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

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

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

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.

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 called Transport 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.

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).

))(/( kLv snth

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.

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.

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..

“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,..)

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Equations of motion (1)

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“Total momentum”

“Generalised Ohm’s Law”

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“Energy”

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Equations solved: reduced forms

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

RTP tokamak: well-diagnosed, revealing subtle features of transport, excellent testing ground

Te(0)

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ECH power deposition radius (Rho/a)

Sawtooth like oscillations

0.5

Hollow Te

Step-like changesin Te(0) “plateaux”whenever deposition radiuscrosses “rational” surfaces!

RTP Experimental Te profiles for different ECH deposition radii

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?

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

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.

Two barrier loops in CUTIEAsymmetric fluxes near mode rational surface

Pressure gradient

Zonal flows modify turbulence-back reacts

Turbulent dynamo, currents

q, dq/dr, j, dj/dr

Driving termsof turbulence

Off-axis ECH

• Ip=80kA, B=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

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

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

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

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.

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.

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.)

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

experiment.

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)

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)

“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.

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.

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!

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.