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, 28 September 2009) 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. 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. 1 The RFP: what and why

5. RFP exist: e.g. 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

6. A dynamic and well-integrated community Stockholm Madison Padova Kyoto RFX-mod EXTRAP T2R RELAX MST

7. The RFP: a tight link with University (all experiments in University environment) and a nursery for the fusion community Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009 2008 IAEA Fusion Energy Conference, Geneva - P. Martin

8. RFP: exploiting the weak field • The distinctive feature of the RFP that motivates its interest as a fusion energy system is the weak applied toroidal magnetic field. • The RFP configuration is similar to a tokamak… • like to the tokamak, the RFP is obtained by driving a toroidal electrical current in a plasma embedded in a toroidal magnetic field  pinch effect. • …..but the applied toroidal field is 10 – 100 times weaker ! Most of the RFP magnetic field is generated by current flowing in the plasma (driven also by a dynamo mechanism) No need for large magnetic coils. DOE Fusion Science, Germantown, MD - 23/07/2008 p.martin

9. Why the RFP ? • A current-carrying low magnetic field configuration as the RFP: • has several technological advantages as a potential reactor configuration and will therefore contribute to the development of a viable reactor concept • has unique capabilities to contribute to fusion energy science and technology research DOE Fusion Science, Germantown, MD - 23/07/2008 p.martin

10. Fusion potential of the low magnetic field • high engineering beta • For configurations like the tokamak the maximum field at the magnet is of order twice the field in the plasma, whereas in the RFP the field at the magnet is less than in the plasma. • The engineering beta in an RFP reactor might be as much as twice the physics beta (up to 26% in present experiments).. • Use of normal(rather than superconducting) coils, • High mass power density, • Efficient assembly and disassembly, • Possibly free choice of aspect ratio DOE Fusion Science, Germantown, MD - 23/07/2008 p.martin

11. A comprehensive understanding of toroidal magnetic confinement, and the possibility of predicting it, implies that plasma behavior would be predictable over a wide range of magnetic field strengths. • The RFP provides new information since it extends our understanding to low field strength, testing the results derived at high field with the tokamak. DOE Fusion Science, Germantown, MD - 23/07/2008 p.martin

12. 2 MHD Equilibrium basics: just to refresh our knowledge

13. 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

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

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

16. 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

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

18. Revisiting stochastic magnetic fields in present day fusion devices • Coils like these are presently under consideration in ITER to produce, by purpose, stochastic magnetic field for ELM suppression (Resonant Magnetic Perturbation) European Ph.D. course . - Garching 29.09.08) p.martin

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

20. 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

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

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

23. 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

24. 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

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

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

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

28. 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

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

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

31. 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

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

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

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