Computations of Vector Potential and Toroidal Flux and Applications to Stellarator Simulations

Size: px
Start display at page:

Download "Computations of Vector Potential and Toroidal Flux and Applications to Stellarator Simulations"

Transcription

1 Computations of Vector Potential and Toroidal Flux and Applications to Stellarator Simulations NIMROD Team Meeting Torrin Bechtel April 30, 2017

2 Outline 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations

3 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations

4 Purpose To study magnetic topology evolution and plasma confinement in stellarators with heating and eventually flow sources. Goals: Study high beta effects in toroidal, not helically symmetric plasmas Studying magnetic geometries with a variety of stability properties Perform rigorous convergence analyses Benchmark with HINT2 code Investigating the effects of plasma flow

5 Beta Limits Have Been Studied with HINT2 Pressure profile is fixed as p = p 0(1 s)(1 s 4 ). At blue circle J B = p can no longer be satisfied on stochastic field lines and pressure profile must be released soft beta limit. At green circle hard beta limit is hit as axis is pushed into separatrix. Y. Suzuki, et al. IAEA FEC 2008, TH/P9-19

6 Equilibrium Beta Limit Observed to Depend on Conduction Anisotropy MHD equilibria are produced by heating from vacuum with zperiod limited Fourier spectrum. Beta limit is observed as time step crash at higher heating. Beta varies strongly with conduction anisotropy.

7 Thermal Conduction is Well Converged Converged reference has 21 modes, 24x24 grid, poly degree = 5. Separate tests have been run with decreased dt, increased nmodes, and increased poly degree. β varies by at most 3% with increased resolution. Tests with eqn model = tonly are consistent.

8 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations

9 Potential is Computed Using iter solve The equations for the potential in the Coulomb Gauge (for uniqueness) A = B, A = 0, are solved in NIMROD s framework by formulating the problem in terms of an artifical time A = c1 ( A) c2 ( A B). t This has the same form as the pertrubed magnetic field advance in NIMROD if the electric field is modified to have the form E = elecd [ B (B eq + ṽ)] with uniform elecd which gives B t = kdivb ( B) elecd [ B (B eq + ṽ)].

10 Equation Must Be Weighted Appropriately for Accuracy and Convergence The choice of coefficients dt, c 1, and c 2 will alter the matrix problem being solved. c = c 1 = c 2 is beneficial for matrix condition. c 1/dt reduces effect of artificial time (mass) but worsens matrix condition. B A is output to ensure sufficient accuracy. Solver has been fully implemented in nimplot mgt.f 90 under compute potential but is currently only used in 3D toroidal flux calculation.

11 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations

12 Toroidal Flux is Integrated With lsode In order to compare with HINT2 we need to know T (ψ). We can compute ψ in 3D geometry using the vector potential, A, and Stokes Theorem. ψ = B ds = ( A) ds = A dl To compute A dl we need a path encircling a poloidal cross section. The differential equations defining a fieldline in 3D geometry are dr = dz = R dφ = dl B R B Z B φ B = dr = r dθ ( = dη ) 2D only. B r B θ B pol Choosing l = ˆθ we can use these equations to convert the integral to B θ ψ = A dˆθ = A θ r dθ = A θ B dl, where the path L is determined from the first 4 fieldline equations and θ is tracked to determine a stopping criteria.

13 Current Implementation Has Issues Toroidal flux should always be zero at the magnetic axis, but for some reason it appears to vary with the axis position.

14 Alternate Toroidal Flux Calculation The value of ψ can be computed by quadrature over triangles bounded by fieldline traces in a poloidal plane. This method also has pitfalls.

15 Comparison Shows... The flux function from fieldline tracing has been shifted down and both have been normalized in the second plot.

16 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations

17 Behavior of Temperature on Closed Flux Surfaces is Not Intuitive On closed flux surfaces we expect, χ eff χ. However, the temperature profile in these regions is affected by changes in χ. This has prompted further investigation.

18 Simple Estimate of Effective Conduction In steady state the heat source, Q, balances the thermal conduction Q = (χ eff P). Integrating and applying the Divergence Theorem Q dv = (χ eff P) dv = (χ eff P) ds. Assuming a uniform heating source and that χ eff can be reduced to a scalar and choosing S to be a poloidal cross section we have χ eff = Q dv. P ds φ

19 Triangle Quadrature is Not Effective

20 Extra: Volume Triangulation

21 Extra: Other Triangulated Surfaces

22 Extra: HINT2 Solves for MHD equilibrium by relaxing initial condition. Toroidal coordinates make no assumption about magnetic geometry. Uses 4th order spatial finite differencing and RK4.

MHD equilibrium calculations for stellarators. Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE)

MHD equilibrium calculations for stellarators. Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE) MHD equilibrium calculations for stellarators Antoine Cerfon, MIT PSFC with F. Parra, J. Freidberg (MIT NSE) March 20, 2012 MAGNETIC FIELD LINE HAMILTONIAN Consider a general toroidal coordinate system

More information

INITIAL EVALUATION OF COMPUTATIONAL TOOLS FOR STABILITY OF COMPACT STELLARATOR REACTOR DESIGNS

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 information

Configuration Optimization of a Planar-Axis Stellarator with a Reduced Shafranov Shift )

Configuration 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 information

The Virial Theorem, MHD Equilibria, and Force-Free Fields

The 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 information

NIMEQ: MHD Equilibrium Solver for NIMROD

NIMEQ: MHD Equilibrium Solver for NIMROD NIMEQ: MHD Equilibrium Solver for NIMOD E.C.Howell, C..Sovinec University of Wisconsin-Madison 5 th Annual Meeting of Division of Plasma Physics Dallas, Texas, Nov. 17-Nov. 1,8 1 Abstract A Grad-Shafranov

More information

Non-Solenoidal Plasma Startup in

Non-Solenoidal Plasma Startup in Non-Solenoidal Plasma Startup in the A.C. Sontag for the Pegasus Research Team A.C. Sontag, 5th APS-DPP, Nov. 2, 28 1 Point-Source DC Helicity Injection Provides Viable Non-Solenoidal Startup Technique

More information

Low 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 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 information

Computations with Discontinuous Basis Functions

Computations with Discontinuous Basis Functions Computations with Discontinuous Basis Functions Carl Sovinec University of Wisconsin-Madison NIMROD Team Meeting November 12, 2011 Salt Lake City, Utah Motivation The objective of this work is to make

More information

DT 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 ) 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 information

Highlights from (3D) Modeling of Tokamak Disruptions

Highlights 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 information

MHD Stabilization Analysis in Tokamak with Helical Field

MHD 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 information

Conservation Laws in Ideal MHD

Conservation 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 information

MHD Simulation of High Wavenumber Ballooning-like Modes in LHD

MHD Simulation of High Wavenumber Ballooning-like Modes in LHD 1 TH/P9-16 MHD Simulation of High Wavenumber Ballooning-like Modes in LHD H. Miura and N. Nakajima National Institute for Fusion Science, 322-6 Oroshi, Toki, Gifu 509-5292, JAPAN e-mail contact of main

More information

NIMROD Boundary Conditions

NIMROD Boundary Conditions NIMROD Boundary Conditions Carl Sovinec University of Wisconsin-Madison and the Two-Fluid and Transport Group Plasma Science and Innovation Center Annual Meeting Seattle Washington, August 13-14, 2007

More information

Modeling of ELM Dynamics for ITER

Modeling of ELM Dynamics for ITER Modeling of ELM Dynamics for ITER A.Y. PANKIN 1, G. BATEMAN 1, D.P. BRENNAN 2, A.H. KRITZ 1, S. KRUGER 3, P.B. SNYDER 4 and the NIMROD team 1 Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015

More information

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

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 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 information

Bunno, M.; Nakamura, Y.; Suzuki, Y. Matsunaga, G.; Tani, K. Citation Plasma Science and Technology (

Bunno, 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 information

Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade. Matthias Hölzl, Isabel Krebs, Karl Lackner, Sibylle Günter

Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade. Matthias Hölzl, Isabel Krebs, Karl Lackner, Sibylle Günter Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade Matthias Hölzl, Isabel Krebs, Karl Lackner, Sibylle Günter Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013

More information

(0,2) L 1 L 2 R (-1,0) (2,0) MA4006: Exercise Sheet 3: Solutions. 1. Evaluate the integral R

(0,2) L 1 L 2 R (-1,0) (2,0) MA4006: Exercise Sheet 3: Solutions. 1. Evaluate the integral R MA6: Eercise Sheet 3: Solutions 1. Evaluate the integral d d over the triangle with vertices ( 1, ), (, 2) and (2, ). Solution. See Figure 1. Let be the inner variable and the outer variable. we need the

More information

NIMROD FROM THE CUSTOMER S PERSPECTIVE MING CHU. General Atomics. Nimrod Project Review Meeting July 21 22, 1997

NIMROD FROM THE CUSTOMER S PERSPECTIVE MING CHU. General Atomics. Nimrod Project Review Meeting July 21 22, 1997 NIMROD FROM THE CUSTOMER S PERSPECTIVE MING CHU General Atomics Nimrod Project Review Meeting July 21 22, 1997 Work supported by the U.S. Department of Energy under Grant DE-FG03-95ER54309 and Contract

More information

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.

MAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant

More information

The RFP: Plasma Confinement with a Reversed Twist

The 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 information

Finite-Orbit-Width Effect and the Radial Electric Field in Neoclassical Transport Phenomena

Finite-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 information

AN UPDATE ON DIVERTOR HEAT LOAD ANALYSIS

AN 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 information

Effect of magnetic reconnection on CT penetration into magnetized plasmas

Effect of magnetic reconnection on CT penetration into magnetized plasmas Earth Planets Space, 53, 547 55, 200 Effect of magnetic reconnection on CT penetration into magnetized plasmas Yoshio Suzuki, Takaya Hayashi 2, and Yasuaki Kishimoto Naka Fusion esearch Establishment,

More information

The 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 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 information

Understanding Edge Harmonic Oscillation Physics Using NIMROD

Understanding Edge Harmonic Oscillation Physics Using NIMROD Understanding Edge Harmonic Oscillation Physics Using NIMROD J. King With contributions from S. Kruger & A. Pankin (Tech-X); K. Burrell, A. Garofalo, R. Groebner & P. Snyder (General Atomics) Work supported

More information

Sawteeth in Tokamaks and their relation to other Two-Fluid Reconnection Phenomena

Sawteeth 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 information

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

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 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 Non-solenoidal startup using point-source DC helicity injectors

More information

Simulation of alpha particle current drive and heating in spherical tokamaks

Simulation of alpha particle current drive and heating in spherical tokamaks Simulation of alpha particle current drive and heating in spherical tokamaks R. Farengo 1, M. Zarco 1, H. E. Ferrari 1, 1 Centro Atómico Bariloche and Instituto Balseiro, Argentina. Consejo Nacional de

More information

Magnetohydrodynamic stability of negative central magnetic shear, high pressure ( pol 1) toroidal equilibria

Magnetohydrodynamic stability of negative central magnetic shear, high pressure ( pol 1) toroidal equilibria Magnetohydrodynamic stability of negative central magnetic shear, high pressure ( pol 1) toroidal equilibria Robert G. Kleva Institute for Plasma Research, University of Maryland, College Park, Maryland

More information

1) H-mode in Helical Devices. 2) Construction status and scientific objectives of the Wendelstein 7-X stellarator

1) H-mode in Helical Devices. 2) Construction status and scientific objectives of the Wendelstein 7-X stellarator Max-Planck-Institut für Plasmaphysik 1) H-mode in Helical Devices M. Hirsch 1, T. Akiyama 2, T.Estrada 3, T. Mizuuchi 4, K. Toi 2, C. Hidalgo 3 1 Max-Planck-Institut für Plasmaphysik, EURATOM-Ass., D-17489

More information

Physics and Operations Plan for LDX

Physics and Operations Plan for LDX Physics and Operations Plan for LDX Columbia University A. Hansen D.T. Garnier, M.E. Mauel, T. Sunn Pedersen, E. Ortiz Columbia University J. Kesner, C.M. Jones, I. Karim, P. Michael, J. Minervini, A.

More information

Lecture # 3. Introduction to Kink Modes the Kruskal- Shafranov Limit.

Lecture # 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 information

Stationary, High Bootstrap Fraction Plasmas in DIII-D Without Inductive Current Control

Stationary, High Bootstrap Fraction Plasmas in DIII-D Without Inductive Current Control Stationary, High Bootstrap Fraction Plasmas in DIII-D Without Inductive Current Control P. A. Politzer, 1 A. W. Hyatt, 1 T. C. Luce, 1 F. W. Perkins, 4 R. Prater, 1 A. D. Turnbull, 1 D. P. Brennan, 5 J.

More information

Non-disruptive MHD Dynamics in Inward-shifted LHD Configurations

Non-disruptive MHD Dynamics in Inward-shifted LHD Configurations Non-disrutive MHD Dynamics in Inward-shifted LHD Configurations.Introduction.MHD simulation 3.DNS of full 3D MHD 4. Summary MIUA H. ICHIGUCHI K. NAKAJIMA N. HAYASHI T. (National Institute for Fusion Science)

More information

Overview of edge modeling efforts for advanced divertor configurations in NSTX-U with magnetic perturbation fields

Overview of edge modeling efforts for advanced divertor configurations in NSTX-U with magnetic perturbation fields Overview of edge modeling efforts for advanced divertor configurations in NSTX-U with magnetic perturbation fields H. Frerichs, O. Schmitz, I. Waters, G. P. Canal, T. E. Evans, Y. Feng and V. Soukhanovskii

More information

Nonlinear Simulation of Energetic Particle Modes in JT-60U

Nonlinear Simulation of Energetic Particle Modes in JT-60U TH/P6-7 Nonlinear Simulation of Energetic Particle Modes in JT-6U A.Bierwage,N.Aiba 2, K.Shinohara 2, Y.Todo 3,W.Deng 4,M.Ishikawa 2,G.Matsunaga 2 and M. Yagi Japan Atomic Energy Agency (JAEA), Rokkasho,

More information

GA 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 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 information

(a) (b) (c) (d) (e) (f) r (minor radius) time. time. Soft X-ray. T_e contours (ECE) r (minor radius) time time

(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 information

Control of Sawtooth Oscillation Dynamics using Externally Applied Stellarator Transform. Jeffrey Herfindal

Control 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 information

Current Drive Experiments in the Helicity Injected Torus (HIT II)

Current Drive Experiments in the Helicity Injected Torus (HIT II) Current Drive Experiments in the Helicity Injected Torus (HIT II) A. J. Redd, T. R. Jarboe, P. Gu, W. T. Hamp, V. A. Izzo, B. A. Nelson, R. G. O Neill, R. Raman, J. A. Rogers, P. E. Sieck and R. J. Smith

More information

Divergence Theorem December 2013

Divergence Theorem December 2013 Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:

More information

McGill University April 20, Advanced Calculus for Engineers

McGill University April 20, Advanced Calculus for Engineers McGill University April 0, 016 Faculty of Science Final examination Advanced Calculus for Engineers Math 64 April 0, 016 Time: PM-5PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer Student

More information

Extension of High-Beta Plasma Operation to Low Collisional Regime

Extension of High-Beta Plasma Operation to Low Collisional Regime EX/4-4 Extension of High-Beta Plasma Operation to Low Collisional Regime Satoru Sakakibara On behalf of LHD Experiment Group National Institute for Fusion Science SOKENDAI (The Graduate University for

More information

Plasmoid Motion in Helical Plasmas

Plasmoid 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 information

Current Profile Control by ac Helicity Injection

Current Profile Control by ac Helicity Injection Current Profile Control by ac Helicity Injection Fatima Ebrahimi and S. C. Prager University of Wisconsin- Madison APS 2003 Motivations Helicity injection is a method to drive current in plasmas in which

More information

Momentum transport from magnetic reconnection in laboratory an. plasmas. Fatima Ebrahimi

Momentum transport from magnetic reconnection in laboratory an. plasmas. Fatima Ebrahimi Momentum transport from magnetic reconnection in laboratory and astrophysical plasmas Space Science Center - University of New Hampshire collaborators : V. Mirnov, S. Prager, D. Schnack, C. Sovinec Center

More information

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem

Divergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:

More information

Abstract. The Pegasus Toroidal Experiment is an ultra-low aspect ratio (A < 1.2) spherical tokamak (ST) capable of operating in the high I N

Abstract. The Pegasus Toroidal Experiment is an ultra-low aspect ratio (A < 1.2) spherical tokamak (ST) capable of operating in the high I N Abstract The Pegasus Toroidal Experiment is an ultra-low aspect ratio (A < 1.2) spherical tokamak (ST) capable of operating in the high I N regime (I N > 12). Access to this regime requires a small centerpost

More information

Effects of stellarator transform on sawtooth oscillations in CTH. Jeffrey Herfindal

Effects 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 information

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.

Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr. 1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line

More information

Derivation of dynamo current drive in a closed current volume and stable current sustainment in the HIT SI experiment

Derivation of dynamo current drive in a closed current volume and stable current sustainment in the HIT SI experiment Derivation of dynamo current drive and stable current sustainment in the HIT SI experiment 1 Derivation of dynamo current drive in a closed current volume and stable current sustainment in the HIT SI experiment

More information

The Field-Reversed Configuration (FRC) is a high-beta compact toroidal in which the external field is reversed on axis by azimuthal plasma The FRC is

The Field-Reversed Configuration (FRC) is a high-beta compact toroidal in which the external field is reversed on axis by azimuthal plasma The FRC is and Stability of Field-Reversed Equilibrium with Toroidal Field Configurations Atomics General Box 85608, San Diego, California 92186-5608 P.O. APS Annual APS Meeting of the Division of Plasma Physics

More information

Real Plasma with n, T ~ p Equilibrium: p = j B

Real 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 information

RFP helical equilibria reconstruction with V3FIT-VMEC

RFP 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 information

arxiv: v1 [physics.plasm-ph] 24 Nov 2017

arxiv: v1 [physics.plasm-ph] 24 Nov 2017 arxiv:1711.09043v1 [physics.plasm-ph] 24 Nov 2017 Evaluation of ideal MHD mode stability of CFETR baseline scenario Debabrata Banerjee CAS Key Laboratory of Geospace Environment and Department of Modern

More information

PLASMA EQUILIBRIUM IN TOKAMAKS

PLASMA 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 information

Internal Magnetic Field Measurements and Langmuir Probe Results for the HIT-SI Experiment

Internal Magnetic Field Measurements and Langmuir Probe Results for the HIT-SI Experiment Internal Magnetic Field Measurements and Langmuir Probe Results for the HIT-SI Experiment (First Evidence of Spheromak Generation and Sustainment) Roger J. Smith Plasma Dynamics Group University of Washington,

More information

Un problème inverse: l identification du profil de courant dans un Tokamak

Un 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 information

Confinement Study of Net-Current Free Toroidal Plasmas Based on Extended International Stellarator Database

Confinement Study of Net-Current Free Toroidal Plasmas Based on Extended International Stellarator Database EX/1-5 IAEA FEC24, Vilamoura, Nov.2, 24 Confinement Study of Net-Current Free Toroidal Plasmas Based on Extended International Stellarator Database H.Yamada 1), J.H.Harris 2), A.Dinklage 3), E.Ascasibar

More information

OVERVIEW OF THE ALCATOR C-MOD PROGRAM. IAEA-FEC November, 2004 Alcator Team Presented by Martin Greenwald MIT Plasma Science & Fusion Center

OVERVIEW OF THE ALCATOR C-MOD PROGRAM. IAEA-FEC November, 2004 Alcator Team Presented by Martin Greenwald MIT Plasma Science & Fusion Center OVERVIEW OF THE ALCATOR C-MOD PROGRAM IAEA-FEC November, 2004 Alcator Team Presented by Martin Greenwald MIT Plasma Science & Fusion Center OUTLINE C-Mod is compact, high field, high density, high power

More information

Problems in Magnetostatics

Problems in Magnetostatics Problems in Magnetostatics 8th February 27 Some of the later problems are quite challenging. This is characteristic of problems in magnetism. There are trivial problems and there are tough problems. Very

More information

Three Dimensional Effects in Tokamaks How Tokamaks Can Benefit From Stellarator Research

Three Dimensional Effects in Tokamaks How Tokamaks Can Benefit From Stellarator Research 1 TH/P9-10 Three Dimensional Effects in Tokamaks How Tokamaks Can Benefit From Stellarator Research S. Günter, M. Garcia-Munoz, K. Lackner, Ph. Lauber, P. Merkel, M. Sempf, E. Strumberger, D. Tekle and

More information

INTRODUCTION TO MAGNETIC NUCLEAR FUSION

INTRODUCTION TO MAGNETIC NUCLEAR FUSION INTRODUCTION TO MAGNETIC NUCLEAR FUSION S.E. Sharapov Euratom/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, UK With acknowledgments to B.Alper for use of his transparencies

More information

MHD. Jeff Freidberg MIT

MHD. 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 information

First plasma operation of Wendelstein 7-X

First plasma operation of Wendelstein 7-X First plasma operation of Wendelstein 7-X R. C. Wolf on behalf of the W7-X Team *) robert.wolf@ipp.mpg.de *) see author list Bosch et al. Nucl. Fusion 53 (2013) 126001 The optimized stellarator Wendelstein

More information

University of Illinois at Chicago Department of Physics

University of Illinois at Chicago Department of Physics University of Illinois at Chicago Department of Physics Electromagnetism Qualifying Examination January 4, 2017 9.00 am - 12.00 pm Full credit can be achieved from completely correct answers to 4 questions.

More information

Fast Ion Confinement in the MST Reversed Field Pinch

Fast Ion Confinement in the MST Reversed Field Pinch Fast Ion Connement in the MST Reversed Field Pinch Gennady Fiksel B. Hudson, D.J. Den Hartog, R.M. Magee, R. O'Connell, S.C. Prager MST Team - University of Wisconsin - Madison Center for Magnetic Self-Organization

More information

Response 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 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 information

16.5 Surface Integrals of Vector Fields

16.5 Surface Integrals of Vector Fields 16.5 Surface Integrals of Vector Fields Lukas Geyer Montana State University M73, Fall 011 Lukas Geyer (MSU) 16.5 Surface Integrals of Vector Fields M73, Fall 011 1 / 19 Parametrized Surfaces Definition

More information

Recent Development of LHD Experiment. O.Motojima for the LHD team National Institute for Fusion Science

Recent Development of LHD Experiment. O.Motojima for the LHD team National Institute for Fusion Science Recent Development of LHD Experiment O.Motojima for the LHD team National Institute for Fusion Science 4521 1 Primary goal of LHD project 1. Transport studies in sufficiently high n E T regime relevant

More information

REPORT. Numerical Investigation of Transients in the SSPX Spheromak. University of Wisconsin-Madison, Madison, Wisconsin

REPORT. Numerical Investigation of Transients in the SSPX Spheromak. University of Wisconsin-Madison, Madison, Wisconsin UNIVERSITY OF WISCONSIN CENTER FOR PLASMA THEORY AND COMPUTATION REPORT Numerical Investigation of Transients in the SSPX Spheromak C. R. Sovinec, 1 B. I. Cohen, 2 G. A. Cone, 1 E. B. Hooper, 2 and H.

More information

Hybrid Kinetic-MHD simulations with NIMROD

Hybrid Kinetic-MHD simulations with NIMROD in NIMROD simulations with NIMROD Charlson C. Kim 1 Dylan P. Brennan 2 Yasushi Todo 3 and the NIMROD Team 1 University of Washington, Seattle 2 University of Tulsa 3 NIFS, Toki-Japan December 2&3, 2011

More information

Magnetic Materials. 1. Magnetization 2. Potential and field of a magnetized object

Magnetic Materials. 1. Magnetization 2. Potential and field of a magnetized object Magnetic Materials 1. Magnetization 2. Potential and field of a magnetized object 3. H-field 4. Susceptibility and permeability 5. Boundary conditions 6. Magnetic field energy and magnetic pressure 1 Magnetic

More information

Generating of fusion plasma neutron source with AFSI for Serpent MC neutronics computing Serpent UGM 2015 Knoxville, TN,

Generating of fusion plasma neutron source with AFSI for Serpent MC neutronics computing Serpent UGM 2015 Knoxville, TN, Generating of fusion plasma neutron source with AFSI for Serpent MC neutronics computing Serpent UGM 2015 Knoxville, TN, 14.10.2015 Paula Sirén VTT Technical Research Centre of Finland, P.O Box 1000, 02044

More information

Time-Varying Systems; Maxwell s Equations

Time-Varying Systems; Maxwell s Equations Time-Varying Systems; Maxwell s Equations 1. Faraday s law in differential form 2. Scalar and vector potentials; the Lorenz condition 3. Ampere s law with displacement current 4. Maxwell s equations 5.

More information

Level-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation

Level-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation Level-Set Minimization of Potential Controlled Hadwiger Valuations for Molecular Solvation Bo Li Dept. of Math & NSF Center for Theoretical Biological Physics UC San Diego, USA Collaborators: Li-Tien Cheng

More information

Motion of Charged Particles in Fields

Motion of Charged Particles in Fields Chapter Motion of Charged Particles in Fields Plasmas are complicated because motions of electrons and ions are determined by the electric and magnetic fields but also change the fields by the currents

More information

Theory 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 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 information

Hybrid Kinetic-MHD simulations Status and Updates

Hybrid Kinetic-MHD simulations Status and Updates in NIMROD simulations Status and Updates Charlson C. Kim 1,2 Yasushi Todo 2 and the NIMROD Team 1. University of Washington, Seattle 2. National Institute for Fusion Science NIMROD Team Meeting Austin,

More information

Simulations of Sawteeth in CTH. Nicholas Roberds August 15, 2015

Simulations 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

The Levitated Dipole Experiment: Experiment and Theory

The Levitated Dipole Experiment: Experiment and Theory The Levitated Dipole Experiment: Experiment and Theory Jay Kesner, R. Bergmann, A. Boxer, J. Ellsworth, P. Woskov, MIT D.T. Garnier, M.E. Mauel Columbia University Columbia University Poster CP6.00083

More information

Toroidal confinement of non-neutral plasma. Martin Droba

Toroidal confinement of non-neutral plasma. Martin Droba Toroidal confinement of non-neutral plasma Martin Droba Contents Experiments with toroidal non-neutral plasma Magnetic surfaces CNT and IAP-high current ring Conclusion 2. Experiments with toroidal non-neutral

More information

Active MHD Control Needs in Helical Configurations

Active 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 information

McGill University April 16, Advanced Calculus for Engineers

McGill University April 16, Advanced Calculus for Engineers McGill University April 16, 2014 Faculty of cience Final examination Advanced Calculus for Engineers Math 264 April 16, 2014 Time: 6PM-9PM Examiner: Prof. R. Choksi Associate Examiner: Prof. A. Hundemer

More information

Introduction to Fusion Physics

Introduction to Fusion Physics Introduction to Fusion Physics Hartmut Zohm Max-Planck-Institut für Plasmaphysik 85748 Garching DPG Advanced Physics School The Physics of ITER Bad Honnef, 22.09.2014 Energy from nuclear fusion Reduction

More information

Nonlinear 3D MHD physics of the helical reversed-field pinch

Nonlinear 3D MHD physics of the helical reversed-field pinch Nonlinear 3D MHD physics of the helical reversed-field pinch Daniele Bonfiglio* Consorzio RFX, Euratom-ENEA Association, Padova, Italy *In collaboration with: S. Cappello, L. Chacón (Oak Ridge National

More information

Equilibrium and transport in Tokamaks

Equilibrium and transport in Tokamaks Equilibrium and transport in Tokamaks Jacques Blum Laboratoire J.-A. Dieudonné, Université de Nice Sophia-Antipolis Parc Valrose 06108 Nice Cedex 02, France jblum@unice.fr 08 septembre 2008 Jacques Blum

More information

Implicit kinetic relaxation schemes. Application to the plasma physic

Implicit kinetic relaxation schemes. Application to the plasma physic Implicit kinetic relaxation schemes. Application to the plasma physic D. Coulette 5, E. Franck 12, P. Helluy 12, C. Courtes 2, L. Navoret 2, L. Mendoza 2, F. Drui 2 ABPDE II, Lille, August 2018 1 Inria

More information

Simulation Research on disruptions and runaway electrons

Simulation Research on disruptions and runaway electrons 第 20 回 NEXT 研究会 Simulation Research on disruptions and runaway electrons A. Matsuyama, M. Yagi, H. Nuga, N. Aiba, and Y. Ishii JAEA, Rokkasho, Japan Acknowledgement: Y. Shibata, A. Ito, M. Furukawa, S.

More information

Oscillating Field Current Drive on MST

Oscillating Field Current Drive on MST Oscillating Field Current Drive on MST John Sarff A. Blair, K. McCollam, P. Nonn, J. Anderson, D. Brower 1, D. Craig, B. Deng 1, D. Den Hartog, W. Ding 1, F. Ebrahimi, D. Ennis, G. Fiksel, S. Gangadhara,

More information

Progress in the Plasma Science and Innovation Center

Progress in the Plasma Science and Innovation Center 1 THP/2-02 Progress in the Plasma Science and Innovation Center U. Shumlak 1, C. Akcay 1, A. H. Glasser 1, C. J. Hansen 1, E. D. Held 2, T. R. Jarboe 1, J.-Y. Ji 2, C. Kim 1, W. Lowrie 1, V. S. Lukin 4,

More information

SMR/ Summer College on Plasma Physics. 30 July - 24 August, Introduction to Magnetic Island Theory.

SMR/ 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 information

Gyrokinetic simulations of magnetic fusion plasmas

Gyrokinetic 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 information

Edge Rotational Shear Requirements for the Edge Harmonic Oscillation in DIII D Quiescent H mode Plasmas

Edge Rotational Shear Requirements for the Edge Harmonic Oscillation in DIII D Quiescent H mode Plasmas Edge Rotational Shear Requirements for the Edge Harmonic Oscillation in DIII D Quiescent H mode Plasmas by T.M. Wilks 1 with A. Garofalo 2, K.H. Burrell 2, Xi. Chen 2, P.H. Diamond 3, Z.B. Guo 3, X. Xu

More information

Neoclassical Tearing Modes

Neoclassical Tearing Modes Neoclassical Tearing Modes O. Sauter 1, H. Zohm 2 1 CRPP-EPFL, Lausanne, Switzerland 2 Max-Planck-Institut für Plasmaphysik, Garching, Germany Physics of ITER DPG Advanced Physics School 22-26 Sept, 2014,

More information

Simple examples of MHD equilibria

Simple 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 information

Bounce-averaged gyrokinetic simulations of trapped electron turbulence in elongated tokamak plasmas

Bounce-averaged gyrokinetic simulations of trapped electron turbulence in elongated tokamak plasmas Bounce-averaged gyrokinetic simulations of trapped electron turbulence in elongated tokamak plasmas Lei Qi a, Jaemin Kwon a, T. S. Hahm a,b and Sumin Yi a a National Fusion Research Institute (NFRI), Daejeon,

More information

Rotation and Neoclassical Ripple Transport in ITER

Rotation 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 information