1. European Ph.D. course. - Garching 29.09.08)p.martin 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) Reversed Field Pinch: equilibrium, stability and transport

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

3. European Ph.D. course. - Garching 29.09.08)p.martin Outline of the lecture 1)MHD equilibrium basics 2)1d examples 1)Q-pinch 2)Z-pinch 3)Screw pinch 3)RFP equilibrium basics 4)RFP Stability 5)RFP dynamics and the dynamo. 6)Effects on transport

4. The RFP: what and why 1

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

6. A dynamic and well-integrated community RFX-modEXTRAP T2RRELAXMST Stockholm Padova Kyoto Madison

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

8. DOE Fusion Science, Germantown, MD - 23/07/2008p.martin 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. 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 ! RFP: exploiting the weak field

9. DOE Fusion Science, Germantown, MD - 23/07/2008p.martin 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

10. DOE Fusion Science, Germantown, MD - 23/07/2008p.martin 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

11. DOE Fusion Science, Germantown, MD - 23/07/2008p.martin 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

26. Z-pinch

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

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

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

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

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

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

33. The general screw pinch

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