托卡马克的平衡计算

1. 托卡马克的平衡计算 李国强 2013.12.18 四室学术报告

2. Introduction • Decompose the physics problem by the orders (time order and space order) • Traditional decomposition of plasma physics (by time order): equilibrium, stability and transport • Equilibrium is the basis for other problem • Here the equilibrium means the state of force equilibrium

3. Introduction (cont.) • Force balance equation (static momentum equation) • Force balance equation in 2D form → Grad-Shafranov (G-S) equation (For axis symmetric, in (R,z) coordinate): • Then the solution of the G-S equation describes the properties of the equilibrium

4. Equilibrium and poloidal field coils • Poloidal field coils induct the ohmic plasma current and control the plasma shape • On EAST • PF1-PF6, center solenoid, mainly for the ohmic current • PF7/9, elongation • PF11,PF13, trianglarity • PF5, PF7/9, PF11, divertor control EAST PF coils and plasma configuration

5. Properties of equilibrium • Plasma configuration • Embedded flux surface • Plasma geometry • Divertor configuration • Profiles (functions of flux surface) • : pressure • : no direct physical meaning, but direct in G-S equation • : safety factor, describe the pitch angle of magnetic field line • : flux surface averaged parallel current • , q and are not independent

6. Fixed boundary and free boundary equilibrium calculation • Fixed boundary • The plasma boundary is given, only calculate the plasma configuration inside the plasma • Easy to calculate, useful for theory study • Free boundary • To calculate the configuration outside the plasma boundary • The current in the PF coils is given • Complicate but sometimes necessary • A third kind • Prescribe a non-fixed plasma boundary

7. Coordinate system • Many kinds of coordinate system in tokamak study • Two major coordinate systems: (R,z) coordinate and magnetic surface coordinate • coordinate system • For free boundary calculation • Can handle the X-point z R Mesh in (R,z) coordinate

8. Coordinate system (cont.) • Flux surface coordinate system • coordinate • Easier, but cannot handle the X-point • can be • Orthogonal • Equal arc length • …… • Some coordinate equivalence • normalized toroidal flux • normalized volume Mesh in flux surface coordinate

9. Equilibrium construction and reconstruction • Construction • Generate an equilibrium from given profiles, plasma shape or current in PF coils, and other parameters • Basis for tokamak design • Basis for many theory study • Reconstruction • Find the experimental equilibrium from the diagnostic data • Basis for experiments analysis

10. Equilibrium reconstruction with EFIT • EFIT is the most popular code for equilibrium reconstruction. Maybe the most popular code in tokamak research area • Assume a polynomial or spline profiles of P’ and FF’, then iteratively find the coefficient to minimize the error quality function

11. Different EFIT reconstruction constraints • At present, EAST only has the magnetic diagnostics and limited kinetic diagnostics • But we can add some constraints to the current profile

12. Magnetic diagnostic constraints • All kinks of magnetic probe and flux loops Strait (2007)

13. kinetic profiles on EAST • Te and ne are from Thomson scattering • Ti is from the XCS, but only central region data are available. So Ti is scale from Te and assume Ti=Te at the edge region • First map the data to space, then fit them with tension spline • Assume flat Zeff=2.5 • At present EAST has no NBI, so the fast ion contribution is neglected Data and fitting profiles for 38300.3900

14. Edge current constraint for H-mode plasma • For H-mode plasma, it is believed that at the edge region, the current is dominated by the bootstrap current • Sauter bootstrap current model is used to calculate the bootstrap current. Bootstrap current calculation relies on the kinetic profiles (Te, Ti, ne, Zeff) • Ohmic current EAST 38300, 3900ms Typical pressure and current profiles of H-mode plasma at edge region Bootstrap current at the edge region

15. Kinetic equilibrium reconstruction on EAST • With the constrains of magnetic diagnostics, pressure profile, edge current profile, we achieved the kinetic equilibrium • The current/q profiles at the central region are not reliable, though we have the global li constrain 38300, 3900ms Pressure, current profiles and configuration from kinetic EFIT and magnetic EFIT

16. Equilibrium construction • Lots of codes for equilibrium construction, most of them are fixed boundary codes • EFIT, CORSICA/TEQ, TOQ, ESC, JSOLVER …… • CORSICA • CORSICA has both direct and inverse solver • Inverse solver: coordinate, solve for R, Z, fixed boundary • Direct solver: coordinate, solve for , free boundary • CORSICA can easily change the plasma shape and profiles

17. Construct self-consistent equilibrium • To construct a self-consistent equilibrium, the self-consistent plasma shape and profiles must be given • Self-consistent profiles: • Bootstrap current dominated edge current • Self-consistent pedestal height and width, EPED model • EPED model (peeling-ballooning model + kinetic ballooning model, ELITE+BALOO) has successfully predict the pedestal height and width • This technic could be useful for EAST and CFETR

18. Thank you