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, 29 September 2008) 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. Relevant bibliography: Freidberg, IDEAL MHD Ortolani, IV Latin American Workshop on Plasma Physics Escande, Martin et al, PRL 2000 and the references therein quoted

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

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

6. 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

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

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

9. 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

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

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

12. 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.

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

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

15. 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

16. 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

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

18. Z-pinch

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

20. 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

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

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

23. 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.

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

25. The general screw pinch

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

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

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

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

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

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

32. TOK to RFP q profile transition

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

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

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

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

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

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

41. External Kink mode

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

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

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

48. Modern technique: real time control of stability with feedback coils

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

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

51. European Ph.D. course. - Garching 29.09.08)p.martin Feedback Control System Architecture on RFX-mod 192 power amplifiers Sensors: b r, b , I coil plasma Digital Controller Each coil is independently controlled Cycle frequency =2.5 kHz inputs outputs To control b r (a) 50 ms thin shell

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

53. European Ph.D. course. - Garching 29.09.08)p.martin Feedback Control System Architecture on RFX-mod 192 power amplifiers Sensors: b r, b , I coil plasma Digital Controller Each coil is independently controlled Cycle frequency =2.5 kHz inputs outputs To control b r (a) 50 ms thin shell

54. European Ph.D. course. - Garching 29.09.08)p.martin MHD stability feedback contro in RFX-modl Full stabilization of multiple resistive wall modes in presence of a thin shell (and RWM physics/code benchmarking) Control and tailoring of core resonant tearing modes – mitigation of mode-locking and smoother magnetic boundary Test of new algorithms and models for feedback control Design of mode controllers

55. European Ph.D. course. - Garching 29.09.08)p.martin RFX-mod contribution to RWM physics and control plasma current m=1,n=-6 mode amplitude t [s] logarithmic mode amplitude mode control Experiments can be designed to measure very precisely growth rate dependencies o Sophisticated algorithms are developed to control single and multiple RWM growth o Error Field Amplification

56. European Ph.D. course. - Garching 29.09.08)p.martin Effect of the active control

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

58. European Ph.D. course. - Garching 29.09.08)p.martin RFP dynamics

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

60. European Ph.D. course. - Garching 29.09.08)p.martin Non-linearity is built-in in RFP physics: an example ► J.M. Reynolds and C.R. Sovinec

61. European Ph.D. course. - Garching 29.09.08)p.martin Electric field in the RFP The RFP is an ohmically driven system : an inductive toroidal electric field, produced by transformer effect, continuously feeds energy into the plasma Ohm’s law mismatch : the electrical currents flowing in a RFP can not be directly driven by the inductive electric field E o..but stationary ohmic RFP are routinely produced for times longer than the resistive diffusion time overdriven underdriven

62. European Ph.D. course. - Garching 29.09.08)p.martin The RFP dynamo electric field An additional electric field, besides that externally applied, is necessary to sustain and amplify the toroidal magnetic flux. self-organized velocity field in the plasma A Lorentz contribution v x B is necessary, which implies the existence of a self-organized velocity field in the plasma. E dynamo

63. European Ph.D. course. - Garching 29.09.08)p.martin The old paradigm: Multiple Helicity (MH) RFP the safety factor q << 1 and the central peaking of the current density combine to destabilize MHD resistive instabilities. For a long time a broad spectrum of MHD resistive instabilities ( m =0 and m =1, variable n ( “multiple helicity” –MH – spectrum ), was considered a high, but necessary, price to pay for the sustainment of the configuration through the “dynamo” mechanism. br spectrum

64. European Ph.D. course. - Garching 29.09.08)p.martin Turbulent dynamo: remarkable self-organization An experimental and numerical database supports the MHD turbulent dynamo theory: the dynamo electric field is produced by the coherent interaction of a large number of MHD modes: Multiple Helicity (MH) dynamo

65. European Ph.D. course. - Garching 29.09.08)p.martin A completely new view eliminates the old paradigm For a long time…. ….a broad spectrum of MHD resistive instabilities, causing magnetic stochasticity, was considered a high, but necessary, price to pay for the sustainment of the configuration through the “MULTIPLE HELICITY dynamo” mechanism ….

66. European Ph.D. course. - Garching 29.09.08)p.martin A completely new view single resistive mode A helical ohmic equilibrium is possible, with a single helicity dynamo, where all the work is done by a single resistive mode (m=1, n=7 - opposite ordering wrt tokamak). Experiments are coming ever closer to the theoretically predicted chaos-free helical ohmic equilibrium This allows to retain the good features of self-organization without the past degradation of confinement.

67. European Ph.D. course. - Garching 29.09.08)p.martin A new approach to RFP dynamo: the Single Helicity Single Helicity (SH): the dynamo is driven by a single m =1 MHD resistive mode and its harmonics: Escande et al., PRL. 85 2000, Bonfiglio et al. PRL 2005 Helical symmetry of the magnetic equilibrium

68. European Ph.D. course. - Garching 29.09.08)p.martin A new approach to RFP dynamo: the Single Helicity Single Helicity (SH): the dynamo is driven by a single m =1 MHD resistive mode and its harmonics: –Helical symmetry of the magnetic equilibrium –Strongly reduced magnetic chaos in comparison to the standard multiple helicity (MH) RFP m=1 mode spectrum MH QSH

69. European Ph.D. course. - Garching 29.09.08)p.martin A new approach to RFP dynamo: the Single Helicity Single Helicity (SH): the dynamo is driven by a single m =1 MHD resistive mode and its harmonics: –Helical symmetry of the magnetic equilibrium –Strongly reduced magnetic chaos in comparison to the standard multiple helicity (MH) RFP – It is expected to have a very strongly improved confinement Two orders of magnitude improvement in numerical loss time of a population of test particles with respect to MH case (Predebon, White et al., PRL 2004) The ohmic helical state retains all the good features of the RFP without the problems connected with the high level of magnetic turbulence typical of the MH scenario

70. European Ph.D. course. - Garching 29.09.08)p.martin Resistive kink mode and dynamo: basic action Plasma is approximated as a current carrying wire placed on the axis of a cylindrical flux conserver where some axial magnetic field B z is present due to the azimuthal current I shell (flowing in the flux container). The wire is in an unstable equilibrium, and a small perturbation leads it to kink Escande et al., PPCF 42, B243, 2000 I BzBzBzBz BBBB I shell BzBzBzBz

71. European Ph.D. course. - Garching 29.09.08)p.martin Resistive kink mode and dynamo: basic action 1. The azimuthal projection of the kinked current I  has the same direction as I shell : growth of instability. 2. Solenoidal effect : B inside the kinked wire increase 3. Flux conservation: B’ outside decreases 4. Continuos growth force I shel l and B’ to reverse. Saturation 5. Final state: B’ in the outer region is reversed! B B’ I shell IIII B B’

72. European Ph.D. course. - Garching 29.09.08)p.martin The Single Helicity state is theoretically predicted and partially understood, but physics in the modeling is still not completed (Bonfiglio et al.PRL 2005) coupling with transport still missing.. Dissipation coefficients (viscosity…) still unknown Toroidal effects… (coupling of m=1 modes and production of m=0) In the experiments we observe Quasi Single Helicity (QSH) states Single and Quasi Single Helicity (QSH) in the experiment

73. European Ph.D. course. - Garching 29.09.08)p.martin Properties of experimental QSH states The n -spectrum of MHD modes is dominated by a single m =1 geometrical helicity Relative amplitudes of m=1 modes QSHMH

74. European Ph.D. course. - Garching 29.09.08)p.martin Properties of experimental QSH states The k -spectrum of MHD modes is dominated by a single m =1 geometrical helicity Dominant mode Secondary modes QSHMH

75. European Ph.D. course. - Garching 29.09.08)p.martin Dynamo electric field is produced in QSH by the dominant mode We are observing the right mechanism! Piovesan et al. PRL 2005 Dynamo electric field toroidal spectrum

76. European Ph.D. course. - Garching 29.09.08)p.martin Helical closed flux surfaces in the QSH plasma core The “secondary” modes have amplitudes still too high for a global improvement of the plasma performance and there is magnetic chaos outside the helical domain: Toroidal coupling m =0 modes T e (eV) SXR

77. European Ph.D. course. - Garching 29.09.08)p.martin Lundquist number scaling is promising S =  R /  A = Dominant mode (m = 1, n = -7) Secondary modes (1,-8 to -15) b dom b secd 5% 0.2% == 25 b/B (%) S S At higher current, when plasma gets hotter, the helical state is more pure

78. European Ph.D. course. - Garching 29.09.08)p.martin X point Topology change at high current: from island to Single Helical Axis Single Helicity states experimentally discovered in 1998 ( ppcf 98, prl 2000 ) Exciting physics result (theoretically predicted), but relatively small volume of plasma involved

79. European Ph.D. course. - Garching 29.09.08)p.martin Topology change at high current: from island to Single Helical Axis X point b dom /b sec increases Magnetic axis

80. European Ph.D. course. - Garching 29.09.08)p.martin X point b dom /b sec increases New helical topology where the orginal axisymmetric axis is replaced by a helical magnetic axis Topology change at high current: from island to Single Helical Axis Extended transport barrier (Escande et al PRL 2000) New Axis

81. European Ph.D. course. - Garching 29.09.08)p.martin X point b dom /b sec increases New helical topology where the orginal axisymmetric axis is replaced by a helical magnetic axis (Single Helical Axis) From island to Single Helical Axis QSH Island SHAx (Escande et al PRL 2000) Lorenzini PRL 2008

82. European Ph.D. course. - Garching 29.09.08)p.martin Experimental confirmation of a helical equilibrium With appropriate reconstruction of the dominant mode eigenfunction, we can build a helical flux  (r,u) = m  (r,u) - nF(r,u) considering the axisymmetric equilibrium and the dominant mode. (r and u = m -n  are flux coordinates). Lorenzini, Martines, Terranova et al, 2008

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