SCIENTIFIC PROGRAMME COMMITTEE
F. Zonca(chair) ENEA, Italy D. Borba IST, Portugal M. Lisak VR, Sweden S. Cappello CNR, Italy V. Naulin Risø, Denmark
F. Castejon CIEMAT, Spain M. Ottaviani CEA, France D. Van Eester TEC, Belgium O. Sauter CRPP, Switzerland th S. Günter IPP, Germany M. Tokar TEC, Germany 11 European J. Heikkinen Tekes, Finland M. Vlad INFLPR, Romania P. Helander UKAEA, UK Fusion Theory Conference
Organised by: Association EURATOM-CEA and Université de Provence IMPORTANT DATES
30 April 2005 Deadline for submission of abstracts 15 June 2005 Information for contributors 30 June 2005 Deadline for registration and hotel reservation 26 -28 September 2005 11th EFTC
LOCAL ORGANISING COMMITTEE
M. Ottaviani (chairman) P. Beyer H. Capes A. Corso-Leclercq T. Hutter (scientific secretary) G. Huysmans (webmaster) R. Stamm
Mailing Address: Mrs. A. Corso-Leclercq 11th EFTC Conference Secretariat 26 -28 September 2005 DRFC/SCCP/G2IC, bât. 513, Conference Centre CEA Cadarache, Aix-en-Provence, France 13108 St-Paul-lez-Durance, France
Website : http://www-fusion-magnetique.cea.fr/eftc11/index.html E-mail : [email protected] Announcement and Call for Abstracts Photos by J-C Carbone SUBJECT MATTER A confirmation of receipt will be sent by e-mail within a week. Submissions printed on paper should be made only if severe problems The 11th European Fusion Theory Conference will be held in Aix-en- arise with electronic submission. The Scientific Committee will select the Provence in the south of France from the 26th to the 28th of September contributions on the basis of the submitted abstracts. Authors will be 2005. The conference will be organised by the Association EURATOM- notified of the acceptance of their contributions by the 15th of June. CEA Cadarache and by the Université de Provence. The aim of the conference is to provide a discussion forum covering all areas of BOOK OF ABSTRACTS magnetic fusion-oriented theoretical activities. A special emphasis will be given to tutorial presentations. The following main topics are included: Each participant will receive a book of abstracts. No Conference proceedings will be published. Presentations will be published on the 1. Basic plasma theory conference website. 2. Macro-instabilities and operational limits 3. Alternative concepts: stellarators, rfp's, spherical tokamaks, etc. REGISTRATION AND ACCOMODATION 4. Turbulent transport and structures: experimental evidence and theoretical basis Registrations and hotel reservations will be handled by the Tourist Office 5. Neoclassical transport: theory and experimental evidence in Aix-en-Provence. The conference fee is 200 euros. The Tourist Office 6. Burning plasmas and fast particles has reserved rooms in several hotels in Aix. In view of the large number 7. Heating, current drive, and wave particle interactions of tourists in September in Aix-en-Provence, it is recommended to return 8. Edge and SOL /divertor physics the reservation form as soon as possible. To benefit from the preferential 9. Computational modelling in plasma physics. rates, the reservation form has to be returned before the 30th of June 2005. After this date, the hotel reservations and the hotel rates are no The conference will consist of invited lectures and posters. The selection longer guaranteed. Full details on registration and accommodation will be of the contributions will be made on the basis of the submitted one-page given on the conference web page. abstracts. For more information please contact the Congress Department: PROGRAMME Florence THURET Phone : 00 33 (0)4 42 16 11 83 / 10 09 The list of invited presentations and the full programme will be available in Email : [email protected] due time on the conference website. HOW TO GET THERE CALL FOR ABSTRACTS The conference centre is in the centre of Aix-en-Provence within walking Contributions to any of the topics are welcome. Authors are invited to distance of the hotels. Aix-en-Provence is close to the Marseille-Provence submit one-page abstracts, describing the most important aspects of the Airport and to a TGV train station. Public transportation is available. work to be reported. They should be suitable for direct reproduction (no faxes please). Instructions and templates are available on the conference website http://www-fusion-magnetique.cea.fr/eftc11/index.html The abstracts and the submission details should be sent by e-mail to [email protected] not later than the 30th of April 2005.
11th European Fusion Theory Conference 26-28 September 2005, Aix-en-Provence, France The 11th European Fusion Theory Conference will be held in Aix-en-Provence in the south of France from the 26th until the 28th of September 2005. The conference will be organized by the Association Euratom/CEA Cadarache
The aim of the conference is to provide a discussion forum covering all areas of magnetic fusion-oriented theoretical activities in Europe. The following main topics are included:
1. Basic plasma theory 2. Macroinstabilities and operational limits 3. Alternative concepts: stellarators, rfp's, spherical tokamaks, etc. 4. Turbulent transport and structures: experimental evidence and theoretical basis 5. Neoclassical transport: theory and experimental evidence 6. Burning plasmas and fast particles 7. Heating, current drive, and wave particle interactions 8. Edge and SOL/divertor physics 9. Computational modelling in plasma physics. The conference will consist of invited lectures and posters. The selection of the contributions has been made on the basis of the submitted one-page abstracts.
Scientific Program Committee F. Zonca(chairman) ENEA, Italy D. Borba IST, Portugal M. Lisak VR, Sweden S. Cappello CNR, Italy V. Naulin Riso, Denmark F. Castejon CIEMAT, Spain M. Ottaviani CEA, France D. Van Eester TEC, Belgium O. Sauter CRPP, Switzerland S. Gunter IPP, Germany M. Tokar TEC, Germany J. Heikkinen Tekes, Finland M. Vlad INFLPR, Romania P. Helander UKAEA, UK Local Organising Committee M. Ottaviani (chairman) P. Beyer H. Capes A. Corso-Leclercq T. Hutter (scientific secretary) G. Huysmans (webmaster) R. Stamm
11th European Fusion Theory Conference Palais des congrès espace Carnot
26/09/2005
08:00 Registration
9:00 Welcome address
Special lecture to celebrate the World Year of Physics, chair Michel CHATELIER
09:30 C. LLEWELLYN SMITH (presentation) Einstein's Legacy
10:30 Coffee Break
Oral session 1, chair Madelina VLAD
11:00 C. Z. CHENG (tutorial) Kinetic Effects on MHD Phenomena in Space and Fusion Plasmas
12:00 Clarisse BOURDELLE (topical) Turbulent Particle Transport in Magnetized Fusion Plasma
12:45 Lunch
Oral session 2, chair Volker NAULIN
14:45 Patrick DIAMOND (topical) Recent developments in the theory of stellar dynamos
Poster session 1 (15:30 - 18:00)
16:00 Coffee Break
27/09/2005
Oral session 3, chair Mikhail Z. TOKAR
09:00 Dmitri D. RYUTOV (tutorial) Effect of the X-point on instabilities of the scrape-off-layer
10:00 Samuli SAARELMA (topical) Edge stability in tokamak plasmas
10:45 Coffee Break
Oral session 4, chair Dirk VAN EESTER
11:15 C CASTALDO (topical) The problem of the spectral gap in lower hybrid current drive solved considering the non- linear wave interaction with the plasma edge 12:00 R SANCHEZ (topical) Modeling plasma transport in the presence of critical thresholds: beyond the diffusive paradigm
12:45 Lunch
Oral session 5, chair Susanna CAPPELLO
14:45 Sacha BRUN (topical) A Turbulent Magnetic Sun
Poster session 2 (15:30 - 18:00)
16:00 Coffee Break
18:00 Departure to the castle of Meyrargues
19:00 Conference dinner
Special evening lecture, chair Fulvio ZONCA
21:00 Jack W. CONNOR (abstract) Magnetic Geometry, Plasma Profiles and Stability
28/09/2005
Oral session 6, chair Francisco CASTEJON
09:00 François WAELBROECK (tutorial) Nonlinear dynamics of magnetic islands
10:00 Fulvio MILITELLO (topical) Saturation of the Neoclassical Tearing Mode Islands
10:45 Coffee Break
Oranl session 7, chair Jukka A. HEIKKINEN
11:15 Sarah L. NEWTON (topical) NEOCLASSICAL MOMENTUM TRANSPORT AND RADIAL ELECTRIC FIELD IN TOKAMAK TRANSPORT BARRIERS
12:00 Paolo ANGELINO (topical) Effects of plasma current on nonlinear interactions of ITG turbulence, zonal flows and geodesic acoustic modes
12:45 Closing remarks
13:00 Closing
Poster Session 1 (Monday 15:30 - 18:00)
1-001 : Philippe LAMALLE THEORY OF THE TOKAMAK PLASMA DIELECTRIC RESPONSE IN CONSTANT-K// COORDINATES 1-002 : Ernesto A LERCHE APPLICATION OF THE 'CONSTANT K//' COORDINATES TO FULL-WAVE ICRF HEATING SIMULATIONS OF NON-MAXWELLIAN TOKAMAK PLASMAS
1-003 : Rémi DUMONT A NEW HAMILTONIAN CODE FOR THE SIMULATION OF ICRF WAVES PROPAGATION AND ABSORPTION IN TOKAMAKS
1-004 : Omar MAJ BEAM TRACING DESCRIPTION OF THE LINEAR AND WEAKLY NONLINEAR PROPAGATION OF WAVEPACKETS
1-005 : Aleksey MERKULOV FOKKER-PLANCK MODELLING OF ELECTRON CYCLOTRON CURRENT DRIVE INCLUDING MAGNETIC DIFFUSION SELF-CONSISTENTLY
1-006 : Olivier AGULLO DYNAMO IN NUMERICAL MULTI-HELICAL MHD FLOWS
1-007 : Silvia Valeria ANNIBALDI GEOMETRY OF THE m=1 MAGNETIC ISLAND IN TOKAMAK
1-008 : Nicolas ARCIS NONLINEAR DYNAMICS OF THE TEARING MODE FOR ANY CURRENT GRADIENT
1-009 : C.V. ATANASIU AN ANALYTICAL MODEL FOR RESISTIVE WALL MODES STABILIZATION
1-010 : Xavier GARBET TURBULENT FLUXES AND ENTROPY PRODUCTION RATE
1-011 : Virginie GRANDGIRARD INTERPLAY BETWEEN DENSITY PROFILE AND ZONAL FLOWS IN DRIFT-KINETIC SIMULATIONS OF SLAB ITG TURBULENCE
1-012 : Gloria Luisa FALCHETTO EFFECT OF ZONAL FLOWS AND GEODESIC ACOUSTIC MODES ON FLUX-DRIVEN ELECTROSTATIC TURBULENT TRANSPORT IN TOKAMAK PLASMA
1-013 : Frederic SCHWANDER TURBULENCE SIMULATIONS OF HIGH TOROIDAL MACH NUMBER NEUTRAL BEAM-DRIVEN MAST DISCHARGES
1-014 : Ingmar SANDBERG GENERATION AND SATURATION OF LARGE SCALE FLOWS IN ELECTROSTATIC TURBULENCE
1-015 : Ingmar SANDBERG EXPLICIT THRESHOLD OF THE TOROIDAL ION TEMPERATURE GRADIENT MODE INSTABILITY
1-016 : Marie FARGE EXTRACTION OF COHERENT BURSTS IN TURBULENT EDGE PLASMA IN MAGNETIC FUSION DEVICES USING ORTHOGONAL WAVELETS
1-017 : Karl Heinz SPATSCHEK A-LANGEVIN APPROACH FOR THE DIFFUSION OF TEST PARTICLES IN STOCHASTIC MAGNETIC FIELDS WITHIN THE CORRSIN APPROXIMATION 1-018 : Andrea CRAVOTTA TAMING CHAOS IN THE REVERSAL REGION OF THE REVERSED-FIELD PINCH
1-019 : Irina VOITSEKHOVITCH MODELLING OF SAWTOOTH-INDUCED MIXING OF TRACE TRITIUM AND DEUTERIUM BEAM WITH THE TRANSP CODE
1-020 : Irina VOITSEKHOVITCH TRITIUM TRANSPORT IN IMPROVED H-MODE JET PLASMAS
1-021 : Guido CIRAOLO CONTROL OF TEST PARTICLE TRANSPORT IN THE SCRAPE OFF LAYER
1-022 : Xavier LEONCINI ANOMALOUS TRANSPORT OF PASSIVE PARTICLES IN HASEGAWA-MINA FLOWS
1-023 : Bhanpersad JHOWRY PLASMA SHAPE AND FINITE BETA EFFECTS ON STABILITY THRESHOLDS OF THE ION TEMPERATURE GRADIENT MODES
1-024 : Giorgio REGNOLI COLLISIONALITY EFFECTS AND ELECTROSTATIC DRFT TURBULENCE IN HIGH DENSITY FTU PLASMAS
1-025 : Kazuo TAKEDA NUSSELT NUMBER SCALING IN TOKAMAK PLASMA TURBULENCE
1-026 : Guillaume DARMET STATISTICAL ANALYSIS OF TURBULENT FRONT PROPAGATION IN A 3D VLASOV-POISSON MODEL
1-027 : Victor I. ILGISONIS SUFFICIENT STABILITY CONDITION FOR AXISYMMETRIC EQUILIBRIUM OF FLOWING MAGNETIZED PLASMA
1-028 : Vladimir P. PAVLENKO LARGE SCALE FLOWS AND NONLINEAR DYNAMICS OF FLUTE MODE TURBULENCE
1-029 : Mehdi HARZCHI NUMERICAL SOLUTIONS OF THE GRAD-SHAFRANOV EQUATION FOR DAMAVAND TOKAMAK GEOMETRIES
1-030 : Yanick SARAZIN Zonal flows governed by density gradients in gyrokinetic simulations of slab ITG turbulence
Poster Session 2 (Tuesday 15:30 - 18:00)
2-001 : Tommy BERGKVIST SELF-CONSISTENT STUDY OF EXCITATION OF ALFVEN EIGENMODES DURING ION CYCLOTRON RESONANCE HEATING
2-002 : Alvaro CAPPA OPTIMUM GAUSSIAN BEAM FOR O-X CONVERSION IN EBW HEATED PLASMAS
2-003 : Francisco CASTEJON ION ORBITS AND ION CONFINEMENT STUDIES ON ECRH PLASMAS IN TJ-II STELLARATOR 2-004 : Dirk VAN EESTER MODELLING RADIO FREQUENCY HEATING IN TOKAMAKS USING A 3-D FOKKER-PLANCK CODE ACCOUNTING FOR DRIFT ORBIT EFFECTS
2-005 : Giovanni LAPENTA 3D Kinetic Simulations of Magnetic Reconnection Onset
2-006 : Dario BORGOGNO LONG TERM EVOLUTION OF 3D COLLISIONLESS MAGNETIC RECONNECTION
2-007 : Susanna CAPPELLO MHD DYNAMO IN RESERSED FIELD PINCH PLASMAS
2-008 : Hugo J DE BLANK TEMPERATURE GRADIENT EFFECTS ON COLLISIONLESS RECONNECTION AND MAGNETIC STOCHASTIZATION
2-009 : Fulvio ZONCA RESONANT AND NON-RESONANT PARTICLE DYNAMICS IN ALFVEN MODE EXCITATIONS
2-010 : Jukka A. HEIKKINEN GYROKINETIC SIMULATION OF PARTICLE AND HEAT TRANSPORT IN THE PRESENCE OF WIDE ORBITS AND STRONG PROFILE VARIATIONS
2-011 : Madelina VLAD TEST PARTICLES, TEST MODES AND SELF-CONSISTENT TURBULENCE
2-012 : Florin SPINEANU A BACKGROUND TREND TO ORDERED STATES IN CONFINED PLASMAS
2-013 : B. WEYSSOW STOCHASTIC MODELLING OF PLASMA EDGE TURBULENCE
2-014 : Volker NAULIN SHEAR FLOW GENERATION AND ENERGETICS IN ELETROMAGNETIC TURBULENCE
2-015 : Laurent VILLARD GYROKINETIC PARTICLE SIMULATIONS OF ITG MODES AND ZONAL FLOWS WITH CANONICAL AND LOCAL EQUILIBRIUM DISTRIBUTIONS
2-016 : Sebastien JOLLIET RECENT ADVANCES FOR NONLINEAR PIC SIMULATIONS IN MAGNETIC COORDINATES
2-017 : Alexander KENDL A LATTICE DRIFT EQUATION FOR MAGNETISED PLASMA DYNAMICS
2-018 : Annika ERIKSSON ELECTROMAGNETIC EFFECTS ON QUASILINEAR TURBULENT PARTICLE TRANSPORT
2-019 : Nicolas DUBUIT IMPURITY TRANSPORT: FLUID SIMULATIONS AND COMPARISON TO QUASILINEAR THEORY
2-020 : Serguey PESCHANY SIMULATION OF CARBON PLASMA TRANSPORT AT ITER-RELEVANT CONDITIONS IN GOL-3 FACILITY 2-021 : Serguey PESCHANY Simulation of ELMs and impurity plasma transport in SOL
2-022 : Elina ASP ELECTRON AND ION STIFFNESS IN THE WEILAND MODEL
2-023 : Costanza ZUCCA CURRENT PROFILE SIMULATIONS DURING eITB FORMATION ON TCV
2-024 : Guillaume FUHR ZERO DIMENSIONAL MODEL FOR TRANSPORT BARRIER OSCILLATIONS IN TOKAMAK EDGE PLASMAS
2-025 : Mikhail Z. TOKAR SELF-SUSTAINED OSCILLATIONS IN A REACTIVE PLASMA-WALL SYSTEM
2-026 : James GUNN QUASINEUTRAL KINETIC SIMULATION OF A COLLISIONLESS SCRAPE-OFF LAYER
2-027 : Eric NARDON ELMs control on ITER with resonant magnetic perturbations
2-028 : Patrick TAMAIN MODELLING OF PARTICLE INJECTION AND EDGE PLASMA FLOWS: DETACHMENT, TRANSPORT AND TURBULENCE
2-029 : Abdelatif TAHRAOUI COLLISIONAL TRANSPORT COEFFICIENTS INLASER-HEATED PLASMAS
2-030 : Tilman DANNERT Gyrokinetic simulation of collisionless trapped-electron mode turbulence turbule
2-031 : Vladimir FUCHS An area-preserving second order integration scheme for use in particle-in-cell c
Printed in France - Credit CEA - 21/10/2005 - T. Hutter
Saturation of the Neoclassical Tearing Mode Islands
F. Militello1, M. Ottaviani2, F. Porcelli1, J. Hastie1
1 Burning Plasma Research Group Politecnico di Torino Italy
2 CEA Cadarache France
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Outline
• NTMs and the generalizations of the Rutherford Equation. • The asymmetric saturation, mathematical technique, nonlinear Theoretical model for solution. Asymmetric saturation (simplified model) • Resistivity models, self-consistent solution, saturated width relations. • The Symmetric Model. • The Code and our Results. Numerical analysis of the Symmetric NTM • Theory and Numerics, do they agree? • Summary and conclusions.
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The NTMs
• NTMs may decrease tokamak performance.
• It is important to have a reliable prediction of the size of the saturated NTMs islands.
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The Generalized Rutherford Equation THE MODEL After Rutherford (PoP ’73)
dw =1.22η[]∆'+ dt L x • The nonlinear solution of the model for the island width, w, can be obtained by using an asymptotic matching procedure.
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The Generalized Rutherford Equation
Hegna & Callen (‘92) Fitzpatrick (‘95)
dw ⎡ v*Cb w ⎤ =1.22η⎢∆'−6.34 2 2 +L⎥ dt ⎣ w + wd ⎦ • Bootstrap current term, drive for the nonlinear instability when ∆’<0
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The Generalized Rutherford Equation
Smolyakov (’89) Waelbroeck et al (’01,’05)ξ
dw ⎡ v*Cbw C pω()ω −ω* ⎤ =1.22η⎢∆'−6.34 2 2 − 3 +L⎥ dt w + w w ⎣ d ⎦ x • Polarization current term, proportional to the magnetic island poloidal rotation frequency, ω
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The Generalized Rutherford Equation
Militello & Porcelli (’04) Escande & Ottavianiξ (’04)
dw ⎡ v*Cb w C pω()ω −ω* ⎤ =1.22η⎢∆'−6.34 2 2 − 3 + 0.41bw +L⎥ dt w + w w ⎣ d x ⎦ • Term related to the shape of the equilibrium current density: d 2 J b ∝ eq (x = x ) dx2 s
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The Generalized Rutherford Equation
Militello & Porcelli (’04) Escande & Ottavianiξ (’04)
dw ⎡ v*Cb w C pω()ω −ω* ⎤ =1.22η⎢∆'−6.34 2 2 − 3 + 0.41bw + ⎥ dt w + w w L ⎣ d x ⎦ • Term related to the shape of the equilibrium current density: Valid only for symmetric 2 equilibria. Corrections are d J eq b ∝ 2 (x = xs ) required in cylindrical dx geometry !!!!
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The Limit of the Asymmetric Saturation
We can do better !!
v*Cbw C pω(ω −ω* ) 0 = ∆'−6.34 2 2 − 3 + 0.41bw +L w + wd w
• Previous investigations on classical asymmetric saturation had two major flaws: 1) limiting model for resistivity, 2) no self-consistency (Ansatz required).
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Simplified Model Equations
• Vorticity equation: ∂U + []ϕ,U = [J,ψ ] ∂t • Ohm’s law: U = ∇2ϕ ∂ψ ⊥ J = −∇2ψ + 2 + []ϕ,ψ = Ez −η(T )J ⊥ ∂t η ∝ T −3/ 2 • Energy equation
2 κ //∇//T + ∇⊥ ⋅κ ⊥∇⊥T = 0
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Mathematical technique
• Following Rutherford, we employ an Asymptotic Matching procedure, justified by the smallness of the island width w compared to the macroscopic length, L: w< Mout(χ)Min(χ) −1 2 w χ ⎛ w ∂ψ 1 ∂ ψ ⎞ ∆'+w()αχlogχ +βχ =− dχ' dξcosξ⎜Jin + + ⎟ ~ ∫−χ ∫ ⎜ 2 2 ⎟ πψ(0) ⎝ r ∂χ rs ∂ξ ⎠ χ = (r − rs ) / w Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Inner Nonlinear Solution • The matching function depends on Jin: −1 2 w χ ⎛ w ∂ψ 1 ∂ ψ ⎞ M in (χ) = − dχ' dξ cosξ⎜ Jin + + ⎟ ~ ∫−χ ∫ ⎜ 2 2 ⎟ πψ (0) ⎝ r ∂χ rs ∂ξ ⎠ • From the averaged Ohm’s law: E J = z in η Resistivity Model ! • The flux surface average is: −1 −1 L = dξ L ∂ χψ dξ ∂ χψ Metric term ! ∫ ∫ A.Thyagaraja, Phys. Fluids 24, 1716 (1981) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Resistivity Models −3/ 2 2 η ∝ T κ //∇//T + ∇⊥ ⋅κ ⊥∇⊥T = 0 • Parallel heat transport is very efficient: κ // >> κ ⊥ • then, the perpendicular transport acts on scale length of order: 1/ 4 ⎛ κ ⎞ ⎜ ⊥ ⎟ wc ~ ⎜ ⎟ ~ 10ρ ⎝ κ // ⎠ • Below this threshold the perturbation of the temperature are smoothed by perpendicular transport. • Where the perp. transport is negligible T=T(ψ) • Cf. R. Fitzpatrick, Phys. Plasmas 2, 825(1995) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Resistivity Models II 1) Small Island case: w << wc << L 2) Non-relaxed Large Island case: wc << w << L, τ <<τ η κ ⊥ 3) Relaxed Large Island case: wc << w << L, τ >>τ ←Core η κ ⊥ Edge→ Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Small Island Case w << wc << L • The shape of the flux surfaces is defined by Ampere’s law, that can ξ be solved by employing a perturbative technique: x E 1 J = z ⇒ −∇2ψ + 2 = in η (r) ⊥ −1 eq ()J eq (rs ) + J 'eq (rs )wχ +L Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Large Island Cases wc << w << L • Now, the Ampere’s law is: E E J = z = z ⇒ −∇2ψ + 2 ∝ T (ψ )3/ 2 in η(ψ ) η ⊥ •where T(ψ) is given by the constrain condition: dT κ (χ) ∂ ψ dξ = w2πκ (r )T ' (r ) dψ ∫ ⊥ χ ⊥ s eq s Metric term again ! Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Small Island Relation • SI Saturation Relation: ⎧ 2 ⎡ ⎛ 1 ⎞ 0.68 ⎤ Aa a ⎫ 0 = ∆'−0.41w⎨a ⎢log⎜ ⎟ + 4.85 − ⎥ − − b − 0.44 ⎬ ⎩ ⎣ ⎝ w ⎠ 2 − s ⎦ 2 rs ⎭ Hastie, Militello, Porcelli PRL (2005) J ' (r ) J '' (r ) eq s ⎛ 2 ⎞ eq s ⎛ 2 ⎞ q'(rs ) a = ⎜1− ⎟ b = ⎜1− ⎟ s = rs Jeq (rs ) ⎝ s ⎠ J eq (rs ) ⎝ s ⎠ q(rs ) Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Small Island Relation • SI Saturation Relation: ⎧ 2 ⎡ ⎛ 1 ⎞ 0.68 ⎤ Aa a ⎫ 0 = ∆'−0.41w⎨a ⎢log⎜ ⎟ + 4.85 − ⎥ − − b − 0.44 ⎬ ⎩ ⎣ ⎝ w ⎠ 2 − s ⎦ 2 rs ⎭ • Thyagaraja: ⎛ 1 ⎞ ∆'= 0.41wa2 log⎜ ⎟ ⎝ w ⎠ •Pletzeret al.: w << e−4.85 ~ 10-2 ⎡ 2 ⎛ 1 ⎞ Aa ⎤ ∆'= 0.41w⎢a log⎜ ⎟ − ⎥ ⎣ ⎝ w ⎠ 2 ⎦ Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Non-Relaxed Large Island Relation • NRLI Saturation Relation: ⎛ 2 a ⎞ 0 = ∆'−w⎜0.8a − 0.27b − 0.09 ⎟ ⎝ rs ⎠ Hastie, Militello, Porcelli PRL (2005) •No log(w) and A contributions! • The thermal boundary layer around the separatrix (where η ≠ η(ψ) ) brings them back (but multiplied by wc). Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Relaxed Large Island Relation • RLI Saturation Relation: ⎧ 2 ⎡ ⎛ 1 ⎞ ⎤ a ⎫ 0 = ∆'−w⎨a ⎢1.15log⎜ ⎟ +1.11⎥ − 0.27b − 0.09 ⎬ ⎩ ⎣ ⎝ w ⎠ ⎦ rs ⎭ • Similar to the NRLI relation • log(w) contribution from the outer solution, not from the inner solution as in the SI case! • The thermal boundary layer around the separatrix (where η ≠ η(ψ) ) would introduce additional log(w) and A terms. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The Complete Model • The full model contains many additional effects. • The Generalized Rutherford Equation is obtained by using strong physical assumptions. • The effect of rotation is not completely clarified. • With numerical investigations it is possible to check the assumptions and shed some light on the relevant physics. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 The 4-Field Model -The model evolves the dU = []J,ψ + µ∇2U 4 fields: dt dψ ∂ψ ∂n = []n,ψ − v* −η(J − J eq − Cb ) ϕ: Stream function dt ∂y ∂x ψ: Magnetic flux dn ∂ϕ 2 2 n: Perturbedx density + v* = ρ* []J,ψ − β[v,ψ ]+ D∇ n dt ∂y v: Parallel ion velocity dv ∂ψ = −[]n,ψ + v + χ∇2v dt * ∂y -2D, slab geometry U = ∇2ϕ -Symmetric equilibrium J = −∇2ψ -Constant η Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Symmetric NTM • Complete model (Small Island Case): v*Cbw C pω(ω −ω* ) ⎧ 2 ⎡ ⎛ 1 ⎞ 0.68⎤ Aa a ⎫ 0 = ∆'−6.34 2 2 − 3 − 0.41w⎨a ⎢log⎜ ⎟ + 4.85 − ⎥ − − b − 0.44 ⎬ +L w + wd w ⎩ ⎣ ⎝ w ⎠ 2 − s ⎦ 2 rs ⎭ • And symmetric case: v*Cbw C pω(ω −ω* ) 0 = ∆'−6.34 2 2 − 3 + 0.41bw +L w + wd w Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Simplifying the Model… • Averaging Ohm’s law: ∂n J = J + P(x, y) − P(x, y) − C eq b ∂x • J is “almost” a function of ψ dU = []J,ψ + µ∇2U [J,ψ ]~ ω J = F(ψ ) + P(x, y) dt Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Simplifying the Model… • Averaging Ohm’s law: ∂n J = J + P(x, y) − P(x, y) − C eq b ∂x • The density equation gives: dn ∂ϕ + v = ρ 2 []J,ψ − β[v,ψ ]+ D∇2n dt * ∂y * -1 From Ohm’s Law J ~ η ()[]n,ψ − v*∂ψ / ∂y Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Simplifying the Model… • Averaging Ohm’s law: ∂n J = J + P(x, y) − P(x, y) − C eq b ∂x • The density equation gives: ∂n ρ 2 Transport Equation ≈ * [][]n,ψ − v ∂ψ / ∂y,ψ + D∇2n ∂t η * Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Numerical Matching • All the terms can be substituted in Min and evaluated numerically. • The islands rotates at ω / ω * << 1 and the Polarization term is small. ∂n J = J − C + P(x, y) − P(x, y) eq b ∂x v*Cbw C pω(ω −ω* ) 0 = ∆'+0.41bw − 6.34 2 2 − 3 +L w + wd w Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Numerical Matching • All the terms can be substituted in Min and evaluated numerically. • The islands rotates at ω / ω * << 1 and the Polarization term is small. ∂n J = J − C + P(x, y) − P(x, y) eq b ∂x v*Cbw C pω(ω −ω* ) 0 = ∆'+0.41bw − 6.34 2 2 − 3 +L w + wd w Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Boundary Conditions Magnetic field – Contour plot Numerical solution: -spectral code, -double periodicity, ξ • The bootstrap current term must be corrected. x Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 v*Cb ws ∆'−0.41bws + 6.34 2 2 +L = 0 ws + wd Cb=1.4 Cb=2 Cb=1 Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 v*Cbws ∆'−0.41bws + 6.34α 2 2 +L = 0 ws + wd Cb=1.4 Cb=2 Cb=1 Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 v*Cbws ∆'−0.41bws + 6.34α 2 2 +L = 0 ws + wd Cb=1.4 Cb=2 Cb=1 Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Conclusions • A systematic investigation of the saturation of the NTMs has been carried out with theoretical and numerical tools. • New terms describing the Three new saturation asymmetric saturation have relations describing been added to the different physical Generalized Rutherford scenarios Equation. • Theoretical models have Good agreement but been compared to the the position of the numerical data obtained with tangent bifurcation is solving the complete not well predicted symmetric model. Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 EINSTEIN’SEINSTEIN’S LEGACYLEGACY Chris Llewellyn Smith Director, UKAEA Culham Division Chairman Consultative committee for Euratom on Fusion z Highlights of Einstein’s Contributions and Legacy – Fundamental Physics – Applied Physics z Unfinished Business: Unification, Meaning (origin) of Quantum Mechanics Einstein’s Contributions and Legacy - Fundamental Physics 1905: Statistical mechanics: Reality of Atomic Brownian motion molecules Physics Bose-Einstein Quantum specific Statistics heats (1906) (1924-5) Quantum Mechanics Photoelectric Spontaneous effect/light quantum emission (1917) (photon) Special relativity (. . ., E = mc2) Equivalence General Modern Principle (1907) relativity (1916) Cosmology Einstein’s Contributions and Legacy - Applied Physics 1905: Statistical mechanics: Reality of Atomic Brownian motion molecules Physics Bose-Einstein Quantum specific Statistics Microelectronics heats (1906) (1924-5) (quantum physics industry) Photoelectric Spontaneous effect/light quantum emission (1917) Lasers (photon) Special ? (parenthesis →) relativity (. . ., E = mc2) Nuclear fission Equivalence General Fusion Principle (1907) relativity (1916) Pedagogical Parenthesis: is E = Mc2 the origin of nuclear energy? z In chemical reactions e.g. Wood + Oxygen → H2O + Ash + Energy – think of energy ~ release of binding energy – could (not useful) think in terms of E = Mc2 (fractional loss of mass ~ 5×10-8) z In nuclear reactions eg 235U + N → 148La + 85Br + 3N + Energy D + T → 4He + N + Energy – sometimes presented ~ E = Mc2 (fractional loss of mass ~10-2) – more profitable to think energy ~ release of binding energy Note 1) In some cases sensible to think in terms of E = Mc2 e.g. proton + anti-proton → few pions; e+e- → Z0 2) Historically, idea that mass defects imply stellar energy derived from nuclear reactions predated knowledge of nuclear structure FUSIONFUSION –– ININ STARSSTARS z Immediately after Aston’s measurement of 4He mass defect, Eddington (1920) voiced his “suspicion . . . that stars are the crucibles in with the lighter atoms . . . are compounded into more complex elements” - resolution of long-standing problem of origin of solar energy; responsible reactions identified in 1936 – 1938 z Was Eddington thinking in terms of E = Mc2? Perhaps – Eddington wrote “the mass of the helium atom is less than the sum of the masses of the four hydrogen atoms which enter into it..there is a loss of mass amounting to 1 part in 120…” But he went on “…mass cannot be annihilated and the deficit can only represent the mass of the electrical energy liberated when helium is made out of hydrogen.” FUSIONFUSION –– ONON EARTHEARTH z Fusion’s time is coming: Environmentally responsible generation of essentially limitless energy: Lithium in one lap-top battery + Deuterium in half a bath of water → 200,000 kWhrs = UK electricity production/population for 30 years Prototype fusion power station could be putting power into grid within 30 years, given recent approval of ITER and assuming Intensified materials testing, with the International Fusion Materials Irradiation Facility (IFMIF) started as soon as possible Einstein’s Contributions and Legacy + Unfinished Business 1905: Statistical mechanics: Reality of Atomic Brownian motion molecules Physics Bose-Einstein Unfinished Quantum specific Statistics Business: heats (1906) (1924-5) Quantum Mechanics Meaning? Photoelectric Spontaneous effect/light quantum emission (1917) (photon) Unified Theory? Special relativity (. . ., E = mc2) Quantum Gravity? Equivalence General Modern Cosmology Principle (1907) relativity (1916) Einstein’sEinstein’s UnfinishedUnfinished BusinessBusiness z Einstein’s text book contributions: 1905 (aged 26) - 1925 z From 1922 to his death (1955), Einstein searched for a unified theory of gravity and electromagnetism special particle-like solutions which would more complete dynamics (→ same predictions as quantum mechanics) What was Einstein’s motivation, and how has physics evolved meanwhile? – Quantum mechanics – Unification Future outlook (Einstein once said physicists would understand him “100 years hence”: could this be true?) QuantumQuantum MechanicsMechanics Maths of QM ~ Waves Observations ~ Things Reconciliation (?): waves → predict probabilities of observations Difficulties? z Predictions statistical? Not what really bothered Einstein (despite “God does not play dice”) z Ambiguity - What is an observation (during which evolution of quantum state ~ Schroedinger equation is suspended: a and b and c . . . → a or b or c . . .)? Without a definition (“observations ~ interactions with large objects” - how large?), the predictions of quantum mechanics are ambiguous (although this does not matter for all practical purposes) z “Spooky” action at a distance - what really bothered Einstein QuantumQuantum MechanicsMechanics PostPost EinsteinEinstein z Until 1960s almost nobody thought there was a problem z Bell’s theorem (1964) showed unavoidable weirdness of QM: Consider (~ Einstein, Rosen, Podolsky) A → a + b: Detector 1 abDetector 2 According to QM: observation 1 ⇒ instantaneously modifies (in some cases determines) observation 2 Could this be like Bertlmann’s socks? No - can find examples in which no classical theory can reproduce QM results (confirmed by experiment) without action at a distance z Increasing number of physicists worried by pink Not ambiguity of QM, and/or troubled by action pink at a distance UnificationUnification z Following successful “geometrization” of gravity, Einstein sought geometrical unification of gravity and electromagnetism*. Hope - special particle-like solutions** - more complete dynamics (→ same predictions at QM) * Only forces then known: weak + nuclear forces ~ 1930s ** Only e, p then known: n, ν - 1930s; p, π, κ . . . 1940s; . . . z Note: Einstein had no good word to say for Quantum Field Theory (QFT), which describes - particles as excitations of fields - forces as due to exchange of particles and has become accepted as the working language of particle physics Now ⇒ steps towards unification in terms of QFT Ingredients of Unified Theories z Quantum Field Theory - forces ~ exchange of particles: 9 experiment particles + interactions ↔ force (not a separate concept) [particles (“matter” and “force carriers”) = fluctuations of fields] z Local (“gauge”) symmetry - conventions to be chosen locally 9 experiment ⇒ existence of force carrying particles: form of interactions + properties fixed ≈ observations, except: Gauge theory → Mass = 0 - only true for photon and graviton z Hidden Symmetry - allows Mass ≠ 0 and unification of 9 presumably electromagnetic and weak forces: requires additional “Higgs” particle(s) z Anti-screening - strong interactions → weaker at short 9 experiment distances; may converge → electroweak force at “Grand Unified” scale z Super-symmetry - connects matter and force carrying particles: helps stabilise theory z Local Super-symmetry - requires existence of gravity! QFT:QFT: AllAll forcesforces duedue toto exchangeexchange ofof particles/carriersparticles/carriers Poor analogy Force carriers z electromagnetic (holds atoms and matter together) - photon: particle of light z weak (radioactive β decay) - W, Z related z strong (holds quarks together) - gluon z gravity (holds us on earth) - graviton Note: Analogy correctly implies: effect of force ~ intrinsic strength (“muscle power”) + mass of carrier (~ range) More professional picture: “Virtual photon” This gets modified by processes such as “Virtual e+ -e- pair” Ingredients of Unified Theories z Quantum Field Theory - forces ~ exchange of particles: 9 experiment particles + interactions ↔ force (not a separate concept) [particles (“matter” and “force carriers”) = fluctuations of fields] z Local (“gauge”) symmetry - conventions to be chosen locally 9 experiment ⇒ existence of force carrying particles: form of interactions + properties fixed ≈ observations, except: Gauge theory → Mass = 0 - only true for photon and graviton z Hidden Symmetry - allows Mass ≠ 0 and unification of 9 presumably electromagnetic and weak forces: requires additional “Higgs” particle(s) z Anti-screening - strong interactions → weaker at short 9 experiment distances; may converge → electroweak force at “Grand Unified” scale z Super-symmetry - connects matter and force carrying particles: helps stabilise theory z Local Super-symmetry - requires existence of gravity! LocalLocal (“gauge”)(“gauge”) SymmetriesSymmetries z Symmetries → conventions, eg origin of coordinates on a sphere Question: can we choose conventions locally, or must they be the same everywhere, for ever? Discrete Symmetries - No Analogy: driving on left or right: local choice → physical effects at frontiers Continuous Symmetries - Yes, provided these are long range forces → signals that allow moving particles to adjust to local conventions Analogy: Choice of origin (midnight) for clocks: sun dials, or clocks linked to GPS, allow local choice of time Problem? Long range forces ↔ massless force carriers 9 for e - m + gravity, but . . . Ingredients of Unified Theories z Quantum Field Theory - forces ~ exchange of particles: 9 experiment particles + interactions ↔ force (not a separate concept) [particles (“matter” and “force carriers”) = fluctuations of fields] z Local (“gauge”) symmetry - conventions to be chosen locally 9 experiment ⇒ existence of force carrying particles: form of interactions + properties fixed ≈ observations, except: Gauge theory → Mass = 0 - only true for photon and graviton z Hidden Symmetry - allows Mass ≠ 0 and unification of 9 presumably electromagnetic and weak forces: requires additional “Higgs” particle(s) z Anti-screening - strong interactions → weaker at short 9 experiment distances; may converge → electroweak force at “Grand Unified” scale z Super-symmetry - connects matter and force carrying particles: helps stabilise theory z Local Super-symmetry - requires existence of gravity! HiddenHidden SymmetrySymmetry Symmetrical solutions may be metastable → theory chooses asymmetrical solutions Earliest example - Buridan’s (1300-c1385) Ass: Analogy for continuous symmetry: ball at centre of mexican hat In QFT: quantum vacuum of universe may not exhibit all underlying symmetries Vacuum interactions of gauge symmetry particles → inertia/mass if symmetry hidden HiddenHidden ElectroweakElectroweak SymmetrySymmetry z Theory starts with four Massless force carriers + a field which chooses asymmetric configuration filling universe ~ “fog” z “Fog” is transparent to photons - which remain massless, but interacts with W±, Z → mass z Fog can support fluctuations → carry energy QM (Higgs) particle(s) MHiggs must be less than 10Mw → Seek at Large Hadron Collider Ingredients of Unified Theories z Quantum Field Theory - forces ~ exchange of particles: 9 experiment particles + interactions ↔ force (not a separate concept) [particles (“matter” and “force carriers”) = fluctuations of fields] z Local (“gauge”) symmetry - conventions to be chosen locally 9 experiment ⇒ existence of force carrying particles: form of interactions + properties fixed ≈ observations, except: Gauge theory → Mass = 0 - only true for photon and graviton z Hidden Symmetry - allows Mass ≠ 0 and unification of 9 presumably electromagnetic and weak forces: requires additional “Higgs” particle(s) z Anti-screening - strong interactions → weaker at short 9 experiment distances; may converge → electroweak force at “Grand Unified” scale z Super-symmetry - connects matter and force carrying particles: helps stabilise theory z Local Super-symmetry - requires existence of gravity! ChargeCharge ScreeningScreening Quantum electrodynamics behaves like this Qeff (∞) ≠ 0 → Q0 = ∞?! because of Virtual electron electron Virtual Virtual photon photon Virtual electron positron ChargeCharge AntiAnti--ScreeningScreening Theories with more than one type of charge (“non-Abelian gauge theories”) exhibit anti screening (→ 2004 Nobel Prize for Physics) eg “quantum chromodynamics” gets modified gluon by processes quark such as This smears out the colour charge which → 0 as r → 0: Qeff ? This behaviour is found experimentally: it opens possibility of Grand Unified Theory (GUT) in which Qstrong → Qe-w for small r r Ingredients of Unified Theories z Quantum Field Theory - forces ~ exchange of particles: 9 experiment particles + interactions ↔ force (not a separate concept) [particles (“matter” and “force carriers”) = fluctuations of fields] z Local (“gauge”) symmetry - conventions to be chosen locally 9 experiment ⇒ existence of force carrying particles: form of interactions + properties fixed ≈ observations, except: Gauge theory → Mass = 0 - only true for photon and graviton z Hidden Symmetry - allows Mass ≠ 0 and unification of 9 presumably electromagnetic and weak forces: requires additional “Higgs” particle(s) z Anti-screening - strong interactions → weaker at short 9 experiment distances; may converge → electroweak force at “Grand Unified” scale z Super-symmetry - connects matter and force carrying particles: helps stabilise theory z Local Super-symmetry - requires existence of gravity! SupersymmetrySupersymmetry (SUSY)(SUSY) Aesthetic motivation - unifies “matter” particles (half integral QM spin; Fermi statistics) with “force carries” (integral QM spin; Bose-Einstein statistics) - last possible symmetry of QFT - local Supersymmetry → requires gravity! For me: more than enough → imperative to explore consequences and see if it is true Pragmatic Motivation -Mw would “like” to be MGUT (or MPlanck): need -13 balancing act to get observed MW ~ 10 MGUT, except in SUSY, provided MSUSY ≤ 10MW Evidence? - SUSY - GUT → Qstrong and Qew converge - Search for SUSY at LHC Unification of coupling constants - works with supersymmetry, but not without: Ingredients of Unified Theories z Quantum Field Theory - forces ~ exchange of particles: 9 experiment particles + interactions ↔ force (not a separate concept) [particles (“matter” and “force carriers”) = fluctuations of fields] z Local (“gauge”) symmetry - conventions to be chosen locally 9 experiment ⇒ existence of force carrying particles: form of interactions + properties fixed ≈ observations, except: Gauge theory → Mass = 0 - only true for photon and graviton z Hidden Symmetry - allows Mass ≠ 0 and unification of 9 presumably electromagnetic and weak forces: requires additional “Higgs” particle(s) z Anti-screening - strong interactions → weaker at short 9 experiment distances; may converge → electroweak force at “Grand Unified” scale z Super-symmetry - connects matter and force carrying particles: helps stabilise theory z Local Super-symmetry - requires existence of gravity! ScoreScore CardCard forfor SUSYSUSY GUTGUT + z correctly incorporates known forces and particles z economical in principles → forces and their form – z baroque in realization, which seems arbitrary: why particular SUSY families? Why bizarre Higgs interactions needed → observed masses? (Could be rectified without fundamentally new ideas?) z does not solve the quantum gravity problem: seems impossible to find theory that yields meaningful predictions (at least in 3 space dimensions) Nevertheless SUSY-GUT represents enormous progress, and will presumably(?) be part of more complete theories. QuantumQuantum GravityGravity [Note - the energy at which quantum gravitational effects become import ~ MGUT; strongly suggesting that gravity plays a key role in unification] Ways to a sensible theory z [Gravity super asymptotically free?? Quantum gravity OK after all in 3 space dimensions, provided theory reformulated/perturbation theory reorganised/summed??] z Modify quantum theory? z More than 3 dimensions? Strings? StringString TheoryTheory Point Particle ⇓ Strings: or Many beautiful properties, including sensible quantum gravity Predicts no of dimensions - 9 Not necessarily wrong* d≈1 d=2 String theory has much in common with real world (bosons, fermions, gauge theories), but – very non-unique (10100+. . . (?) ways to compactify) – has not explained any fact, or solved any cosmological problem but – premature to dismiss – has lead to much interesting mathematics *Kaluza (1919) put together electromagnetism and gravity by introducing one extra dimension, an idea explored by Einstein in 1920s ConcludingConcluding RemarksRemarks z Since Einstein’s death – growing sympathy for his view of quantum mechanisms (which is ambiguous, except for solipsists, and “spooky”) – huge progress with unification (based on QFT, which he rejected, and driven “bottom-up” by facts unknown to him) z Bottom-up approach at a standstill, pending new data - LHC (2007) [quantum mechanics?] z Meanwhile string theorists are tackling what seems biggest problem (quantum gravity) in the hope (in the spirit of Einstein’s work on unification, but incorporating QM) that the solution will resolve all other issues EFTC-11 : Aix-en-Provence NEOCLASSICAL TOROIDAL ANGULAR MOMENTUM TRANSPORT IN A ROTATING IMPURE PLASMA S. Newton P. Helander This work was funded jointly by the UK Engineering and Physical Sciences Research Council and by the European Communities under the contract of Association between EURATOM and UKAEA OVERVIEW • Motivation • Current observations • Neoclassical transport • Current predictions • Impure, rotating plasma • Calculating the transport •Results • Summary MOTIVATION • Internal Transport Barriers (ITBs) observed in tokamaks - steep pressure & temperature gradients - low radial transport - believed to be caused by sheared electric field • Toroidal velocity related to radial electric field ' dVs Er ps B× msns =V−∇φ =ps −+∇ ⋅π s + e+sn...s ()E + Vs ×B + R s dt Bp nsesBp ⇒ Er (r) determined by angular momentum transport CURRENT OBSERVATIONS If turbulence is suppressed in an ITB ⇒ neoclassical angular momentum transport should play key role in ITB formation / sustainment Bulk ion thermal diffusivity - observed at neoclassical level in ITB core plasma - prediction assumes bulk ions in low collisionality, ‘banana’ regime Bulk ion viscosity determines angular momentum confinement -measured toroidal viscosity order of magnitude higher than neoclassical prediction ⇒ angular momentum transport anomalous NEOCLASSICAL TRANSPORT •Gyroradius ρ << perpendicular length scale Lr - follow the motion of the guiding centre - determined by magnetic field structure and Coulomb collisions • Parallel friction ⇒ neoclassical cross-field diffusion - classical transport from perpendicular friction • Transit frequency ωt = vT / qR • Two extreme collisionality regimes - ν >> ωt : Pfirsch-Schlüter regime - ν << ωt : banana regime NEOCLASSICAL TRANSPORT Pfirsch-Schlüter Banana - parallel motion is diffusive - trapped particle orbits form 1/2 - radial drift ⇒ step width - step width ~ ε ρ p - trapped particle effects PS 2 2 D =ν q ρ dominate q2 ρ 2 D B =ν D ε 3/ 2 ωt /ν 1 ~ ε −3/ 2 Parallel friction ⇒ neoclassical cross-field diffusion CURRENT PREDICTIONS 1971, Rosenbluth et al calculate viscosity in pure plasma: - bulk ions in banana regime, slow rotation 2 2 -scales as νii q ρ - expected for Pfirsch-Schlüter regime 1985, Hinton and Wong, 1987, Catto et al: - extended to sonic plasma rotation: - still no enhancement characteristic of banana regime - transport is diffusive, driven by gradient of toroidal velocity - plasma on a flux surface rotates as a rigid body - angular velocity determined by local radial electric field INTERPRETATION Er Vφ = Bp E x B • Trapped particles collide: - change position, toroidal velocity determined by local field - no net transfer of angular momentum • Angular momentum transported by passing particles: - same toroidal velocity as trapped particles due to friction - typical excursion from flux surface ~ q ρ 2 2 ⇒ momentum diffusivity ~ νii q ρ IMPURE ROTATING PLASMA Typically Zeff > 1 - plasma contains heavy, highly charged impurity species - mixed collisionality plasma 1976, Hirshman: particle flux ~ Pfirsch-Schlüter regime heat flux ~ banana regime Rotating Plasma - centrifugal force pushes particles to outboard side of flux surface - impurity ions undergo significant poloidal redistribution - variation in collision frequency around flux surface 1999, Fülöp & Helander: particle flux typical of banana regime IMPURE ROTATING PLASMA Zeff MAST discharge #8321 -1 Vφ ~ 300 km s R[m] Rotating Plasma - centrifugal force pushes particles to outboard side of flux surface - impurity ions undergo significant poloidal redistribution - variation in collision frequency around flux surface 1999, Fülöp & Helander: particle flux typical of banana regime IMPURE ROTATING PLASMA Typically Zeff > 1 - plasma contains heavy, highly charged impurity species - mixed collisionality plasma 1976, Hirshman: particle flux ~ Pfirsch-Schlüter regime heat flux ~ banana regime Rotating Plasma - centrifugal force pushes particles to outboard side of flux surface - impurity ions undergo significant poloidal redistribution - variation in collision frequency around flux surface 1999, Fülöp & Helander: particle flux typical of banana regime CALCULATING THE TRANSPORT Hinton & Wong: - transform kinetic equation to rotating frame - consider effects occurring on different timescales - expansion in δ = ρi / Lr f = f0 + f1 + f2 +... f1 ~ δ f0 • Cross-field transport second order in δ - evaluate using flux-friction relations - relate flux to collisional moment of f1: m Angular momentum flux: Π = − i 〈 d3vm R2v2 C L ()f 〉 2e ∫ i φ i 1 • Linearised collision operator ~ • Separate classical and neoclassical contributions: f1 = f1 + f1 CALCULATING THE TRANSPORT ~ • f1 determined by Hinton & Wong: - valid for any species, independent of form of Ci • f1 obtained from the drift kinetic equation - this can be cast as a variational problem - requires an assumed trial function • Alternative method of solution - subsidary expansion of drift kinetic equation in small ratio of ion collision to bounce frequency - adopt a model for the collision operator - the drift kinetic equation may be solved analytically COLLISION OPERATOR • Neglect ion-electron collisions • Impurity concentration typically ⇒ νii ~ νiz • Explicit form of collision operator: Ci = Cii + Ciz Cii : Kovrizhnykh model operator for self collisions Ciz : disparate masses ⇒ analogous to electron - ion collisions ⎛ m ⎞ ⎜ i ⎟ Ciz =ν iz ()Ψ,θ ⎜ L(f1 )+ v ⋅Vz ()Ψ,θ f0 ⎟ ⎝ Ti ⎠ • Parallel impurity momentum equation used to determine Vz mz nz ()Vz ⋅∇ Vz = −nz ze∇Φ − ∇p + R + nz zeVz ×B TRANSPORT MATRIX Represent the fluxes in matrix form ⎛ Γ ⎞ ⎛ L L L ⎞ ⎛d()ln N T dr ⎞ ⎜ ⎟ ⎜ 11 12 13 ⎟ ⎜ i i ⎟ ⎜ q ⎟ = ⎜ L21 L22 L23 ⎟ ⎜ d(lnTi ) dr ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝Π⎠ ⎝ L31 L32 L33 ⎠ ⎝ d()lnω dr ⎠ ⎛ e ~ m ω 2 R2 ⎞ ⎜ i ⎟ • ni = Ni ()Ψ exp ⎜− Φ + ⎟ ⎝ Ti 2Ti ⎠ dn dT dω • Homogeneous non-rotating plasma: i = i = = 0 dr dr dr - spin-up as a rigid body - centrifugal potential between flux surfaces drives transport • Slow rotation: Ni → ni and Ni Ti → pi TRANSPORT MATRIX Represent the fluxes in matrix form ⎛ Γ ⎞ ⎛ L L L ⎞ ⎛d()ln N T dr ⎞ ⎜ ⎟ ⎜ 11 12 13 ⎟ ⎜ i i ⎟ ⎜ q ⎟ = ⎜ L21 L22 L23 ⎟ ⎜ d(lnTi ) dr ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝Π⎠ ⎝ L31 L32 L33 ⎠ ⎝ d()lnω dr ⎠ • Off-diagonal terms are non-zero due to the presence of impurities ~ • Each L ij is sum of classical, L ij , and neoclassical, L ij , contribution • Restricted to subsonic rotation to calculate neoclassical terms • Angular momentum transport coefficients L31, L32 and L33 • L33 usual measure of toroidal viscosity RESULTS r (θ ) • Not restricted to circular flux surfaces θ - minor radius r(θ ) and inverse aspect ratio ε (θ ) functions of poloidal angle •Zeff = 1 recover: Braginskii - classical contributions to fluxes Hinton & Wong - neoclassical part of L33 R2 R2 • Classical angular momentum flux ~ n + n i B2 z B2 Enhanced transport - larger outboard step size ρi ~ 1/B 2 - larger angular momentum miω R No dominant driving force NEOCLASSICAL COEFFICIENTS Π ⎛ p' T ' ω ' ⎞ Π = i = − ⎜ L i + L i + L ⎟ i,n 2 ⎜ 31 32 33 ⎟ miω R ⎝ pi Ti ω ⎠ ⎡ b2 ⎤ 1− f nr 2 ζ 2 −1 ⎢ 2 c ⎥ pi I τ iz nr n L = ⎢ − f nr 2 − f ⎥ 31 2 ⎢ 2 c t 2 ⎥ mi Ωi b b ⎢ ()1− fcζ ⎥ ⎣ n ⎦ ... - flux surface average n Z R2 B2 n = z ζ = eff r 2 = b2 = 2 2 nz Zeff R B ft - ‘fraction of trapped particles’ • Off-diagonal ⇒ only τiz appears • n within average ⇒ effect of redistribution NEOCLASSICAL COEFFICIENTS Most experimentally relevant limit: - conventional aspect ratio, ε (θ ) << 1 - strong impurity redistribution, ncosθ ~ 1 , n = nz nz 2 2 2 2 2 q ρ ⎛ 2 M ⎞ q ρ L ~ ⎜ M − z ⎟ν L31 ~ L32 ~ν iz 3/ 2 33 ⎜ i ⎟ iz 3/ 2 ε ⎝ z ⎠ ε • Enhancement of ε -3/2 over previous predictions - effectiveness of rotation shear as a drive increased by small factor - radial pressure and temperature gradients dominate ⇒ strong density and temperature gradients sustain strong Er shear NEOCLASSICAL COEFFICIENTS Numerical evaluation using magnetic surfaces of MAST - ε = 0.14 Zeff = 2 L31 • increase with impurity content 0.015 • increase with Mach• Transport number 0.010 Zeff =1.5 as~ impurity 10 times redistributionprevious increasespredictions 0.005 previous level - weak effect Zeff =1 for small ε : 0.00024 0.1 0.2 0.3 0.4 0.5 ncosθ ~ 0.04 ion Mach number NEOCLASSICAL COEFFICIENTS m I Π = − i 〈 d3vm R2v2 C L (f )〉 = m ω R2 R + Π ()2 2e ∫ i φ i 1 eB i zi|| • Large aspect ratio and strong impurity redistribution Π (2) ∝ ε 2 - convective angular momentum transport dominates • Angular momentum transport enhanced by enhanced particle flux I Γ = R eB zi|| •Small ω - impurities rotate poloidally to minimise friction: Rzi|| = 0 •Large ω - poloidal impurity rotation reduced due to centrifugal localisation ⇒ enhanced friction NEOCLASSICAL COEFFICIENTS •Large L31 , L32 ⇒ spontaneous toroidal rotation may arise: ω ' 1 ⎛ p' T ' ⎞ ⎜ ⎟ = − ⎜ L31 + L32 ⎟ ω L33 ⎝ p T ⎠ • Rotation direction depends on edge boundary condition • L23 relates heat flux to toroidal rotation shear: Co-NBI ⇒ shear ⎛ Γ ⎞ ⎛ L11 L12 L13 ⎞ ⎛d(ln pi ) dr⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⇒ heat pinch ⎜ q ⎟ = ⎜ L21 L22 L23 ⎟ ⎜ d(lnTi ) dr ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Sub-neoclassical heat ⎝Π⎠ ⎝ L31 L32 L33 ⎠ ⎝ d()lnω dr ⎠ transport… SUMMARY • Experimentally, angular momentum transport in regions of neoclassical ion thermal transport has remained anomalous • In a rotating plasma impurities will undergo poloidal redistribution • Including this effect, general expressions for particle, heat and angular momentum fluxes derived for mixed collisionality plasma • At conventional aspect ratio, with impurities pushed towards outboard side, angular momentum flux seen to increase by a factor of ε -3/2 ⇒ now typical of banana regime • Radial bulk ion pressure and temperature gradients are the primary driving forces, not rotation shear ⇒ strong density and temperature gradients sustain strongly sheared Er • Spontaneous toroidal rotation may arise in plasmas with no external angular momentum source REFERENCES [1] S. I. Braginskii, JETP (U.S.S.R) 6, 358 (1958) [2] P. J. Catto et al, Phys. Fluids 28, 2784 (1987) [3] T. Fülöp & P. Helander, Phys. Plasmas 6, 3066 (1999) [4] C. M. Greenfield et al, Nucl. Fusion 39, 1723 (1999) [5] P. Helander & D. J. Sigmar, Collisional Transport in Magnetized Plasmas (Cambridge U. P., Cambridge, 2002) [6] F. L. Hinton & S. K. Wong, Phys. Fluids 28, 3082 (1985) [7] S. P. Hirshman, Phys. Fluids 19, 155 (1976) [8] W. D. Lee et al, Phys. Rev. Lett. 91, 205003 (2003) [9] M. N. Rosenbluth et al, Plasma Physics & Controlled Nuclear Fusion Research, 1970, Vol. 1 (IAEA, Vienna, 1971) [10] J. Wesson, Nucl. Fusion 37, 577 (1997), P. Helander, Phys. Plasmas 5, 1209 (1998) Evening Talk at EFTC-11 MAGNETICMAGNETIC GEOMETRY,GEOMETRY, PLASMAPLASMA PROFILESPROFILES ANDAND STABILITYSTABILITY J W Connor UKAEA/EURATOM Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK A story about mode structures with the ‘Universal Mode’ providing a ‘Case History’ 1. INTRODUCTION The Universal Mode ● Anomalous transport associated with micro-instabilities such as the electron drift wave ● Early theory for a homogeneous plasma slab or cylinder (with dn/dx = constant) showed that in a shear free situation one could always find an unstable mode – driven by electron Landau resonance, with long parallel wavelengths to minimise ion Landau damping ω VTi < < VTe k|| – the so-called Universal Mode ● This talk will explore how the stability and mode structure responds to more realistic magnetic geometry and radial profiles – leads to ballooning theory and more recent developments in this topic Geometry Cylinder rB z ● n(r), q(r): q = RBθ ● Fourier analyse: φ = φ(r)e-i(mθ-nz/R) 1 ● k =−()m nq(r) || Rq ● For electron Landau drive and to minimise ion Landau damping ω VTi < < VTe ⇒ long parallel wavelength k|| r dq ● Shear s = ≠ 0 ⇒ mode localised around resonant surface q dr r0 : m = nq(r0) (nq) (r - r0 ) 1s ⇒=k|| − , = rLssLRq Axisymmetric Torus, B(r,θ) ● φ = φ (r, θ)einζ ● 2D (r,θ) – periodic in θ ; different poloidal m coupled ● High-n – simplify with eikonal ~ φ(r,θ) ~ φ(r,θ)einS(r,θ) ● Problem is to reconcile this with small k|| and periodicity B⋅∇S = 0 ⇒ S = ζ − q(r)θ but not periodic! ; shear ⇒ q(r) ≠ q(r0) Preview of Model Eigenvalue Equation ⎧⎫2 2 ⎪⎪∂∂22⎛⎞ ⎛ ∂⎞ ⎨⎬2 −σ ⎜⎟+iX −κ12X−κ X −2ε⎜cosθ+issinθ ⎟+iγe−Λ φ(X,θ) =0 ⎩⎭⎪⎪∂X ⎝⎠∂θ ⎝ ∂X⎠ FLR Ion Sound Radial variation Toroidicity Electron Eigenvalue due to ω* or ΩE Drive Λ(Ω) - X = nq′ (r – r0) • Resonant surfaces M M+1 M+2 M+3 X • Different cases, depending on magnitudes of ε, κ1, κ2 2. SHEARED SLAB/CYLINDER ● Include magnetic shear, s ≠ 0, and density profile n(r) ● Eigenvalue equation 2 ⎧⎫d 22 2 ⎨⎬2 + σ−XXκ2e+iγ(X)−Λ(Ω)φ(X)=0 ⎩⎭dX FLR shear density electron Landau eigenvalue profile drive Potential Q(X) ● Treat γe as perturbation MARSHALL ROSENBLUTH – 1927-2003 Low shear (or rapidly varying n(r)) ● Localised in potential well (Krall-Rosenbluth) – ImΩ∝γe : UNSTABLE Increased shear ● Well becomes ‘hill’: no localised mode – shear stabilisation criterion? LL s <⇒STABLE L niρ Outgoing waves ● But, Pearlstein-Berk realised these are acceptable solutions (wave propagates to ion Landau damping region where it becomes evanescent)! 2 φ = e−ioX , | X | → ∞ iL ⇒ δΩ = -iσ = - n Ls ● Radiative or shear damping φ Outgoing wave (L /L )1/2 X s n Ls/Ln Q Potential ‘hill’ ● Balancing against γe gives stability criterion L n >γe ⇒SLTAB E L s ● ‘Confirmed’ by erroneous numerical calculations! Aside ● But later numerical (Ross-Mahajan) and analytic (Tsang-Catto et al; Antonsen, Liu Chen) treatments with full Plasma Dispersion function Z for electrons showed UNIVERSAL MODEL ALWAYS STABLE! ● Traced to fact that Z has a crucial effect on mode structure in narrow region m X ≤ e mi affecting γe ⇒ perturbation treatment inadequate! Quasi-modes ● In periodic cylinder k|| ∝ m –nq(r) ● Fix n, different m have resonant surfaces rm: n = q(rm) ● For large n, m they are only separated by ∆X1 ~ <<1 ⇒ κ 0( 1 ) 1 rnq′ 2 n 2 ● Then each radially localised ‘m-mode’ ‘looks the same’ about its own resonant surface ● Each mode has almost same frequency – ie almost degenerate and satisfies k ρ || ~ i <<1 k⊥ Ln JOHN BRYAN TAYLOR ● Roberts and Taylor realised it was possible to superimpose them to form a radially extended mode – the twisted slicing or quasi-mode which maintains k|| << k⊥ BORIS KADOMTSEV – 1928-1998 3. TOROIDAL GEOMETRY ● In a torus two new effects arise from inhomogeneous magnetic fields – new drives from trapped electrons (Kadomtsev & Pogutse): DESTABILISES universal mode: trapped particle modes – changes in mode structure from magnetic drifts: affects shear damping B ● B ~ 0 ⇒ a vertical drift ζ R Z ζ mv 2 V ~ D eBR B r θ R R 0 ● Doppler shift: ω→ω- k . vD ⎛ ∂ ⎞ r εT ⎜cosθ + issin θ ⎟ ; εT = <<1 ⎝ ∂x ⎠ R normal geodesic ● φ (r) →φ(r,θ): two-dimensional 1 ⎛ i∂ ⎞ ⎛ i∂ ⎞ ⇒ k|| → ⎜− + m − nq(r)⎟ ∝ ⎜− + X⎟ Rq ⎝ ∂θ ⎠ ⎝ ∂θ ⎠ ● Eigenvalue equation (absorb γe into Λ) 2 ⎡ ∂2 ⎛ ∂ ⎞ ⎛ ∂ ⎞ ⎤ − σ2 + iX − 2ε cosθ + issin θ − κ X2 − Λ(Ω) φ = 0 ⎢ 2 ⎜ ⎟ ⎜ ⎟ 2 ⎥ ⎣⎢∂X ⎝ ∂θ ⎠ ⎝ ∂X ⎠ ⎦⎥ ε ~ 0(1) Suppression of Shear Damping (Taylor) Reflect ∆X = 1 Xm Xm+1 −imθ ● Seek periodic solution φ = Σmum (X)e ● Approximate translational invariance of rational surfaces ⇒κ2 << 1 ● Model equation: ignore geodesic drift ⎡ d2 ⎤ + σ2 (X − m)2 − Λ u − ε(u + u ) = 0 ⎢ 2 ⎥ m m+1 m−1 ⎣⎢dX ⎦⎥ ● Cylindrical limit: ε = 0 −iσ(X−m)2 / 2 ⇒ um = Ame – independent shear damped Fourier modes ● Strong toroidal coupling limit: ε > 1 ⇒ um slowly varying ‘functions’ of m 2 dum 1 d um um±1 ≅ um ± + dm 2 dm2 −σ(X−m)2 / ε ⇒ um = Ae – each um is localised about x = m: no shear damping 2 ⇒ φ ~ Ae−iXθ− εθ / σ – decays before θ ~ 2π, so periodicity not an issue ● φ is a quasi-mode – ‘balloons’ in θ – ‘m’ varies with X – radially extended ⇒ determine slowly varying radial envelope A(m) by reintroducing κ2 << 1 – seen in gyrokinetic simulations (eg W Lee) ● Arbitrary ε X −iXθ+i∫ kdX Try φ ~ φ(θ)e ⎪⎧ ∂2 ⎪⎫ σ2 + (θ − k)2 + 2ε[cosθ + s(θ − k)sin θ] + Λ(Ω) + κ X2 φ = 0 ⎨ 2 2 ⎬ ⎩⎪ ∂θ ⎭⎪ κ2 << ε - one-dimensional equation in θ But solution must be periodic in θ over 2π - must reconcile with secular terms! ● Solve problem with Ballooning Transformation (Connor, Hastie, Taylor) ∞ ~ φ = e−imθ dηe−i(X−m−k)ηφ(η,X;k) ∑ ∫−∞ m 1424 434 u m (X) Aside ● The ballooning transformation was first developed for ideal MHD ballooning modes, leading to the s-α diagram ● Interesting to contrast published and actual diagrams! s 1 0.8 0.6 0.4 0.2 a 0.25 0.5 0.75 1 1.25 1.5 1.75 2 ● Lack of trust in – numerical computation (confirmed by two-scale analysis for low s) nd – ability to find access to 2 stability (eg raise q0) • Alternatively (Lee, Van Dam) Translational invariance limit, κ2 → 0 ⇒ system invariant under X → X + 1 , m → m + 1 imk Bloch Theorem ⇒ um(x) = e u(x – m) Fourier Transform of u leads to ballooning equation • Return to drift wave: 2 ⎧⎫22∂ 2 ⎨⎬σ 2 +(η−k) +2ε(cosη+s(η−k)sin η) +Λ+κ2X φ%(η,X,k) =0 ⎩⎭∂η ● Potential Q(η) - ∞ < η < ∞⇒no longer need periodic solution! ● Often consider ‘Lowest Order’ equation, κ2 =0 – Schrődinger equation with potential Q(η) with k a parameter, Λ(Ω) a ‘local’ eigenvalue – k usually chosen to be 0 or π (to give most unstable mode) – increasing ε removes shear damping; vanishes for ε > 4: more UNSTABLE Potential Q 00 Marginal γs -0.5 Shear damping -1.0 -1.3 1.0 2.0 3.0 4.0 5.0 ε 4. RADIAL MODE STRUCTURE ● ‘Higher order’ theory: determines radial envelope A(X) (Taylor, Connor, Wilson) ● Reintroduce κ1,2 << 1 ⇒ k = k(X) – yields radial envelope in WKB approximation X i ∫ k(X)dX A(X) = e kdX = ( +π1 ) – eigenvalue condition ∫ l 2 Conventional Version: ω* has a maximum ● Lowest order theory gives ‘local’ eigenvalue 2 Ω = Ω*(0) − κ2X + iγs (k) 1424 434 123 'local' ω* (x) shear damping ● Expand Ω about k0 where shear damping is minimum 2 2 Ω = Ω(0) − κ2X + iγkk (k − k0 ) (γkk ~ ε) ⇒ k(X,Ω) (a) (b) Quadratic profile Linear profile