Tokamak: perfect magnetic confinement

1. nearly Tokamak: perfect magnetic confinement Edge plasma is determined by balance of parallel (simple) and radial (turbulent) transport r

2. What radial transport? Number of particles crossing a particular surface Convection Due turbulence, but how to quantify it? Diffusion is weak collisions radial toroidal Experiment

3. Motivation • Radial particle transport is much higher than due to collisions – anomalous • Edge modeling of tokamaks is done by non-turbulent codes that use ad-hoc values for the radial transport. • Turbulence is claimed to be responsible but no model was demonstrated yet to really quantitatively agree with experiment

4. TCV tokamak geometry

5. Destroyed by ECRH X2! Probe head Vfl1 Is,Te B-field 4mm Vfl4 Experimental set-up for diagnosing edge turbulence in tokamak TCV • Reciprocating Langmuir probe = the first ever diagnostic to explore plasma • Measures fast and locally electric potential, plasma density, temperature, turbulence-driven radial particle flux, parallel flows, velocities, … Plasma Nobel prize 1932 Merci: R. Pitts, Ph. Marmillod, P. Lavanchy, J.-C. de Giorgi, X. Llobet, P. Conti, P. Gorgerat, R. Chavan, O. Bartolomeoli

6. L-mode Density fluctuations Slowed down 120x Discovered: ELM followed by an MHD mode linked to sawtooth! H-mode potential Exploring a wonderland! Jan,stop it! Let’s do something useful! The only diagnostic inserted inside ~350 discharges, 100MB / discharge

7. First edge turbulence studies on TCV Started in 2003 Various discharges (ne,B<>0,Ip, L/H-mode, Z, D/He) Confirmed many observations from other tokamaks, e.g.: • Self-similar (fractal) behaviour • Intermittency • Time-asymmetric bursts New discovery: • Flux scales solely with density • Universality of PDF on wall • Density PDF described by Gamma

8. Two-parameter Gamma PDF: • mean <n> • fluctuation level A = <n>/sn • A determines the shape Gamma distribution describes density PDF everywhere Graves et al. PPCF 47, L1 (2005) Horacek et al. Czech J. Phys. (2004) Analogy with a sandpile Ask the Danish! Interpretation? Hmm, well … density Jan,what about the Gamma distribution? radial Local sandpile height

9. Analogy with a sandpile model • Self-organized criticality • Jon introduced the statistics Interpretation? Hmm, well …

10. Gr Gr Ez Experiment vs. interchange model Take existing 2D fluid ESEL model based on interchange motions: Curvature and BxB drift  vertical charge separtion  Generation of Ez  ExB drift outwards  Unstable at LFS due p Risø made the simulations, we compared with experiment. + + - - B Garcia, PoP’05

11. Sinks Parallel damping Diffusion - Particle conservation n Energy conservation Vorticity conservation Curvature operator, Advective derivative The ESEL model • Electrostatic 2D fluid model solves selfconsistently turbulence in n,Te,W. No neutrals. • Scalar measurable inputs: TLCFS,nLCFS,BLCFS,R+a,L|| determine the sinks (W. Fundamenski et al, Physics of Plasma 2006) • Simplifications: drift approximation, finite rLi effects neglected, thin layer approximation (dn/n<<1,dT/T<<1), only LFS, parallel losses in the 3rd dimension: linear damping. Garcia et al., PRL 92, 165003 (2004); PoP 12, 062309 (2005)

12. ESEL simulation geometry

13. 30mm LCFS wall 30mm ESEL simulation Qualitatively consistent with all experimental observations and theories. Is it so also quantitatively? ESEL 116, particle density Experiment S.J. Zweben et al, Nucl. Fusion 44,134 (2004)

14. Comparison with experiment • The thesis demonstrates that nearly all statistics of density, flux (and temperature) match perfectly the experiment • Most importantly, absolute level of flux matches => it describes the anomalous transport! O … Model + … Experiment

15. Summary • First parallel flow study on TCV. • 80% of Midplane M|| is accounted by: • Pfirsch-Schlütter (dominant) • + • ballooning (turbulence-driven) • + • divertor sink • First edge turbulence studies on TCV • Build large database, found self-similarity, universality in PDF (Gamma distribution) • Demonstrated for the first time that experimental anomalous transport quantitatively agrees with interchange model (Denmark), due (BxB)xB drifts in p at LFS, dominated by rare convective blobs of ~cm size and vr~km/s. • This opens a way for quantitative predictions of plasma-wall interactions. • More scientifically in the seminar at CRPP on Wednesday 17 May, 830. Questions?

16. 2D fluid turbulence • Re = speed* *dimension / viscosity.

17. Various analytical distributions • Gamma: in systems with clustering, e.g. sand-piles with avalanches [Graves PoP 2002] • Lognormal: for Boltzmann-distributed electrons, neexp(-f/Te) and Gaussian f[Sattin, PoP 2004] • BHP: describes self-organized critical systems [van Milligen, PoP 2005] • Gumbel: PDF of extreme values • Gaussian: most frequent in nature, sum of independent random processes Gamma Lognormal

18. Te correlated with ne at a fixed position Density Temperature Gamma PDF match best TCV & ESEL Functional dependence of statistical moments defines a particular PDF. A = <n>/sn AT = <T>/sT Skewness Kurtosis Gamma: S=2/A Log-Normal S=3/A+A-3 BHP: S=0.9 Gumbel: S=1.14 Gaussian: S=0

19. Density • Gradients, time-scales, turbulence levels and statistical moments match

20. Inside LCFS experiment not reliable due pins too far Flux • Cross-field turbulence-driven ExB particle flux • Gradients, turbulence levels and statistical moments match for flux

21. t- t+ Coherently averaged density bursts match • Isolate large bursts, normalize, average them and fit by exp(-t/t+-) • Time-scales and asymmetry match • Inter-burst period match => even blob generation is well modelled => no additional mechanism needed

22. Dissipation and parallel loss estimates • Transport coefficients based on • neo-classical collisional perpendicular transport: D┴n~D┴T~D┴W~10-3m2s-1. • classical parallel transport: sT~sn=sW~Lc/cs~1/250ms • Taken to be constants in space and time with abrupt changes at LCFS and wall • W. Fundamenski et al, Physics of Plasma 2006