Analytical and computational paradigms for plasma turbulence-II

1. Analytical and computational paradigms for plasma turbulence-II • A Thyagaraja • UKAEA/EURATOM Fusion Association • Culham Science Centre, Abingdon, OX14 3DB, UK • Trieste Plasma School, October, 2003

2. Acknowledgements • Peter Knight,Terry Martin, Jack Connor, Chris Lashmore-Davies (Culham) • Marco de Baar, Erik Min, Hugo de Blank, Dick Hogeweij, Niek Lopes Cardozo (FOM) • Xavier Garbet, Paola Mantica, Luca Garzotti (EFDA/JET) • Nuno Loureiro (Imperial College) • Michele Romanelli (Frascati) • Dan McCarthy (USEL) • EPSRC (UK)/EURATOM

3. Synopsis • What is plasma turbulence? • What are the key problems to be addressed? • Main ideas of the approach. • Typical results: long-time evolution and modulated ECH in RTP as examples • Conclusions

4. What is plasma turbulence? • In principle, a plasma can be maintained (driven) by sources against collisional (dissipative)losses. • Resulting current/pressure profiles are strongly unstable. • Instability spontaneously breaks symmetry in space& time. • Growing modes nonlinearly saturate, leading to turbulent fluxes, spectral cascades and anomalous transport. • Equilibrium and turbulence cross-talk on a range of scales, especially in the mesoscales.

5. Why is turbulence important? • Usually, though not invariably, turbulent losses are more severe than neoclassical. • Magnetic shear (q’) and E x B flow shear seem to play key roles in formation and dynamics of high gradient regions calledTransport Barriers (ETB’s or ITB’s) identified in experiments. • Understanding and control crucial to power plant issues: economics, divertor loading, ash removal etc. • Difficult unsolved problem. Much recent progress through complementary approaches, close theory/expt interaction.

6. Characteristics of tokamak turbulence • “Universal”, electromagnetic (dn/n and dj/j comparable!), between system size and ion gyro radius; between confinement (s) and Alfvén (ns) times: • Plasma is “self-organising”, like planetary atmospheres (Rossby waves=Drift waves). • Transport barriers connected with sheared flows, rational q’s, inverse cascades/modulational instabilities (Hasegawa). • Analogous to El Nino, circumpolar vortex, “shear sheltering” (J.C.R Hunt et al).

7. Key Concepts: q and zonal flow • “Mode rational surface” when m=nq; long wave length MHD modes may occur. “Magnetic shear” dq/dr, an important stability parameter;dynamo effects. • Plasma knows “number theory”, resonances analogous to Saturn’s rings occur -KAM theory • Radial electric field associated with sheared zonal flow (from ExB drifts); influences stability: Taylor flow analogy! • Inverse and direct cascades determine turbulent saturation and transport.

8. Challenges for Theory • Explain observations, scalings, thresholds. • Predict phenomena (ITB’s, transitions, sawteeth, ELM’s, impurity behaviour, pinches..) • Calculate with adequate accuracy, faster than experiment, consistent with both qualitative and quantitative facts. • Suggest new diagnostics, improved performance, better engineering design.

9. Challenges for Experiments • Comprehensive, time-space resolved diagnostics of T, n, q, E, Z needed. • Measurements of turbulent spectra (high & low k). • Transients: pellets, modulated heating. • Adequate inter and intra machine comparisons. • Only starting to be met in JET, ASDEX, TORE-Supra, DIIID, MAST, NSTX, JT-60U, TEXTOR, FTU..

10. “Arithmetizing” tokamak turbulence:CUTIE • Global, electromagnetic ( ), two-fluid (electrons/ions) code.Co-evolves turbulence and equilibrium-”self-consistent” transport. • “Minimalist” approach to tokamak turbulence: 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). • Question: What, if anything, do the solutions of such a model tell us about experiments? (Long-time evolution, q, zonal flows,..)

11. Equations of motion (1) S “Continuity” “Total momentum” “Generalised Ohm’s Law”

12. Equations of motion (2) “Energy” “Pre-Maxwell”

13. Equations solved: reduced forms

14. The mean equations have a generic structure: turbulent and collisional fluxes add to give total flux. Radial/temporal “corrugations” allowed!

15. 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

16. RTP ExperimentalTe profiles for different ECH deposition radii

17. What is the physics? • RTP has well-defined phenomenology: “q-comb model” to fit data (thermal diffusivity lowered at rational q surfaces). Explanation? • Why do barriers have a preference to form near simple rationals? • What is the role of zonal flows in RTP? • What is the cause of the outward advection producing off-axis Te maxima? • What drives off-axis MHD?

18. CUTIE/RTP scenarios • Instability -linear growth-nonlinear saturation- “corrugated profiles” • The corrugations feed-back on the turbulence: nonlinear co-evolution of turbulence and profiles essential. • Calculations start from an (arbitrary) Ohmic initial condition, after which the off-axis Electron Cyclotron Heating (ECH) is switched on. • “Switch-off”, modulated ECH (MECH), and pellets have also been studied

19. Barriers and q • CUTIE produces barriers near the simple rationals in q.(only global codes can do this!) • Mechanism: due to heating, a mode forms. Gives rise to turbulent fluxes which locally steepen pressure gradient resulting in, zonal flows and dynamo effects which tend to reduce turbulence and flatten q. • Two barrier loops operate in CUTIE! The loops interact in synergy.

20. Two barrier loops in CUTIE Asymmetric fluxes near mode rational surface Driving terms of turbulence Pressure gradient Turbulent dynamo, currents Zonal flows modify turbulence-back reacts q, dq/dr, j, dj/dr

21. Off-axis ECH • Ip=80kA, Bf=2.24T, qa=5.25 • neav ~ 2.7e19 m-3 • PECH 350 kW, P approximately 100 kW • PECH deposited at r/a = 0.55 • We present the transition from monotonic (qmin ~ 1) to reversed shear (qo>3) and the associated changes in the profiles

22. Central safety factor and temperature evolution: off-axis ECH (350 kW) in RTP switch-on simulation; black-central, green-heating radius.

23. Comparing final state Te profiles with Expt.Solid line is simulation whilst expt. is triangles.

24. Final state turbulence in a poloidal plane: dVr(ExB) , dB(pol)/B contours

25. Outbound heat flow and ears • Off-axis ECH-power enhances the fluctuation level within the deposition radius. • The interplay of the EM-and ES-component of these fluctuations gives rise to an outward heat-flow. • This flow is sufficient for supporting pronounced off-axis Te maxima in CUTIE. • The ears are quite comparable to the experimental observations.

26. Modulated ECH in RTP (Mantica et al): CUTIE has simulated transients in many machines. Present results for Case A’, high duty cycle simulation vs expt.

27. MECH in RTP (Mantica et al): CUTIE has simulated transients in many machines. Present results for Case A’, low duty cycle (simulation- note differences from hdc.)

28. MECH in RTP (Mantica et al): average Te(keV) solid line, simulation, squares, experiment.

29. MECH in RTP (Mantica et al): simulated harmonics (keV)and phases: I harmonic, red, II, green, III, blue. Symbols-expt., lines-simulation (high duty cycle)

30. MECH in RTP (Mantica et al): simulated harmonics (keV)and phases: I harmonic, red, II, green, III, blue. symbols-expt., lines-simulation. (low duty cycle)

31. “Ear” choppers (MHD events) • CUTIE produces MHD events like experiments. • CUTIE shows that these events are both radially and poloidally localised, tending to flatten off-axis maxima (“ears”). • Normally they show modest amplitudes with respect to the experiment. • Transport involves “avalanching” and “bursts”; intermittency in certain locations. • Qualitative features agree with experiment.

32. Zonal Flows • Poloidal E x B flows, driven by turbulent Reynolds stresses: “Benjamin-Feir” type of modulational instability, “inverse cascade” recently explained in Generalized Charney Hasegawa Mima Equation. • 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. • Confines turbulence to low shear zones.

33. Discussion • Some other results from CUTIE not presented here: • “Ohmic bifurcations” (cf. Marco de Baar, et al. RTP) • Cold pulses and pellet experiments in particle transport (cf. Mantica, Garbet, Garzotti-JET, Romanelli-FTU,Min et al RTP) • Minimalist model can be used globally to get a synoptic description of a range of dynamic phenomena involving turbulence and transport. • Trapped particles, kinetic (finer-scale) dynamics and full geometrical atomic physics effects remain future challenges!

34. Conclusions • “Minimalist CUTIE model” reproduces qualitatively many RTP phenomena: 1) Barriers near rational q surfaces. 2) Off-axis maxima and outward heat convection (“ears”) 3) n,Te reflecting episodic q evolution (switch-on/off studies). 4) Zonal flow plays a role outside the ECH power deposition radius. 6) MHD modes (“ear choppers”); MECH (and pellet) behaviour. • CUTIE/CENTORI applied to JET, MAST, FTU, ASDEX, TEXTOR. • Approach complementary to gyrokinetics: more suited to long-term evolutionary features and global phenomena.