Bulletin of the American Physical Society
APS April Meeting 2012
Volume 57, Number 3
Saturday–Tuesday, March 31–April 3 2012; Atlanta, Georgia
Session S1: Poster Session III: Sherwood (2:00 pm - 5:00 pm) |
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Sponsoring Units: APS Room: Grand Hall West |
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S1.00001: SHERWOOD POSTERS II |
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S1.00002: Convective transport of fast particles in dissipative plasmas near an instability threshold Boris Breizman, Matthew Lilley We demonstrate that a marginally unstable energetic particle population in a dissipative plasma can change globally due to the act of a single wave-particle resonance. The resonance serves as a seed for the continuous production of nonlinear holes and clumps, whose convective motion in phase space results in substantial flattening of the fast particle distribution function. A bump-on-tail instability is considered as an example in a single-mode limit as well as in the quasilinear regime. The convective hole-clump transport mechanism tends to be more efficient near the instability threshold than quasilinear diffusion. [Preview Abstract] |
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S1.00003: Energetic Particle Effects on Resistive MHD Instabilities with Varying Aspect Ratio Michael Halfmoon, Dylan Brennan, Charlson Kim Advances in computational analysis have made it possible to simulate toroidal MHD equilibrium and stability, including physics outside of the MHD description, with a high degree of precision. Such simulations have recently begun to explore the interaction between non-Maxwellian particle distributions and resistive MHD stability. This project consists of an analysis of the interaction of a slowing down distribution of energetic particles with a m/n=2/1 resistive instability as a function of varying aspect ratio. Using the TOQ, PEST-III and NIMROD codes we initially characterize the MHD stability of the 2/1 mode without particles, and examine the particle effects with the NIMROD code. The equilibrium has a circular cross section and a Bessel function current derived from a linear perturbation in flux, allowing comparison to an analytic result at infinite aspect ratio. Methods and motivations for developing energetic particle effects into the PEST-III code are also discussed. [Preview Abstract] |
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S1.00004: Interaction Between Neutral-Beam-Injected Ions and Tearing Modes in Reversed Field Pinch V.V. Mirnov, C.C. Kim A long pulse (20 ms) high current (40 A) energetic (25 keV) neutral beam used for plasma heating now operates routinely on the Madison Symmetric Torus (MST) Reversed Field Pinch (RFP). This motivates our interest in studying fast particles and their interaction with tearing modes in the RFP. An analytic model has been developed recently in the asymptotic limit of large Larmor orbits[1]. The contribution from the fast particles is expressed locally in terms of the hot ion density gradient on the mode rational surface. Major MST core resonant modes (1,6) and (1,7) are localized far away from the axis and, correspondingly, weakly interact with the injected ions. In some nearly non-reversed MST regimes, an additional resonant tearing mode (1,5) is excited which is localized around the axis and, therefore, effectively interacts with the fast ions. An improved version of the model\footnote{V.V. Mirnov et al., Int. Sherwood Fusion Theory Conf. (2010), IAEA Fusion Energy Conf., THS/P5-11 (2010).} is developed that includes effects of mode asymmetry and a non constant-$\psi $ approximation. This model is applied to the above MST regimes with realistic density profiles for the high energy component and geometry of the (1,5) mode. The theory predicts reduction of the growth rate that is in reasonable agreement with the experimental observations. These effects are also analyzed with the use of NIMROD by extending a hybrid drift-kinetic-MHD model to include fast ion motion. Work supported by DOE and NSF. [Preview Abstract] |
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S1.00005: Simulation of the effects of energetic ions on tearing modes Huishan Cai, Guoyong Fu The physics of energetic ions and tearing modes are two critical problems in ITER. It is important to understand the physics of the influences of energetic ions on tearing modes, and to explore the control method for tearing modes by energetic ions in ITER and future tokamaks. Recently, it was found that tearing instability can be enhanced by counter circulating energetic ions (CEI) and can be reduced by co-CEI [1]. In this work, the simulation of the effects of energetic ions on linear tearing modes by the M3D code is performed. We focus on the kinetic effects of energetic ions on tearing modes in the low beta plasmas. It is found that the effects of energetic ions on linear tearing modes depend on the magnitude of fast ion gyro-radius and pressure, but don't depend on the velocity. For co-CEI, the growth rate of tearing modes decreases with gyroradius and decreases with beta below a critical value, while increases with beta above the critical value. For counter-CEI, the growth rate increases with gyroradius and beta. For the large gyroradius and low beta, the simulation results are qualitatively consistent with the analytical results [1]. It is also found that tearing mode instability can be enhanced by trapped ions. [Preview Abstract] |
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S1.00006: Simulation of Non-resonant Internal kink mode with Toroidal Rotation in NSTX Feng Wang, Guoyong Fu, Josh Breslau Plasmas in spherical tokamak with a safety factor above unity and weakly reversed magnetic shear may be unstable to an ideal, non-resonant internal kink mode. This mode, termed the ''long-lived mode'' (LLM) in MAST [1], will saturate and persist, exhibiting a strong m/n=2/1 component in NSTX. The resulting magnetic islands are capable of seeding neoclassical tearing modes (NTMs) [2]. Experimental results show that coupled 1/1 and 2/1 kink/tearing modes can also limit the sustained plasma beta. In this work, we perform nonlinear MHD simulations of the behavior of the non-resonant internal kink using M3D code initialized with measured NSTX equilibrium profiles. In particular, the effects of toroidal rotation are investigated systematically. The results show that when the rotation velocity is near the experimental level, its effect of equilibrium and linear stability is small. The nonlinear saturation level of the 1/1 mode is also weakly affected. However, the rotation is observed to have significant effects on the 2/1 island even at small value. With finite rotation, the 2/1 island width exhibits oscillations in the initial evolution before final steady state saturation. The width of the saturated island is reduced greatly as compared to that of non-rotating case. [1] ~I. Chapman et al Nuclear Fusion 50 (2010) 045007 [2] ~J. Breslau et al Nuclear Fusion 51 (2011) 063027 [Preview Abstract] |
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S1.00007: Radial Phase Shearing of Alfven Eigenmodes driven by Energetic Particles Guoyong Fu It is known that ideal MHD mode structure of Alfven eigenmodes such as TAE or RSAE has up-down symmetry for an up-down symmetric tokamak equilibrium. However, it has also been observed in many numerical simulations that such symmetry is broken for energetic particle-driven Alfven modes. The driven modes exhibit radial phase shearing of their 2D mode structure in a polodal plane. This radial phase shearing has clearly been observed by ECEI for beam-driven RSAE in the recent DIII-D experiments [1]. In this work, an analytic theory has been developed to show that energetic particle resonant drive can indeed break the ideal MHD symmetry and induce the radial phase shearing. Furthermore, the phase shearing is shown to be independent of the direction of plasma current. The phase shearing changes direction as the toroidal magnetic field is reversed. These analytic results agree with experimental observation as well as numerical simulations.\\[4pt] [1] B. J. Tobias \textit{et al.}, Phys. Rev. Lett. \textbf{106}, 075003 (2011); [Preview Abstract] |
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S1.00008: Double adiabatic theory of driven collisionless geodesic acoustic modes (GAMs) in toroids Adil Hassam, Robert Kleva, Wrick Sengupta The GAM is an axisymmetric oscillation of a toroidal magnetically confined plasma, resulting from an interplay between poloidal plasma rotation and perpendicular flux tube compression from the B field gradient. The frequency is super-parallel-sonic, ie, $\omega \quad \sim $ (ion thermal speed)/R, greater than the parallel acoustic mode which is lower by a factor of q. Consequently, collisionless geodesic acoustic modes in tokamaks can be described by the Chew-Goldberger-Low double-adiabatic fluid closures. This allows a simpler nonlinear formulation. We use these equations to study driven, collisionless GAMs in tokamaks. The motivation for this study is a proposal by Hallatschek and McKee to drive GAMs on the D3D tokamak at resonance. The drivers in the CGL theory include external magnetic forces to effect flux surface displacements as well as sources to provide modulated non-axisymmetric ion heating. We show that the linear mode frequency from CGL theory agrees with previous kinetic results. Comparisons will be made between different approaches to resonate the mode. Nonlinear effects will be evaluated. Results of a 2D toroidal numerical simulation of driven GAMs are described [Preview Abstract] |
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S1.00009: Exploration of turbulent optimization in stellarators {\&} tokamaks H. Mynick, N. Pomphrey, P. Xanthopoulos, M. Lucia A method\footnote{H.E. Mynick, N. Pomphrey, P. Xanthopoulos, Phys. Rev. Letters, 105, 095004 (2010).}$^,$\footnote{H.E. Mynick, N. Pomphrey, P. Xanthopoulos, Phys. Plasmas, 18, 056101 (2011).} recently developed for evolving toroidal configurations to ones with reduced turbulent transport, using the STELLOPT optimization codes and the GENE gyrokinetic code, is being applied and extended. The growing body of results has found that the effectiveness of the current proxy measure Q$_{prox}$ used by STELLOPT to estimate transport levels depends on the class of toroidal device considered. The present proxy works well for quasi-axisymmetric stellarators and tokamaks, modestly for quasi-helically symmetric designs, but not for the W7X quasi-omnigenous/quasi-isodynamic design. We are exploring the origin of this variation, and improving the dependence of the proxy on key geometric factors, extending the proxy to apply to transport channels other than the ITG turbulence it was originally developed for, and are also examining the relative effectiveness of different search algorithms. To help in these efforts, we have adapted STELLOPT to provide a new capability for mapping the topography of the cost function in the search space. [Preview Abstract] |
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S1.00010: The effects of 3-D shaping on ITG stability Mordechai Rorvig, Chris Hegna In this work we seek to understand how 3-D shaping can be used to improve ion temperature gradient stability. Part of the difficulty in deducing the role of 3-D shaping is the generation of 3-D MHD equilibria necessary for the calculations. In this work, MHD equilibrium surfaces are generated using local 3-D magnetostatic equilibrium theory [1]. We distinguish three different types of toroidal magnetic surface shaping: axisymmetric shaping, toroidal rotation of the cross section, and toroidal translation of the magnetic axis. We study these types of shaping independently and in combination to look for improvements. Linear growth rates for ITG modes are calculated using the gyrokinetics code GENE [2]. The geometric interface package GIST [3] accepts the equilibrium input data from the local equilibrium calculation. Growth rates for both axisymmetric and 3-D equilibrium calculations are presented. \\[4pt] [1] C. C. Hegna, Physics of Plasmas {\bf 7}, 3921 (2000).\\[0pt] [2] F. Jenko, W. Dorland, M. Kotschenreuther, and B. N. Rogers, Physical Review Letters {\bf 7}, 1904 (2000).\\[0pt] [3] P. Xanthopoulos, W. A. Cooper, F. Jenko, Yu. Turkin, A. Runov, and J. Geiger, Physics of Plasmas {\bf 16}, 082303 (2009). [Preview Abstract] |
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S1.00011: Simulation of Internal Kink Mode using GTC Joseph McClenaghan, Zhihong Lin Magnetohydrodynamic (MHD) instabilities excited by equilibrium current in toroidal fusion devices play important roles in plasma stability and confinement. Kinetic effects are important in the excitation and saturation of the MHD modes, as well as resulting transport. In this work, we have applied Gyrokinetic Toroidal Code (GTC) to study kinetic effects in current-driven MHD modes. As the first step, we have performed GTC simulation of the n=m=1 internal kink mode, which has been studied extensively in tokamak experiments, theory and MHD simulations. We will compare the dispersion relation and mode structure from the simulation to the ideal MHD theory in a low beta, large aspect ratio limit to verify the gyrokinetic simulation of current-driven MHD modes. [Preview Abstract] |
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S1.00012: Modeling of Nonlinear Beat Signals of TAE's Bo Zhang, Herbert Berk, Boris Breizman, Linjin Zheng Experiments on Alcator C-Mod reveal Toroidal Alfven Eigenmodes (TAE) together with signals at various beat frequencies, including those at twice the mode frequency. The beat frequencies are sidebands driven by quadratic nonlinear terms in the MHD equations. These nonlinear sidebands have not yet been quantified by any existing codes. We extend the AEGIS code to capture nonlinear effects by treating the nonlinear terms as a driving source in the linear MHD solver. Our goal is to compute the spatial structure of the sidebands for realistic geometry and q-profile, which can be directly compared with experiment in order to interpret the phase contrast imaging diagnostic measurements and to enable the quantitative determination of the Alfven wave amplitude in the plasma core [Preview Abstract] |
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S1.00013: Turbulent Impurity Transport Modeling for C-Mod Xiangrong Fu, Wendell Horton, William Rowan, Igor Bespamyatnov, Sadruddin Benkadda, Catherine Fiore Turbulent particle transport is investigated by analyzing boron impurity transport experiments in the Alcator C-Mod transport experiments with a quasilinear theory. Eigenvalue problems for sets of reduced fluid equations for the multi-component plasmas are solved to get the fluctuating field vector composed of the electric potential $\phi$, the main ion density $\delta n_i$, the impurity density $\delta n_{z}$ and the ion temperature fluctuation $\delta T_i$(for ITG). For Alcator C-Mod parameters, we investigate three drift waves models (1) the usual drift waves driven by density gradients, (2)impurity drift waves supported by the impurity density gradients and (3)turbulence driven by ITG mode. With turbulent spectrum obtained from simulations or nonlinear theories, we calculate particle transport coefficients and compare with the experiment and the neoclassical theory. This procedure results in a fast code that could run in real-time on the transport time scale to give the particle fluxes as a function of the state of the plasma. The code may be extended to include multiple modes for a more complete description of plasmas. Examples for the particle fluxes are given for C-Mod in the H modes and newly discovered I modes. Recent experiments reported on LHD are briefly discussed. [Preview Abstract] |
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S1.00014: Theory of external geodesic acoustic mode excitation Klaus Hallatschek, George R. McKee It is extremely appealing to be able to externally excite geodesic acoustic modes in a tokamak, either for diagnostic purposes, since the GAM frequency is dependent on the ion and electron temperature as well as the flux surface shapes, or, provided sufficiently large amplitude is achievable, to artificially reduce the turbulent transport, since GAMs are theoretically expected to impact the transport. It can be shown that injection of momentum by neutral particle beams or various plasma waves tends to be extremely inefficient. In contrast resonant excitation by magnetic perturbations induced by external coils is a viable and potentially efficient method. (In principle, this could already be done at present, e.g., in the DIII-D tokamak with the in-vessel RWM stabilization coils [I-coils]). The action of external coil currents on the interior flux surfaces of the plasma has been studied by means of a novel dynamic equilibrium code as well as analytically, deriving a surprisingly simple transfer function. The response of the turbulent plasma to the external driving has been studied using nonlocal NLET code runs. The results offer several control knobs to influence the drive effectivity and aid in designing a GAM drive antenna. [Preview Abstract] |
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S1.00015: Convective motion in low frequency trapped electron mode turbulence Yong Xiao Collisionless trapped electron mode (CTEM) is a prominent candidate for electron turbulent transport in burning plasmas. Global gyrokinetic simulation of CTEM turbulence finds both diffusive and convective motion using a Lagrangian analysis of the self-consistent electron orbits. A resonance broadening model fits well the diffusive and convective electron motion. The kinetic origin of the convective motion is identified to arise from the conservation of the second invariant when trapped electrons lose energy to the drift wave by toroidal precessional resonance. Strong correlation is found between the convection velocity and the kinetic energy loss rate, as shown by the 2D phase diagram. The connection between the convection velocity and the kinetic energy loss of trapped electrons are further verified by comparing analytic theory and simulation. The conservation of second invariant is found to act as a powerful constraint in low frequency turbulent transport, which can induce a convective motion by losing/gaining energy. This mechanism comes from single particle dynamics and is robust no matter whether the underlying process is quasilinear or not. The discovery has extensive applications in many fusion-related scenarios. (Phys. Plasmas 18 (letter), 110703, 2011) [Preview Abstract] |
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S1.00016: Neoclassical impurity transport in stellarator geometry J.M. Garc\'Ia-Rega\~na, C.D. Beidler, R. Kleiber, Y. Turkin, H. Maa\ss berg, P. Helander, K. Kauffmann The appearance of a (neoclassical) inward radial electric field in stellarators is known to cause, under certain plasma conditions, the accumulation of impurities in the core, and sometimes the subsequent plasma radiative collapse. Quantitatively neoclassical theory has barely covered the impurity transport due to the conventional neglect of the assumed first order electrostatic potential and density, $\Phi_{1}$ and $n_{1}$ respectively, in the drift kinetic ordering. This practice, which ignores the fulfilment of the quasi-neutrality condition, carries intrinsically the assumption $Z|e|\Phi_1/k_{\rm {B}}T\ll 1$, with $Z$ the atomic number, $|e|$ the unit charge, $k_{\rm {B}}$ the Boltzmann constant and $T$ the temperature. This inequality, valid for the bulk plasma, is violated by high $Z$ impurities. In this work the $\delta f$ PIC Monte Carlo code EUTERPE [1] together with the GSRAKE code [2] are used to obtain the first numerical output of neoclassical impurity dynamics retaining $\Phi_{1}$ and $n_{1}$ in the drift kinetic equation. The case of the LHD stellarator is considered.\\[4pt] [1] V. Kornilov {\it et al}, \textit{Nucl. Fusion} \textbf{45} 238, 2005.\\[0pt] [2] D. Beidler and W. D. D'haeseleer, \textit{Plasma Phys. Control. Fusion} \textbf{37} 463, 1995. [Preview Abstract] |
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S1.00017: Linear gyrokinetic studies in NCSX, W7-AS, and W7-X stellarators using GS2 J.A. Baumgaertel, W. Guttenfelder, G.W. Hammett, D.R. Mikkelsen, P. Xanthopoulos, J. Geiger, W. Dorland, E. Belli The GS2 gyrokinetic code is being used to study microinstabilities and turbulence in non-axisymmetric flux-tube geometries. Non-axisymmetric systems, such as stellarators, have a number of interesting features, like natural reversed magnetic shear and a large number of shaping parameters. These offer possibilities for reducing microturbulence and improving performance. The NCSX, W7-AS, and W7-X designs were partially optimized for neoclassical transport; however, the turbulent transport has not been studied in detail. We will present studies of gyrokinetic instabilities in NCSX and W7-X equilibria, including important geometry and linear benchmarks between GS2 and GENE, a gyrokinetic code from IPP. Specifically, we have investigated effects of plasma beta and magnetic shear on linear instabilities in NCSX geometry. In addition, we have conducted studies of the microstability of realistic W7-AS plasmas. Finally, we will discuss improvements to the GS2 trapped particle treatment and a new computational grid generator for GS2. This work was supported by the SciDAC Center for the Study of Plasma Microturbulence and Department of Energy Contract DE-AC02-09CH11466. [Preview Abstract] |
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S1.00018: Gyrokinetic Studies of Turbulence Reduction with Reverse Shear ETG Transport Barriers or Lithium Walls G.W. Hammett, J.L. Peterson, E.M. Granstedt, R. Bell, W. Guttenfelder, S. Kaye, B. LeBlanc, D.R. Mikkelsen, D.R. Smith, H.Y. Yuh, J. Candy The National Spherical Torus Experiment (NSTX) can achieve high electron confinement regimes that are super-critically unstable to the electron temperature gradient (ETG) instability. These electron internal transport barriers (e-ITBs) occur when the magnetic shear becomes strongly negative. Using the gyrokinetic code GYRO, the first nonlinear ETG simulations of NSTX e-ITB plasmas demonstrate reduced turbulence consistent with this observation. This is qualitatively consistent with a secondary instability picture of reduced ETG turbulence at negative shear (Jenko and Dorland PRL 2002). Local simulations identify a strongly upshifted nonlinear critical gradient for thermal transport that depends on magnetic shear. Global simulations show that ETG-driven turbulence outside of the barrier is large enough to be experimentally relevant, but cannot propagate very far into the barrier. We also use GYRO to study turbulence in regimes that might be expected in the Lithium Torus eXperiment (LTX). While lithium has experimentally been shown to raise the edge temperature and improve performance, there can still be some turbulence from density-gradient-driven trapped electron modes, and a temperature pinch is found in some cases. (Supported by DOE.) [Preview Abstract] |
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S1.00019: The quench rule, Dimits shift, and eigenmode localization by small-scale zonal flows Sumire Kobayashi We perform gyrokinetic simulations in a simple Z-pinch geometry to study the physics of small scale, entropy-mode-driven zonal flows. The entropy-modes create radial $E\times B$ streamers, which become unstable to the Kelvin-Helmholz (KH) instability at the point of nonlinear saturation. Sufficiently close to marginal entropy-mode stability, the break-up of the streamers by the KH mode generates zonal flows that produce a nearly static, low transport state (the Dimits shift). The flows in this state have a preferred, automatically maintained level, typically several times stronger than the quench-rule threshold, that sits at a critical point of the linear mode-structure: the radial streamers of the entropy-modes become, at about the preferred shearing rate, radially localized to the regions where the shearing rate of the zonal flows pass through zero. Coincident with the localization, the linear growth rates drop to smaller but usually finite levels. [Preview Abstract] |
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S1.00020: Suppression of turbulence and subcritical fluctuations in differentially rotating gyrokinetic plasmas Alexander Schekochihin, Edmund Highcock, Steven Cowley Differential rotation suppresses linear instabilities in fusion plasmas. However, subcritical fluctuations that grow transiently can lead to sustained turbulence and transport. Here fluctuations driven by the parallel velocity gradient (PVG) and ITG in the presence of perpendicular velocity shear and zero magnetic shear are considered. There are no growing eigenmodes, so all excitations are transient. In the PVG-dominated regime, the maximum amplification factor is $e^N$ with $N\propto q/\epsilon$, the maximally amplified wavenumbers satisfy $k_y\rho_i\approx (\epsilon/q)^{1/3}k_\parallel v_{thi}/S$, where $S$ is the ${\bf E}\times{\bf B}$ shear. In the ITG-dominated regime, $N$ is independent of $k$ and $N\propto v_{thi}/(L_T S)$. For intermediate ITG-PVG regimes, $N$ is calculated vs. $q/\epsilon$, $L_T$ and $S$. Analytical results are supplemented by linear gyrokinetic numerical tests. Regimes with $N<1$ for all $k$ are possible for small enough $q/\epsilon<7$; ion-scale turbulence is expected to be fully suppressed in such regimes. For cases when it is not suppressed, an elementary heuristic theory of subcritical PVG turbulence and a scaling of the ion heat flux with $q$, $\epsilon$, $S$ and $L_T$ is proposed; the transport is much less stiff than in the ITG regime. [Preview Abstract] |
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S1.00021: The Zero Turbulence Region in a Toroidal Plasma Edmund Highcock, Alexander Schekochihin, Steven Cowley, Michael Barnes, Felix Parra, Colin Roach Turbulence in the presence of a strongly sheared equilibrium flow can be thought of as the result of a three-way competition between underlying drives such as a background temperature gradient, the suppression of turbulence due to the gradient of the component of the velocity perpendicular to the magnetic field, and the additional driving of turbulence by the gradient of the parallel velocity component. We present a thorough nonlinear investigation of these three effects, in the limit of zero magnetic shear, to show that there is a large region in parameter space where turbulence is completely suppressed, and that, at sufficiently low ratios of the magnetic safety factor to the inverse aspect ratio (which determines the ratio of the parallel and perpendicular components of the velocity) this zero turbulence region extends to extremely high temperature gradients. Indeed, the critical temperature gradient, R/LTc, can exceed 20, where LTc is the critical temperature gradient scale length and R the major radius. Thus, we show that the limit of zero magnetic shear, low magnetic safety factor, low aspect ratio and high toroidal flow is a promising regime for tokamak operation. [Preview Abstract] |
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S1.00022: Investigation of equilibrium and nonlinear stability of high beta 3-D configurations M. Schlutt, C.C. Hegna, C.R. Sovinec, E.D. Held, S.E. Kruger 3D MHD straight stellarator equilibria are generated with NIMROD. A vacuum equilibrium helical magnetic field is loaded in cylindrical geometry. This magnetic field is initialized to have a continuous symmetry or spoiled symmetry by adding 3D magnetic perturbations with helicities that are disproportionate with the dominant harmonic. These perturbations alter the magnetic spectrum, and produce magnetic islands and stochastic regions. Finite beta equilibria are generated via numerical simulation by introducing a heating source and employing self-consistent anisotropic pressure transport. A variety of magnetic configurations, including helically symmetric and spoiled symmetry cases, are investigated. To study the stability properties of 3D equilibria, finite beta equilibria are created which are helically symmetric. If the equilibrium is linearly unstable, MHD modes are triggered. The nonlinear consequences of violating MHD stability are simulated. These cases are compared to simulations of heated cases with fully 3D equilibrium fields, where symmetry-spoiling harmonics are added to the helically symmetric system and the degree of symmetry-spoiling is varied. Cases are shown where large differences are observed in the nonlinear time evolution of symmetric configurations compared with spoiled-symmetry configurations. The similarity between the effect of the magnitude of the symmetry-spoiling harmonics and the effect of varying the degree of anisotropic transport is studied. [Preview Abstract] |
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S1.00023: Comparisons of Linear and Nonlinear Plasma Response Models for Non-Axisymmetric Perturbations A.D. Turnbull, N.M. Ferraro, V.A. Izzo, M.J. Lanctot, L.L. Lao, E.A. Lazarus, Y.Q. Liu, A. Reiman, J.-K. Park The plasma response to nonaxisymmetric field perturbations can be treated as an initial value (or dynamic) stability problem or from a nearby perturbed equilibrium approach. The approaches are quite different in principle and examples for specific DIII-D discharges are discussed in which representative codes, both linear and nonlinear, are used to validate the solutions. While the different approaches yield similar answers in some cases, in others they are significantly different. The plasma response can be large and linear models can break down even when the external field is small. In general, comparisons of the differences in the solutions between the approaches then provides improved understanding of the response and yields additional insights into some of the physics issues. For the nearby equilibrium approach, constraints need to be imposed on the profiles to obtain the physically accessible equilibrium solution. These can be informed using full nonlinear extended MHD calculations that include sufficient physics and experimentally realistic dissipation. Constraints on the pressure can be found from transport and a set of constraints on the current may be obtained from considering magnetic helicity, which has a physical interpretation in terms of field line linkage. [Preview Abstract] |
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S1.00024: Impurity Mixing Due to MHD in MGI Simulations of DIII-D V.A. Izzo Simulations of massive gas injection (MGI) into DIII-D are performed with the NIMROD code. An impurity source that is strongly localized to the edge is used to initiate the simulated MGI shutdown, based on observations that gas jets do not penetrate deeply into the plasma. When the neutral impurity source is poloidally symmetric, it is found that the distribution of impurity ions does not remain poloidally symmetric as the ions diffuse toward the core; instead 2D flows tend to concentrate the impurities on the low-field-side (LFS). Later, 3D flows associated with an m=1/n=1 MHD instability, efficiently mix the impurities into the core, causing a rapid increase in the central electron density simultaneous with a rapid decrease in the central temperature. Further simulations are carried out in which the impurity source is concentrated only on the LFS or the high-field-side (HFS). The effects of varying the poloidal impurity distribution on the efficiency of the MHD mixing are investigated. [Preview Abstract] |
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S1.00025: Extended-MHD modeling of diamagnetic-drift tearing instabilities Jacob King, Scott Kruger We use analytics and computations with the NIMROD code to examine tearing stability in large-guide-field slab cases with a nonzero equilibrium pressure gradient. A well known result from drift-reduced MHD is the diamagnetic drift associated with the pressure gradient has a stabilizing influence were the dispersion relation becomes $(\gamma+i\omega_{*e})^{3}\gamma(\gamma+i\omega_{*i})=\gamma_{rMHD}^{5}$ [1]. Here $\omega_{*i}$ and $\omega_{*e}$ are the ion- and electron-diamagnetic frequencies and $\gamma_{rMHD}$ is the tearing growth rate with a resistive-MHD model. Preliminary computational results with an unreduced extended-MHD model do not produce the expected drift-reduced result. For moderate values of $\omega_{*i}$ ($\omega_{*i}\leq3\gamma_{rMHD}$), the computations follow the dispersion relation that would result if the $\nabla{p}_{e}$ term were not included in the drift-reduced parallel Ohm's law: $(\gamma+i\omega_{*e})^{4}(\gamma+i\omega_{*i})=\gamma_{rMHD}^{5}$. Analytics, guided by computational diagnostics, are used to examine the significant terms in the flux evolution equation and investigate the discrepancy with the drift-reduced result.\\[4pt] [1] For example Coppi, PoF 7, 1501 (1964); Biskamp, NF 18, 1059 (1978). [Preview Abstract] |
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S1.00026: Small perturbations in magnetically confined plasmas Linda Sugiyama Two types of small perturbation theories exist for plasmas confined in strong toroidal magnetic fields. Both are well developed and provide useful insights into plasma behavior. The MHD plasma equations can be linearized to predict exponential growth rates for small disturbances.\footnote{I.B.~Bernstein et al., \emph{Proc.~Roy.~Soc.~ London, Ser.~A} \textbf{244} 1765 (1958).} The toroidal magnetic field can be described as a Hamiltonian system with two degrees of freedeom, as long as the toroidal field component is nonzero. Small perturbations produce magnetic island chains at low order rational surfaces and characteristic stochasticity around magnetic X-points (homoclinic or heteroclinic tangles around hyperbolic saddle points in Hamiltonian dynamics), The X-points can be induced by the perturbations or exist in the equilibrium configuration). The two descriptions appear to predict different behavior for small plasma perturbations. The explanation of this apparent paradox has important implications for linear and nonlinear small plasma perturbations and for plasma models, that extend beyond MHD. The cases of magnetic tearing modes on interior flux surfaces and edge instabilities in plasmas with X-poinhts on the plasma separatrix are discussed. [Preview Abstract] |
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S1.00027: Investigating the Interactions Between MHD Instabilities and Microturbulence in Magnetized Plasmas S.D. James, D.P. Brennan, C. Holland The effects of small-scale drift-wave microturbulence on the evolution of MHD instabilities are known to be significant. Also important are the influences MHD instabilities have on turbulent fields. Existing codes used to study the behavior of MHD instabilities are typically not capable of investigating their interactions with microturbulence due to the wide range of spatial and temporal scales involved. We present progress towards the development of a new code solving a four field fluid model in slab geometry, which simultaneously models drift-wave turbulence and slow growing MHD instabilities. Our initial motivations are focused on the interactions between an unstable resistive tearing mode and turbulent fields, and the magnitudes and directionality of different nonlinear energy transfer channels. In particular we aim to compare approximate analytic forms for the effects turbulent viscosity and turbulent resistivity in the MHD evolution to the nonlinear simulations. [Preview Abstract] |
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S1.00028: Supersonic regime of the Hall-MHD tearing instability J.J. Ramos, E. Ahedo An earlier analysis of the Hall-MHD tearing instability [E. Ahedo and J.J. Ramos, Plasma Phys. Controlled Fusion 51, 055018 (2009)] has been extended to cover the regime where the growth rate becomes comparable or exceeds the sound frequency. Like in the previous subsonic work, a resistive, two-fluid Hall-MHD model with massless electrons and zero-Larmor-radius ions is adopted and a linear stability analysis about a force-free equilibrium in slab geometry is carried out. The most salient feature of the supersonic regime is that the mode eigenfunctions become intrinsically complex, but the growth rate remains purely real. More surprisingly, the dispersion relations remain of the same form as in the subsonic regime for any beta value, provided only that the ion inertial length is sufficiently small for the mode ion inertial layer width to be smaller than the macroscopic length scales (an extremely generous bound scaling like a positive power of the Lundquist number) thus allowing a boundary layer analysis. These results will allow further verification studies in very low beta regimes, including the zero-beta limit where there is disagreement among some older analytic results. [Preview Abstract] |
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S1.00029: Shock Waves in Hall-MHD George Hagstrom, Eliezer Hameiri Hall-MHD is a partial differential equation of degenerate parabolic type that describes the dynamics of an ideal two fluid plasma with massless electrons. We study shock waves and discontinuities in this system. We characterize planar travelling wave solutions and find solutions with discontinuities in the hydrodynamic variables. These solutions, which correspond to the ion-acoustic wave, arise due to the presence of hydrodynamic real characteristics in Hall-MHD. We demonstrate finite-time discontinuity formation for certain types of initial data with discontinuous derivatives and study the shock structure under different regularizations. We also explore the possible existence of solutions with discontinuous magnetic field. A non-algebraic, non-local set of jump conditions is derived under the assumption of $[B]\neq0$. These conditions are used to study the contact discontinuity and it is shown that massless electrons crossing the surface of discontinuity may enter and leave at different locations. These conditions suggest the possible existence of mathematically novel shocks in Hall-MHD. [Preview Abstract] |
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S1.00030: Magnetic Reconnection ``Crisis'' in Weakly Collisional Regimes P. Buratti, B. Coppi Experiments to identify the phase velocities of reconnecting modes in weakly collisional plasma regimes have focused on measuring the Doppler shift associated with the plasma local rotation. [1] The relevant phase velocity direction in the laboratory frame has been tentatively found to be that of the ion diamagnetic velocity both for ion beam and ohmic heated plasmas, the former having higher temperatures and lower densities. The expectations were that two classes of reconnecting mode could emerge: drift-tearing modes [2] and ``inductive modes'' [3] having opposite phase velocity directions, both classes requiring the pre-excitation of a different kind of mode [4] in order to become unstable in weakly collisional regimes. A theoretical justification for the observed mode characteristics is given.\\[4pt] [1] P. Buratti, Paper EXS/P5-02 (23rd IAEA Conf., 2010).\\[0pt] [2] B. Coppi, \textit{Phys. Fluids} \textbf{8}, 2273 (1965).\\[0pt] [3] B. Coppi, \textit{Bull. APS} \textbf{45}, 366 (2000).\\[0pt] [4] B. Coppi, ``Collective Phenomena'' Eds. G. Bertin \textit{et al.}, \textit{Publ. World Scientific}, \textbf{59} (2007). [Preview Abstract] |
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S1.00031: Spreading of Magnetic Reconnection X-lines in Three Dimensions Paul Cassak, Lucas Shepherd Naturally occurring magnetic reconnection often begins in a spatially localized region and spreads in the out-of-plane direction in time. A number of authors have studied this problem for magnetotail applications such as substorms and bursty bulk flows, for which the out-of-plane (guide) field is typically small. However, spreading also occurs in laboratory experiments and two-ribbon solar flares (such as the Bastille Day flare), and is inferred to occur at the dayside magnetopause. The reconnection site in each of these settings is known or thought to have a significant guide field. With no guide field, it was shown that the reconnection spreading is controlled by the species that carries the current. However, laboratory experiments with a large guide field (Katz et al., Phys. Rev. Lett., 104, 255004, 2010) revealed that spreading takes place in both directions at the Alfven speed based on the guide field. This implies a qualitative change of behavior as the guide field varies. We present a scaling argument for the condition on the guide field at which the nature of the spreading switches from being caused by current carriers to Alfven waves. Further, we show results of three-dimensional two-fluid simulations that agree with the theory. We discuss applications to observations. [Preview Abstract] |
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S1.00032: Global Modeling of Magnetic Reconnection Using Full Particle Simulations Vadim Roytershteyn, Homa Karimabadi Magnetic reconnection is the dominant mechanism for the solar wind mass and energy entry into the Earth's magnetosphere. In the collisionless plasma it is a complex multi-scale process, driven by macroscopic dynamics, but crucially dependent upon microscopic physics on the electron kinetic scales. Previous theoretical investigations have necessarily focused on either small scales, correctly describing the physics of reconnection in highly idealized small systems (such as Harris current sheets), or on macroscopic evolution without regard for the microphysics of reconnection (e.g., relying on numerical or imposed resistivity in global MHD codes). We present the first results of a simulation study that analyzes reconnection in a self-consistent global model that a) is adequately resolved to describe electron physics of reconnection and b) incorporates several key features of reconnection in the magnetosphere such as the manner in which the system is driven, absence of artificially imposed boundary conditions, presence of self-generated anisotropies and associated fluctuations, flows past the reconnection site, asymmetry of the current sheet, some aspects of the geometry, etc. [Preview Abstract] |
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S1.00033: A global normal form for two-dimensional mode conversion David Johnston, Eugene Tracy, Allan Kaufman Mode conversion is a phenomenon that is of interest as a method for heating in fusion reactors. A magnetosonic wave with dispersion relation $D_{MS}$ propagates toward the interior of the plasma, where it excites an ion-hybrid wave with dispersioon relation $D_{IH}$ and thereby transfers energy to the plasma. We wish to study this process using ray-based methods. The 2 $\times $2 dispersion matrix $D$, which is, in general, a function of the phase space variables $(x,y,k_x ,k_y )$, must be put into \textit{normal form}, in which the diagonals of $D$ are identified as the \textit{uncoupled} dispersion relations, $D_{MS}$ and $D_{IH}$, with the off-diagonals as the coupling constants. Once in this form, the dynamics of the mode conversion are analyzed using the technique of Tracy, et al. (E. R. Tracy, A. N. Kaufman, and A. J. Brizard, Phys. Plasmas 10, 2147 (2003)). The focus of our work is putting the dispersion matrix into normal form. We are considering a two-dimensional model of the polodial cross section of a tokamak reactor with a DT plasma with a density ratio of one-to-one. It is possible to put the dispersion matrix into normal form, \textit{locally}, everywhere on the dispersion surface (E. R. Tracy and A. N. Kaufman, Phys. Rev. Lett. 91, 130402 (2003)). However, using numerical methods, we put the matrix into normal form \textit{globally}. Once in normal form, it is possible to compute the rays and follow their trajectories using the notion of \textit{rooms} (E. R. Tracy, A. J. Brizard, D. Johnston, A. N. Kaufman, A. S. Richardson, and N. Zobin, Communications in Nonlinear Science and Numerical Simulation 17, 5, 2161 (2011)). [Preview Abstract] |
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S1.00034: Induced electric field by ion cyclotron wave heating Nong Xiang, John R. Cary Ion cyclotron resonant frequency (ICRF) heating has been widely used to heat ions in fusion devices. As the ion cyclotron wave passes the (lower hybrid or hybrid) resonance, the incident electromagnetic wave converts to an electrostatic wave and the wave amplitude reaches its maximum. Meanwhile, the parametric decay may be triggered for a sufficiently large input power. Thus,the wave amplitude forms a peak near the resonance with a width of a few tens of ion gyro-radius. For typical ICRF heating parameters, it is found that the ponderomotive force induced is very significant. As a result, the ions are expelled from the region while the electrons are pulled in. Therefore an ambipolar electric field is produced. Our 1D PIC simulations show that the electric field induced could be of order of 10kV/m, comparable to the typically observed radial electric field. It is believed that the resulted electric field should be important to particle transport and plasma rotation. [Preview Abstract] |
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S1.00035: Efficiency of a Traveling Wave Direct Energy Converter with High-Density Beam for Applications to Aneutronic Fusion Experiments Alfonso Tarditi Due to the appeal of aneutronic fusion, a variety of reactor concepts have been proposed in past. In most cases, to achieve a positive net power balance these reactor concepts rely on a significant re-circulation of the energy produced to maintain a non-equilibrium configuration (unlike ignited plasmas). The availability of a direct conversion process with high efficiency is then critical for determining the feasibility of a reactor (particularly when the ``almost true aneutronic'' reaction like $p-^{11}B$ is considered). A Traveling Wave Direct Energy Converter (TWDEC, [1]) is considered for the energy conversion of a high-density beam formed by the fusion products (MeV-range $\alpha $-particles). As in [2], a PIC code is utilized for a realistic beam model. The study is focused on the possibility of obtaining high-efficiency coupling between a modulated high-density ``bunched'' beam, accounting also for a neutralizing electron environment, and the TWDEC electrode collector structure.\\[4pt] [1] Momota\textit{ et al.} (1999) Fus. Tech., 35, 60\\[0pt] [2] Y.Yasaka \textit{et al.} (2009), Nucl. Fus., 49, 075009 [Preview Abstract] |
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S1.00036: Integrated Plasma Simulation of Lower Hybrid Current Drive in Tokamaks P.T. Bonoli, J.C. Wright, R.W. Harvey, D.B. Batchelor, L.A. Berry, C.E. Kessel, S.C. Jardin It has been shown in Alcator C-Mod that the onset time for sawteeth can be delayed significantly (up to 0.5 s) relative to ohmically heated plasmas, through the injection of off-axis LH current drive power [1]. We are simulating these experiments using the Integrated Plasma Simulator (IPS) [2], where the driven LH current density profiles are computed using a ray tracing component (GENRAY) and Fokker Planck code (CQL3D) [3] that are run in a tightly coupled time advance. The background plasma is evolved using the TSC transport code with the Porcelli sawtooth model [4]. Predictions of the driven LH current profiles will be compared with simpler ``reduced'' models for LHCD such as the LSC code which is implemented in TSC and which is also invoked within the IPS. \\[4pt] [1] C. E. Kessel \textit{et al}, Bull. of the Am. Phys. Soc. \textbf{53}, Poster PP6.00074 (2008). \\[0pt] [2] D. Batchelor \textit{et al}, Journal of Physics: Conf. Series \textbf{125}, 012039 (2008). \\[0pt] [3] R. W. Harvey and M. G. McCoy, Proc. of the IAEA Tech. Comm. Meeting on Simulation and Modeling of Therm. Plasmas, Montreal, Canada (1992). \\[0pt] [4] S. C. Jardin \textit{et al}, J. Comp. Phys. \textbf{66}, 481 (1986). [Preview Abstract] |
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S1.00037: Nonlinear kinetic simulation of radio frequency wave in fusion plasmas Animesh Kuley, Zhixuan Wang, Zhihong Lin We are looking into a new nonlinear kinetic simulation model to study the radio frequency heating and current drive of fusion plasmas using toroidal code GTC. In this model ions are considered as fully kinetic (FK) particles using Vlasov equation and the electrons are treated as drift kinetic (DK) particles using drift kinetic equation. This scheme is particularly suitable for plasma heating and current drive with wave frequencies lower than the electron cyclotron frequency, ranging from fast wave and ion cyclotron wave to lower hybrid wave. This model also can handle physics with realistic electron-to-ion mass ratio and nonlinear dynamics in the full torus simulation. [Preview Abstract] |
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S1.00038: Neoclassical and Initial Divertor-Geometry Tests of COGENT R.H. Cohen, M. Dorf, J.C. Compton, M. Dorr, T.D. Rognlien, P. Colella, P. McCorquodale, J. Angus, S. Krasheninnikov COGENT is a full-f continuum kinetic code being developed for study of edge physics phenomena in tokamaks. The code is distinguished by 4th order conservative discretization and mapped multiblock grid technology to handle the geometric complexity of the tokamak edge. We discuss a number of recent neoclassical results in closed-flux-surface geometry, in particular self-consistent neoclassical simulations with increasingly complete collision operators (Lorentz, full test-particle, and adding model momentum- and energy-conserving terms). We also examine the effects of strong radial electric fields on neoclassical transport and decay of geodesic acoustic modes (GAM's). The code is being upgraded to full single-null divertor geometry, with numerical geometric coefficients imported from an external MHD equilibrium calculation. We discuss several initial tests of the divertor code: advection of phase-space blobs through the x-point region, and neoclassical transport and flows in the presence of divertor losses. We also summarize progress on code-development activities needed to complete the divertor code. [Preview Abstract] |
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S1.00039: Effects of Increasing Edge Current Density on Pedestal Instabilities P. Zhu, C.C. Hegna, C.R. Sovinec Recent resistive MHD computations find stabilizing effects of the edge parallel current density on the low-$n$ instabilities localized in the tokamak pedestal region. Here $n$ is the toroidal mode number. Stabilization is prominent when the magnetic shear of high edge current density region becomes negative. A moderate stabilizing effect of the edge current density has also been observed in regimes of weakened positive magnetic shear. The reduced or reversed magnetic shear is self-consistently generated with an increase in current density. In these computations, the region between the foot of pedestal and the conducting wall is described as in a ``cold-halo'' model, where the plasma has a much lower temperature, Spitzer conductivity and current density. Both the hot plasma inside the pedestal region and the cold plasma in the halo region are governed by the same set of resistive MHD equations. These results appear to be consistent with previous analytic theory on peeling modes, even though the theory was based on the ideal MHD model where the hot plasma inside pedestal is considered as a free boundary fluid surrounded by an external vacuum. We further investigate the effects of edge current density ramping on the nonlinear evolution of edge localized instabilities in simulations. [Preview Abstract] |
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S1.00040: Micro-Instabilities of Tokamak Edge Pedestal Weigang Wan, Scott Parker, Yang Chen We study micro turbulence of the tokamak edge pedestal with global and flux tube gyrokinetic particle simulations using the electromagnetic particle code GEM. In global simulations, three different sets of DIII-D H-mode experimental profiles are used and the simulation results exhibit quite similar characteristics. The dominant instabilities appear to be two kinds of modes with comparable linear growth rates: a low n, high frequency mode that propagates in the electron diamagnetic direction and a high n, low frequency mode that propagates in the ion direction. The global results of the high n mode agree with our flux tube simulations, which are well benchmarked against other codes in the same study, including GYRO, GTC and HD7. These results are important for the EPED model which is developed to predict the maximum height and width of the edge pedestal. [Preview Abstract] |
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S1.00041: Shifting the CFC/W transition point on the first ITER divertor target plates: equilibrium considerations R.A. Kolesnikov, R.H. Bulmer, L.L. LoDestro, T.A. Casper, R.A. Pitts In the 2007 ITER Design Review, the CFC/W transition point on the first divertor target plates was lowered by 10 cm to allow some experience to be gained in the non-active phases of vertical target operation with strike points on W surfaces, in preparation for a full W divertor in the nuclear phase. For operation on W just above the transition point, we use the CORSICA code to investigate the range of possible H- and L-mode equilibria, with emphasis on the maximum plasma current, achievable shapes, etc. We then investigate the operational space as the transition is lowered still further (both L- and H-mode), while still ensuring sufficient carbon vertical target extent to fulfill the requirements of the non-active phase program. The primary aim of this study is to determine if the current transition point, which can still be modified within some range if required, is optimized with respect to gaining early operational experience on an all-metal target before the nuclear phases begin. In our previous work [1], we investigated the size of feasible $\beta _{p}$--li space for both reference and elevated strikes for operation at 14 MA (both L- and H-mode) as well as 12 MA (H-mode) currents. In this paper we present new results on the maximum achievable plasma current as a function of strikes locations. Also, we study plasma self-inductance, volt-second consumption and the vertical instability over the range of the new equilibria. [1] R.A. Kolesnikov, et al., 53$^{rd}$ APS/DPP, Salt Lake City (2011) [Preview Abstract] |
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S1.00042: 3D Blob Theory and Modelling Justin Angus, Maxim Umansky, Sergei Krasheninnikov Most of the work to date on plasma blobs is limited to 2D theory and simulations which ignore the variation of blob parameters along the magnetic field line. However, 3D features can have an important impact on blob dynamics. For example, if the time scale of the blob motion is on the order of the time scale of unstable drift waves, then drift wave turbulence (DWT) can drastically alter the dynamics of the blob from that predicted by 2D theory. The DWT depletes the density gradients and leads to a much more diffuse blob with reduced radial convection. Furthermore, an initially varying blob density profile along the field line can lead to blob spinning, which reduces the net charge that causes the blob to propagate radially outward. This is similar to the spinning that arises due to a radial temperature profile in 2D sheath connected blobs. Other 3D effects such as parallel flows and ExB shearing are also discussed. [Preview Abstract] |
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S1.00043: Dust in the divertor sheath: a problem or a possible solution to a problem? Gian Luca Delzanno, Xianzhu Tang In this work, we will present results on dust transport in the magnetized sheath near the divertor plate for micron-sized dust. We consider conditions relevant to present short-pulse tokamak machines as well as conditions for long-pulse ITER/DEMO reactors. We solve the dust charging equation, the dust equation of motion and the equations for dust heating and mass loss in the magnetized sheath. We present parametric studies changing the divertor plasma conditions and the angle of the equilibrium magnetic field relative to the wall. Our main result is that, for conditions relavant to DEMO, the stronger heat flux to the wall severely limits the dust survivability and mobility. We discuss the implications of this result for the divertor plates of long-pulse fusion reactors. We will also discuss two fusion technology solutions to DEMO PFC, the dust patch and the dust shield, based on externally introduced solid particulates to patch areas of net erosion and to provide the primary heat exhaust for the divertor. [Preview Abstract] |
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S1.00044: Equilibrium and convective stability of a plasma with interlinked toroidal and weak poloidal magnetic fields Dmitri Ryutov This study is related to the physics of a snowflake divertor [1]. One of its features is a large size of the zone of a very weak poloidal magnetic field in the divertor area. In this zone, the poloidal plasma beta (defined as a ratio of the plasma pressure to the pressure of a poloidal magnetic field) significantly exceeds unity. This brings up some interesting new features to the equilibrium and convective stability of the divertor plasma. It has recently been suggested [2] that convective instability can spread the heat over a larger area than normally assumed and ``activate'' all four divertor legs of a snowflake divertor, thereby significantly reducing the divertor heat load, especially during the ELM events. A quantitative analysis of the convective instability and/or loss of equilibrium at high poloidal betas will be presented and related to the heat flux reduction for a simplified geometry. Experimentally detectable signatures of plasma convection in the divertor zone will be considered. Work performed for U.S. DoE by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344\\[4pt] [1] D.D. Ryutov, Phys. Plas., 14, 064502, 2007;\\[0pt] [2] D.D. Ryutov et al., Contrib. Plasma Phys., to be published, 2012. [Preview Abstract] |
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S1.00045: Theoretical and computational studies of the sheath of a planar wall Martina Giraudo, Enrico Camporeale, Gian Luca Delzanno, Giovanni Lapenta We present an investigation of the stability and nonlinear evolution of the sheath of a planar wall. We focus on the electrostatic limit. The stability analysis is conducted with a fluid model where continuity and momentum equations for the electrons and ions are coupled through Poisson's equation. The effect of electron emission from the wall is studied parametrically. Our results show that a sheath instability associated with the emitted electrons can exist. Following Ref. [1], it is interpreted as a Rayleigh-Taylor instability driven by the favorable combination of the sheath electron density gradient and electric field. Fully kinetic Particle-In-Cell (PIC) simulations will also be presented to investigate whether this instability indeed exists and to study the nonlinear effect of electron emission on the sheath profiles. The simulations will be conducted with CPIC, a new electrostatic PIC code that couples the standard PIC algorithm with strategies for generation and adaptation of the computational grid. \\[4pt] [1] G.L. Delzanno, ``A paradigm for the stability of the plasma sheath against fluid perturbations,'' Phys. Plasmas 18, 103508 (2011). [Preview Abstract] |
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S1.00046: Edge Plasma Mode as the Signature of the EDA H-Regime B. Basu, B. Coppi, T. Zhou A deep $E_r$-well observed in the EDA H-Regime produced by the Alcator C-Mod machine is considered as inducing a sharp localization of the Quasi-Coherent Mode proposed as the signature of this regime. This has a phase velocity in the ion diamagnetic velocity direction, in the plasma reference frame [1], and is a ``viscous-ballooning-mode'' driven by the combined effects of the plasma pressure gradient and the magnetic field curvature that would be stable due to FLR effects in the absence of a finite viscosity $\mu^i_\bot$. The frequency is near $\omega_{di}\equiv k_\phi v_{di}$ where $v_{di}\equiv-[c/(e nB_\theta)]d p_i/dr$ and $k_\phi = n^0/R$. The relevant model dispersion relation [2] obtained for a plane geometry in which the magnetic field curvature is simulated by an effective gravity is $(\omega+i\gamma_\mu)(\omega-\omega_{di})=-\gamma^2_{IC}+ k^2_\parallel v_A^2/[1+i k^2_\bot D_m/(\omega-\overline{\omega}_{*e})]$, where $\gamma_\mu\propto\mu^i_\bot$, $\gamma_{IC}$ is the interchange growth rate, $D_m\equiv c^2\eta_\parallel/(4\pi)$, $\eta_\parallel$ is the longitudinal resistivity and $\overline{\omega}_{*e}\equiv[ck_\phi T_e/(enB_\theta)](dn/dr)(1+1.71d\ln T_e/d\ln n_e)$.\\[4pt] [1] I. Cziegler (2010).\\[0pt] [2] B. Coppi, et al., Ann. Phys. 121,1 (1979) [Preview Abstract] |
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S1.00047: Modeling Edge Plasma Response to Non-Axisymmetric Fields in Tokamaks N.W. Ferraro, L.L. Lao, D.M. Orlov, R.A. Moyer, R.J. Buttery The application of non-axisymmetric fields to H-mode tokamak plasmas is observed to have significant effects on transport, rotation, and edge-localized mode (ELM) stability. Here, a two-fluid model is used to calculate the edge plasma response in order to elucidate the physical mechanism of these effects. This model self-consistently includes rotation, two-fluid physics, resistivity, and viscosity, as well as the full plasma and scrape-off layer in diverted toroidal geometry. The M3D-C1 code is used to perform the two-fluid modeling. The plasma response to applied $n=3$ fields of both ELMing and ELM-suppressed DIII-D discharge fields is considered, with a focus on the stochastization of the edge and the penetration of islands near the top of the pedestal. Empirically, shifts in the density and temperature pedestal location of more than one centimeter are observed, far in excess of the predictions of vacuum modeling. The results of two-fluid calculations compare favorably with data from Thomson scattering and beam emission spectroscopy diagnostics for several DIII-D discharges. The electromagnetic and neoclassical toroidal viscous torques on the plasma are estimated. [Preview Abstract] |
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S1.00048: Edge Profile Evolving Plasma Turbulence Simulations Bo Li, Darin Ernst We have developed a fluid edge turbulence code for the simulation of scrape off layer turbulence in a tokamak device. The fluid model is based on simplified drift-ordered Braginskii equations. The code is radially global, field-line following, and flux driven. The simulations evolve the density, temperature and electric potential within an initially simplified fluid framework. No separation is made between perturbations and equilibrium. Plasma profiles are evolved self-consistently in response to the heat and particle sources, subject to the cross-field transport produced by plasma instabilities, and plasma losses at the sheath resulting from sonic scrape-off layer flows. The simulation domain includes both closed and open field line regions. A strong curvature-driven interchange instability is observed on the low field side of pressure profile, which develops turbulent structures in the nonlinear phase resembling blobs. [Preview Abstract] |
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S1.00049: Modeling of the ``death-ray'' phenomenon in tokamak edge M.V. Umansky, D. Brunner, B. LaBombard, T.D. Rognlien In the ``death-ray'' regime often seen in tokamak edge plasma experiments, the downstream electron pressure, as measured by Langmuir probes at the divertor plate, exceeds the upstream values by nearly a factor of 2 over a narrow radial region at the strike point [1,2]. However, recent studies on Alcator C-Mod indicate that the death-ray over-pressure may be a result of local plasma perturbation by the negative probe bias [3]. We investigate the effects of probe perturbation of the plasma using the tokamak edge fluid code UEDGE. The code models a slab-like configuration roughly matching the basic dimensions and characteristics of edge plasma in Alcator C-Mod near detachment, where the death-ray is often observed. In the code setup, a small axisymmetric segment of target plate is biased, which mimics a plate-mounted Langmuir probe. It is observed in the simulations that at sufficiently large negative bias voltage the probe substantially modifies the local plasma characteristics. Moreover, the simulations reproduce the overpressure along the field line, similar to the experimental death-ray; pointing to the interplay of ion-neutral momentum exchange and the sheath boundary conditions [4].\\[4pt] [1] D. Brunner et al., APS DPP 2010, poster TP9.00069;\\[0pt] [2] B. LaBombard et al., J. Nucl. Mater. 241-243, 149-166 (1997);\\[0pt] [3] A. Loarte et al., Nucl. Fusion 38, 331 (1998);\\[0pt] [4] M.V. Umansky et al., Contrib. Plasma Phys., accepted (2011). [Preview Abstract] |
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S1.00050: Recent progress on a discontinuous Galerkin method for Vlasov-like systems Y. Cheng, I.M. Gamba, P.J. Morrison A discontinuous Galerkin (DG) method for integrating the Vlasov-Poisson system [1] is described and generalized. Higher order polynomials on basis elements are used in recent calculations and an extensive error analysis has been performed. In particular, recurrence properties have been examined in detail. The method is conservative and a limiter is designed to preserve positivity of the distribution function. Several linear and nonlinear examples will be presented that elucidate the DG methods ability to resolve filamentation and obtain high resolution BGK states. Progress on adaption of the algorithm to a kinetic ITG model and the Maxwell-Vlasov system will be discussed. \\[4pt] [1] R.E.~Heath, I.M.~Gamba, P.J.~Morrison, and C.~Michler, J.~Comp.~Phys.~{\bf 231} 1140 (2012). [Preview Abstract] |
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S1.00051: Determination of broken KAM surfaces in Hamiltonian systems Roscoe White The destruction of KAM surfaces in a Hamiltonian conservative system is an important topic in nonlinear dynamics, and in particular in the theory of particle orbits in toroidal magnetic confinement systems. In this paper we compare three different methods for the detection of the loss of stability of orbits in the dynamics of charged particles in a toroidal magnetic confinement device in the presence of time dependent magnetic perturbations. Only through resonance can orbits be significantly modified in an integrable conservative dynamical system. Without resonance the trajectories in phase space occupy Kolmogorov Arnold Moser (KAM) surfaces that prohibit diffusion. Often individual perturbations of an integrable system are so small that the system is very far from Chirikov overlap, and thus the destruction of the last KAM surface is not a good paradigm for the examination of diffusive transport. It can be necessary to understand the domains of broken KAM surfaces for each perturbation separarately, and to find the resulting transport due to the action of all modes together. We compare methods of determining the stability of orbits in a Hamiltonian system,the method of phase vector rotation, that of the finite Lyapunov indicator and that of frequency analysis. [Preview Abstract] |
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S1.00052: Tensor product decomposition methods for plasmas physics computations D. del-Castillo-Negrete Tensor product decomposition (TPD) methods are a powerful linear algebra technique for the efficient representation of high dimensional data sets. In the simplest 2-dimensional case, TPD reduces to the singular value decomposition (SVD) of matrices. These methods, which are closely related to proper orthogonal decomposition techniques, have been extensively applied in signal and image processing, and to some fluid mechanics problems. However, their use in plasma physics computation is relatively new. Some recent applications include: data compression of 6-dimensional gyrokinetic plasma turbulence data sets,\footnote{D. R. Hatch, D. del-Castillo-Negrete, and P. W. Terry. Submitted to Journal Comp. Phys. (2011).} noise reduction in particle methods,\footnote{R. Nguyen, D. del-Castillo-Negrete, K. Schneider, M. Farge, and G. Chen: Journal of Comp. Phys. 229, 2821-2839 (2010).} and multiscale analysis of plasma turbulence.\footnote{S. Futatani, S. Benkadda, and D. del-Castillo-Negrete: Phys. of Plasmas, 16, 042506 (2009)} The goal of this presentation is to discuss a novel application of TPD methods to projective integration of particle-based collisional plasma transport computations. [Preview Abstract] |
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S1.00053: Parallel heat transport in reversed shear magnetic field configurations D. Blazevski, D. del-Castillo-Negrete Transport in magnetized plasmas is a key problem in controlled fusion, space plasmas, and astrophysics. Three issues make this problem particularly challenging: (i) The extreme anisotropy between the parallel (i.e., along the magnetic field), $\chi_\parallel$, and the perpendicular, $\chi_\perp$, conductivities ($\chi_\parallel/\chi_\perp$ may exceed $10^{10}$ in fusion plasmas); (ii) Magnetic field lines chaos; and (iii) Nonlocal parallel transport. We have recently developed a Lagrangian Green's function (LG) method to solve the local and non-local parallel ($\chi_\parallel/\chi_\perp \rightarrow \infty$) transport equation applicable to integrable and chaotic magnetic fields.\footnote{D. del-Castillo-Negrete, L. Chac\'on, PRL, {\bf 106}, 195004 (2011); D. del-Castillo-Negrete, L. Chac\'on, Phys. Plasmas, APS Invited paper, submitted (2011).} The proposed method overcomes many of the difficulties faced by standard finite different methods related to the three issues mentioned above. Here we apply the LG method to study transport in reversed shear configurations. We focus on the following problems: (i) separatrix reconnection of magnetic islands and transport; (ii) robustness of shearless, $q'=0$, transport barriers; (iii) leaky barriers and shearless Cantori. [Preview Abstract] |
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S1.00054: Asymptotic-preserving Lagrangian approach for modeling anisotropic transport in magnetized plasmas Luis Chacon, Diego del-Castillo-Negrete Modeling electron transport in magnetized plasmas is extremely challenging due to the extreme anisotropy between parallel (to the magnetic field) and perpendicular directions (the transport-coefficient ratio $\chi_{\parallel}/\chi_{\perp} \sim 10^{10}$ in fusion plasmas). Recently, a novel Lagrangian Green's function method has been proposed\footnote{D. del-Castillo-Negrete, L. Chac\'on, \emph{PRL}, {\bf 106}, 195004 (2011); D. del-Castillo-Negrete, L. Chac\'on, \emph{Phys. Plasmas}, submitted (2011)} to solve the local and non-local \emph{purely} parallel transport equation in general 3D magnetic fields. The approach avoids numerical pollution, is inherently positivity-preserving, and is scalable algorithmically (i.e., work per degree-of-freedom is grid-independent). In this poster, we discuss the extension of the Lagrangian Green's function approach to include perpendicular transport terms and sources. We present an asymptotic-preserving numerical formulation, which ensures a consistent numerical discretization temporally and spatially for {\em arbitrary} $\chi_{\parallel}/\chi_{\perp}$ ratios. We will demonstrate the potential of the approach with various challenging configurations, including the case of transport across a magnetic island in cylindrical geometry. [Preview Abstract] |
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S1.00055: Lattice Boltzmann formulation for Braginskii magnetohydrodynamics Paul Dellar We present a lattice Boltzmann formulation of the Braginskii magnetohydrodynamic equations that describe large-scale motions in strongly magnetised plasmas. Fluid quantities, density, velocity and stress, are represented by a finite set of distribution functions associated with particles moving on a square or cubic lattice. Equilibrium distributions are constructed from Hermite moment expansions, so slowly varying solutions of the discrete kinetic equation exactly satisfy the Navier--Stokes or MHD momentum equations. Electromagnetic quantities are represented by a second kinetic equation for a set of vector-valued distribution functions. Maxwell's equations and the resistive MHD induction equation may be recovered from slowly varying solutions using different scalings. The resulting algorithm, comprising only local operations at grid points and data copying between adjacent points, readily lends itself to large-scale parallel computations. We modify the collision operator to apply different relaxation times to components of the stress parallel and perpendicular to the local magnetic field, simulating a form of the Braginskii MHD equations encountered in astrophysics. Large shears develop in simulations where the fluid velocity perpendicular to the field lines reverses. [Preview Abstract] |
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S1.00056: Hermite expansions with hypercollisionality for velocity space degrees of freedom in ion-temperature-gradient driven instabilities Joseph Parker, Paul Dellar We study a 1+1 dimensional model for instabilities driven by ion density and temperature gradients. We use a truncated Hermite expansion in velocity space, and an iterated Fokker--Planck collision operator. This selectively damps the highest terms in the Hermite expansion, analogous to the hyperviscous dissipation used in Fourier spectral simulations of Navier--Stokes turbulence. Our approach accurately captures the full range of growing and decaying modes with only a few tens of Hermite coefficients in velocity space, and is insensitive to parameter values in the collision operator. Without hypercollisions, hundreds of Hermite coefficients are required to capture the growing modes. Decaying modes (due to Landau damping in the original kinetic equation) are completely absent without some form of collisionality, or other source of dissipation. We also derive an partial differential equation for the flow of relative entropy in Hermite space. Solutions of this equation are in excellent agreeement with computed Hermite spectra. The approach developed here extends to more general kinetic equations, and should improve the accuracy of large-scale gyrokinetic simulations, for which only modest numbers of degrees of freedom in velocity space are computationally feasible. [Preview Abstract] |
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S1.00057: Numerical calculation of neoclassical distribution functions in an axisymmetric torus B.C. Lyons, S.C. Jardin, J.J. Ramos We solve for stationary, axisymmetric distribution functions ($f$) in the conventional banana regime for both ions and electrons using a set of drift-kinetic equations (DKEs) with complete Landau collision operators. Solubility conditions on the DKEs determine the relevant non-Maxwellian pieces of $f$ (called $f_{NM}$). We work in a 4D phase space in which $\psi$ defines a flux surface, $\theta$ is the poloidal angle, $v$ is the total velocity, and $\lambda$ is the pitch angle parameter. We expand $f_{NM}$ in finite elements in both $v$ and $\lambda$. The Rosenbluth potentials, $\Phi$ and $\Psi$, which define the collision operator, are expanded in Legendre series in $\cos \chi$, where $\chi$ is the pitch angle, Fourier series in $\cos \theta$, and finite elements in $v$. At each $\psi$, we solve a block tridiagonal system for $f_{NMi}$ (independent of $f_e$), then solve another block tridiagonal system for $f_{NMe}$ (dependent on $f_i$). We demonstrate that such a formulation can be accurately and efficiently solved. Results will be benchmarked against other codes (e.g., NEO) and could be used as a kinetic closure for an MHD code (e.g., M3D-C1). Results will also include the lowest-order collisionality correction and the use of generalized Grad-Shafranov equilibria. [Preview Abstract] |
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S1.00058: Are the electric field resonances in stellarator transport real? Matt Landreman Numerical calculations of the transport coefficients in stellarators predict that the radial transport becomes large for ``resonant'' values of the radial electric field. These resonances are understood to occur due to particles that do not experience an effective rotational transform due to the sum of parallel plus $E \times B$ motion. However, here it is argued that the resonances are an artifact of the ``monoenergetic and local'' approximation used in the calculations, in which the kinetic energy rather than total energy (kinetic plus electrostatic potential) is taken as a constant of particle motion. Due to the nonzero width of particle orbits, potential energy is not constant when a radial electric field is present, and the resulting variation in $v_{||}$ prevents particles from staying in resonance. We prove this point by explicitly calculating particle orbits without making the monoenergetic approximation. We also analytically calculate the radial ion heat transport in several idealized stellarator fields to show that when the monoenergetic approximation is not made, no resonance appears. [Preview Abstract] |
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S1.00059: Intrinsic rotation driven by neoclassical departure from a Maxwellian Peter J. Catto, Michael Barnes, Felix I. Parra Intrinsic rotation is the rotation observed in tokamak plasmas in the absence of momentum injection. Turbulence redistributes toroidal angular momentum within the core leading to differentially rotating plasmas that may have better stability and transport properties. To self-consistently calculate the momentum transport that leads to observed intrinsic rotation profiles, it is necessary to account for all the mechanisms that provide a preferred direction for the plasma rotation. A fundamental symmetry of the fluctuations shows that to lowest order in an expansion in the ratio of the gyroradius over scale length, turbulent momentum transport cannot give rise to intrinsic rotation, and higher order effects must be considered. This feature is an added difficulty, because it implies going to higher order, but also an advantage since it helps identify which mechanisms will lead to intrinsic rotation. In this poster, we consider for the first time the effect of the neoclassical departures from a Maxwellian on the turbulence, showing that it is an appreciable contribution to intrinsic rotation. We will study how this intrinsic rotation drive changes with temperature gradient and collisionality. [Preview Abstract] |
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S1.00060: Bootstrap current for a deuterium-tritium mixture Felix I. Parra, Michael Barnes, Peter J. Catto The parallel current is calculated analytically for tokamak plasmas with two ion species and electrons in the large aspect ratio limit using the full Fokker-Planck collision operators. Two collisionality limits are considered: the plateau and the banana regime. This calculation gives the bootstrap current for a deuterium-tritium mixture. The bootstrap current not only depends on the total pressure gradient and the electron and ion temperature density gradients, but it also contains a term proportional to the difference between the deuterium density gradient and the tritium density gradient. The consequences for a reactor will be discussed. [Preview Abstract] |
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S1.00061: The complete set of Casimirs in Hall-MHD Yohei Kawazura, Eliezer Hameiri A procedure to determine all Casimir constants of motion in MHD\footnote{E. Hameiri, Phy. Plasmas, {\bf 11}, 3423 (2004).} is extended to Hall-MHD. We obtain differential equations for the variational derivatives of all Casimirs which must be satisfied for any dynamically accessible motion of Hall-MHD. In an extension of the more commonly considered model, we also include the electron fluid entropy. The most interesting case, usually true for axisymmetric configurations, is when both the electron and ion entropy functions form families of nested toroidal surfaces. The Casimirs are then three functions of each of the entropies, involving fluxes of certain vector fields and the number of particles contained in each torus. If any of the species loses its nested tori, the number of the associated Casimirs is much larger (but physically less relevant). [Preview Abstract] |
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S1.00062: Reference Magnetic Coordinates (RMC) for toroidal confinement systems Leonid Zakharov, Egemen Kolemen, Samuel Lazerson Because of intrinsic anisotropy of high temperature plasma with respect to magnetic field, use of proper coordinates is of high priority for both theory and numerical methods. While in axisymmetric case, the poloidal flux function $Y(r,z)=const$ determines proper flux coordinates, in 3-D, such a function does not exist. The destruction of nested magnetic surfaces even by small 3-D perturbations leads to a sudden change of topology of magnetic field. As a result, the coordinate systems can no longer be based on tracing the magnetic field lines resulting in difficulties for theory and 3-D numerical simulations. The RMC coordinates $\hat{a},\hat{\theta},\hat{\zeta}$ presented here (introduced in 1998 but not really used) are nested toroidal coordinates, which are best aligned with an ergodic confinement fields. In particular, in RMC the vector potential of the magnetic field has an irreducible form $$ {\bf A} = \bar{\Phi}_{00}(\hat{a})\nabla\hat{\theta} +\left[Y_{00}(\hat{a}) +\bar{\psi}^*(\hat{a},\hat{\theta},\hat{\zeta})\right]\nabla\hat{\zeta} ,$$ where 3-D function $\bar{\psi}^*$ contains only resonant Fourier harmonics of angle coordinates. RMC can be generated and advanced using a fast (Newton) algorithm not involving the field line tracing. [Preview Abstract] |
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S1.00063: SABR Fusion-Fission Hybrid Studies Chris Stewart The Subcritical Advanced Burner Reactor (SABR) concept is a fast reactor comprised of a tokamak fusion neutron source based on ITER surrounded by an annular fission core adapted from Integral Fast Reactor designs. Previous work has examined SABR used to help close the nuclear fuel cycle by fissioning the transuranics from spent nuclear fuel. One focus of the present work is a SABR Breeder Reactor to achieve tritium self-sufficieny and a Pu breeding ratio significantly above 1 in order to provide fuel for SABR as well as for MOX-fueled LWR's and other fast reactors. Another focus of this research is the dynamic safety simulation of lloss-of-flow loss-of-heat-sink, loss-of-power, and positive reactivity accidents in the TRU fuel SABR burner reactor. The reactivity effect of thermal-induced bowing of fuel pins has been modeled, which is expected to provide passive safety. [Preview Abstract] |
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S1.00064: BOUT Simulations of Edge Turbulence in the DIII-D Tokamak Bruce Cohen, Maxim Umansky, William Nevins, Mike Makowski, Jose Boedo, Dmitry Rudakov, Chris Holland, George McKee, Zheng Yan Progress is reported on simulations of electromagnetic drift-resistive ballooning turbulence in realistic single-null tokamak geometry using the BOUT three-dimensional fluid code [1] that solves Braginskii-based fluid equations [2]. The simulation domain models the actual magnetic geometry of the DIII-D tokamak. The simulations follow unstable drift-resistive ballooning turbulence in the edge region to saturation. Fluctuation amplitudes, fluctuation spectra, and particle and thermal fluxes are compared to experimental probe and beam-emission-spectroscopy data for a well-characterized L-mode discharge in DIII-D. Post-processing of the simulation data using synthetic diagnostics facilitates the comparisons. The simulations are comprised of a suite of runs in which the physics model evolves to include more fluid fields and physics terms. The relative agreement of the simulation results with the experimental data improves as more physics is included. \\[4pt] [1] X. Q. Xu, and R. H. Cohen, Contrib. Plasma Phys. 36 (1998) 158. \\[0pt] [2] S. Braginskii, ``Transport Processes in a Plasma,'' in Rev. Plas. Phys., Vol 1, ed. M. A. Leontovich (Consltnts. Bureau, New York, 1965), p. 205. [Preview Abstract] |
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S1.00065: GAM damping and neoclassical rotation in a tokamak A.G. Elfimov, R.J.F. Sgalla, R.M.O. Galvao, A. Smolyakov Geodesic Acoustic Modes (GAM) are linear eigen-modes of poloidal plasma rotation supported by plasma compressibility in toroidal geometry and linearly coupled to drift-waves via toroidal side-bands of plasma pressure. These modes are expected to play an important role in dynamics of drift-wave turbulence. In this talk we describe the mechanisms of GAM damping via Landau collisionless wave-particle interaction and ion-ion collisions. GAM intrinsically involve anisotropic perturbations of plasma pressure (corresponding to parallel viscosity). This underlies the intrinsic relation of GAM modes with neoclassical rotation in a tokamak. It is shown that GAMs and standard equilibrium (neoclassical) plasma rotation represent two limit cases of poloidal plasma rotation: high frequency rotational mode (GAM) and the low frequency over damped (damping is larger than the real part of the frequency) mode of the neoclassical equilibrium rotation. New regimes of global GAM modes will be reported. These regimes occurs as a result of the parallel kinetic response of electrons which has not been included previously. It is shown that in certain regimes (corresponding to global modes), the electron response becomes strongly electromagnetic and GAM have significant electromagnetic components that affect mode damping. Resulting modifications in the mode dispersion and mode damping will be presented. [Preview Abstract] |
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S1.00066: Symplectic representation of higher-order guiding-center theory Alain Brizard, Natalia Tronko Two representations of guiding-center theory are possible depending on whether the guiding-center Poisson bracket (i.e., the symplectic structure) or the Hamiltonian contains higher-order corrections due to the nonuniformity of the magnetic field. By combining the guiding-center parallel hierarchy with the symplectic representation, the guiding-center equations of motion are derived with second-order corrections included in the symplectic structure without the need of carrying out the guiding-center transformation to second order. Guiding-center polarization and magnetization are thus shown to arise naturally from higher-order guiding-center theory within the context of a two-step derivation of nonlinear gyrokinetic theory.\footnote{A.~J.~Brizard and T.~S.~Hahm, Rev.~Mod.~Phys.~{\bf 79}, 421 (2007).} [Preview Abstract] |
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