EFTC10 Oral Presentation

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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
  Valovic, 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 wavesDrift waves).
• Transport barriers connected with sheared flows,
  rational qs, 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, ELMs,
  ITBs,..) 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
Nonlinearity phase mixing by flows Alfven waves
Direct cascade
ExBjxB
Zonal flows
Random phases
Streamers
Turbulent diffusion
Dynamo currents
Mesoscale
Microscale
Macroscale
Inverse cascade
Modulational Instabilities beating
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Arithmetizing two-fluid plasma turbulenceCUTIE

• Global, electromagnetic, two-fluid
  code.Co-evolves turbulence and equilibrium-self-c
  onsistent transport.
• Minimalist plasma climatology Conservation
  Laws and Maxwells equations for 7-fields, 3-d,
  pseudo spectralradial 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)

• Ip80 kA, Bf2.24 T, qa5.0, Hydrogen plasma
• neav 3.0 E19 m-3 PECH350 kW, P? 80 kW
• PECH deposited at r/a 0.55
• Resolution 100x32x16 dt25 ns simulated for
  gt50 ms

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Initial and Averaged ProfilesTe,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 gt modegt asymmetric turbulent
  fluxesgt zonal flow and dynamo effectsgt reduce
  high-k turbulence and flatten qgtlocal reduction
  of advection
• gthigher pressure gradientsgtrelaxation 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/a0, 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!
With dynamo
No dynamo
Volume averaged magnetic turbulence measure and
loop voltage No "precursors" but "postcursors" in
magnetic turbulence
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High resolution study of Ohmic sawteeth ELMs
?!

• Ip90 kA, Bf2.24 T, qa5.0, Hydrogen plasma
• neav 3.0 E19 m-3 P? 90 kW Zeff 2-4 Edge
  source
• Resolution 100x64x16 dt25 ns simulated for
  gt25 ms
• Movies of profiles ne, Te, V(zonal) -cEr/B,
  j(dynamo), j(bs)
• Contours Te, radial ExB, A-parallel fluctuations

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Ohmic m1 sawteeth edge instability V-loop,
Beta
Te(0)800 eV (CUTIE) close to RTP760 eV
monotonic ne(0) 4.0 E19 (CUTIE) RTP 5.0 E19
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Ohmic RTP caseaveraged 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? Occams 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, ITBs
• 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 Earths 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 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
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The Advection-Diffusion Equation
Sheared velocity in combination with diffusion
changes spectrum
Reynolds number measures shear/diffusion
Damping rate is proportional to
Spectrum discrete, direct cascade due to phase
mixing
Jets in velocity lead to ghetto-isation/confine
ment 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
flattening Corrugated dynamo current (both
signs!) localization
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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
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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 forms
Continuity
Energy
Parallel momentum
Potential vorticity
Quasi-neutrality
OhmFaraday