1. 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 15 th December – 17 th December, 2003 Y. Nakamura and Y. Suzuki Graduate School of Energy Science, Kyoto University, JAPAN 1-D Transport Simulation for a 3-D MHD Equilibrium

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

4. 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 minimize  1 st variation of =0 MHD equilibrium

5. Vacuum flux surfaces & fixed boundary equilibrium Vacuum flux surfaces obtained by KMAG code (field line tracing) Fixed boundary equilibrium obtained by the VMEC (  axis ~ 1%) M  M  0.8 1.6 0.8 1.6 R (m) VMEC (  axis =1%) VMEC (  axis =1%) inner wall surface HF coil B=1T  B=0.2T 

6. free boundary equilibrium (  axis ~1%) fixed boundary equilibrium (  axis ~1%)  = 4/7

7. PIES code (PPPL) Direct MHD equilibrium calculation by the iterative method update pressure distribution by field line tracing update magnetic field vector  construction of (quasi) flux coordinates 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 3-D MHD equilibrium code without assuming nested surfaces Poisson eq.

8. “virtual equilibrium” by the VMEC background coordinates control surface = vacuum vessel vacuum flux surfaces by the KMAG-PIES  0 ~ 1.5% equlibrium by the PIES (k=51, m=30/34, n=20/24, niter~100; SX-6: 6.5GB) ~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. Equilibrium in standard configuration of Heliotron J (HINT) Initial pressure profile is  0 ~1.0%  0 ~1.5%

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

12. Results (at the Basic Concept phase of LHD): (1) High heating power into low density plasma(~2.5x10 19 m -3 )  electron root (E r >0) & hot ion mode (T i (0)>10keV) (2) High density  ion root (E r >0), high n , T e ~T i (0)<5keV (3) weak  h & B dependence small  h  high n  for ion root (4) larger  h  electron root R=(4 - 5)m, a=(0.5 - 0.6)m, B=4T,  h (a)=0.1 - 0.2 P NBI =20MW Device parameter (at the Basic Concept phase of LHD) low density  Ti(0) ~ 10keV high density  n  E ~ 10 19 m -3 sec Target (at the Basic Concept phase)

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. effect of multi-helicity GIOTA VMEC 3-D MHD equilibrium ripple transport NBI deposition equilibrium  bootstrap current  magnetic island 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 ？ low beta high beta conventional treatment of TF ripple 2-D MHD equilibrium + TF ripple produced by TF coils non-axisymmetric equilibrium current? 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 passing banana ripple trapped  loss transition (ripple trapping)  loss