Plasma Instabilities and Turbulence-II: Fusion Plasmas, particularly tokamaks

1. Plasma Instabilities and Turbulence-II: Fusion Plasmas, particularly tokamaks A Thyagaraja UKAEA/EURATOM Fusion Association Culham Science Centre, Abingdon, OX14 3DB, UK QUAMP Lecture, Durham, September 18, 2006

2. Plasmas: what are they and why study them? • Fully or partially ionized collections of charges; dominated by collective, long-range Coulomb forces; ubiquitous in the Universe • Debye’s theory of electrolytes; electron gas in metals are “cold plasmas”. Technologically important plasmas are discussed by Professor Braithwaite. “Classical”, high temperature, fully ionized plasmas common to Astrophysics and Fusion studies. • Fusion can only occur at high temperature: Coulomb repulsion of positively charged nucleii vs. nuclear forces. • Individual charged particle motions in E,B fields understood; “collective behaviour” very complex many-body problem- “Final frontier” of classical physics!

3. What is a Plasma? Plasma is the fourth state of matter At high temperature, the atoms in a gas dissociate into nuclei and electrons: a “plasma”. Plasmas are electrically conducting, and can therefore be controlled by magnetic fields. Most of the Universe consists of plasma. Picture coutesy of Stephen Haigh, Culham Electromagnetics and Lightning

4. Plasmas Around Us

5. The DT fusion reaction cycle The DT fusion reaction produces He and a neutron. The latter reacts with Lithium producing Tritium, which is re-cycled.

7. Kinetic picture of plasma dynamics: Fokker-Planck/Maxwell system with sources

8. Fokker-Planck/Maxwell system with sources • Rather formidable system with extremely different scales! • “Collisionless kinetics”-Vlasov-Maxwell system. Leads to plasma waves and “Landau (collisionless) damping” by wave-particle interactions and phase-mixing. • Quasi-neutrality applies to all length-scales much larger than the Debye shielding length and frequencies smaller than plasma frequency.

9. Individual charged-particle behaviour in specified E and B fields • Neglecting inter particle interactions, individual charged particle motions in given E and B fields well-described by Newton-Lorentz equations. • Charged particles with charge e and mass m “gyrate” about field lines with Larmor frequency (eB/m) and a Larmor (or gyro) radius (mv/eB). In the presence of “slowly varying” fields, they possess an adiabatic invariant, the magnetic moment μ. • The perpendicular kinetic energy is μB : This leads to “mirror force” and particle “trapping” in inhomogeneous B-fields. From these we can distinguish “trapped particle” and “passing particle” regions in velocity space • The particles “feel” various drifts: “E x B”, “grad B”, “curvature”, “inertial” etc. • “Collective behaviour” involves “back reaction” by particles on fields via their currents and charge densities (usually negligible); very complicated!

10. Magnetically confined Fusion... Picture courtesy of NASA/GSFC.

11. Two-fluid model: evolution equations

12. Characteristics of inhomogeneous turbulence • Driven, dissipative systems with very large numbers of degrees of freedom exhibit “turbulent evolution”. • Energy from the driving sources is redistributed among the unstable and stable degrees of freedom and dissipated/transported out of the system. • Free energy redistribution occurs in both real space (“turbulence spreading”) and wave number space (“cascading”). Enstrophy (vorticity) cascades to high-k! • Typically excitation occurs at macro/meso scales. • Dissipation usually occurs at very short length scales.

13. Nature of plasma instabilities • Plasma waves of importance in tokamak: Alfvén waves in which the “tension” of magnetic field provides the restoring force. Sound waves due to compressibility of plasma. Drift waves - sound waves modified by magnetic field structure and gradients of density and temperature • The plasma stores energy in the magnetic field, pressure and possibly flows. These sources of free energy are tapped by a wide variety of “current” and “pressure” and “entropy gradient/curvature” -driven linear and nonlinear instabilities (tearing, ballooning, drift collisional and collisionless, trapped-particle, sawteeth, disruptions, ELM’s NTM’s, RWM’s,…). • This zoology of unstable plasma modes are due both to velocity and position space effects. The most dangerous ones are big MHD modes. After “taming” them with careful manipulation of current and field (safety factor q) residual losses are due to electromagnetic turbulence in the “mesoscale”.

14. Tokamaks: equilibrium, macro-stability, transport • Equilibrium: “force balance” and field distribution; flows; electric field: “macro-structure” sustained by sources • Macro-stability against large-scale “disruptive” instabilities (MHD, ELM’s, NTM’s, RWM’s,…) • Strictly “collisionless” systems do not exist! Collisional dissipation sets lower limit to losses. • “Micro-structure”: enormous range; fundamentally nonlinear though may be “triggered” linearly.

15. The Tokamak Invented by Igor Tamm and Andrei Sakharov in the 1950's. Has dominated magnetic fusion research since the early 1970's.

16. A Plasma Inside JET

17. A MAST plasma MAST SphericalTokamak Temperature ~ 15million oC ~ 3m

18. The Next Step - ITER Produce significant amounts of fusion power (at least ten times the power required to heat the plasma up). Plasma duration ~30 minutes Aim at demonstrating steady-state operation. Develop fusion reactor relevant technologies • Project involves Europe, Japan, USA, Russia, China, S Korea. India • Approved July, 2005, start-up 2015? Construction at Cadarache, France

19. Tokamaks: spectral transfer mechanisms • Electromagnetic turbulence is due to linear/nonlinear instability and spontaneous symmetry breaking-results in both direct and inverse spectral cascades. • Sheared flows and Alfvén waves cascade (particularly enstrophy) to high radial k. Landau and other damping “kill” fine-scale structures smaller than collisionless skin depth (if they exist, “where are they?”) • Two high-k linearly growing modes can “beat” to populate the low-k and can also decay strongly by modulational instability: a fundamental “inverse spectral cascade” process [ cf. Lashmore-Davies et al PoP (2005).] • Powerful means to “self-generate” equilibrium flows, currents and populate low-k spectral regions “forming condensates”.

20. Characteristics of tokamak “plasma climatology” • Universal electromagnetic turbulence (dn/n and dj/j comparable!), < system size and >ion gyro radius; <confinement (s) and >Alfvén (ns) times. • Strong interactions between large and small scales. • Plasma “self-organizes”, like planetary atmospheres (Rossby waves=Drift waves). • Transport “barriers” connected with sheared flows, rational q’s, inverse cascades/modulational instabilities (Hasegawa). • Analogous to, “shear sheltering” (J.C.R Hunt et al):

21. Spectral transfer mechanisms: schema Nonlinearity; phase mixing by flows & Alfven waves Direct cascade ExB;jxB Inverse cascade Zonal flows Streamers Dynamo currents Mesoscale Microscale Macroscale Modulational Instabilities; Wave beating

22. A basic turbulence characteristic • The free energy flux into the unstable modes is redistributed by the nonlinearity (saturation). • All turbulent transport occurs with “effective diffusivity” (turbulent Prandtl number O(1)) of order:

23. Plasma turbulence • Turbulence provides a typical “decorrelation-rate”, (effective collision frequency) via the saturated RMS Ex B flow vorticity [1/T]. Poloidal gyro-radius (meso lengthscale) Turbulent RMS vorticity ( meso frequency scale)

24. Tokamaks: profile-turbulence interactions • All instability, linear or nonlinear caused by thermal disequilibrium in a driven-dissipative system-sources drive transport! • Profiles and turbulence cross-talk: turbulence corrugates profiles; latter saturate turbulence. Both electrostatic and magnetic components interact strongly and play a role (cf. NSTX, MAST,..) • Macroscale phenomena (pellets, sawteeth, ELM’s, ITB’s,..) influence and are influenced by mesoscale turbulence (possibly also micro scale): nonlinear self-organization • Momentum/angular momentum exchanges between turbulence and “mean profiles” result in dynamo currents(electrons) and zonal flows (ions). • No real “scale separation”-a continuum of scales in time and space

25. Zonal flows • Poloidal E x B flows, driven by Reynolds/Maxwell stresses against collisional damping. • Modulational instability, “inverse cascade” , eg. Generalized Charney Hasegawa Mima Equation(cf. Lashmore-Davies et al, PoP, 2005). • Highly sheared transverse flows “phase mix” and lead to a “direct cascade” in the turbulent fluctuations. • Enhances diffusive damping and stabilizes turbulence linearly and nonlinearly. • Confinesturbulence to low shear zones. (“turbulence sequestration”)

26. Typical advection-diffusion equation Sheared velocity in combination with diffusion changes spectrum “Reynolds number” Damping rate: Spectrum discrete, “direct cascade due to phase mixing” “Jets” in velocity “ghettoized” to low shear regions

27. Zonal flow dynamics “Poloidal momentum” “Lorentz force” “Reynolds stress” “Radial Ohm’s Law”

28. Neoclassical expressions(cf. Rosenbluth and Hinton, 1999)

29. Zonal flow experimental analysis LHS can be measured! (in principle) Higher the Z, easier it is to get Er. Estimation of non-neoclassical poloidal torque needs background flow measurements.

30. Mean poloidal field evolution “Faraday’s Law” “Generalised Ohm’s Law” “Ampère’s Law” “Dynamo current”

31. Time-averaged Zonal Flow (-Er/B) and Current density components (RTP)

32. Sawtooth like oscillations A A’ A” B C D E 0.5 ECH power deposition radius (Rho/a) RTP tokamak: well-diagnosed, revealing subtle features of transport, excellent testing ground Step-like changes in Te(0) “plateaux” whenever deposition radius crosses “rational” surfaces! Te(0) Hollow Te

33. RTP ExperimentalTe profiles for different ECH deposition radii

34. “Arithmetizing” two-fluid plasma turbulence:CUTIE • Global, electromagnetic, two-fluid code.Co-evolves turbulence and equilibrium-”self-consistent” transport. • “Minimalist plasma climatology” : evolve Conservation Laws and Maxwell’s equations for 7-fields, 3-d, pseudo spectral+radial finite-differencing, semi-implicit predictor-corrector, fully nonlinear. • Periodic cylinder model, but field-line curvature treated; describes mesoscale, fluid-like instabilities, but no kinetics or trapped particles (but includes neoclassics). • Very simple sources/boundary conditions.

35. Equations solved: reduced forms Continuity Energy Parallel momentum Potential vorticity Quasi-neutrality Ohm+Faraday

36. Off-axis ECH in RTP[Phys Rev Letts.- de Baar et al, 94, 035002, (2005)] • Ip=80 kA, Bf=2.24 T, qa=5.0, Hydrogen plasma • neav ~ 3.0 E+19 m-3 PECH~350 kW, P ~80 kW • PECH deposited at r/a = 0.55 • Resolution: 100x32x16; dt=25 ns ; simulated for >50 ms

37. Zonal flow (-Er/B) evolution (RTP): corrugations

38. Barriers and q Off-axis Sawteeth simulated by CUTIE: Te, q at r/a=0, 0.55

39. Sawtooth details and Magnetic and Electrostatic turbulence evolution in CUTIE (will be clearly shown in movies!) Volume averaged magnetic and electrostatic turbulence measures Note "precursors", compound crash, amplitude levels and phasing

40. No dynamo, no sawteeth! With dynamo No dynamo Volume averaged magnetic turbulence measure and loop voltage No "precursors" but "postcursors" in magnetic turbulence

41. Plasma turbulence:density fluctuations in R-Z plane and poloidal mode spectrum(Thyagaraja et al, Phys. Plasmas,12, 090907, 2005)

42. “Ear choppers”: CUTIE vs. Expt.

43. Discussion of CUTIE's strengths and weaknesses • Have shown only a small selection of results for many machines • Some limitations: missing are, full toroidal geometry, trapped particle physics, kinetic (finer-scale) dynamics, atomic physics effects, proper source terms, "real time" (ie fast!) calculations. • Higher resolution in space (with correct physics!) would also be helpful to answer worries about missing out on the "microscale". • CUTIE's "minimalist" model can be used globally to get a synopticdescription of a range of dynamic phenomena involving turbulence and transport. • Provides broad, qualitative account of macro/mesoscale experimental phenomena.

44. Conclusions • CUTIE results bear a qualitative resemblance to experiments (RTP, MAST, COMPASS, JET) • Is there any quantitative agreement? Yes, partially! • What have we learned from CUTIE simulations? Profile-turbulence, electromagnetic effects are not negligible • What are the limitations of minimalismand how can one proceed further? Many defects, some addressed in CENTORI • What are the lessons (if any) for the future? Mesoscale evolution is crucial; trapped particles, electron-inertia relevant

45. Conclusions Terrestrial, man-made fusion offers the promise of limitless energy supply with no environmental impact and no safety issues. JET has achieved everything it set out to do, to the point where we can build ITER with confidence and design power-plant scenarios. JET is a truly European experiment that operates very effectively. It is run along the lines of a user facility - operated by UKAEA for teams of visiting European scientists. It is part of a wide-ranging international fusion research programme. The world is ready to take the next step towards commercial fusion power by building ITER. This work was funded by EURATOM and the UK OST.