MHD equilibrium calculations for stellarators. Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE)
|
|
- Luke Underwood
- 5 years ago
- Views:
Transcription
1 MHD equilibrium calculations for stellarators Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE) March 20, 2012
2 MAGNETIC FIELD LINE HAMILTONIAN Consider a general toroidal coordinate system (r, θ, φ) θ poloidal angle, φ toroidal angle Using the gauge freedom for the vector potential A, it is always possible to construct Ψ and χ such that B = φ Ψ + χ θ Ψ and χ not necessarily poloidal and toroidal flux In (χ, θ, φ) coordinates, the field line trajectories are given by dχ dφ = Ψ θ dθ dφ = Ψ χ Hamilton s equations with H Ψ, x θ and p χ The time φ is periodic, so this is a 1.5 degree of freedom Hamiltonian
3 FUNDAMENTAL COMPLICATIONS IN 3D Pendulum Energy conserved Energy not conserved Magnetic field surfaces Axisymmetry No Symmetry
4 THE WORK HORSE: VMEC (1) Based on an idea by Kruskal & Kulsrud; Numerical scheme first proposed by F. Bauer, O. Betancourt, and P. Garabedian; Implemented and developed as VMEC by S. Hirschman et al. Minimization of the plasma potential energy ( ) B 2 E = + ργ dv = E B + E int 2µ 0 γ 1 Ω Only allowed virtual displacements ξ satisfy δρ = ρ ξ δ B = ( ξ B ) Assuming there is a nested family of flux surfaces s = cst, define B ds = FT (s 0 ) B ds = FP (s 0 ) ρd 3 x = M(s 0 ) s s 0 s s 0 s s 0 F T, F P and M are given functions of s, held fixed during variation
5 THE WORK HORSE: VMEC (2) Turn constrained minimization into unconstrained minimization: Flux and solenoidal constraints incorporated by writing B = s G with G = F T (s)u + F P (s)v + λ(s, u, v) and with λ(s, u, v) a periodic function of u and v Mass constraint obtained from Hölder s inequality M(s) = m(s) = ρ dsdvdu J ρ J 1 γ 1 J γ 1 γ, M (s) m(s) = ( ρ γ dvdu J dvdu ρ (s, u, v) dvdu, J = J (x, y, z) ) 1 γ ( dvdu J ) γ 1 γ Minimum of E int when ρ = ρ(s) = m(s)/( dudv/j ) p = p(s)
6 THE WORK HORSE: VMEC (3) Choose v s.t. v = φ, φ usual toroidal angle Write E = D D D2 3 dsdvdu 2µ 0 D + 1 γ 1 D m(s) γ ( Ddvdu ) γ 1 ds (R, Z) (s, u), D (ψ, R) 1 (u, v), D (ψ, Z) 2 Rψ u, D 3 (u, v) R = R mn (s)cos(mu + nv) Z = Z mn (s)cos(mu + nv) Find R mn and Z mn corresponding to minimum of E using the Steepest Descent Method SIESTA can be used in conjunction with VMEC to compute equilibria with islands
7 A DIRECT SOLVER WITH ISLANDS: PIES Solves the MHD equilibrium equations iteratively 1. B p = 0 2. B p J = B 2 3. B ( ) J = J B 4. B = J Eq.(1) is used to compute magnetic coordinates This step is where most of the action is complications with rational surfaces, stochasticity... Eq.(2)-(3) are then solved analytically, mode by mode, in magnetic coordinates Eq.(4) is then solved for the updated magnetic field, in lab (toroidal) coordinates
8 ASYMPTOTIC APPROACHES: GREENE-JOHNSON (1961) ATF LHD Assumes strong toroidal guide field B 0 Expansion based on the small parameter δ B hel /B 0 1 Parameter Symbol Scaling Beta β β t δ 2 Inverse aspect ratio a/r 0 ɛ δ 2 Number of helical periods N 0 1/δ 2 Poloidal helicity l 1 Rotational transform ι/2π 1 NOTE: β ɛ
9 CONSEQUENCES OF THE G-J ORDERING R = R 0 + aρcosθ Z = aρsinθ φ = φ N 0 To lowest order, B ψ = 0 ɛn 0 B 0 ψ φ Greene-Johnson stellarators are perturbed tokamaks! a = + = ρ eρ + 1 e θ ρ θ 1 = ɛn 0 e φ 1 + ɛρcosθ φ 1 = 0 ψ = ψ(ρ, θ)
10 A NEW ORDERING W7-X HSX NCSX Same expansion parameter as Greene-Johnson: δ B hel /B 0 1 Parameter Symbol G-J Scaling New scaling Beta β β t δ 2 δ Inverse aspect ratio ɛ δ 2 δ Number of helical periods N 0 1/δ 2 1 Poloidal helicity l 1 1 Rotational transform ι/2π 1 1 NOTE: In both expansions, β ɛ
11 CONSEQUENCE OF THE NEW ORDERING To lowest order, B ( ψ = 0 ɛn 0 B 0 φ + B p ) ψ = 0 Equilibria can have a non-planar magnetic axis Our expansion: O(1) O(δ) B = B0 eφ + ( ) B φ1 ɛxb eφ 0 + B p1 J = Jφ1 eφ + J p1 p = p 1 (ρ, θ, φ) Greene-Johnson expansion: O(1) O(δ) O(δ 2 ) O(δ 3 ) B = B0 eφ + B 1 (ρ, θ, φ) + (B φ2 ɛxb 0 ) e φ + B p2 + B 3 (ρ, θ, φ) J = J2 (ρ, θ) + J 3 (ρ, θ, φ) p = p 2 (ρ, θ) +p 3 (ρ, θ, φ)
12 THE NEW EXPANSION (1) The B = 0 equation To lowest order, this is B p1 = 0. Introduce A 1 (ρ, θ, φ) s.t. ( B = 1 ɛx + B ) φ1 eφ + A 1 e φ B 0 B 0 The µ 0 J = B equation µ 0 a J 1 B 0 = e φ B φ1 B 0 e φ 2 A 1 The J = ( B p)/b 2 equation ( β1 2 + B ) φ1 = 0 B φ1 = β 1 B 0 B 0 2 θ-pinch pressure balance relation β 1 = 2µ 0p 1 B 2 0
13 THE NEW EXPANSION (2) The B p = 0 equation ( ɛn 0 φ e φ A 1 ) β 1 = 0 The J = 0 equation ( ɛn 0 φ e φ A 1 ) 2 A 1 = ɛ e Z β 1 ( φ + e φ A ) β = 0 ( φ + e φ A ) 2 A = e Z β A = A 1 ɛn 0 β = 1 ɛn 2 0 2µ 0 p 1 B 2 0 = β 1 ɛn 2 0
14 THE NEW EXPANSION: RESULTS ( φ + e φ A ) β = 0 ( φ + e φ A ) J φ = e Z β 2 A = J φ A = A 1 β = 1 ɛn 0 ( B = 1 ɛx β 1 B 0 2 ɛn 2 0 2µ 0 p 1 B 2 0 = β 1 ɛn 2 0 J φ = µ 0aJ 1φ ɛn 0 B 0 ) eφ + A 1 µ 0 a J 1 e φ = 1 eφ β 1 e φ 2 B 0 2 A 1 Equations agree with the Greene-Johnson expansion in the right limit Equations are in a convenient form for iterations
15 ITERATION STEPS Start with given A (0) for k = 1, 2,... ( φ + e φ A (k 1) ) β (k) = 0 ( φ + e φ A (k 1) ) J (k) φ = e Z β (k) 2 A (k) = J (k) φ until a stopping criterion holds
16 HYPERBOLIC EQUATIONS AND INITIAL CONDITIONS ( φ + e φ A ) β = 0 How to choose the I.C. at φ = 0 such that β is periodic in φ?
17 DICK AND JANE CALCULATE 3D EQUILIBRIA ( φ + e φ A ) β = 0 Given A, the field line trajectories are given by the characteristics: dρ dφ = 1 A ρ θ dθ dφ = 1 A ρ ρ Given a starting point (ρ 0, θ 0, φ 0 ), we can easily integrate these equations ρ(φ), θ(φ) Integrate for many turns (very large φ) to sample the whole magnetic surface If starting point is always of the form (ρ 0, 0, 0), ρ 0 is a unique label for each magnetic surface
18 EXTRACTING INFORMATION FROM CHARACTERISTICS Through this process, we have ρ = ρ(ρ 0, θ, φ) ( ρ ez β = dβ dρ 0 ρ 0 ) 1 [ sinθ cosθ ρ ] ρ θ
19
20
21
22 CLOSING THE ITERATION LOOP ( φ + e φ A ) J φ = e Z β We already know the characteristics for this equation, and the RHS of this equation on the characteristics Solving for J φ is a straightforward 1D (Lagrangian) integral We can then evaluate A/ ρ and A/ θ required for the next iteration: [ A 1 (ρ, θ, φ) = dρ dθ ρ ρ ρ cos(θ θ ] ) ρ 2π ρ 2 + ρ 2 2ρρ cos(θ θ J φ (ρ, θ, φ) ) A 1 (ρ, θ, φ) = dρ dθ ρρ 2 cos(θ θ ) θ 2π ρ 2 + ρ 2 2ρρ cos(θ θ ) J φ(ρ, θ, φ)
23 PRELIMINARY RESULTS LAST ISSUE? Start with vacuum field (in this case, dominant l = 2 component, some l = 3 and l = 4) β(ρ 0 ) = 0.01(1 ρ 2 0 )2
24 PRELIMINARY RESULTS LAST ISSUE? Flux surfaces after 1st iteration Need to calculate the new location of the magnetic axis to evaluate new β profile; is there a fast numerical scheme to do that?
25 REFERENCES Variational methods M.D. Kruskal and R.M. Kulsrud, Phys. Fluids (1958) F. Bauer, O. Betancourt and Paul Garabedian in A Computational Method in Plasma Physics (Springer-Verlag, New York, 1978) S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983) S. P. Hirschman, R. Sanchez, and C.R. Cook, Phys. Plasmas 18, (2011) PIES A.H. Reiman, H.S. Greenside, Comput. Phys. Commun. 43, 157 (1986) A.H. Reiman, H.S. Greenside, J. Comput. Phys (1988)
INITIAL EVALUATION OF COMPUTATIONAL TOOLS FOR STABILITY OF COMPACT STELLARATOR REACTOR DESIGNS
INITIAL EVALUATION OF COMPUTATIONAL TOOLS FOR STABILITY OF COMPACT STELLARATOR REACTOR DESIGNS A.D. Turnbull and L.L. Lao General Atomics (with contributions from W.A. Cooper and R.G. Storer) Presentation
More informationFrom tokamaks to stellarators: understanding the role of 3D shaping
Under consideration for publication in J. Plasma Phys. From tokamaks to stellarators: understanding the role of 3D shaping Samuel A. Lazerson and John C. Schmitt 2 Princeton Plasma Physics Laboratory,
More informationComputations of Vector Potential and Toroidal Flux and Applications to Stellarator Simulations
Computations of Vector Potential and Toroidal Flux and Applications to Stellarator Simulations NIMROD Team Meeting Torrin Bechtel April 30, 2017 Outline 1 Project Goals and Progress 2 Vector Potential
More information(a) (b) (c) (d) (e) (f) r (minor radius) time. time. Soft X-ray. T_e contours (ECE) r (minor radius) time time
Studies of Spherical Tori, Stellarators and Anisotropic Pressure with M3D 1 L.E. Sugiyama 1), W. Park 2), H.R. Strauss 3), S.R. Hudson 2), D. Stutman 4), X-Z. Tang 2) 1) Massachusetts Institute of Technology,
More informationLecture # 3. Introduction to Kink Modes the Kruskal- Shafranov Limit.
Lecture # 3. Introduction to Kink Modes the Kruskal- Shafranov Limit. Steve Cowley UCLA. This lecture is meant to introduce the simplest ideas about kink modes. It would take many lectures to develop the
More informationConfiguration Optimization of a Planar-Axis Stellarator with a Reduced Shafranov Shift )
Configuration Optimization of a Planar-Axis Stellarator with a Reduced Shafranov Shift ) Shoichi OKAMURA 1,2) 1) National Institute for Fusion Science, Toki 509-5292, Japan 2) Department of Fusion Science,
More informationGeneralized Solovev equilibrium with sheared flow of arbitrary direction and stability consideration
Generalized Solovev equilibrium with sheared flow of arbitrary direction and stability consideration D.A. Kaltsas and G.N. Throumoulopoulos Department of Physics, University of Ioannina, GR 451 10 Ioannina,
More informationStellarators. Dr Ben Dudson. 6 th February Department of Physics, University of York Heslington, York YO10 5DD, UK
Stellarators Dr Ben Dudson Department of Physics, University of York Heslington, York YO10 5DD, UK 6 th February 2014 Dr Ben Dudson Magnetic Confinement Fusion (1 of 23) Previously... Toroidal devices
More informationAN UPDATE ON DIVERTOR HEAT LOAD ANALYSIS
AN UPDATE ON DIVERTOR HEAT LOAD ANALYSIS T.K. Mau, A. Grossman, A.R. Raffray UC-San Diego H. McGuinness RPI ARIES-CS Project Meeting June 4-5, 5 University of Wisconsin, Madison OUTLINE Divertor design
More informationMHD Equilibrium and Stability of Tokamaks and RFP Systems with 3D Helical Cores
15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 MHD Equilibrium and Stability of Tokamaks and FP Systems with 3D Helical Cores W. A. Cooper Ecole Polytechnique Fédérale de
More informationLow Beta MHD Equilibrium Including a Static Magnetic Island for Reduced MHD Equations in a Straight Heliotron Configuration
Low Beta MHD Equilibrium Including a Static Magnetic Island for Reduced MHD Equations in a Straight Heliotron Configuration Kinya SAITO 1,a), Katsuji ICHIGUCHI 1,2) and Ryuichi ISHIZAKI 1,2) 1) The Graduate
More informationMHD. Jeff Freidberg MIT
MHD Jeff Freidberg MIT 1 What is MHD MHD stands for magnetohydrodynamics MHD is a simple, self-consistent fluid description of a fusion plasma Its main application involves the macroscopic equilibrium
More informationFinite-Orbit-Width Effect and the Radial Electric Field in Neoclassical Transport Phenomena
1 TH/P2-18 Finite-Orbit-Width Effect and the Radial Electric Field in Neoclassical Transport Phenomena S. Satake 1), M. Okamoto 1), N. Nakajima 1), H. Sugama 1), M. Yokoyama 1), and C. D. Beidler 2) 1)
More informationThe Linear Theory of Tearing Modes in periodic, cyindrical plasmas. Cary Forest University of Wisconsin
The Linear Theory of Tearing Modes in periodic, cyindrical plasmas Cary Forest University of Wisconsin 1 Resistive MHD E + v B = ηj (no energy principle) Role of resistivity No frozen flux, B can tear
More informationInnovative Concepts Workshop Austin, Texas February 13-15, 2006
Don Spong Oak Ridge National Laboratory Acknowledgements: Jeff Harris, Hideo Sugama, Shin Nishimura, Andrew Ware, Steve Hirshman, Wayne Houlberg, Jim Lyon Innovative Concepts Workshop Austin, Texas February
More informationRotation and Neoclassical Ripple Transport in ITER
Rotation and Neoclassical Ripple Transport in ITER Elizabeth J. Paul 1 Matt Landreman 1 Francesca Poli 2 Don Spong 3 Håkan Smith 4 William Dorland 1 1 University of Maryland 2 Princeton Plasma Physics
More informationHOW THE DEMO FUSION REACTOR SHOULD LOOK IF ITER FAILS. Paul Garabedian and Geoffrey McFadden
HOW THE DEMO FUSION REACTOR SHOULD LOOK IF ITER FAILS Paul Garabedian and Geoffrey McFadden 1. Summary Runs of the NSTAB equilibrium and stability code show there are many 3D solutions of the advanced
More informationRFP helical equilibria reconstruction with V3FIT-VMEC
RFP helical equilibria reconstruction with V3FIT-VMEC D. Terranova 1 J.D. Hanson 2, S.P. Hirshman 3, L. Marrelli 1 1 Consorzio RFX, Associazione EURATOM-ENEA sulla Fusione, Padova, Italy 2 Auburn University,
More informationThe RFP: Plasma Confinement with a Reversed Twist
The RFP: Plasma Confinement with a Reversed Twist JOHN SARFF Department of Physics University of Wisconsin-Madison Invited Tutorial 1997 Meeting APS DPP Pittsburgh Nov. 19, 1997 A tutorial on the Reversed
More informationI. INTRODUCTION ISLAND COORDINATE SYSTEM
Analytical theory of the shear Alfvén continuum in the presence of a magnetic island C. R. Cook 1, a) and C. C. Hegna 1 Departments of Physics and Engineering Physics, University of Wisconsin-Madison,
More informationSMR/ Summer College on Plasma Physics. 30 July - 24 August, Introduction to Magnetic Island Theory.
SMR/1856-1 2007 Summer College on Plasma Physics 30 July - 24 August, 2007 Introduction to Magnetic Island Theory. R. Fitzpatrick Inst. for Fusion Studies University of Texas at Austin USA Introduction
More informationBRIEF COMMUNICATION. Near-magnetic-axis Geometry of a Closely Quasi-Isodynamic Stellarator. Greifswald, Wendelsteinstr. 1, Greifswald, Germany
BRIEF COMMUNICATION Near-magnetic-axis Geometry of a Closely Quasi-Isodynamic Stellarator M.I. Mikhailov a, J. Nührenberg b, R. Zille b a Russian Research Centre Kurchatov Institute, Moscow,Russia b Max-Planck-Institut
More informationMHD Stabilization Analysis in Tokamak with Helical Field
US-Japan Workshop on MHD Control, Magnetic Islands and Rotation the University of Texas, Austin, Texas, USA AT&T Executive Education & Conference Center NOVEMBER 3-5, 8 MHD Stabilization Analysis in Tokamak
More informationIterative optimisation of auxiliary coils for stellarators B.F. McMillan a, B.D. Blackwell b and J.H. Harris b a Department of Theoretical Physics, RS
Iterative optimisation of auxiliary coils for stellarators B.F. McMillan a, B.D. Blackwell b and J.H. Harris b a Department of Theoretical Physics, RSPhysSE, The Australian National University, Canberra,
More informationTheory for Neoclassical Toroidal Plasma Viscosity in a Toroidally Symmetric Torus. K. C. Shaing
Theory for Neoclassical Toroidal Plasma Viscosity in a Toroidally Symmetric Torus K. C. Shaing Plasma and Space Science Center, and ISAPS, National Cheng Kung University, Tainan, Taiwan 70101, Republic
More informationThe Virial Theorem, MHD Equilibria, and Force-Free Fields
The Virial Theorem, MHD Equilibria, and Force-Free Fields Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 10 12, 2014 These lecture notes are largely
More informationActive MHD Control Needs in Helical Configurations
Active MHD Control Needs in Helical Configurations M.C. Zarnstorff 1 Presented by E. Fredrickson 1 With thanks to A. Weller 2, J. Geiger 2, A. Reiman 1, and the W7-AS Team and NBI-Group. 1 Princeton Plasma
More information3D toroidal physics: testing the boundaries of symmetry breaking
3D toroidal physics: testing the boundaries of symmetry breaking Donald A. Spong Oak Ridge National Laboratory Abstract - Toroidal symmetry is an important concept for plasma confinement; it allows the
More information1 EX/P5-9 International Stellarator/Heliotron Database Activities on High-Beta Confinement and Operational Boundaries
1 International Stellarator/Heliotron Database Activities on High-Beta Confinement and Operational Boundaries A. Weller 1), K.Y. Watanabe 2), S. Sakakibara 2), A. Dinklage 1), H. Funaba 2), J. Geiger 1),
More informationarxiv: v2 [physics.plasm-ph] 8 Mar 2018
Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum G.G. Plunk 1 and P. Helander 1 1 Max-Planck-Institut für Plasmaphysik, EURATOM Association, 17491 Greifswald, Germany arxiv:1801.02990v2
More information0 Magnetically Confined Plasma
0 Magnetically Confined Plasma 0.1 Particle Motion in Prescribed Fields The equation of motion for species s (= e, i) is written as d v ( s m s dt = q s E + vs B). The motion in a constant magnetic field
More informationIssues in Neoclassical Tearing Mode Theory
Issues in Neoclassical Tearing Mode Theory Richard Fitzpatrick Institute for Fusion Studies University of Texas at Austin Austin, TX Tearing Mode Stability in Tokamaks According to standard (single-fluid)
More informationQUARK EFFECT ON H-ATOM SPECTRA(1S)
QUARK EFFECT ON H-ATOM SPECTRA1S March 8, 18 1 A QUARK MODEL OF THE PROTON The purpose of this paper is to test a theory of how the quarks in the proton might affect the H-atom spectra. There are three
More informationPLASMA EQUILIBRIUM IN TOKAMAKS
PLAMA EQUILIBRIUM IN TOKAMAK H.J. de Blank FOM Institute DIFFER Dutch Institute for Fundamental Energy Research, Association EURATOM-FOM, P.O. Box 27, 343 BE Nieuwegein, The Netherlands, www.differ.nl.
More informationLecture 8 Phase Space, Part 2. 1 Surfaces of section. MATH-GA Mechanics
Lecture 8 Phase Space, Part 2 MATH-GA 2710.001 Mechanics 1 Surfaces of section Thus far, we have highlighted the value of phase portraits, and seen that valuable information can be extracted by looking
More informationIntroduction to MagnetoHydroDynamics (MHD) Antoine Cerfon, Courant Institute, New York University
Introduction to MagnetoHydroDynamics (MHD) Antoine Cerfon, Courant Institute, New York University Email: cerfon@cims.nyu.edu SULI Introductory Course in Plasma Physics, June 6, 2016 PART I: DESCRIBING
More informationChapter 1 Hamiltonian Representation of Magnetic Field
Chapter 1 Hamiltonian Representation of Magnetic Field In this chapter we present the Hamiltonian formulation of the equations for magnetic field lines. We specifically consider the magnetic field corresponding
More informationPRINCETON PLASMA PHYSICS LABORATORY PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY
PREPARED FOR THE U.S. DEPARTMENT OF ENERGY, UNDER CONTRACT DE-AC02-76CH03073 PPPL-3517 UC-70 PPPL-3517 Physics Issues in the Design of Low Aspect-Ratio, High-Beta, Quasi-Axisymmetric Stellarators by M.C.
More informationPhysics Considerations in the Design of NCSX *
1 IAEA-CN-94/IC-1 Physics Considerations in the Design of NCSX * G. H. Neilson 1, M. C. Zarnstorff 1, L. P. Ku 1, E. A. Lazarus 2, P. K. Mioduszewski 2, W. A. Cooper 3, M. Fenstermacher 4, E. Fredrickson
More informationFlow, current, & electric field in omnigenous stellarators
Flow, current, & electric field in omnigenous stellarators Supported by U.S. D.o.E. Matt Landreman with Peter J Catto MIT Plasma Science & Fusion Center Oral 2O4 Sherwood Fusion Theory Meeting Tuesday
More informationApproximate Solutions of the Grad-Schlüter-Shafranov Equation
Approximate Solutions of the Grad-Schlüter-Shafranov Equation Gerson Otto Ludwig Associated Plasma Laboratory, National Space Research Institute 17-010, São José dos Campos, SP, Brazil ludwig@plasma.inpe.br
More informationVolume 114 Number 4 July-August Journal of Research of the National Institute of Standards and Technology
[J. Res. Natl. Inst. Stand. Technol. 114, 229-236 (2009)] Design of the DEMO Fusion Reactor Following ITER Volume 114 Number 4 July-August 2009 Paul R. Garabedian Courant Institute, New York University,
More informationJP Sta,onary Density Profiles in Alcator C Mod
JP8.00072 Sta,onary Density Profiles in Alcator C Mod 1 In the absence of an internal particle source, plasma turbulence will impose an intrinsic relationship between an inwards pinch and an outwards diffusion
More informationMHD Linear Stability Analysis Using a Full Wave Code
US-Japan JIFT Workshop on Progress of Extended MHD Models NIFS, Toki,Japan 2007/03/27 MHD Linear Stability Analysis Using a Full Wave Code T. Akutsu and A. Fukuyama Department of Nuclear Engineering, Kyoto
More informationRecent Developments in Theory for W7-X
Recent Developments in Theory for W7-X J. NÜHRENBERG, M. DREVLAK, R. HATZKY, R. KLEIBER, A. KÖNIES, P. MERKEL, D. MONTICELLO +,C.NÜHRENBERG, A. REIMAN +,T.M.TRAN? Max-Planck-Institut für Plasmaphysik,
More informationNonlinear hydrid simulations of precessional Fishbone instability
Nonlinear hydrid simulations of precessional Fishbone instability M. Faganello 1, M. Idouakass 1, H. L. Berk 2, X. Garbet 3, S. Benkadda 1 1 Aix-Marseille University, CNRS, PIIM UMR 7345, Marseille, France
More informationPlasmoid Motion in Helical Plasmas
Plasmoid Motion in Helical Plasmas Ryuichi ISHIZAKI and Noriyoshi NAKAJIMA National Institute for Fusion Science, Toki 509-5292, Japan (Received 12 December 2009 / Accepted 18 May 2010) In order to explain
More informationIdeal MHD Equilibria
CapSel Equil - 01 Ideal MHD Equilibria keppens@rijnh.nl steady state ( t = 0) smoothly varying solutions to MHD equations solutions without discontinuities conservative or non-conservative formulation
More informationBunno, M.; Nakamura, Y.; Suzuki, Y. Matsunaga, G.; Tani, K. Citation Plasma Science and Technology (
Title The Finite Beta Effects on the Toro Tokamak Plasma Author(s) Bunno, M.; Nakamura, Y.; Suzuki, Y. Matsunaga, G.; Tani, K. Citation Plasma Science and Technology (2013 Issue Date 2013-02 URL http://hdl.handle.net/2433/173038
More informationFundamentals of Magnetic Island Theory in Tokamaks
Fundamentals of Magnetic Island Theory in Tokamaks Richard Fitzpatrick Institute for Fusion Studies University of Texas at Austin Austin, TX, USA Talk available at http://farside.ph.utexas.edu/talks/talks.html
More informationGauge Invariant Variables for SU(2) Yang-Mills Theory
Gauge Invariant Variables for SU(2) Yang-Mills Theory Cécile Martin Division de Physique Théorique, Institut de Physique Nucléaire F-91406, Orsay Cedex, France. Abstract We describe a nonperturbative calculation
More informationLectures on basic plasma physics: Hamiltonian mechanics of charged particle motion
Lectures on basic plasma physics: Hamiltonian mechanics of charged particle motion Department of applied physics, Aalto University March 8, 2016 Hamiltonian versus Newtonian mechanics Newtonian mechanics:
More informationSimple examples of MHD equilibria
Department of Physics Seminar. grade: Nuclear engineering Simple examples of MHD equilibria Author: Ingrid Vavtar Mentor: prof. ddr. Tomaž Gyergyek Ljubljana, 017 Summary: In this seminar paper I will
More informationMultifarious Physics Analyses of the Core Plasma Properties in a Helical DEMO Reactor FFHR-d1
1 FTP/P7-34 Multifarious Physics Analyses of the Core Plasma Properties in a Helical DEMO Reactor FFHR-d1 J. Miyazawa 1, M. Yokoyama 1, Y. Suzuki 1, S. Satake 1, R. Seki 1, Y. Masaoka 2, S. Murakami 2,
More informationStudy of neoclassical transport and bootstrap current for W7-X in the 1/ν regime, using results from the PIES code
INSTITUTE OF PHYSICS PUBLISHING Plasma Phys. Control. Fusion 46 (2004) 179 191 PLASMA PHYSICS AND CONTROLLED FUSION PII: S0741-3335(04)65723-X Study of neoclassical transport and bootstrap current for
More informationGyrokinetic simulations of magnetic fusion plasmas
Gyrokinetic simulations of magnetic fusion plasmas Tutorial 2 Virginie Grandgirard CEA/DSM/IRFM, Association Euratom-CEA, Cadarache, 13108 St Paul-lez-Durance, France. email: virginie.grandgirard@cea.fr
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 15: Alternate Concepts (with Darren Sarmer)
22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 15: Alternate Concepts (with Darren Sarmer) 1. In todays lecture we discuss the basic ideas behind the main alternate concepts to the tokamak.
More informationEnergetic Particle Stability and Confinement Issues in 3D Configurations
Energetic Particle Stability and Confinement Issues in 3D Configurations D. A. Spong Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Collaborators: Y. Todo, M. Osakabe, B. Breizman, D. Brower,
More informationReal Plasma with n, T ~ p Equilibrium: p = j B
Real Plasma with n, T ~ p Equilibrium: p = j B B lines must lie in isobaric surfaces. Since B = 0, only possible if isobaric surfaces are topological tori. Magnetic field lines must form nested tori. Equilibrium
More informationSawteeth in Tokamaks and their relation to other Two-Fluid Reconnection Phenomena
Sawteeth in Tokamaks and their relation to other Two-Fluid Reconnection Phenomena S. C. Jardin 1, N. Ferraro 2, J. Chen 1, et al 1 Princeton Plasma Physics Laboratory 2 General Atomics Supported by the
More informationGA A27857 IMPACT OF PLASMA RESPONSE ON RMP ELM SUPPRESSION IN DIII-D
GA A27857 IMPACT OF PLASMA RESPONSE ON RMP ELM SUPPRESSION IN DIII-D by A. WINGEN, N.M. FERRARO, M.W. SHAFER, E.A. UNTERBERG, T.E. EVANS, D.L. HILLIS, and P.B. SNYDER JULY 2014 DISCLAIMER This report was
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 16
ECE 6345 Spring 5 Prof. David R. Jackson ECE Dept. Notes 6 Overview In this set of notes we calculate the power radiated into space by the circular patch. This will lead to Q sp of the circular patch.
More informationIssues of Perpendicular Conductivity and Electric Fields in Fusion Devices
Issues of Perpendicular Conductivity and Electric Fields in Fusion Devices Michael Tendler, Alfven Laboratory, Royal Institute of Technology, Stockholm, Sweden Plasma Turbulence Turbulence can be regarded
More informationEffects of stellarator transform on sawtooth oscillations in CTH. Jeffrey Herfindal
Effects of stellarator transform on sawtooth oscillations in CTH Jeffrey Herfindal D.A. Ennis, J.D. Hanson, G.J. Hartwell, E.C. Howell, C.A. Johnson, S.F. Knowlton, X. Ma, D.A. Maurer, M.D. Pandya, N.A.
More informationShear Flow Generation in Stellarators - Configurational Variations
Shear Flow Generation in Stellarators - Configurational Variations D. A. Spong 1), A. S. Ware 2), S. P. Hirshman 1), J. H. Harris 1), L. A. Berry 1) 1) Oak Ridge National Laboratory, Oak Ridge, Tennessee
More informationIon plateau transport near the tokamak magnetic axis. Abstract
Ion plateau transport near the tokamak magnetic axis K.C. Shaing and R.D. Hazeltine Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712 (December 2, 1997) Abstract Conventional
More informationand in each case give the range of values of x for which the expansion is valid.
α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else P Kraft Further Maths A (MFPD)
More informationNonlinear MHD Stability and Dynamical Accessibility
Nonlinear MHD Stability and Dynamical Accessibility Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University Philip J. Morrison Department of Physics and Institute
More informationOverview of Gyrokinetic Theory & Properties of ITG/TEM Instabilities
Overview of Gyrokinetic Theory & Properties of ITG/TEM Instabilities G. W. Hammett Princeton Plasma Physics Lab (PPPL) http://w3.pppl.gov/~hammett AST559: Plasma & Fluid Turbulence Dec. 5, 2011 (based
More informationAn homotopy method for exact tracking of nonlinear nonminimum phase systems: the example of the spherical inverted pendulum
9 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -, 9 FrA.5 An homotopy method for exact tracking of nonlinear nonminimum phase systems: the example of the spherical inverted
More informationA theory for localized low-frequency ideal MHD modes in axisymmetric toroidal systems is generalized to take into account both toroidal and poloidal
MHD spectra pre-history (selected results I MHD spectra pre-history (selected results II Abstract A theory for localized low-frequency ideal MHD modes in axisymmetric toroidal systems is generalized to
More informationAMSC 663 Project Proposal: Upgrade to the GSP Gyrokinetic Code
AMSC 663 Project Proposal: Upgrade to the GSP Gyrokinetic Code George Wilkie (gwilkie@umd.edu) Supervisor: William Dorland (bdorland@umd.edu) October 11, 2011 Abstract Simulations of turbulent plasma in
More informationHelicity fluctuation, generation of linking number and effect on resistivity
Helicity fluctuation, generation of linking number and effect on resistivity F. Spineanu 1), M. Vlad 1) 1) Association EURATOM-MEC Romania NILPRP MG-36, Magurele, Bucharest, Romania spineanu@ifin.nipne.ro
More informationControl of Sawtooth Oscillation Dynamics using Externally Applied Stellarator Transform. Jeffrey Herfindal
Control of Sawtooth Oscillation Dynamics using Externally Applied Stellarator Transform Jeffrey Herfindal D.A. Ennis, J.D. Hanson, G.J. Hartwell, S.F. Knowlton, X. Ma, D.A. Maurer, M.D. Pandya, N.A. Roberds,
More informationMain Results of Vector Analysis
Main Results of ector Analysis Andreas Wacker Mathematical Physics, Lund University January 5, 26 Repetition: ector Space Consider a d dimensional real vector space with scalar product or inner product
More informationIdeal Magnetohydrodynamics (MHD)
Ideal Magnetohydrodynamics (MHD) Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 1, 2016 These lecture notes are largely based on Lectures in Magnetohydrodynamics
More informationPlasma spectroscopy when there is magnetic reconnection associated with Rayleigh-Taylor instability in the Caltech spheromak jet experiment
Plasma spectroscopy when there is magnetic reconnection associated with Rayleigh-Taylor instability in the Caltech spheromak jet experiment KB Chai Korea Atomic Energy Research Institute/Caltech Paul M.
More informationBohr & Wheeler Fission Theory Calculation 4 March 2009
Bohr & Wheeler Fission Theory Calculation 4 March 9 () Introduction The goal here is to reproduce the calculation of the limiting Z /A against spontaneous fission Z A lim a S. (.) a C as first done by
More informationEigenvalue problems for Beltrami fields arising in a three-dimensional toroidal magnetohydrodynamic equilibrium problem
Eigenvalue problems for Beltrami fields arising in a three-dimensional toroidal magnetohydrodynamic equilibrium problem S. R. Hudson, M. J. Hole, and R. L. Dewar Citation: Physics of Plasmas (1994-present)
More informationHighlights from (3D) Modeling of Tokamak Disruptions
Highlights from (3D) Modeling of Tokamak Disruptions Presented by V.A. Izzo With major contributions from S.E. Kruger, H.R. Strauss, R. Paccagnella, MHD Control Workshop 2010 Madison, WI ..onset of rapidly
More informationPLASMA PHYSICS AND CONTROLLED NUCLEAR FUSION RESEARCH
.- " Reprint from PLASMA PHYSICS AND CONTROLLED NUCLEAR FUSION RESEARCH 1984 PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON PLASMA PHYSICS AND CONTROLLED NUCLEAR FUSION RESEARCH HELD BYTHE INTERNATIONAL
More informationHeat Transport in a Stochastic Magnetic Field. John Sarff Physics Dept, UW-Madison
Heat Transport in a Stochastic Magnetic Field John Sarff Physics Dept, UW-Madison CMPD & CMSO Winter School UCLA Jan 5-10, 2009 Magnetic perturbations can destroy the nested-surface topology desired for
More informationof HIGH betan*h SIMULATIONS TOKAMAK ADVANCED IN DIII-D WITH DISCHARGES 3D NONLINEAR CODE NFTC. THE N.N.Popova A.M.Popov, State University Moscow La Ha
of HIGH betan*h SIMULATIONS TOKAMAK ADVANCED IN DIII-D WITH DISCHARGES 3D NONLINEAR CODE NFTC. THE N.N.Popova A.M.Popov, State University Moscow La Haye, A.D.Turnbull V.S.Chan,R.J. Atomics General M.Murakami
More informationMeasurements of rotational transform due to noninductive toroidal current using motional Stark effect spectroscopy in the Large Helical Device
REVIEW OF SCIENTIFIC INSTRUMENTS 76, 053505 2005 Measurements of rotational transform due to noninductive toroidal current using motional Stark effect spectroscopy in the Large Helical Device K. Ida, a
More informationELM control with RMP: plasma response models and the role of edge peeling response
ELM control with RMP: plasma response models and the role of edge peeling response Yueqiang Liu 1,2,3,*, C.J. Ham 1, A. Kirk 1, Li Li 4,5,6, A. Loarte 7, D.A. Ryan 8,1, Youwen Sun 9, W. Suttrop 10, Xu
More informationMagnetic Control of Perturbed Plasma Equilibria
Magnetic Control of Perturbed Plasma Equilibria Nikolaus Rath February 17th, 2012 N. Rath (Columbia University) Magnetic Control of Perturbed Plasma Equilibria February 17th, 2012 1 / 19 The HBT-EP Tokamak
More informationCalculation of alpha particle redistribution in sawteeth using experimentally reconstructed displacement eigenfunctions
Calculation of alpha particle redistribution in sawteeth using experimentally reconstructed displacement eigenfunctions R. Farengo, H. E. Ferrari,2, M.-C. Firpo 3, P. L. Garcia-Martinez 2,3, A. F. Lifschitz
More informationDT Fusion Ignition of LHD-Type Helical Reactor by Joule Heating Associated with Magnetic Axis Shift )
DT Fusion Ignition of LHD-Type Helical Reactor by Joule Heating Associated with Magnetic Axis Shift ) Tsuguhiro WATANABE National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan (Received
More informationResponse of a Resistive and Rotating Tokamak to External Magnetic Perturbations Below the Alfven Frequency
Response of a Resistive and Rotating Tokamak to External Magnetic Perturbations Below the Alfven Freuency by M.S. Chu In collaboration with L.L. Lao, M.J. Schaffer, T.E. Evans E.J. Strait (General Atomics)
More informationUn problème inverse: l identification du profil de courant dans un Tokamak
Un problème inverse: l identification du profil de courant dans un Tokamak Blaise Faugeras en collaboration avec J. Blum et C. Boulbe Université de Nice Sophia Antipolis Laboratoire J.-A. Dieudonné Nice,
More informationTESTING ADS/CFT. John H. Schwarz STRINGS 2003
TESTING ADS/CFT John H. Schwarz STRINGS 2003 July 6, 2003 1 INTRODUCTION During the past few years 1 Blau et al. constructed a maximally supersymmetric plane-wave background of type IIB string theory as
More informationConservation Laws in Ideal MHD
Conservation Laws in Ideal MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 3, 2016 These lecture notes are largely based on Plasma Physics for Astrophysics
More informationEquations of linear stellar oscillations
Chapter 4 Equations of linear stellar oscillations In the present chapter the equations governing small oscillations around a spherical equilibrium state are derived. The general equations were presented
More informationTokamak/Helical Configurations Related to LHD and CHS-qa
9TH WORKSHOP ON MHD STABILITY CONTROL: "CONTROL OF MHD STABILITY: BACK TO THE BASICS" NOVEMBER 21-23, 2004, PRINCETON PLASMA PHYSICS LABORATORY Tokamak/Helical Configurations Related to LHD and CHS-qa
More informationFormation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas )
Formation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas ) Yasutomo ISHII and Andrei SMOLYAKOV 1) Japan Atomic Energy Agency, Ibaraki 311-0102, Japan 1) University
More informationCurrent-driven instabilities
Current-driven instabilities Ben Dudson Department of Physics, University of York, Heslington, York YO10 5DD, UK 21 st February 2014 Ben Dudson Magnetic Confinement Fusion (1 of 23) Previously In the last
More informationModels for Global Plasma Dynamics
Models for Global Plasma Dynamics F.L. Waelbroeck Institute for Fusion Studies, The University of Texas at Austin International ITER Summer School June 2010 Outline 1 Models for Long-Wavelength Plasma
More information- Effect of Stochastic Field and Resonant Magnetic Perturbation on Global MHD Fluctuation -
15TH WORKSHOP ON MHD STABILITY CONTROL: "US-Japan Workshop on 3D Magnetic Field Effects in MHD Control" U. Wisconsin, Madison, Nov 15-17, 17, 2010 LHD experiments relevant to Tokamak MHD control - Effect
More informationD.J. Schlossberg, D.J. Battaglia, M.W. Bongard, R.J. Fonck, A.J. Redd. University of Wisconsin - Madison 1500 Engineering Drive Madison, WI 53706
D.J. Schlossberg, D.J. Battaglia, M.W. Bongard, R.J. Fonck, A.J. Redd University of Wisconsin - Madison 1500 Engineering Drive Madison, WI 53706 Concept Overview Implementation on PEGASUS Results Current
More informationSimulations of Sawteeth in CTH. Nicholas Roberds August 15, 2015
Simulations of Sawteeth in CTH Nicholas Roberds August 15, 2015 Outline Problem Description Simulations of a small tokamak Simulations of CTH 2 Sawtoothing Sawtoothing is a phenomenon that is seen in all
More information