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Profile-turbulence interactions, MHD relaxations and transport in Tokamaks

A Thyagaraja*, P.J. Knight*,

M.R. de Baar†, G.M.D. Hogeweij† and E.Min†

*UKAEA/EURATOM Fusion Association

Culham Science Centre, Abingdon, OX14 3DB, UK

†Assoc. EURATOM-FOM, Trilateral Euregio Cluster, P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands

IAEA Meeting, Trieste, Mar 2-4, 2005

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Acknowledgements

• Jack Connor, Jim Hastie, Chris Gimblett, Martin Valovič, Ken McClements, Terry Martin, Chris Lashmore-Davies (Culham)

• Niek Lopes Cardozo (FOM)

• Xavier Garbet, Paola Mantica, Luca Garzotti (EFDA/JET)

• EPSRC (UK)/EURATOM

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Synopsis

• Role of profile-turbulence interactions and spectral transfer processes in tokamak turbulence and transport

• The key concepts: spectral cascades, profile-turbulence interactions, nonlinear self-organization, dynamos, zonal flows

• Some typical simulation results from CUTIE and comparisons with experiment

• Conclusions

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Characteristics of tokamak “plasma climatology”

• Universal, electromagnetic turbulence, between system size and ion gyro radius; confinement (s) and Alfvén (ns) times.

• Strong interactions between large and small scales; inhomogeneity of turbulence.

• Plasma is strongly “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):

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Profile-turbulence interactions • All plasma instability, linear or nonlinear, caused by thermal

disequilibrium in a driven-dissipative system

• Profile-turbulence cross-talk: turbulence corrugates profiles; latter saturate turbulence. Both electrostatic and magnetic components interact strongly and play a role

• 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

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Spectral Transfer Mechanisms

Direct cascade ExB;jxB

Inverse cascade

Nonlinearity; phase mixing by flows & Alfven waves

Modulational Instabilities; beating

Macroscale Mesoscale Microscale

Zonal

flows

Streamers

Dynamocurrents

Random phases

Turbulent diffusion

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“Arithmetizing” two-fluid plasma turbulence:CUTIE

• Global, electromagnetic, two-fluid code.Co-evolves turbulence and equilibrium-”self-consistent” transport.

• “Minimalist plasma climatology” : 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; no kinetics or trapped particles (but includes neoclassics).

• Very simple sources/boundary conditions (overly simple perhaps?!)

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Off-axis ECH in RTP [Phys Rev Letts.- de Baar et al, 94, 035002, (2005)]

• Ip=80 kA, B=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

0.1;007.0;16.0(%) **

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Initial and Averaged Profiles:Te,Ti,ne,q (Squares-experiment; solid line-CUTIE)

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Power density and Electron advective Heat flux Profiles

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Time-averaged Zonal Flow (-cEr/B) and Current density components

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Zonal Flows

• Poloidal E x B flows, turbulent Reynolds stresses: “Benjamin-Feir” type of modulational instability, “inverse cascade” recently explained in Generalized Charney Hasegawa Mima Equation

• McCarthy et al. PRL, 93, 065004, 2004

• 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.

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Zonal Flow Evolution

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Current/q Profile Evolution

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Barriers and q

• CUTIE naturally tends to produce barriers near the simple rationals in q.(only global codes can do this!)

• Mechanism: heating > mode> asymmetric turbulent fluxes> zonal flow and dynamo effects> reduce high-k turbulence and flatten q>local reduction of advection

• >higher pressure gradients>relaxation oscillation

• Two barrier loops operate in CUTIE! The loops interact in synergy.

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Outbound heat flow and "ears"

• Off-axis ECH-power enhances the MHD level near the deposition radius.

• The interplay of the EM-and ES-component of these fluctuations gives rise to an outward heat-flow.

• This is sufficient for supporting pronounced off-axis Te maxima in CUTIE, comparable with expt.

• The ears are quite comparable to the experimental observations.

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Barriers and q

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

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“Ear choppers”: CUTIE vs. Expt.

CUTIE RTP

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Sawtooth details and Magnetic and Electrostatic turbulence evolution in CUTIE

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Off-axis sawteeth: comparison with RTP

• CUTIE produces MHD events (as in experiment) associated with profile-turbulence interactions, zonal "jets", "elbows" in the q profile; relaxations called “ear choppers”.

• CUTIE Period (~3 ms), RTP (~1.5-2 ms)• CUTIE Amplitude (~150-200 eV) RTP (~100 eV) • CUTIE Crash time (~0.3 ms) RTP (~0.2-0.5 ms)• CUTIE Conf. time (~3-4 ms) RTP (~3 ms)

• “Avalanching” and “bursts”; intermittency outside heating radius.

• Qualitative agreement with experiment.

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No dynamo, no sawteeth!

Volume averaged magnetic turbulence measure and loop voltageNo "precursors" but "postcursors" in magnetic turbulence

With dynamo

No dynamo

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High resolution study of Ohmic sawteeth [& ELM’s ?!]

• Ip=90 kA, B=2.24 T, qa=5.0, Hydrogen plasma

• neav ~ 3.0 E+19 m-3 P~90 kW; Zeff= 2-4; Edge source

• Resolution: 100x64x16; dt=25 ns ; simulated for >25 ms

• Movies of profiles: ne, Te, V(zonal)= -cEr/B, j(dynamo), j(bs)

• Contours: Te, radial ExB, A-parallel fluctuations

6.0;0067.0;14.0(%) **

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Ohmic m=1 sawteeth & edge instability: V-loop, Beta

Te(0)~800 eV (CUTIE) close to RTP~760 eV; monotonic

ne(0) 4.0 E+19 (CUTIE) RTP 5.0 E+19

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Ohmic RTP case:averaged Te,Ti,ne,q (Squares-experiment; solid line-CUTIE)

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Movie!

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Question: What does this model predict?

• Do CUTIE results bear a qualitative resemblance to experiments (RTP, MAST, JET, FTU,..)? (Conditional “yes”!)

• Is there any quantitative agreement? (in some cases and fields)

• What have we learned from CUTIE simulations? (profile-turbulence interaction crucial)

• What are the limitations of minimalism and how can one proceed further? (many effects omitted; do they matter? Occam’s Razor!)

• What are the lessons (if any) for the future? (go from “large” to “small” scale)

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Conclusions-I

• “Minimalist CUTIE model” applied to RTP, JET, MAST, FTU, TEXTOR, T-10

• First "turbulence code" to describe ”on and off-axis sawteeth" dynamically in experimental conditions • Describes self-organization caused by profile-turbulence interactions • Insight into spectral transfer & spontaneously generated zonal flows and dynamo currents in tokamaks

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Conclusions-II

• Illuminates role of turbulence in shaping large-scale behaviour & demonstrates features of experiment: 1) key role of rational q surfaces and electromagnetic modes

2) off-axis maxima and outward heat advection (“ears”)

3) role played by “corrugated” zonal flows, MHD relaxation 4) deep and shallow pellet behaviour in JET(with ITB's)

• Complementary to gyrokinetics: better suited to long-term evolutionary studies (“plasma climatology”) and global, electromagnetic, meso plasma dynamics.

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Discussion

• CUTIE's "minimalist" model used globally, provides synoptic description of a range of dynamic phenomena involving turbulence and transport: MECH, pellets, MHD relaxation, ITB’s

• Limitations/ short-comings: • Geometry• Trapped particle physics, kinetic effects• Atomic physics effects, radiation, impurities• Proper source terms• ”Real time" (ie fast!) calculations and effective predictions to guide

experiments, diagnostics and design.• Higher resolution in space (with correct physics!)• Worries about missing "microscale” physics. (Is the Earth’s climate

influenced by air turbulence on a 10x10x10 m grid?)

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Spectral transfer mechanisms• Electromagnetic turbulence due to linear/nonlinear instability:

spontaneous symmetry breaking-results in spectral cascades (both direct and inverse).

• Sheared flows and Alfven waves cascade (particularly enstrophy) to high radial k. Landau damping/phase-mixing “kills” fine-scale structures (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” (Hasegawa, Lashmore-Davies et al, Benjamin-Feir)

• Powerful means to “self-generate” equilibrium flows & currents and populate low-k spectrum forming “condensates”

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Generic Transport Equation & Flux

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Equations of Motion (in brief!)

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Equations of Motion (2)

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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

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)((x)vy x

fD

xy

f

t

f

The Advection-Diffusion Equation

Sheared velocity in combination with diffusion changes spectrum

“Reynolds number” measures shear/diffusion: DxR Lxy/))(

2'v(

Damping rate is proportional to RD yy

3/13/13/2

'v'v

Spectrum discrete, “direct cascade due to phase mixing”

“Jets” in velocity lead to “ghetto-isation/confinement” to low shear regions

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Zonal Flow (-cEr/B) Evolution: corrugations

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Total current density and dynamo current density evolution

Current is expelled from core and strong profile flatteningCorrugated dynamo current (both signs!); localization

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

Te(0)

A

A’

A”B

CD E

ECH power deposition radius (Rho/a)

Sawtooth like oscillations

0.5

Hollow Te

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

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RTP Experimental Te profiles for different ECH deposition radii

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Zonal flow (-cEr/B) and bootstrap current density

Negative values of zonal flow indicate ion diamagnetic flow values; note corrugations in both fields (j-bs is typically positive)

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Equations solved: reduced formsContinuity

Energy

Parallel momentum

Potential vorticity

Quasi-neutrality

Ohm+Faraday