THEORY AND THEORY-BASED MODELS FOR THE PEDESTAL, EDGE STABILITY AND ELMs IN TOKAMAKS

Transcription

1 THEORY AND THEORY-BASED MODELS FOR THE PEDESTAL, EDGE STABILITY AND ELMs IN TOKAMAKS P. N. GUZDAR 1, S. M. MAHAJAN, Z. YOSHIDA 3 W. DORLAND 1, B. N. ROGERS 4 G. BATEMAN 5, A. H. KRITZ 5, A. PANKIN 5, I. VOITSEKHOVITCH 5, T. ONJUN 5, S. SNYDER 5, J. MCELHENNY 5, C. MACDONALD 5 0 th IAEA,Fusion Enegy Confeence Vilamoua, Potugal 1-6 Novembe INSTITUTE FOR RESEARCH IN ELECTRONICS AND APPLIED PHYSICS, UNIV. MARYLAND INSTITUE FOR FUSION STUDIES, UNIV. TEXAS 3 GRADUATE SCHOOL OF FRONTIER SCIENCES, UNIV. TOKYO 4 DEPT. PHYSICS AND ASTRONOMY, DARTMOUTH COLLEGE, HANOVER, NEW HAMPSHIRE 5 LEHIGH UNIVERSITY PHYSICS DEPT., BETHLEHEM PA

2 MOTIVATION Afte L-H Tansition, the plasma develops a steep pessue gadient Enhancement facto H in enegy confinement fo dischage in vey stongly coelated with height of pessue pedestal What is the maximum sustainable pessue gadient? Pape addesses Equilibium of H-mode (Guzda, Mahajan, Yoshida) Stability of pedestal to non-cuvatue diven modes (Doland, Roges) Integated modeling of pedestal and ELMs (Bateman, Kitz, Pankin, Voitsekhovitch, Onjun, Snyde, McElhenny, MacDonald)

3 THEORY FOR H-MODE EQUILIBRIUM TWO-FLUID HALL-MHD EQUATIONS S. M. Mahajan and Z. Yoshida, PoP 7, 635, 000 n + ( nv) = 0 Continuity t A B p = φ = t n n e V B Ohm's Law (me 0) Hall tem ( + ) = ( + ) +φ + t n V pi V A V B V Ion Momentum p / n γ = 1 Equation of State Nomalizations B B/B 0, n n/n 0, x x/λ i, t tv A / λ i V V/V A, A A/λ i B 0, φ φc/v A λ i B 0

4 EQUILIBRIUM- B= B + B B B s 0 A e/i 0 s V ~ ~ β<< 1 V (VxB 0) = 0 VxB0 = Πi B B (V xb ) = V xb = 0 V = Π n n Π potential functions s s e 0 0 e A t s B = V n s B s ( φ + Π e p ) n e t ( V + A ) = V ( B + V) s s V + φ Π i p + n i

5 EQUILIBRIUM-3 ( Uj j) 0 t j = 1, Ohm's Law (electons) Ω j Ω = j =, Ion momentum Genealized voticities Ω = B, Ω = B + V 1 s s Genealized flow B s U1 = V U = V n Double Beltami conditions Ω j =µu j j Such a condition fo fluid voticity was deived by Beltami. In plasma physics foce-fee magnetic field states ae the Woltje-Chandasheka states( ) o Taylo states (1974)

6 EQUILIBRIUM-4 Double Beltami conditions B =µ (nv B ) s 1 s B + V =µ nv s Genealized Benoulli condition p+ B B = C s 0 p= p + p e Equation of state i 1 D adial (x) equations y poloidal, z tooidal dbs,y 1 nvz + Bs,z = 0 dx µ dbs,z 1 + nvy Bs,y = 0 dx µ dv dx y + B µ nv = s,z z dvz B s,y +µ nv y = 0 dx γ p/n = 1 + = 1/ γ n Bs,z 0 Complete system of steady state equations

7 EQUILIBRIUM-5 Edge pofiles fo the magnetic fields, B y, B z, flows, v y, v z, density, tempeatue Pessue, plasma cuents J y, J z and adial electic field E x γ 1 5/3 3 n=const

8 EQUILIBRIUM-6 Fo constant density case x xb + c xb + c B = 0 ( ) s 1 s s c = (1/ µ µ ), c = 1 µ / µ (cul Λ )(cul Λ )B = 0 + s Λ ± = µ m +µ 4 µ 1 µ 1 1/ πx 4 πx B = B Cos B = B 1+ Sin s,z * s,y * λi π λi B = πx λ * Vy Sin π i B B s,z 0 p + = 0 4π

9 EQUILIBRIUM-7 8 q R dp π B 0 dx =α c V = β c ~ V ped y s z =λ ped α β = π c i ped λ= i i ρ β pi pi λ qr 1/ 3 /3 /3 α ρε c pi T e β ped = 1 + q T i π R 1/ 3 1/ 3 /3 ped π T ρ e piε = 1+ R αc Ti R ped cex π βped 4 πx = cs 1 Sin B0 4 + π λi (m) = 0.03 Z A n H ped0 α A B(T) T (kev) T (kev) 0.36 Zq R(m)n 1/ c H T e,ped + i,ped = 3/ ped0

10 EQUILIBRIUM-8 COMPARISON WITH DATA FROM JT-60U Kamada et al., PPCF 41, 1371 (1999) H. Uano et al., 7 th EPS CCFPP, Budapest 1-16 June 000, ECA Vol 4B (000) Coutesy Uano and Kamada JT-60U I p =1.8 MA, B T =3.0 T, κ= , δ= T e T i T e +T i (T e +T i ) Th ped banana λi n ped n ped (T e ped +T i ped )=10.5/(n ped ) 1.5 a c /q =0. q=3.5, a c =.7 (low elongation and tiangulaity)

11 EQUILIBRIUM-9 COMPARISON WITH DATA FROM JET M Sugihaa et al., 6 th EPS CCFPP, Maasticht June 1999, ECA Vol 3J (1999) Data fom JET, Lingetat et al., J. Nucl. Mat, 66-69, 14 (1999) T i =T e (assumed) dischages have with high elongation and tiangulaity T α q e c ped Fo = 0.9 q =.5 (kev ) α c = = 3 / 19 3 n e (10 m ped )

12 EQUILIBRIUM-10 COMPARISON WITH DATA FROM DIII-D Snyde et al., Nuc. Fusion 44, 30 (004) DIII-D, B T = T, I p =1.5 MA, R=1.685, a=0.603, κ=1.77, δ= T q e α c ped q = 3, (kev) = 0.31 α c = = 3/ 19 3 ne (10 m ped )

13 STABILITY TO NON-CURVATURE DRIVEN MODES IN THE PEDESTAL Main Focus: What ae the dominant linea instabilities in the H-mode edge pedestal? Main Results: Stong E shea of H-mode pedestal is not always stabilizing: Even without magnetic cuvatue, thee ae at least thee linea instabilities that ae potentially Destabilizing by ExB shea and plasma gadients These modes all equie finite cuvatue in V E and/o plasma gadients to be unstable Potentially stonge than cuvatue diven modes since L p << R Not pesent in simulations with spatially constant plasma gadients and/o spatially constant ExB shea

14 PEDESTAL STABILITY- Methods of Analysis Gyokinetic GS simulations and analytic calculations of a simple slab model fo H-mode that includes: -- spatially vaying ExB shea and plasma gadients with typical magnitudes and scales fo pedestal V E ~ V *e ~ -V *I and pedestal widths ~(10-0)ρ i -- Magnetic shea -- Electomagnetic and kinetic effects (eg Landau damping, FLR) But excludes (fo now) -- Magnetic cuvatue -- Poloidal vaiation and othe tooidal geomety effects --Paallel flows

15 Summay: Thee Main Modes-I 1. Kelvin-Helmholtz Instability PEDESTAL STABILITY-3 -- Diven by shea in ExB velocity V E -- Magnetic shea and V *i ae stabilizing Vey nea maginal stability fo typical H-mode paametes. Moe detailed analysis that includes tooidal effects is cuently undeway k y ρ i =0., T i =T e, s=0, β e =10-4 k y ρ i =0., T i =T e, q=3.5, φ 0 =1.5x10-3, γ 0 gowth ate fo s=0 KH mode γ vs k y /k x KH mode γ vs s KH mode γ vs V *i

16 Summay: Thee Main Modes-II. Tetiay Mode PEDESTAL STABILITY-4 -- An adiabatic, electostatic mode with high k and -- Diven by dt i dx -- Insensitive to magnetic shea -- FLR effects stabilizing k > 1/ Also nea maginal stability fo typical H-mode pedestals due to FLR effects. Futhe wok on to include tooidal effects in pogess k y ρ i =0., T i =T e γ 0 = t V E / Tetiay Mode γ vs (a) k y fo η i = (squaes), η i =3.8 (tiangles) and (b) t ~(Lρ s ) 1/

17 Summay: Thee Main Modes-III 3. Dift-wave PEDESTAL STABILITY-5 Aside fom the ExB shea, simila to the mode discussed in liteatue decades ago -- Diven by plasma gadients and (fo typical H-mode paametes) electon Landau damping -- k ρ s ~ (0. 5) and k ~ β k ρ s / -- Requies spatial vaiations in ExB shea o plasma gadients (like those typical of the pedestal) to be linealy unstable in the pesence of magnetic shea -- Electomagnetic effects ae stabilizing Robustly unstable fo fo all paametes elevant to edge pedestals. Stong candidate fo diving tanspot in the H-mode edge fo tooidal and linea devices (ω,γ)v th /R γv th /R γv th /R k x ρ i =0., T i =T e β e =4x10-4, V E =.4 V *in =-399.6, V *it =-16.5 k y ρ s k y ρ s β e

18 Objective: Pedestal and ELM Models Used in Integated Modeling Develop model fo integated simulations, which includes pedestal and coe Height, width and shape of the pedestal Fequency and effect of ELMs Effect of pedestal and ELMs on the coe plasma pofiles Pogession of inceasingly sophisticated models that have been developed and/o used by Lehigh goup: Pedestal height model developed and used to povide bounday conditions in simulations of evolution of tempeatue and density pofiles in H-modes Model calibated against pedestal data Resulting density and tempeatue pofiles compaed with expeimental pofiles Pedestal and ELM model used to investigate physics of the ELM cycle Results compaed with JET data Pedestal and ELM model developed to investigate pedestal fomation and evolution of pofiles duing the ELM cycle Model calibated with DIII-D data and esults compaed with DIII-D and JET data

19 Static Pedestal Model Used to Povide Bounday Conditions Sepaate models applied fo the coe and the H-mode pedestal PEDESTAL model used is available as NTCC Module Libay Module Models descibed in papes by T. Onjun et al. Phys. Plasmas 9 (00) 5018 and G. Codey et al., 00 IAEA, Nucl. Fusion 43 (003) 670 Multi-Mode o GLF3 model used fo coe tanspot These simulations do not have time-dependent model fo ELM cycle Simulations with pedestal model bounday conditions have been caied out and compaed with data Pofiles agee with data as well as when expeimental data bounday conditions ae used G. Bateman et al., Phys. Plasmas 10 (003) 4358 Simulations of buning plasma expeiments Pedicted pefomance of buning plasma expeiment depends on the choice of coe and pedestal models G. Bateman, et al., Plasma Phys. Contol. Fusion 45 (003) J. Kinsey, et al., Nucl. Fusion 43 (003) 1845.

20 Model in which MHD Citeia ae Used to Initiate ELM cashes Implemented by V. Paail et al. in the JETTO code and used by the Lehigh goup and eseaches at JET V. Paail et al., Plasma Physics Repots 9 (003) 539 Two citeia to tigge ELMs: Pessue-diven ballooning mode and cuent-diven peeling mode ELM tiggeed by ballooning mode if nomalized pessue gadient, α, exceeds the citical pessue gadient, α c, anywhee in pedestal µ q dp α 0 > α ( s, κ, δ ) c ε BT dρ ELM tiggeed by peeling mode if the cuent density, J, satisfies 1 4D + C < 1+ 1 π q' Details in aticle by H. R. Wilson et al., Nucl. Fusion, 40 (000) 713 Citeia calibated using HELENA and MISHKA MHD stability codes Tigge citeia affect fequency of ELMs µ 0J B dl R B m 3 p

21 Results of Dynamic Pedestal and ELM Model Neoclassical tanspot used in the pedestal Detemines height of tempeatue and density pedestal Tansition fom 1 st to nd stability egion obseved in tiangulaity scan T. Onjun et al., Phys. Plasmas 11 (004) 3006 Pedestal height inceases with heating powe T. Onjun et al., Phys. Plasmas 11 (004) 1469 Tansition fom ballooning to peeling mode ELM cash tigge obseved in simulations of some dischages T. Onjun et al., to appea in Phys. Plasmas (005) Effect of isotope mass on H-mode pedestal and ELMs In collaboation with V. Paail and JET Team Pedestal Pessue (kpa) ELM Fequency (Hz) Deuteium Titium JET Dischage Simulation 1 14 Expeiment Simulation 9 3 Expeiment

22 Pedestal and ELM Model Developed at Lehigh Single tanspot model used in plasma coe and pedestal Local self-consistent calculation of plasma pofiles fom magnetic axis to the base of the pedestal Cuently implemented in the ASTRA code Model pedicts the height, width, and shape of the H-mode pedestal and fequency of the ELMs Stated with model developed by G. Pache, H. Pache, G. Janeschitz, et al., Nucl. Fusion 43 (003) 188 Initial development and application of model by A. Pankin, I. Voitsekhovitch et al., submitted to Plasma Physics and Contolled Fusion (005) Tanspot model consists of a combination of ITG/TEM ion modes (Weiland) and esistive ballooning (Guzda/Dake) ETG diven by shot wavelength electon dift modes (Hoton/Jenko) Neoclassical tanspot (NCLASS) Flow shea stabilization of local anomalous tanspot Modes with diffeent wavelengths stabilized at diffeent ates ELM cashes tiggeed by pessue-diven ballooning o cuent-diven peeling mode analytic instability citeia

23 Tanspot Model Used fo Coe and Pedestal Ion dift mode tubulence within the pedestal suppessed by ExB flow shea, but not the shot wavelength ETG mode tubulence χ i i i ITG/TEM RB χ = C ITG / TEM ( ωe Bτ ITG/ TEM ) 1+ CRB( ωe Bτ RB) χ e e e ITG/TEM RB χ = C ITG / TEM ( ωe Bτ ITG/ TEM ) 1+ CRB( ωe Bτ RB) χ χ χ χ i neoclassical e ETG χ i,e χ ITG/TEM χ RB ω E B τ ITG/TEM τ RB C ITG/TEM,RB χ i neoclassical χ e ETG ion o electon themal diffusivities ion dift mode (ITG/TEM) diffusivity without flow shea (Weiland) esistive ballooning mode diffusivity without flow shea flow shea ate tubulence coelation time fo ion dift mode tubulence coelation time fo esistive ballooning mode calibation coefficients neoclassical ion themal diffusivity (NCLASS) ETG mode electon themal diffusivity

24 Scaling of Pedestal Tempeatue and ELM Fequency Tanspot model calibated using expeimental data fo DIII-D povided by Tom Osbone at GA Noise educed by ovelaying data fom consecutive ELM cycles Once flow shea lage enough, the citical nomalized pessue gadient fo tiggeing ELM cashes (α c ) is most impotant calibation constant Simulations follow cuent pofile diffusion, including bootstap cuent Pescibed paticle density and powe density pofiles in these simulations Model in ASTRA code has been used to study scaling of pedestal tempeatue and ELM fequency Many of the tends ae consistent with expeimental data Pedestal tempeatue deceases with pedestal density: T ped ~ 1/n α ped ; α >1 Pedestal tempeatue inceases slowly with inceasing tooidal magnetic field (with fixed plasma cuent) ELM fequency deceases with magnetic field

25 Effect of Heating Powe on Pedestal and ELMs Simulations caied out fo scan of dischages with diffeent auxiliay heating powe ELM fequency and pedestal tempeatue incease with powe Consistent with data [J.G. Codey et al., Nucl. Fusion 43 (003) 670] Effect caused by changes in pedestal cuent density and esulting magnetic shea Pedestal tempeatue inceases with heating powe because of time-dependent inductive effects as cuent density ebuilds in pedestal between ELM cashes

26 Simulation of T i and T e Pofiles Lehigh model fo pedestal and ELMs used in ASTRA simulations of DIII-D and JET dischages

27 CONCLUSIONS Equilibium Double-Beltami H-mode equilibium with constaint fo pessue gadient detemined by ideal ballooning mode stability yields unique pesciption fo pessue pedestal width and height. Compaison with expeimental data shows pomise Stability Linea dift-wave eigenmodes obustly unstable in the H-mode edge egion despite ExB and magnetic shea effects, hence potentially main dive of tanspot in the (Hmode) edge of tooidal o linea devices. Linea modes not stabilized by magnetic shea (as is widely believed) because eigenmode well-localized adially by ExB shea and pofile vaiation. Result veified by GS(gyokinetic) simulations and analytic calculations and consistent with past wok on the univesal instability in the liteatue going back many decades. Integated Modeling The Lehigh Univesity Fusion Reseach Goup developed and tested theoy-based edge models in the JETTO and ASTRA codes to pedict the height of the H-mode pedestal as well as the fequency of ELM cashes. Simulations using these edge models, validated using expeimental data fom JET and DIII-D, epoduce expeimentally obseved scalings as a function of heating powe, magnetic field stength, plasma density, tiangulaity, and isotope mass