestal gradients become large enough to satisfy the peeling and ballooning mode instability conditions.4 Peeling mode instability is driven by the parallel current density, while ballooning modes are driven by the pedestal pressure gradient.5 The coupled peeling-ballooning modes are driven by both parallel current and pressure gradient. Several different types of ELM exist. The peeling-ballooning stability model has been validated for the tokamak plasmas with type I ELMs, which are the most dangerous for the tokamak operation and which are considered in this paper.
In this study, the XGC0 code, which employs a neoclassical model, is combined with the Edge Localized Instabilities in Tokamak Equilibria (ELITE) code, which applies the peeling-ballooning stability conditions used in the EPED model, in order to study the H-mode pedestal dependence with the plasma elongation and triangularity. Previously, these two codes were combined in an investigation of ELM crash dynamics.6 Neoclassical models and models that are based on the MHD constrains, such as EPED,7 can independently reproduce many experimentally observed features of the H-mode pedestal. In the EPED model, the pedestal height is limited by the stability conditions of the peeling and ballooning modes and the pedestal slope is controlled by the kinetic ballooning modes. During the H-mode pedestal recovery after an ELM crash, both the pedestal width and height increase following the constraint from the kinetic ballooning model. The slope of the H-mode pedestal typically evolves slowly according to the changes in the kinetic ballooning stability conditions. The pedestal width and height continue to grow until the peeling-ballooning instability condition is satisfied and next ELM crash occurs. The model provides good agreement with the experimental data from a number of tokamaks.8 However, the EPED model alone does not provide the details of the pedestal buildup dynamics and does not distinguish among the widths of the electron temperature, ion temperature, and plasma density pedestals. In experiments, these widths are usually different. In particular, the ion temperature pedestal width is often significantly wider than the electron temperature and plasma density pedestal widths.9 The EPED model is capable of taking into account the different widths observed in experiments but does not predict the separation in the pedestal width. There is also the question of how to take into account the effects of plasma collisionality in the pedestal model and if the existing pedestal models can be used for discharges with very low beta where the kinetic ballooning modes are expected to be stable.10
The neoclassical XGC0 code11,12 has been successfully used for the modeling of the pedestal buildup in several tokamaks. The comparison of XGC0 simulations for several DIII-D and Alcator C-Mod discharges13,14 show reasonable agreement with the experimental profiles. The XGC0 code has adjustable parameters that describe the anomalous transport and the auxiliary sources.14 Moreover, simulations carried out using the XGC0 code, which do not include MHD stability constraints to limit the growth of the pedestal, can be used only for modeling of discharges with the pedestal profiles that reach semi-steady state and that evolve slowly towards an ELM crash.10,13 Typically, these discharges have rather large elongation and triangularity.
The effects of plasma shaping are included in the XGC0 model and in the peeling-ballooning stability conditions computed with the ELITE code.5 While the elongation and triangularity effects can be investigated with either model, it is useful to understand the effects that comes from the interaction of neoclassical and MHD stability physics. In the studies described in this paper, the anomalous transport is intentionally reduced to a negligibly small level and the kinetic ballooning constraint is not considered. In the coupled XGC0/ELITE simulations, the height of the pedestal is determined by the peeling-ballooning stability conditions, and the pedestal slope is determined by the neoclassical physics. The anomalous transport is usually strongly reduced in the H-mode pedestal region as a consequence of the strong quenching effect resulting from flow shear. However, anomalous transport is still expected to play an important role in the H-mode pedestal buildup. In this paper, the effects that cannot be robustly described by the XGC0 and ELITE codes are excluded. These effects include the edge kinetic turbulence physics, core turbulence and extended MHD physics, details of impurity transport, sputtering, energetic particle influx from core, and 3D magnetic field effects.
Studies are carried out to illustrate the dependence of H-mode pedestals, in tokamaks that have type I ELMs, on plasma elongation and triangularity. The objective of this study is to obtain the dependence of pedestal width and height on plasma shaping that can be tested in experiments. Three DIII-D discharges that represent a scan with respect to plasma shaping are considered: a discharge with high elongation and low triangularity, a discharge with low elongation and low triangularity, and a discharge with low elongation and high triangularity. Dependence of the pedestal width and height is developed as a function of the scanned plasma parameters. The effect of neutral temperature on the pedestal width and height is examined. Differences in the dependence of the widths of electron temperature and ion temperature pedestals are investigated.
The manuscript is organized as follows: Sec. II contains a description of experimental data for the three DIII-D discharges with different elongation and triangularity. In Sec. III, End-to-end Framework for fusion Integrated Simulation (EFFIS) integrated modeling framework data flow for automatic coupling of XGC0-ELITE-M3D is described. A brief description of XGC0 kinetic neoclassical code and ELITE ideal MHD stability code is also presented. In Sec. IV, simulation results for the dependence of the pedestal width and height is presented as a function of the scanned plasma parameters and the results are summarized in Sec. V.
II. EXPERIMENTAL DIII-D DATA
Experimental data from three DIII-D H-mode discharges15 that represent a scan with respect to plasma shaping are considered. The values of elongation, κ, triangularity, δ, major radius, R, minor radius, a, magnetic field strength at the axis, B0, plasma current, , safety factor at the edge, , and total plasma energy, , are shown in Table I.
The DIII-D discharges analyzed in this study and their plasma parameters.
Tokamak discharge type . | DIII-D 136674 high elongation . | DIII-D 136693 low elongation . | DIII-D 136705 high triangularity . |
---|---|---|---|
time (s) | 1.41 | 1.39 | 1.21 |
κ | 1.71 | 1.25 | 1.26 |
δ | 0.12 | 0.03 | 0.30 |
R (m) | 1.70 | 1.70 | 1.70 |
a (m) | 0.59 | 0.59 | 0.59 |
B0 (T) | 2.00 | 2.00 | 2.00 |
(MA) | 1.20 | 0.88 | 0.94 |
5.65 | 4.27 | 5.56 | |
(MJ) | 2.56 | 2.52 | 4.31 |
Tokamak discharge type . | DIII-D 136674 high elongation . | DIII-D 136693 low elongation . | DIII-D 136705 high triangularity . |
---|---|---|---|
time (s) | 1.41 | 1.39 | 1.21 |
κ | 1.71 | 1.25 | 1.26 |
δ | 0.12 | 0.03 | 0.30 |
R (m) | 1.70 | 1.70 | 1.70 |
a (m) | 0.59 | 0.59 | 0.59 |
B0 (T) | 2.00 | 2.00 | 2.00 |
(MA) | 1.20 | 0.88 | 0.94 |
5.65 | 4.27 | 5.56 | |
(MJ) | 2.56 | 2.52 | 4.31 |
The discharge 136674 is the highly elongated discharge () which is typical for most DIII-D discharges, but with rather low triangularity (). The discharge 136693 has nearly circular geometry () with triangularity almost zero (). The discharge 136705 is also nearly circular () but has the highest triangularity () among the three discharges. The discharges 136674 and 136693 correspond to the part of experimental plasma elongation campaign in which elongation is varied by almost 50%. While the discharges 136693 and 136705 correspond to the part of experimental plasma triangularity campaign in which triangularity is varied by a factor of 10. The flux surfaces computed using the equilibrium reconstruction for these three DIII-D discharges are shown in Fig. 1.
Contours of equilibrium flux surfaces with different plasma elongation and triangularity are shown in (R, Z) plane for DIII-D discharges 136674 at time t = 1.405 s (left panel), 136693 at time t = 1.385 s (middle panel), and 136705 at time t = 1.205 s (left panel).
Contours of equilibrium flux surfaces with different plasma elongation and triangularity are shown in (R, Z) plane for DIII-D discharges 136674 at time t = 1.405 s (left panel), 136693 at time t = 1.385 s (middle panel), and 136705 at time t = 1.205 s (left panel).
The energy confinement time increases from 50 ms in the low κ case to 80 ms in the high κ case.15 It is found that twice the heating power (two neutral beam sources) was required in the low κ case to maintain profiles similar to those obtained with a single neutral beam source in the high κ case. Also, note that there is a substantial change in the plasma current Ip. This change is necessary in order to maintain consistency in the safety factor.
III. SIMULATION FRAMEWORK
The End-to-end Framework for fusion Integrated Simulation (EFFIS) framework6 for automatic coupling of the XGC0, ELITE, and M3D codes is used to carry out simulations of the development of H-mode pedestal and the pedestal ELM cycle. Initial equilibria that are based on the interpretation of experimental results with Equilibrium Fitting (EFIT) equilibrium code are used to initialize the XGC0 code.11,12 Relaxed experimental electron and ion temperature, and plasma density profiles are used as initial profiles in XGC0. The new plasma profiles are advanced in time in XGC0 and transferred to the MHD equilibrium analysis code M3D-OMP code.16 The new eqdsk files are generated with M3D-OMP code for every second ion transit period. The new eqdsk files are then used in the ideal MHD stability ELITE code5 to verify peeling-ballooning triggering conditions. The data flow diagram of the EFFIS integrated modeling framework is shown in Fig. 2. One million ions and the one million electrons are used in the XGC0 simulations. The temperature of neutrals is varied from 20 eV to 70 eV.
The data flow diagram of the EFFIS integrated modeling framework is shown.
The XGC0 simulations do not include the effects associated with the anomalous transport. However, the anomalous transport is taken to be a small residual level through the whole edge region for all three DIII-D discharges that are investigated in this study. The anomalous electron and ion thermal diffusivity is set equal to 0.02 in the pedestal region and 0.4 in the scrape-off-layer (SOL) region, and the anomalous particle diffusivity is set equal to in the pedestal region and in the SOL region. The motivation of the selection of anomalous transport coefficients at a small residual level is that the focus of this study is the neoclassical and MHD effects on the development of the H-mode pedestal structure. However, a residual anomalous transport is still required in these simulations. In the absence of the anomalous transport, the plasma density becomes too low in the near outer separatrix region. The plasma density depletion leads to unrealistic ion and electron temperature profiles. These changes to the plasma profiles might occur before the H-mode pedestal develops enough to trigger the peeling or ballooning instabilities that would lead to an ELM crash. Without a rigorous model for anomalous transport, there is no mechanism to correctly describe the transport in the plasma edge as well the transport from the plasma core, where the neutral beam injection is mostly applied. It is not expected that the plasma profiles predicted from the coupled XGC0/ELITE simulations reproduce the experimental results. The objective of this study is to utilize in the XGC0 code the magnetic equilibria that correspond to different DIII-D plasma shapings in order to derive a scaling for the pedestal width and height from the coupled kinetic neoclassical/MHD stability simulations with respect to the plasma elongation and triangularity.
A. XGC0 code
In the XGC0 neoclassical code,11,12 the gyrokinetic plasma ion and electron guiding centers evolve in time within a five-dimensional phase space using realistic magnetic equilibrium and limiter geometry. The code uses cylindrical coordinates so that the separatrix and X-point region can be easily included in the simulation domain. XGC0 implements collisions among multiple species (ions, impurities, and electrons) and neutrals using a Monte Carlo approach and employing several models for the source of neutrals at the wall.
A combination of cylindrical and field-line-following magnetic coordinates is used in the XGC0 code. Magnetic coordinates have the advantage of accurately treating physical phenomena that are aligned with the magnetic field lines. Cylindrical coordinates have the advantage of being able to treat the open magnetic surfaces of the scrape-off-layer and the closed magnetic surfaces of the plasma core with equal accuracy and without the divergence associated with magnetic coordinates at the separatrix.
As a full-f code, XGC0 can include sources and sinks to account for heat and torque input. Neutral particle effects can be studied using a simple Monte-Carlo transport model that has been verified against a more advanced neutral DEGAS2 code.17 XGC0 is also capable of simulating self-consistent magnetic perturbations using M3D as Ampéres law solver. The anomalous transport is introduced using a modified random walk algorithm that enables different diffusivities for electron and ion thermal transport, as well as for particle transport.14
B. ELITE code
ELITE,5,18 a highly efficient MHD stability code, is used to calculate quantitative stability constraints on the pedestal, including constraints on the pedestal height. ELITE solves a eigenvalue problem for a set of 2D equations that are based on a rigorous expansion in inverse toroidal mode number, n, retaining all terms up to O(1/n) and some higher order terms to preserve the Hermitian nature of ideal MHD dynamics. ELITE uses a finite-n extension of ballooning theory that allows accurate and highly efficient study peeling ballooning modes. The physically relevant range for ELMs in existing tokamaks is typically .
ELITE employs a Fourier representation in the poloidal direction and utilizes numerical methods that allow very efficient stability calculations, facilitating its use in the large number of stability calculations that are needed to characterize pedestal stability constraints as a function of mode wavelength, pedestal width, plasma shape, collisionality, and safety factor. ELITE has been developed to allow efficient evaluation of stability bounds, growth rates, and mode structures of intermediate wavelength instabilities in the pedestal region. ELITE calculations allow quantitative study of stability constraints on the pedestal and together with analytic insight lead to a model of various types of small and large ELM cycles. Stability calculations on reconstructed experimental equilibria from multiple machines consistently find that peeling-ballooning mode growth rates rise to significant values just before ELMs occur. A number of effects, including flows, a separatrix, and a scrape-off layer plasma, as well as nonlinear dynamics, are not currently available in the ELITE code.
IV. SIMULATION RESULTS
In Secs. IV A–IV E, the effects of elongation, triangularity, and neutral temperature on pedestal width and height are presented. The density and electron temperature pedestal width and height are compared with respect to plasma shaping. The peeling ballooning mode growth rates as a function of toroidal mode number are shown.
A. Effect of high elongation on H-mode pedestal width and height
In Fig. 3, the simulated plasma density, electron temperature, and ion temperature profiles as functions of normalized poloidal flux (ψ), for the high elongation DIII-D discharge 136674, are shown for different ion transit periods. The ion transit period is the time that takes a single ion to travel around the torus. It is defined as , where R is the major radius and vi is the ion thermal speed. For the discharges considered in this study, the ion transit period is approximately 77 μs. The initial plasma density and temperature profiles are shown for time t = 2 ion transit period. The plasma density and electron temperature profiles demonstrate development of H-mode pedestal in kinetic XGC0 simulations. It is found that the pedestal width decreases from about 4% (initial profile) to about 3% (profile that satisfy ELM stability conditions, prior to t = 54 ion transit periods, see Fig. 5) for plasma density profile, and from about 4% to about 1% for the electron temperature profile. In the simulations, it is noticed that the effect of increased elongation manifests itself in the plasma density profile first and then in the temperature profiles. It is also found that the pedestal width for the ion temperature profile is much wider than the pedestal width for the electron temperature and density profiles. The normalized ion temperature gradients in this region are found below than the normalized electron temperature gradients. This difference between the ion and electron temperature gradients agrees with the experimental measurements from DIII-D [see Fig. 1(b) in Ref. 9]. It is interesting to note that the normalized electron and ion temperature gradients are approximately the same in the pedestal region of AUG. In Ref. 9, this difference between electron and ion temperature gradients in DIII-D and AUG tokamaks is attributed to higher collisionalities in AUG and a stronger coupling of electron and ion heat channels. It would be interesting to analyze the dynamics of the AUG discharges in the coupled XGC0/ELITE simulations, because such simulation will allow additional validation with respect to the separation of electron and ion temperature gradients. However, these studies are outside the scope of this paper.
Simulation results for the plasma density (upper left panel), electron temperature (upper right panel), ion temperature (lower left panel), and bootstrap current density (lower right panel) as functions of normalized poloidal flux (ψ) are shown for the high elongation DIII-D discharge 136674 for different ion transit periods.
Simulation results for the plasma density (upper left panel), electron temperature (upper right panel), ion temperature (lower left panel), and bootstrap current density (lower right panel) as functions of normalized poloidal flux (ψ) are shown for the high elongation DIII-D discharge 136674 for different ion transit periods.
Simulation results for the plasma temperatures, densities, and bootstrap current density profiles as functions of normalized poloidal flux (ψ) are shown in Fig. 3. It is found that the density and temperatures and bootstrap current density continue to grow if the MHD stability constrains are not considered. The slope of the density and temperature profiles do not change much at the end of the simulation. In particular, it is true for the normalized temperature gradients in the H-mode pedestal region. The temperature profiles shift to larger values, but the normalized gradients remain relatively constant after 40 ion transit times. The development of the density pedestal looks different. The normalized density gradients continue to slowly increase after 40 ion transit times and the density pedestal width continue to decrease. The normalized density gradient averaged in the H-mode pedestal region as a function of time is shown in Fig. 4. Similar to the experimental observations, the ion temperature pedestal width is much wider than the pedestal widths of the electron temperature and plasma density profiles. The free energy in the large pressure gradient and the resulting bootstrap current in the pedestal can drive ELM instabilities. The MHD stability is a necessary condition to study physics relevant scaling of the H-mode pedestal width and height. The peeling-ballooning mode growth rate normalized to the diamagnetic drift frequency () versus toroidal mode number is plotted for different ion transit periods in Fig. 5. The MHD edge modes are considered unstable if their maximum growth rate exceeds ().19 However, due to peculiarities of diamagnetic frequency computations in the particular version of ELITE code used in this study, an additional averaging is necessary. This additional averaging reflects the facts that the maximum diamagnetic frequency in the plasma edge region is used. The location of the maximum diamagnetic frequency does not necessarily coincide with the location of the maximum growth rates. The edge mode instability criterion used in this research is (). The ELITE analysis of plasma profile indicates that H-mode pedestal becomes ELM unstable after 52 ion transit times. The most unstable toroidal mode number is found to be 8.
The normalized density gradient averaged in the H-mode pedestal region of the DIII-D discharge 136674 as a function of normalized time. Time is normalized by the ion transit time.
The normalized density gradient averaged in the H-mode pedestal region of the DIII-D discharge 136674 as a function of normalized time. Time is normalized by the ion transit time.
MHD stability analysis of plasma profiles computed using the XGC0 code for the high elongation DIII-D discharge 136674. The peeling-ballooning mode growth rate normalized to the diamagnetic drift frequency () are shown as functions of toroidal mode numbers for different ion transit periods.
MHD stability analysis of plasma profiles computed using the XGC0 code for the high elongation DIII-D discharge 136674. The peeling-ballooning mode growth rate normalized to the diamagnetic drift frequency () are shown as functions of toroidal mode numbers for different ion transit periods.
B. Effect of low elongation on the H-mode pedestal width and height
The plasma pressure and bootstrap current density profiles as functions of normalized poloidal flux are shown in Fig. 6 in the low elongation () DIII-D discharge 136693 for different ion transit periods. It is found that the low elongation DIII-D discharge is more MHD unstable compared with high elongation DIII-D discharge shown in Fig. 5. The value of pedestal pressure at which ELMs are triggered is about 5 times lower in low elongation DIII-D discharge as compared to the high elongation discharge. The maximum value of bootstrap current in the pedestal region at the time of an ELM crash is found to be about two times lower in the low elongation discharge. These results are consistent with a previous MHD studies.20 The low toroidal modes () at time t = 18 ion transit periods are found to become unstable first. Note that in the high elongation case, shown in Fig. 5, the mode became unstable at time t = 54 ion transit periods. The pedestal heights for the low elongation discharge are significantly lower, and the resulting ELM crashes are more frequent. The pedestal stores less energy and it is likely that the ELMs carry less energy.
Simulation results for the plasma pressure (left panel) and bootstrap current density (right panel) as functions of normalized poloidal flux (ψ) are shown in the low elongation () DIII-D discharge 136693 for different ion transit periods.
Simulation results for the plasma pressure (left panel) and bootstrap current density (right panel) as functions of normalized poloidal flux (ψ) are shown in the low elongation () DIII-D discharge 136693 for different ion transit periods.
C. Effect of high triangularity on the H-mode pedestal width and height
In Fig. 7, simulation results of the plasma pressure and bootstrap current density are shown for the high triangularity () and low elongation () DIII-D discharge 136705 at different ion transit periods. The plasma profiles in the high triangularity DIII-D discharge are found to be less unstable compared with low elongation and low triangularity DIII-D discharge 136693, but more unstable compared with high elongation and low triangularity DIII-D discharge 136674. The pedestal pressure is about 1.5 times higher than in low elongation discharge 136693 and 3 times lower than in high elongation discharge 136674. The most unstable modes correspond to low toroidal mode numbers () which is similar to the case of low elongation and low triangularity DIII-D discharge 136693. However, the mode become unstable after time t = 40 ion transit periods, which is unstable later in time than the low elongation case. The ELM frequency for this discharge is between the ELM frequencies of high and low elongation DIII-D discharges analyzed in Secs. IV A and IV B. Similar to the results for the low elongation and low triangularity DIII-D discharge 136705, neither the pedestal height nor the pedestal width reaches saturation states.
Simulation results of the plasma pressure (left panel) and bootstrap current density (right panel) as functions of normalized poloidal flux (ψ) are shown for the high triangularity () and low elongation () DIII-D discharge 136705 at different ion transit periods.
Simulation results of the plasma pressure (left panel) and bootstrap current density (right panel) as functions of normalized poloidal flux (ψ) are shown for the high triangularity () and low elongation () DIII-D discharge 136705 at different ion transit periods.
D. Effect of neutral temperature on MHD controlled H-mode pedestal
Neutral particle temperature is one of few free parameters in the XGC0 code that might affect the dynamics of H-mode pedestal development. The effect of neutral temperature on the pedestal width and height is studied for the high elongation DIII-D discharge 136674. The pressure profile and bootstrap current density as a function of normalized poloidal flux (ψ) are shown in Fig. 8 for different values of neutral particle temperature at different ion transit periods. The neutral particle temperature values are set , and The rate of H-mode pedestal development depends on neutral temperature. The pedestal that satisfied ELM stability conditions is developed after 54 ion transit times for , after 52 ion transit times for , and after 46 ion transit times for It means that the mode becomes unstable earlier in the ion transit periods with increasing neutral temperature. However, the final electron and ion temperatures profiles (and plasma density not shown here) are almost independent of the selected values of neutral temperature.
The pressure profiles (left panel) and bootstrap current density (right panel) as functions of normalized idal flux (ψ) are shown in the high elongation DIII-D discharge 136674 for different values of neutral particle temperature at different ion transit periods.
The pressure profiles (left panel) and bootstrap current density (right panel) as functions of normalized idal flux (ψ) are shown in the high elongation DIII-D discharge 136674 for different values of neutral particle temperature at different ion transit periods.
E. Comparison of H-mode pedestal widths in the DIII-D discharges with different plasma shaping
In order to compare the differences in the H-mode pedestal widths for the DIII-D discharges with different plasma shaping, the plasma density and electron temperature profiles are normalized to bring them to the same scale. In Fig. 9, the normalized plasma density and normalized temperature profiles for all three DIII-D discharges with different elongations and triangularities are presented. It is found that the pedestal width for the plasma density profiles is the smallest for the high elongation discharge, 136674, and the pedestal width for the electron density profile is the smallest for the high triangularity discharge, 136693. The pedestal width for the plasma density profiles is found to be decreasing with increasing elongation (compare a curve indicating 136674 with 136693 and 136705 curves) and is almost independent of triangularity (compare a curve indicating 136693 with a curve indicating 136705). In the discharges considered in this study, the effect of increased elongation manifests itself in the plasma density profiles first, while the effect of increased triangularity manifests itself in the electron temperature profiles first. The pedestal height is found to be significantly larger in the discharges with larger elongation. It has also been found that the pedestal width for the electron temperature profiles is found to be almost independent of elongation and show a weak dependence on the triangularity. The pedestal width for ion temperature profiles is found to be much wider than the pedestal width for the plasma density and electron temperature profiles (not shown here). Neoclassical effects together with the MHD stability conditions that are verified using the ideal stability ELITE code can explain some experimental trends such as the dependence of the ELM frequency on the plasma elongation21 and the dependence of the H-mode pedestal height on triangularity.22 The question about the contributions of the anomalous transport in the pedestal and SOL regions remains. The role of anomalous transport will be addressed in future studies.
Normalized plasma density and normalized temperature profiles as functions of normalized poloidal flux (ψ) for DIII-D discharges with low and high elongations and with low and high triangularities. Here, , and are the plasma densities and electron temperatures at the separatrix and at 95% of the normalized poloidal flux, respectively.
Normalized plasma density and normalized temperature profiles as functions of normalized poloidal flux (ψ) for DIII-D discharges with low and high elongations and with low and high triangularities. Here, , and are the plasma densities and electron temperatures at the separatrix and at 95% of the normalized poloidal flux, respectively.
V. SUMMARY
The neoclassical effects are combined with the peeling-ballooning stability conditions in order to study the dependence of H-modes pedestal on plasma elongation and triangularity. The anomalous thermal and particle transport is included at a small residual level in the H-mode pedestal region. The growth of the pedestal is limited by an ELM instability criterion computed using the ELITE MHD stability code. The kinetic neoclassical XGC0 code and ELITE ideal MHD stability code coupling is automated in the EFFIS computer science framework. Three DIII-D ELMy H-mode discharges with high elongation and low triangularity, with low elongation and low triangularity, and with low elongation and high triangularity are considered. It is found that the H-mode pedestal in simulations continues to grow and pedestal width continues to decrease in the absence of MHD stability constraints. MHD constrains do not affect the dynamics of an ELM crash. They set the final time, beyond which the plasma profiles become MHD unstable and cannot be reached in the experiments. At the time of an ELM crash, the slope of the H-mode pedestal reaches a semi-steady state and evolves slowly for higher elongation and higher triangularity geometry. The dynamics of maximum bootstrap current densities as function of maximum pressure gradients for all three discharges are shown in Fig. 10. The arrows show the direction of time and large dots mark the conditions that would lead to an ELM crash. The ELM crashes in the discharges with weak shaping occur at lower plasma bootstrap current densities and lower plasma pressure gradients. This is consistent with a reduced stable region for the discharges with weak shaping (for example, see Fig. 2(b) in Ref. 20). For lower elongation and lower triangularity, the plasma profiles in the H-mode pedestal evolve rapidly at the time of crash. The ELM frequency is the largest and the H-mode pedestal height is the smallest for the discharge with the low elongation and low triangularity. The ELM frequency is the smallest and the pedestal height is the largest for the high elongation discharge. The discharge with low elongation and high triangularity has an average ELM frequency and H-mode pedestal height. The effect of neutral temperature on the pedestal width and height is found to be weak. However, the peeling-ballooning mode becomes unstable earlier in the ion transit periods with increasing neutral temperature.
The time histories of maximum bootstrap current densities as functions of maximum pressure gradients in the H-mode pedestal region for three DIII-D discharges with different elongations and triangularities. The arrows show the direction of time and large dots indicate the conditions for triggering of ELM crashes.
The time histories of maximum bootstrap current densities as functions of maximum pressure gradients in the H-mode pedestal region for three DIII-D discharges with different elongations and triangularities. The arrows show the direction of time and large dots indicate the conditions for triggering of ELM crashes.
The elongation and triangularity affect the pedestal density differently than they affect the pedestal temperature. The width of the pedestal density is found to decrease with increasing elongation and is almost independent of triangularity. The effect of increased elongation manifests itself in the plasma density profiles first, while the effect of increased triangularity manifests itself in the electron temperature profiles first.
The anomalous transport included in this study is at a small residual level and is independent of the plasma parameters. In order to investigate the role of the anomalous transport on the H-mode pedestal buildup in the tokamak discharges with different plasma shaping, coupled gyro-kinetic XGC1/MHD stability simulations will be considered in future. It should be also noted that the experimental equilibria have been used to investigate the dependencies of the pedestal width and height on the elongation and triangularity. In the experimental equilibria, there are factors other than elongation and triangularity that can affect the pedestal width and height. In particular, the plasma currents in the high and low elongation discharges in this research differ by approximately 25%. There also differences in q-profiles. In the EPED model, the differences in plasma currents can explain larger pedestal gradients for the discharge with the larger elongation. The plasma current might have a strong effect on the pedestal width and height.23 While it is expected that the neoclassical XGC0 code captures many effects associated with the differences in the plasma currents,14 it is difficult to separate if the changes in the predicted pedestal width and height come from the differences in the elongation and triangularities or other differences in the experimental equilibria. In order to isolate possible effects from these differences, an additional study for a set of analytical equilibria will be useful and is also considered. Such a scan is outside the scope of this paper that has a focus on the experimental equilibria and can be a subject of future research. However, even though there are possible effects of anomalous transport and plasma current, the coupled neoclassical-MHD stability simulations yield valuable insights regarding the effects of elongation and triangularity on the pedestal width of different plasma profiles. The results of the simulations carried out can provide additional leverage in controlling the pedestal buildup, the bootstrap current in the pedestal region, and the stability of the H-mode pedestal and are not obtained with either the XGC0 or ELITE codes when they used separately. The discovered dependence of the pedestal width confirms the importance of the coupled neoclassical/MHD stability dynamics in the description of the H-mode pedestal.
ACKNOWLEDGMENTS
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. This work was also supported by the U.S. Department of Energy, Office of Science, under Award Nos. DE-SC0012174, DE-SC0008605, DE-FG02-92ER54141, and DE-SC0013977. The authors appreciate Dr. C. Holland for providing experimental data of DIII-D discharges analyzed in this paper. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.