Bulletin of the American Physical Society
49th Annual Meeting of the Division of Plasma Physics
Volume 52, Number 11
Monday–Friday, November 12–16, 2007; Orlando, Florida
Session BI2: Simulation of Magnetic Fusion Plasmas |
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Chair: William Nevins, Lawrence Livermore National Laboratory Room: Rosen Centre Hotel Salon 3/4 |
Monday, November 12, 2007 9:30AM - 10:00AM |
BI2.00001: Closure of computational fluid models with evolving-background \textit{$\delta$f} kinetics Invited Speaker: D.C. Barnes A new method of applying simulation particles to close implicit time-dependent nonlinear extended-MHD modeling has been formulated, analyzed, and tested. The new method has three important features that will likely prove useful for any evolving-background \textit{$\delta $f} simulation. First, the fluid equations should be closed with particle information at the momentum-density level to minimize statistical noise from closure terms. Second, the particle motion is described by a particular velocity that represents dynamics without thermal forces, separating the kinetic dynamics from the fluid dynamics. With the use of this particular velocity, there exists symmetry between the \textit{$\delta $f }weight evolution equation and the fluid closure. Third, an optimal prescription for particle shape in velocity space can be derived using Hermite polynomials. The symmetry and optimal shaping together ensure that the numerical kinetic distortion acquires no low-order moments, analogous to the analytical Chapman-Enskog-like approach. They also lead to a conserved energy integral for the discrete nonlinear system, and the r.m.s. particle weight is bounded. With this advance in computation, combined particle-fluid simulation of low-frequency extended-MHD dynamics with majority ion kinetics is now possible. The new method has been implemented for kinetic ion dynamics with fluid electron modeling in the 2D code IMP2. The method successfully reproduces dynamics where the electric field is perpendicular to the magnetic field, including kinetic stabilization of the isothermal g-mode in a slab. Extensions to include temperature gradient and arbitrary polarization are described. Co-author: W. D. Nystrom, \textit{Coronado Consulting}, Lamy, NM [Preview Abstract] |
Monday, November 12, 2007 10:00AM - 10:30AM |
BI2.00002: Fully 3D RWM and Feedback Stabilization Studies for ITER and AUG Invited Speaker: Erika Strumberger A high $\beta$-limit is a necessary condition for a working power plant. However, instabilities associated with ideal internal and external modes limit the plasma beta. External kink modes of MHD equilibria can be stabilized by a perfectly conducting wall sufficiently close to the plasma. In case of a real wall with non-zero resistivity the modes become unstable and grow on the resistive timescale of magnetic field diffusion through the wall. The growth rates of resistive wall modes (RWMs) are typically orders of magnitude smaller than of kink modes in the no-wall case so that the stabilization of RWMs by an active feedback system becomes feasible. Some axisymmetric approaches already exist which deal with this problem numerically. Nevertheless, because of experimental needs a realistic external wall has a complex three-dimensional shape. Usually, it is a multiply-connected structure. Besides the resistive wall also the feedback coils violate the axisymmetry of a tokamak configuration. Therefore, a three-dimensional, numerical treatment of the feedback stabilization problem is necessary. For this reason, starting from a stellarator code (CAS3D code) we developed the fully three-dimensional stability code STARWALL, and the feedback optimization code OPTIM. With these codes, we are able to compute the growth rates of resistive wall modes in the presence of non-axisymmetric, multiply-connected wall structures (i.e. with holes), and to model the active feedback stabilization of these modes. Analogue to the axisymmetric approaches, the problem is divided into two parts. In the open-loop part, the complete set of eigenvalues and eigenfunctions of the plasma-resistive-wall system without feedback currents is determined. Then, in the closed-loop part an initial value problem is formulated for the time evolution of the RWMs and the currents in the feedback coils. The feedback logics controlled by a set of free parameters specifies the interaction between the feedback currents and the RWMs. After choosing their values, the effectiveness of the feedback can be studied by solving the characteristic equation of the closed-loop system. The procedure has been implemented numerically (STARWALL code) and applied to resistive wall configurations for ITER and ASDEX Upgrade. For an optimal choice of the feedback parameters, the OPTIM code has been developed which optimizes the stability of a truncated closed-loop system under variations of the free parameters. [Preview Abstract] |
Monday, November 12, 2007 10:30AM - 11:00AM |
BI2.00003: Scalable algorithms for 3D extended MHD. Invited Speaker: Luis Chacon In the modeling of plasmas with extended MHD (XMHD), the challenge is to resolve long time scales while rendering the whole simulation manageable. In XMHD, this is particularly difficult because fast (dispersive) waves are supported, resulting in a very stiff set of PDEs. In explicit schemes, such stiffness results in stringent numerical stability time-step constraints, rendering them inefficient and algorithmically unscalable. In implicit schemes, it yields very ill-conditioned algebraic systems, which are difficult to invert. In this talk, we present recent theoretical and computational progress that demonstrate a scalable 3D XMHD solver (i.e., $CPU \sim N$, with $N$ the number of degrees of freedom). The approach is based on Newton-Krylov methods, which are preconditioned for efficiency. The preconditioning stage admits suitable approximations without compromising the quality of the overall solution. In this work, we employ optimal ($CPU \sim N$) multilevel methods on a parabolized XMHD formulation, which renders the whole algorithm scalable. The (crucial) parabolization step is required to render XMHD multilevel-friendly. Algebraically, the parabolization step can be interpreted as a Schur factorization of the Jacobian matrix, thereby providing a solid foundation for the current (and future extensions of the) approach. We will build towards 3D extended MHD\footnote{L. Chac\'on, \emph{Comput. Phys. Comm.}, {\bf 163} (3), 143-171 (2004)}$^,$\footnote{L. Chac\'on et al., {\em 33rd EPS Conf. Plasma Physics}, Rome, Italy, 2006} by discussing earlier algorithmic breakthroughs in 2D reduced MHD\footnote{L. Chac\'on et al., {\em J. Comput. Phys}. {\bf 178} (1), 15- 36 (2002)} and 2D Hall MHD.\footnote{L. Chac\'on et al., {\em J. Comput. Phys.}, {\bf 188} (2), 573-592 (2003)} [Preview Abstract] |
Monday, November 12, 2007 11:00AM - 11:30AM |
BI2.00004: Nonlinear Hybrid Simulations of Multiple Energetic Particle driven Alfven Modes in Toroidal Plasmas Invited Speaker: Guoyong Fu Understanding of nonlinear behavior of energetic particle-driven instabilities in tokamaks is of fundamental importance for burning plasmas. Here we report recent advances in self-consistent nonlinear simulations of fast beam ion-driven Alfven modes in NSTX and DIII-D using the extended MHD code M3D [1]. In the hybrid model, the thermal electrons and ions are treated as an ideal fluid while the energetic species is described by either drift-kinetic equation or gyrokinetic equation. The effects of energetic particles are coupled to the MHD equations via the stress tensor term in the momentum equation. The hybrid code has been recently applied to study nonlinear dynamics of fishbone instability [2]. The code was also used to simulate nonlinear evolution of a single beam-driven TAE mode in NSTX. The result showed a weak frequency chirping about 20{\%} consistent with experimental measurement [3]. In this work, we use the M3D code to simulate beam ion driven Alfven modes in NSTX plasmas with multiple unstable Alfven modes. It is found that mode saturation level of each mode can be enhanced significantly by presence of other unstable modes indicating strong nonlinear interaction between different modes. It is also found that a linearly stable n=2 mode can be nonlinearly driven by an n=1 mode at significant mode amplitude. These results together with simulation results of beam ion-driven Alfven modes in DIII-D reversed shear plasmas [4] will be presented. \newline \newline [1] W. Park, E.V. Belova, G.Y. Fu et al., Phy. Plasmas 6, 1796 (1999) \newline [2] Fu, GY, Park, W, Strauss HR et al. PHYSICS OF PLASMAS 13, 052517 (2006) \newline [3] Fu GY, Breslau J, Fredrickson E et al. ,Proceedings of 2004 IAEA Fusion Energy Conference, Vilamoura, Portugal, Paper TH/P4-38. \newline [4] M.A., Van Zeeland, G.J. Kramer, M.E. Austin et al., Phys. Rev. Lett. 97, 135001 (2006). [Preview Abstract] |
Monday, November 12, 2007 11:30AM - 12:00PM |
BI2.00005: Coarse-graining phase-space in delta-f Particle-in-Cell (PIC) simulation Invited Speaker: Yang Chen Particle weights in a $\delta \! f$ PIC simulation slowly grow in time even after the turbulent fluxes reach a steady state. This can eventually lead to discrete particle noise at long times. We present a solution to this problem. Growing weights are the result of balancing the flux in a truly collisionless simulation. Collisions as currently implemented in PIC models via Monte-Carlo do not reduce the growing weight problem \footnote{Y.~Chen and S.~E.~Parker, J. Comput. Phys. 220 839 (2007)}. Since weight growth is manifestation of phase-space filamentation, the weight level can be reduced by simply periodically coarse-graining the distribution function in phase space. A five-dimensional phase-space grid is used to facilitate the coarse-graining procedure (CGP). $\delta \! f$ is periodically deposited on the 5-D grid, then re-evaluated at the particle position using interpolation. Any discontinuity of $\delta \! f$ in time arising from CGP is reduced by resetting only a small fraction of the particle weight. CGP effectively introduces dissipation into the otherwise dissipation-less PIC method. An estimate of the numerical diffusion due to this smoothing procedure is provided in the limit of large particle number. CGP is demonstrated to effectively suppress the long-term increase of the particle weights in ITG simulations, while keeping the turbulent flux unchanged. Spectral analysis indicates that the reduction in particle weights is mainly due to the elimination of the short scale structures in the density fluctuations. Large scale density fluctuations that account for the turbulent fluxes are not affected. We have implemented CGP in the GEM code and applied it to the study of beta-scaling of turbulent transport in core plasmas, as well as, the simulation of edge plasmas with strong density and temperature gradients. Both of these problems were previously difficult to model in flux-tube simulations. [Preview Abstract] |
Monday, November 12, 2007 12:00PM - 12:30PM |
BI2.00006: First Transport Code Simulations using the TGLF Model Invited Speaker: J.E. Kinsey The first transport code simulations using the newly developed TGLF theory-based transport model [1,2] are presented. TGLF has comprehensive physics to approximate the turbulent transport due to drift-ballooning modes in tokamaks. The TGLF model is a next generation gyro-Landau-fluid model that includes several recent advances that remove the limitations of its predecessor, GLF23. The model solves for the linear eigenmodes of trapped ion and electron modes (TIM, TEM), ion and electron temperature gradient (ITG, ETG) modes and finite beta kinetic ballooning (KB) modes in either shifted circle or shaped geometry [1]. A database of over 400 nonlinear GYRO gyrokinetic simulations has been created [3]. A subset of 140 simulations including Miller shaped geometry has been used to find a model for the saturation levels. Using a simple quasilinear (QL) saturation rule, we find remarkable agreement with the energy and particle fluxes from a wide variety of GYRO simulations for both shaped or circular geometry and also for low aspect ratio. Using this new QL saturation rule along with a new ExB shear quench rule for shaped geometry, we predict the density, temperature, and toroidal rotation profiles in a transport code and compare the results against experimental data in the ITPA Profile Database. We examine the impact of the improved electron physics in the model and the role of elongation and triangularity on the predicted profiles and compare to the results previously obtained using the GLF23 model. \newline \newline [1] G.M. Staebler, J.E. Kinsey, and R.E. Waltz, Phys. Plasmas \textbf{12}, 102508 (2005). \newline [2] G.M. Staebler, J.E. Kinsey, and R.E. Waltz, to appear in Phys. Plasmas, May(2007). \newline [3] The GYRO database is documented at fusion.gat.com/theory/gyro. [Preview Abstract] |
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