Magnetic Fluctuations and Transport X. Garbet CEA Cadarache Thanks to: M. Bécoulet, C. Bourdelle, P. Beyer, L.Colas, P. Diamond, P. Ghendrih, F. Jenko, B.Labit, M.Ottaviani, Y.Peysson, S. Prager, P. Terry, R.Waltz, X.L.Zou. TTF Meeting Madison 2-5 April 2003 X.Garbet 1
Motivation Nature of electron transport. Dependence of transport on β. Nature of small scale fluctuations. Ergodic divertor and stellarator edge. RFP's Renewed interest in electromagnetic turbulence and consequences for turbulent transport TTF Meeting Madison 2-5 April 2003 X.Garbet 2
Theory Outline - transport associated to magnetic fluctuations - stability : effect of β (large and smal scale fluctuations) - non linear effects Experiment - transport - measurement of magnetic fluctuations - transport of fast electrons TTF Meeting Madison 2-5 April 2003 X.Garbet 3
Effect of a Perturbed Magnetic Field Main component B =rot(δa // b eq ) (however δb // affects stability at high β) Electric potential φ never negligible 3 main ingredients: - Magnetic flutter displacement due to perturbed magnetic field, B. =B eq. + B. - Ohm's law ηj // =- t A // - // φ + // p e /n e e + electron inertia - Ampère equation 2 A // =-µ 0 j // TTF Meeting Madison 2-5 April 2003 X.Garbet 4
Alfvén waves Ion inertia vs field line bending.j=0 n i m i d t 2 φ= // j // + curvature terms Alfvén velocity v A =c s /β e 1/2 β comes into play Two limits β=0 ω<<k // v A Boltzmann // φ T e // n e /n e e Drift waves β= ω k // v A MHD t A // - // φ Ballooning modes β drive is quantified in several ways, e.g. α=-q 2 Rdβ/dr or β p* =β (qr/l p ) 2 gives weight to the edge. TTF Meeting Madison 2-5 April 2003 X.Garbet 5
Transport of test particles Magnetic flutter δv r (δb/b) v // + random walk χ m = πqr δb B 2 v // δb/b 10-4 produces a χ m 1m 2 s -1 Rechester-Rosenbluth 78 Further refinements by Krommes 83, Isichenko 91, Vlad 03,... Test particle approximation is a stringent limitation Appropriate for fast electrons, if orbit average effects are included Mynick 78. TTF Meeting Madison 2-5 April 2003 X.Garbet 6
Fluid approach Self-consistent transport q r = δq // δb B Can be very different from RR expression due to shielding effect by the electric potential. Example (quasi-linear): δq // = d 3 v mv2 T v // δf Im(δf) = πn ψ F eq δ( ω k // v // )( φ v // A // ) =0 for an MHD mode ωa // =k // φ Generalized in several ways, e.g. Terry 86 Very persistent feature of electro-magnetic turbulence. TTF Meeting Madison 2-5 April 2003 X.Garbet 7
Comparison with E B Convective Transport Test Particles Fluid Transport channels Electrostatic χ es (K θ δφ/b) 2 τ c q r = 3/2<δpδv E > all Magnetic χ m (K θ δa // /B) 2 L c v // q r = <δq // δb/b> Electron heat χ es χ m in the MHD limit : ω k // v A and t A // - // φ TTF Meeting Madison 2-5 April 2003 X.Garbet 8
Effect of β Linear stability, k ρ i <1 δb/b increases with β Waltz 85 ITG/TEM modes are stabilised. Onset of kinetic ballooning modes above a critical β. Rewoldt 87 Rewoldt 87, Zonca 01, Falchetto 02 Shafranov shift is stabilising at large β. β (%) TTF Meeting Madison 2-5 April 2003 X.Garbet 9 γ KBM TEM/ITG 2nd stability: Shafranov shift
Dependence on β - k ρ i <1 Non linear simulations : χ e,χ i ultimately increase with β Due to the onset of kinetic ballooning modes. Snyder 01 Transport dominated by ExB turbulent convection. Waltz 02 Non linear flutter No flutter Linear flutter Camargo 96, Scott 01 Reference TTF Meeting Madison 2-5 April 2003 X.Garbet 10 15 10 5 0 0.0 χ i /(c s ρ s 2 /L n ) simulation mixing length 0.5 β(%) 1.0 Snyder 01
Transport associated to magnetic flutter is significant close to the MHD limit Jenko 01 χ e Magnetic flutter Jenko 01 χ em Not clear why shielding is less effective. ExB convection TTF Meeting Madison 2-5 April 2003 X.Garbet 11 t
Collisonality plays an important role in the edge β stabilizing at low collisionality ν* L-H transition? destabilizing at high ν* density limit? β Rogers 98 1/ν* TTF Meeting Madison 2-5 April 2003 X.Garbet 12
k ρ i >1 modes with electrostatic parity. Horton 88, Drake 88 Kinetic simulations consistent with Okhawa scaling Jenko 01, χ e = ρ e 2 v Te β e qr but not fluid simulations Labit 03 Small scale (1): ETG modes Labit 03 TTF Meeting Madison 2-5 April 2003 X.Garbet 13
Modes with tearing parity and negative ', unstable in the linear collisional regime Hazeltine 75 Weakly unstable in the linear collisionless regime. Farengo 83, Lau 90 Self-sustained in non linear regime k ρ i >1 Garbet 88, Chatenet 96 Unstable in low R/a device Dorland 98, e.g. in NSTX Small scales (2): microtearing modes Bourdelle 03 1 2 3 4 TTF Meeting Madison 2-5 April 2003 X.Garbet 14 3.0 2.5 2.0 1.5 1.0 β p */k θ δ c ν c /ω
More non-linear effects... Alfvén waves: cancellation between Reynolds and Maxwell stress tensors Diamond 91 Reduces Zonal Flow generation 1 t V θ =. v Er v Eθ +. B µ 0 n i m r B θ i Generation of Zonal Fields Diamond 00 - similar to the generation of Zonal Flows Smolyakov 02, Kaw 02,... - connected to dynamo effect Gruzinov 03, Thyagaraja 00 Turbulent reconnection : may control saturation Zeiler 00 Magnetic fluctuations are prone to inverse cascade Biskamp 89, Horton 88 TTF Meeting Madison 2-5 April 2003 X.Garbet 15
Conclusion (theory part) The transport related to self-consistent magnetic fluctuations calculated with a test particle approximation is too large :"shielding" effects. Transport is often dominated by E B turbulent convection. However linear and NL effects are important for β>m e /m i. The observable consequence is a variation of transport with β. Close to the β limit, the transport due to magnetic flutter could be as large as E B convection. TTF Meeting Madison 2-5 April 2003 X.Garbet 16
Measurements of magnetic fluctuations - Effect on Transport Magnetic fluctuations in core fusion plasmas are not well documented. Some experimental results however: Externally imposed magnetic fluctuations. Global scaling laws - Density limit. Tokamaks. RFP's. Fast Electrons. TTF Meeting Madison 2-5 April 2003 X.Garbet 17
Externally imposed magnetic fluctuations. Ergodic Divertor : the transport follows the Rechester- Rosenbluth prediction. Mc Cool 89, Ghendrih 96 Same in stochastic edge of stellarators. 500 400 300 200 100 0 T e (ev) TEXT 0.7 0.8 0.9 EML Limiter r/a 1.0 T e (kev) 2 0.1 2 0.01 0.0 TTF Meeting Madison 2-5 April 2003 X.Garbet 18 1 2 4 4 Limiter Divertor 0.4 Tore Supra 0.8 ρ (r/a ED )
Electromagnetic effects seem to be important in the plasma edge Most of global scaling laws depend on β once recast in a dimensionless form, for instance τ E,elmyH ρ * -2.7 β -1 Most of this β dependence comes from the edge in H mode (type I ELM's) Cordey 02, confirmed in DIII-D? A degradation with β is also observed in the L mode: also comes from the edge? No β effect in the core : at fixed ρ * and ν *, χ β 0 Petty 98 TTF Meeting Madison 2-5 April 2003 X.Garbet 19
Is the density limit a transport limit? Greenwald 01 β Is the density limit a β limit? Suttrop 99 α=-q 2 Rdβ/dr=α c + neutral penetration Greenwald limit. The L-H transition part of the model was tested in DIII-D Carlstrom 99, AUG Suttrop 99, JET,... 1/ν* TTF Meeting Madison 2-5 April 2003 X.Garbet 20
Measurement of magnetic fluctuations Diagnostic Measurement Devices Coils δb all CPSD δb,δn Tore Supra Bolometer + coils δb,δq //,q r MST Polarimetry δb MST X Ray Camera fast electrons AUG, JET, Tore Supra, PBX... TTF Meeting Madison 2-5 April 2003 X.Garbet 21
Magnetic fluctuations are small in the far edge, but increase rapidly in the core Usually very small in the SOL δb/b 10-5 but grows rapidly in the plasma core. 10 5 δb/b Not correlated with energy confinement time in Asdex. Giannone 89 Ritz 89 TTF Meeting Madison 2-5 April 2003 X.Garbet 22
Stochastic magnetic field lines may play an important role during relaxation events Field line stochastization by ELM's is an option M.Becoulet 03 Same for sawteeth Lichtenberg 80, Baty 91,.. M.Becoulet 03 TTF Meeting Madison 2-5 April 2003 X.Garbet 23
The level of magnetic fluctuations radially increases Measured with Cross-Polarizing Scattering diagnostics Signal radially localized at the cut-off layer. Measurement at k=1250m -1, extrapolated with a box-type spectrum. δb/b 3.10-5 10-4 0 TTF Meeting Madison 2-5 April 2003 X.Garbet 24 (δbr/b) 2 7 6 5 4 3 2 1 10-9 Colas 98 Ip=0.7MA Ip=1.0MA Ip=1.3MA 0.2 0.4 0.6 0.8 r c /a
Rechester Rosenbluth diffusivity agrees with the value calculated from power balance analysis (δb/b) 2 is large : could justify in itself χ e with RR formula Colas 97 Well correlated with confinement. However Rechester Rosenbluth is known to overestimate the diffusivity... χe (r/a=0,5) (m 2 /s) 2.5 2 1.5 1 0.5 0 ICRH LH Colas 98 0 0.5 1 1.5 χ e mag (r/a=0,55) (m 2 /s) TTF Meeting Madison 2-5 April 2003 X.Garbet 25
Heat flux is driven by magnetic fluctuations in the core of RFP's Heat flux driven by magnetic fluctuations agrees with the source integral in the core of MST Fiksel 94, Prager 99 Lower than Rechester Rosenbluth prediction Terry 96 TTF Meeting Madison 2-5 April 2003 X.Garbet 26
Current drive improves the confinement in RFP's New diagnostic: fast polarimetry Well correlated with a decrease of δb when confinement is improved. Ding 03 TTF Meeting Madison 2-5 April 2003 X.Garbet 27
Transport of Runaways Sensitive to (δb/b) 2 : easily verified with an Ergodic Divertor. Orbit average corrections are needed. δb/b 2 10-4 in Asdex may affect χ Te Kwon 88 Too small in TEXT to justify χ Te Myra 92 τ Kwon 88 thermal runaway TTF Meeting Madison 2-5 April 2003 X.Garbet 28
Energy in the range of 100keV: less sensitive to orbit corrections, but small slowing-down time scale. Fast electrons are usually well confined in Tokamaks Peysson 93 Questioned recently in TCV (but ECCD) Coda 03 Transport of Suprathermal Electrons (LHCD) D st Peysson 93 n e TTF Meeting Madison 2-5 April 2003 X.Garbet 29
Conclusions Transport often dominated by E B turbulent convection, but magnetic fluctuations play an important role via non linear effects. Some effects (stress tensor, Zonal Fields, turbulent reconnection,...) are still under theoretical investigation. Unfortunately, magnetic fluctuations in core fusion plasmas are not well documented. The issue deserves a larger effort: - the level of δb/b was found to be significant in several devices (10-5 10-4 ) but the parametric dependence is unclear. - the wavenumber spectrum is not known, nor the time dynamics. - even less information on Zonal Fields. TTF Meeting Madison 2-5 April 2003 X.Garbet 30
Conclusions (cont.) Electromagnetic effects should affect the dependence of confinement on β : results are contradictory, both in theory and experiment. Nature of β limit. Transport associated to magnetic flutter could be dominant at high β : issue to be clarified in turbulence simulations. Is there a contradiction with coil measurements in the edge? Probably important in tokamak edge : density limit, edge confinement. Magnetic flutter could also be important during relaxation events: ELM's and sawteeth. TTF Meeting Madison 2-5 April 2003 X.Garbet 31
Non linear flutter is crucial. Magnetic effects important when β>m e /m i. Camargo 96, Scott 01 Non linear flutter No flutter Reference Linear flutter ITG, edge TTF Meeting Madison 2-5 April 2003 X.Garbet 32
Contour of constant A // Turbulent Reconnection Controls the saturation of electromagnetic ITG modes. Occurs above a critical (low) value of β. Zeiler 00 TTF Meeting Madison 2-5 April 2003 X.Garbet 33
Cross-Polarization Scattering Polarising mirror effect. Amplification of the incident beam Spatial localisation of the measurements y (cm) 100 80 60 40 20 0-20 -40-60 emitting antenna Scattering volume inner wall O mode cut-off n e =4.47x10 19 m -3 Plasma Scattered X-mode Probing O-mode -80 receiving ~ O+B X antenna -100-100 -50 0 50 100 x (cm) k s k i ~ k= k s -k i Zou 91 TTF Meeting Madison 2-5 April 2003 X.Garbet 34
Consistent with Magnetic and Density Fluctuation Measurements The behavior of (δn/n) 2 and (δb/b) 2 with dt e /T e dr is consistent with the existence of a threshold. Agrees with the value determined from transport analysis. TTF Meeting Madison 2-5 April 2003 X.Garbet 35 (δbr/b) 2 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0-9 Colas 98 10 ICRH LH 0 2 4 6 8 T e at r/a=0,55 (kev/m)