1-D Transport Simulation for a 3-D MHD Equilibrium

1. 1-D Transport Simulation for a 3-D MHD Equilibrium Presented by Yuji NAKAMURA at US-Japan JIFT Workshop “Theory-Based Modeling and Integrated Simulation of Burning Plasmas” and 21COE Workshop “Plasma Theory” Kyodai-Kaikan, Kyoto, JAPAN 15th December – 17th December, 2003 Y. Nakamura and Y. Suzuki Graduate School of Energy Science, Kyoto University, JAPAN

2. Outline • 1. Predictive Transport Simulation for Helical Systems • Introduction • 3-D MHD equilibrium code VMEC • 3-D MHD equilibrium code without assuming nested flux surfaces, HINT and PIES • 1-D transport simulation for a current-free helical system • 1-D transport simulation with 3-D MHD equilibrium • 2. Effects of Non-Axisymmetric MHD Equilibrium • on the Transport Simulation for Burning Tokamak Plasmas • 3-D MHD equilibrium calculation for a tokamak plasma with TF ripples • Finite beta effects on the TF ripples • Finite beta effects on the collisionless ripple loss

3. 1. Predictive Transport Simulation for Helical Systems Introduction helical systems (or stellarators) Heliotron E (L=2/M=19 heliotron) LHD (L=2/M=10 heliotron) non-axisymmetric torus 3-D MHD equilibrium M>>1  stellarator approximation lowest order flux surfaces are axisymmetric Heliotron J (L=1/M=4 helical-axis heliotron)

4. minimize  1st variation of =0 MHD equilibrium 3-D MHD equilibrium code VMEC VMEC : 3-D Inverse solver based on the variational principle S. P. Hirshman (ORNL) • assume existence of nested flux surfaces • conserve toroidal flux & pressure • Basically, fixed boundary equilibrium Calculate a solution closest to the equilibrium state under given constraints (weak solution) . minimize plasma potential energy using descent path equation R(s, θ,φ), Z(s, θ,φ) can be obtained as a function of (s, θ,φ) (inverse solver) Free boundary calculation can be possible boundary shape  variational approach for the pressure balance at the boundary

5. Vacuum flux surfaces & fixed boundary equilibrium f =0 M B=1T b VMEC ( =1%) axis D B=0.2T HF coil 1.6 0.8 R (m) inner wall surface f =-p M b VMEC ( =1%) axis HF coil D B=1T B=0.2T 1.6 0.8 R (m) Fixed boundary equilibrium obtained by the VMEC (baxis ~ 1%) Vacuum flux surfaces obtained by KMAG code (field line tracing)

6. fixed boundary equilibrium (baxis~1%) i/2p = 4/7 free boundary equilibrium (baxis~1%) i/2p = 4/7

7. 3-D MHD equilibrium code without assuming nested surfaces PIES code (PPPL) Direct MHD equilibrium calculation by the iterative method update pressure distribution by field line tracing  construction of (quasi) flux coordinates update magnetic field vector Poisson eq. Magneto-differential eq. --- (quasi) flux coordinates Poisson eq. --- background coordinates * separates external field and the field produced by plasma current * virtual casing method free boundary equilibrium

8. “virtual equilibrium” by the VMEC b0 ~ 1.5% equlibrium by the PIES (k=51, m=30/34, n=20/24, niter~100; SX-6: 6.5GB) vacuum flux surfaces by the KMAG-PIES background coordinates control surface = vacuum vessel ~1500 field periods

9. HINT code (NIFS) step A ; distribution of pressure on 3D grid points with fixed magnetic field vector (relaxation method or field line tracing method) step B; relaxation calculation of magnetic field vector on 3D grid points with fixed pressure distribution (relaxation process using time evolution of the dissipative MHD equations) Eulerian coordinates --- rotating helical coordinate system # boundary condition at the computational boundary : fixed boundary  we use a large “box” so that the size of the box does not affect the result. “Free boundary” calculation

10. b0~1.0% b0~1.5% Equilibrium in standard configuration of Heliotron J (HINT) Initial pressure profile is

11. 1-D Transport simulation for current-free helical systems 1-D Transport equations “Basic Concept of the Next Large Helical Device Project” (Green Book) (1) (2) (3) NBI: FIFPC (Fokker-Planck), neutral: AURORA (Monte Carlo) 1987 March ambipolar condition :

12. Results (at the Basic Concept phase of LHD): (1) High heating power into low density plasma(~2.5x1019m-3)  electron root (Er>0) & hot ion mode (Ti(0)>10keV) (2) High density  ion root (Er>0), high nt, Te ~Ti(0)<5keV (3) weak eh & B dependence small eh  high ntT for ion root (4) larger eh  electron root Device parameter (at the Basic Concept phase of LHD) R=(4 - 5)m, a=(0.5 - 0.6)m, B=4T, eh(a)=0.1 - 0.2 PNBI=20MW Target (at the Basic Concept phase) low density  Ti(0) ~ 10keV high density  ntE ~ 1019m-3sec

13. 1-D Transport simulation with 3-D MHD equilibrium Plasma Transport Simulation modelling for Helical Confinement System, K. Yamazaki & T.Amano, Nucl. Fusion Vol.32 (1992) 633

14. 3-D MHD equilibrium ripple transport NBI deposition effect of multi-helicity equilibrium  bootstrap current  magnetic island GIOTA VMEC transport simulation?

15. ？ 2. Effects of Non-Axisymmetric MHD Equilibrium on the Transport Simulation for Burning Tokamak Plasmas 3-D MHD equilibrium calculation for a tokamak plasma with TF ripples conventional treatment of TF ripple 2-D MHD equilibrium + TF ripple produced by TF coils non-axisymmetric equilibrium current? high beta low beta fully 3-D MHD equilibrium calculation is necessary!

16. J.L.Johnson & A.H.Reiman; Nucl. Fusion 28 (1988) 1116.

17. “Finite beta effects on the troidal field ripple in three dimensional tokamak equilibria”, Yasuhiro Suzuki, Yuji Nakamura, and Katsumi Kondo Nuclear Fusion, Vol.43 (2003) 406. 3-D MHD equilibrium calculation with free-boundary constraint by the VMEC • number of TF coils ;20 • major radius of TF coils ;3m • minor radius of TF coils ; 1.5m • plasma major radius ; 2.8m • plasma minor radius ; 0.8m • limiter position ; 3.6m • plasma aspect ratio ; 3.5 almost circular cross section fixed p(s) and q(s) profile

19. Finite beta effects on the TF ripples

20. Results

21. finite beta effects on the ripple well depth

23. Finite beta effects on the collisionless ripple loss ripple trapped  loss passing banana transition (ripple trapping)  loss