Piero Martin Consorzio RFX- Associazione Euratom-ENEA sulla fusione, Padova, Italy

1. Reversed Field Pinch: equilibrium, stability and transport Piero Martin Consorzio RFX- Associazione Euratom-ENEA sulla fusione, Padova, Italy Department of Physics, University of Padova Notes for the lecture at the European Ph.D. Course (Garching, 29 September 2008) European Ph.D. course . - Garching 29.09.08) p.martin

2. Note for users These slides are intended only as tools to accompany the lecture. They are not supposed to be complete, since the material presented on the blackboard is a fundamental part of the lecture. Relevant bibliography: Freidberg, IDEAL MHD Ortolani, IV Latin American Workshop on Plasma Physics Escande, Martin et al, PRL 2000 and the references therein quoted European Ph.D. course . - Garching 29.09.08) p.martin

3. Outline of the lecture • MHD equilibrium basics • 1d examples • Q-pinch • Z-pinch • Screw pinch • RFP equilibrium basics • RFP Stability • RFP dynamics and the dynamo. • Effects on transport European Ph.D. course . - Garching 29.09.08) p.martin

4. A reversed field pinch exists: RFX-mod a=0.459 m, R=2 m, plasma current up to 2 MA The largest RFP in the world, located in Padova, Italy A fusion facility for MHD mode control European Ph.D. course . - Garching 29.09.08) p.martin

5. MHD equilibrium basics European Ph.D. course . - Garching 29.09.08) p.martin

6. The MHD equilibrium problem • Time-indpendent form of the full MHD equations with v=0 European Ph.D. course . - Garching 29.09.08) p.martin

7. Linear vs. toroidal configurations European Ph.D. course . - Garching 29.09.08) p.martin

8. Magnetic flux surfaces European Ph.D. course . - Garching 29.09.08) p.martin

9. Current, magnetic and pressure surfaces • The angle between J and B is in general arbitrary European Ph.D. course . - Garching 29.09.08) p.martin

10. Rational, ergodic and stochastic European Ph.D. course . - Garching 29.09.08) p.martin

11. Surface quantities European Ph.D. course . - Garching 29.09.08) p.martin

12. One-dimensional configurations • Even if the magnetic configurations of fusion interest are toroidal, some physical intuition can be obtained by investigating their one-dimensional, cylindrically simmetric versions. • This separates: • Radial pressure balance • Toroidal force balance • For most configurations, once radial pressure balance is established, toroidicity can be introduced by means of an aspect ratio expansion, from which one can then investigate toroidal force balance. European Ph.D. course . - Garching 29.09.08) p.martin

13.  pinch European Ph.D. course . - Garching 29.09.08) p.martin

14. A simple example: -pinch • Configuration with pure toroidal field European Ph.D. course . - Garching 29.09.08) p.martin

15. A simple example: -pinch • The sum of magnetic and kinetic pressure is constant throughout the plasma • The plasma is confined by the pressure of the applied magnetic field European Ph.D. course . - Garching 29.09.08) p.martin

16. Experimental -pinch • Experimental -pinch devices among the first experiments to be realized • End-losses severe problem • A -pinch is neutrally stable, and can not be bent into a toroidal equilbrium • Additional field must be added to provide equilibrium European Ph.D. course . - Garching 29.09.08) p.martin

17. European Ph.D. course . - Garching 29.09.08) p.martin

18. Z-pinch European Ph.D. course . - Garching 29.09.08) p.martin

19. Z-pinch • Purely poloidal field • All quantities are only functions of r European Ph.D. course . - Garching 29.09.08) p.martin

20. Z-pinch • In contrast to the -pinch, for a Z-pinch it is the tension force and not the magnetic pressure gradient that provides radial confinement of the plasma • The Bennet pinch satisfies the Z-pinch equilibrium European Ph.D. course . - Garching 29.09.08) p.martin

21. Bennet Z-pinch • Tension force acts inwards, providing radial pressure balance. European Ph.D. course . - Garching 29.09.08) p.martin

22. Experimental Z-pinch European Ph.D. course . - Garching 29.09.08) p.martin

23. Z-machine • The Z machine fires a very powerful electrical discharge (several tens million-ampere for less than 100 nanoseconds) into an array of thin, parallel tungsten wires called a liner. • Originally designed to supply 50 terawatts of power in one fast pulse, technological advances resulted in an increased output of 290 terawatts • Z releases 80 times the world's electrical power output for about seventy nanoseconds; however, only a moderate amount of energy is consumed in each test (roughly twelve megajoules) - the efficiency from wall current to X-ray output is about 15% • At the end of 2005, the Z machine produced plasmas with announced temperatures in excess of 2 billion kelvin (2 GK, 2×109 K), even reaching a peak at 3.7 billion K. European Ph.D. course . - Garching 29.09.08) p.martin

24. European Ph.D. course . - Garching 29.09.08) p.martin

25. The general screw pinch European Ph.D. course . - Garching 29.09.08) p.martin

26. General Screw Pinch • Though the momentum equation is non-linear, the Q-pinch and Z-pinch forces ad as alinear superposition, a consequence of the high degree of symmetry European Ph.D. course . - Garching 29.09.08) p.martin

27. RFP equilibrium European Ph.D. course . - Garching 29.09.08) p.martin

28. Tokamak and RFP profiles European Ph.D. course . - Garching 29.09.08) p.martin

29. safety factor profiles in tok and RFP European Ph.D. course . - Garching 29.09.08) p.martin

30. RFP B profile European Ph.D. course . - Garching 29.09.08) p.martin

31. European Ph.D. course . - Garching 29.09.08) p.martin

32. TOK to RFP q profile transition European Ph.D. course . - Garching 29.09.08) p.martin

33. The reversed field pinch • Pinch configuration, with low magnetic field The toroidal field is 10 times smaller than in a tokamak with similar current Reactor issues: normal magnets, low force at the coils, high mass power density, no additional heating European Ph.D. course . - Garching 29.09.08) p.martin

34. Kruskal Shafranov limit for tokamak European Ph.D. course . - Garching 29.09.08) p.martin

35. The reversed field pinch • Pinch configuration, with low magnetic field • Bp and Bthave comparable amplitude and Btreverses direction at the edge • Modes in RFP : • low m (0-2) • high n (2*R/a) Safety factor European Ph.D. course . - Garching 29.09.08) p.martin

36. The reversed field pinch • Pinch configuration, with low magnetic field • Bp and Bthave comparable amplitude and Btreverses direction at the edge • Most of the RFP magnetic field is generated by current flowing in the plasma Magnetic self-organization European Ph.D. course . - Garching 29.09.08) p.martin

37. ..something on stability European Ph.D. course . - Garching 29.09.08) p.martin

38. European Ph.D. course . - Garching 29.09.08) p.martin

39. European Ph.D. course . - Garching 29.09.08) p.martin

40. European Ph.D. course . - Garching 29.09.08) p.martin

41. External Kink mode European Ph.D. course . - Garching 29.09.08) p.martin

42. RFP stability diagram for m=1 modes European Ph.D. course . - Garching 29.09.08) p.martin

43. RFP linear stability European Ph.D. course . - Garching 29.09.08) p.martin

44. European Ph.D. course . - Garching 29.09.08) p.martin

45. European Ph.D. course . - Garching 29.09.08) p.martin

46. European Ph.D. course . - Garching 29.09.08) p.martin

47. European Ph.D. course . - Garching 29.09.08) p.martin

48. Modern technique: real time control of stability with feedback coils European Ph.D. course . - Garching 29.09.08) p.martin

49. m=1, n =-5 m=1, n =-6 m=1, n=-7 m=1, n=-8 m=1, n=-9 m=0, all n Multi-mode control is a requirements for the RFP q (r) Resistive Wall Modes Tearing Modes Resistive Wall Modes m=1, n > 0 r (m) European Ph.D. course . - Garching 29.09.08) p.martin

50. RFX-mod: 192 active saddle coils, covering the whole plasma surface Each is independently driven (60 turns) and produces br from 50 mT (DC) to 3.5 mT (100 Hz) Power supply: 650 V x 400 A European Ph.D. course . - Garching 29.09.08) p.martin