Session UO1: Kinetic Edge Simulation, Edge Physics

Kinetic simulation of Plasma Flows in the Pedestal/Scrape-off region

We have obtained from the edge PIC code XGC-1, for the first time, comprehensive kinetic macroscopic flow solutions in the pedestal/scrape-off region. As indicated by many experiments [1], the simulation shows negative electrostatic potential in the H-mode layer where a strong plasma pedestal exists, and a positive potential in the scrape-off plasma. The simulation also shows formation of the co-current flow in the scrape-off layer. The flow in a steep pedestal shoulder region is always in the co-current direction, indicating a co-rotation source for the core plasma. However, the parallel flow in the vicinity of the separatrix appears sensitive to the neutral density, making the flow shear to be stronger at lower neutral density. Another significant new physics found from XGC is the existence of the global ExB convective flow pattern in the scrape-off region, which may have an important implication to the divertor design. \newline \newline [1] B. Labombard, et al., Nucl. Fusion 44, 1047 (2004)   

Coupling of the XGC and M3D codes and corresponding XGC-based linear and nonlinear ELM simulations

Linear and nonlinear simulations with the M3D extended MHD code are being carried out by coupling with the XGC edge kinetic code, which is able to simulate the edge plasma density and pressure pedestal buildup. Currently, the XGC-0 version, which includes a turbulence diffusion model as well as neoclassical effects, is considered for coupling work. Kinetic pedestal information from the XGC code, comprising density, pressure, and bootstrap current profiles across the pedestal, is imported into the M3D input and added to an initial DIIID EQDSK equilibrium data to construct a new consistent equilibrium and associated field-aligned mesh, which can be subsequently utilized for the XGC and M3D nonlinear ELM simulations. Several existing neoclassical bootstrap current formulas can be tested against the XGC simulation results and embedded in it. Different kinds of ELMs (i.e., pressure-driven ballooning and current-driven peeling modes) and their evolution could be obtained from the XGC kinetic input by carefully controlling initial profile characteristics such as pedestal height and bootstrap current strength, etc. Additionally, XGC-based ELMs are compared with the other ones which use different models of ELM instability drive such as the fluid bootstrap current model. We will present briefly our initial linear and nonlinear simulation results which are based on the combined implementation of M3D and XGC codes.   

Issues in Kinetic Edge Turbulence Simulation

Simulations of trapped electron modes are underway using GEM [1,2], a global electromagnetic gyrokinetic delta-f simulation with collisions. We report results with no temperature gradient so that ITG and ETG instabilities are not present. For typical weak density gradient core values, the CTEM is dominant. However, for steeper density gradient edge values, higher k drift-waves are most unstable [J. Lang this mtg.]. For the weaker density gradient core case, nonlinear simulations using GEM are routine. For the steeper gradient edge case, the nonlinear fluctuations are very high and a stationary state has not been obtained. More physics, e.g. profile variation and equilibrium ExB shear flow should be significantly stabilizing, and may make such simulations feasible using standard delta-f techniques. These features are fully implemented in GEM and research is ongoing. One approach to addressing the high fluctuation levels in the edge turbulence regime is the particle-continuum method [3]. A new scheme that periodically resets the particle weights, using a Maxwellian particle load is being tested in GEM [Y. Chen this mtg.] and will be discussed. [1] Y. Chen, S. Parker, J. Comput. Phys. 189 463 (2003). [2] Y. Chen, S. Parker, accepted, available on-line, J. Comput. Phys. (2006). [4] S. Vadlamani, S. Parker, Y. Chen and C. Kim, Comput. Phys. Comm. 164 209 (2004).   

Simulating Coulomb Collisions in Particle Codes

A deterministic algorithm for simulating changes in particle velocities, due to Coulomb collisions, has been developed. This algorithm is similar to a Monte-Carlo algorithm, but differs by using averages defined as sums over given lattice points instead of random sample means. Noise due to sampling errors does not occur, although errors occur due to the finite number of points included in the sums. This use of deterministic quadratures is a generalization of a method due to B.J. Albright, et. al [1], who pointed out that far fewer quadrature points should be needed, compared with the number of M-C samples required, for given accuracy. The Fokker-Planck friction vector and diffusion tensor which are needed are obtained from the Rosenbluth potentials, which are determined by a ``field solve'' in velocity space. Methods for achieving momentum and energy conservation will be discussed, as well as the possibility of using lattice symmetry to achieve higher order accuracy in the time step. \newline \newline [1] IEEE Transactions on Plasma Science 31, 19 (2003).   

Overview of Edge Simulation Laboratory (ESL)

The ESL is a new collaboration to build a full-f electromagnetic gyrokinetic code for tokamak edge plasmas using continuum methods. Target applications are edge turbulence and transport (neoclassical and anomalous), and edge-localized modes. Initially the project has three major threads: (i) verification and validation of TEMPEST, the project's initial (electrostatic) edge code which can be run in 4D (neoclassical and transport-timescale applications) or 5D (turbulence); (ii) design of the next generation code, which will include more complete physics (electromagnetics, fluid equation option, improved collisions) and advanced numerics (fully conservative, high-order discretization, mapped multiblock grids, adaptivity), and (iii) rapid-prototype codes to explore the issues attached to solving fully nonlinear gyrokinetics with steep radial gradiens. We present a brief summary of the status of each of these activities.   

Verification of TEMPEST with neoclassical transport theory

TEMPEST is an edge gyro-kinetic continuum code developed to study boundary plasma transport over the region extending from the H-mode pedestal across the separatrix to the divertor plates. For benchmark purposes, we present results from the 4D (2r,2v) TEMPEST for both steady-state transport and time-dependent Geodesic Acoustic Modes (GAMs). We focus on an annular region inside the separatrix of a circular cross-section tokamak where analytical and numerical results are available. The parallel flow velocity and radial particle flux are obtained for different collisional regimes and compared with previous neoclassical results. The effect of radial electric field and the transition to steep edge gradients is emphasized. The dynamical response of GAMs is also shown and compared to recent theory.   

Studies of Gyrokinetic Turbulence Models for Edge Plasmas

Gyrokinetic computational models are developed for studying tokamak edge plasmas. A 5D $\delta f$ Eulerian gyrokinetic code which uses ($\vec{R}$, $\mu$, $v_\parallel$) coordinates has been developed and benchmarked with the GS2 gyrokinetic code in the linear, collisionless, electrostatic limit, including trapped electron dynamics. Various collisional and numerical dissipation algorithms for the ($\mu$, $v_\parallel$) velocity space formulation with nonlinear dynamics are explored. Extensions of the $ \delta f$ gyrokinetic formulation to full $F (F=F_0 + \delta f)$ are also presented. We discuss studies of turbulence and transport in the tokamak edge/scrape-off region, where $\delta f \sim F_0$ so $O\left(\rho_*^2\right)$ effects neglected for core plasma simulations, such as the parallel nonlinearity, may now be important.   

5D Tempest simulations of kinetic edge turbulence

Results are presented from the development and application of TEMPEST, a nonlinear five dimensional (3d2v) gyrokinetic continuum code. The simulation results and theoretical analysis include studies of H-mode edge plasma neoclassical transport and turbulence in real divertor geometry and its relationship to plasma flow generation with zero external momentum input, including the important orbit-squeezing effect due to the large electric field flow-shear in the edge. In order to extend the code to 5D, we have formulated a set of fully nonlinear electrostatic gyrokinetic equations and a fully nonlinear gyrokinetic Poisson's equation which is valid for both neoclassical and turbulence simulations. Our 5D gyrokinetic code is built on 4D version of Tempest neoclassical code with extension to a fifth dimension in binormal direction. The code is able to simulate either a full torus or a toroidal segment. Progress on performing 5D turbulence simulations will be reported.   

Plasma equilibrium in the vicinity of X-point

In [1] it was demonstrated that a strong coupling of ExB and parallel flows can result in a large pressure variation along the magnetic flux surfaces. Experiment studies also show [2] a strong inhomogeneity of plasma pressure in the vicinity of X-point. Moreover, similar to the conclusions of Ref. 1, UEDGE modeling of edge plasma in DIII-D tokamak with ExB drift effects [2] back up experimental observations and also highlights an importance of drifts. The effects of parallel and ExB flow coupling can be important for many physical applications including physics of tokamak plasma transport in X-point region. In this paper we extend the results of Ref. 1 to more general equilibrium magnetic configurations. We adopt ``cylindrical'' tokamak geometry with a strong ``toroidal'' magnetic field in z-direction corresponding to the vicinity of X-point and assume that there is no z-dependence of the plasma parameters. We consider continuity, parallel plasma momentum balance (including ExB drifts), and parallel Ohm's law equations. We find analytically two general classes of plasma equilibrium: one, which has the essential features of inhomogeneity seen in the DIII-D experiments and another one, which can be considered as the extension of the results of Ref. 1 on more general magnetic configuration. [1] S. I. Krasheninnikov, D. J. Sigmar, and P. N. Yushmanov, Phys. Plasmas \textbf{2} (1995) 1972 [2] M. J. Schaffer, B. D. Bray, J. A. Boedo, et al., Phys. Plasmas \textbf{8} (2001) 2118   

Atomic processes in edge plasmas

Atomic processes play a number of key roles in both the physics of edge plasmas and in their diagnostics. We will provide a brief overview of a number of electron-impact and heavy-particle atomic collision calculations and the associated evaluated databases that are pertinent to edge modeling. Examples will include a large, well tested set of elastic and related transport cross sections as well as generalized collisional-radiative coefficients for all ion stages of Li and Be. We will also report on recent work that has re-evaluated widely assumed scaling relations for electron-impact ionization of excited states of hydrogen-like ions and how this affects the effective ionization rate coefficient used in a wide range of models. Finally, novel calculations of chemical sputtering, sticking, and reflection of D and D$_2$ incident upon deuterated carbons surfaces (amorphous and graphite), in the energy range from about one eV to hundreds of eV, will be described. New and unique features of these simulations in comparison to the previous ones include the surface preparation, enhanced statistics enabled by ultrascale computer resources, and use of the most recent, improved hydrocarbon potentials.   

Status of BOUT fluid turbulence code: improvements and verification

BOUT is an electromagnetic fluid turbulence code for tokamak edge plasma [1]. BOUT performs time integration of reduced Braginskii plasma fluid equations, using spatial discretization in realistic geometry and employing a standard ODE integration package PVODE. BOUT has been applied to several tokamak experiments and in some cases calculated spectra of turbulent fluctuations compared favorably to experimental data. On the other hand, the desire to understand better the code results and to gain more confidence in it motivated investing effort in rigorous verification of BOUT. Parallel to the testing the code underwent substantial modification, mainly to improve its readability and tractability of physical terms, with some algorithmic improvements as well. In the verification process, a series of linear and nonlinear test problems was applied to BOUT, targeting different subgroups of physical terms. The tests include reproducing basic electrostatic and electromagnetic plasma modes in simplified geometry, axisymmetric benchmarks against the 2D edge code UEDGE in real divertor geometry, and neutral fluid benchmarks against the hydrodynamic code LCPFCT. After completion of the testing, the new version of the code is being applied to actual tokamak edge turbulence problems, and the results will be presented. [1] X. Q. Xu et al., Contr. Plas. Phys., 36,158 (1998). *Work performed for USDOE by Univ. Calif. LLNL under contract W-7405-ENG-48.   

Simulation of ELM Instabilities using Extended MHD

Studies of ELM instabilities in the edge region of high temperature tokamaks have been carried out in realistic configurations, using the M3D initial value code. The simulations are intended to be combined with kinetic calculations, eg XGC, to study edge stability in fusion plasmas. MHD results for different scenarios, including DIII-D and ITER, are compared. Initial results for two-fluid effects will be presented. The near-vacuum region surrounding the plasma is modeled as a high resistivity, low density plasma, bounded by a rigid conducting wall at the vacuum vessel. Good initial equilibria and high spatial resolution in the affected region are essential for accurate modeling. The massively parallel M3D code can handle a large number of toroidal harmonics, with corresponding resolution in the poloidal plane. The spectrum can be studied and convergence accelerated by first enforcing toroidal periodicity at a given mode number $n$, then increasing the number of poloidal planes and independently decreasing the periodicity, finally feeding into the full simulation.   

Benchmarking the UEDGE and SOLPS edge plasma transport codes in DIII-D and JET geometries.

A program to benchmark the 2-D tokamak edge plasma transport codes UEDGE, SOLPS and EDGE2D-NIMBUS, that are widely used for interpretive and predictive edge plasma simulations, began with a comparison of the latter two codes. Here UEDGE and SOLPS are benchmarked against each other using grids based on DIII-D and JET lower single-null discharges. Neutral transport is described by fluid models using flux limiters for kinetic effects. In addition, plasma parameters calculated using the fluid neutrals models are compared to results from SOLPS using EIRENE (kinetic Monte Carlo) neutrals. Two DIII-D plasmas are used: L-mode from discharge 119919 and H-mode from 110223. The JET case is based on a grid for the high X-point-clearance discharge 50401. The comparisons have allowed identification and resolution of some model differences that can have significant impact on the solutions. Final comparisons will be discussed together with lessons learned about the process.   

Modeling of ITER ELM Dynamics

Stability analyses are carried out for a series of ITER equilibria that are generated with the TEQ and TOQ equilibrium codes. The H-mode pedestal pressure and parallel component of plasma current density are systematically varied to include the relevant parameter space for a specific ITER discharge. Ideal MHD stability codes, DCON, ELITE, and BALOO, are employed to determine whether or not each ITER equilibrium profile is unstable to peeling or ballooning modes in the pedestal region. Several equilibria, close to the marginal stability boundary for peeling and ballooning modes, are tested with the NIMROD non-ideal MHD code. It is found that the peeling-ballooning stability threshold is very sensitive to the resistivity and viscosity profiles, which vary dramatically over a wide range near the separatrix. Due to the effects of finite resistivity and viscosity, the peeling-ballooning stability threshold is shifted compared to the ideal threshold. NIMROD simulations are continued into the nonlinear stage for several ITER equilibria that are marginally unstable to peeling or ballooning modes. The differences in dynamics for ELMs triggered by ballooning instabilities and by peeling instabilities are described. Also, the formation of the H-mode pedestal, recovery of plasma profiles after ELM crashes, and effects of ELM crashes on the H- mode pedestal height and width are examined for ITER using the ASTRA code.