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MHD Stability Analysis Consistent to Transport Properties presented by T.Ozeki Thanks to N.Hayashi, S.Tokuda, N.Aiba JAERI US/Japan JIFT Workshop on Integrated Modeling PPPL, September 21-24 2004 Introduction For estimations and analyses of burning plasmas in ITER, analysis and simulation codes for JT-60U experiments are being reconstructed and developed based on the transport code of TOPICS in JAERI. Plan of the reconstruction and development of the codes - Fusion research grid - Burning Plasma Simulation Code Cluster - Results of stability analysis based on the transport code are discussed. - NTM, Beta-collapse, ELM Fusion Research Grid Establish the remote research environment using the IT technology • Remote Experiments, Remote diagnostics, and Remote collaborative analysis • Communication, information shearing and much presence with high security JT-60U lTER Operation system Remote Experiments Diagnostic system Data analysis system Experimental Remote experiments massive data base Experimental and monitoring massive data base operations Remote Diagnostics Exp. Device Remote Analysis Super Comp. Remote control of Massive data of diagnostics and data Collaborative Analysis simulations and acquisition of experiments and experiments simulations For the collaboration to BPSI, plug-in modules of the element of the code is proposed and investigated. Plug-in IMP TRN NBI ECH MHD module Chosen TRN IMP NBI ECH module Plasma simulation platform: Interface D, Xe, Xi Metric pitch, Lp Burning Plasma Simulation Code Cluster in JAERI Tokamak Preduction and Interpretation Code Time dependent/Steary state analyses Transport code TOPICS 1D transport and 2D equilibrium Matrix Inversion Method for NeoClassical Trans. ECCD/ECH (Ray tracing, Relativistic Current Drive F-P), NBCD(1 or 2D F-P) 1D transport for each impurities, Impurity Transport Radiation: IMPACT Edge Pedestal Perp. and para. transport in SOL and Divertor, Neutral particles, Impurity transport on SOL/Div. : Divertor SOLDOR, NEUT2D, IMPMC Tearing/NTM, High-n ballooning, MHD Low-n: ERATO-J, Low and Mid.-n MARG2D High Energy Behaviour OFMC MHD Stability and Modeling MHD Behavior Stability Modeling Ideal/Resistive Sawtooth Kadomtsev Model m/n=1/1 mode Island Evolution Tearing/NTM Modiﬁed Rutherford Eq. Beta Limits/ low n kink ERATO-J Disruption high n ballooning Ballooing Eq. Medium n modes MARG2D ELM high n ballooning Ballooning Eq. High energy induced TAE/EAE/EPM... Under instability & particles Particles loss consideration NTM StabilityAnalysis 1. Purpose of NTM Analysis The Neoclassical tearing mode (NTM) emerges by the luck of the bootstrap current inside the island and it induces the transport degradation. Therefore, NTM is important issue to improve beta limit. Also, NTM stabilization is crucial in ITER. Local current driven by electron cyclotron wave (ECCD) is one of effective methods to stabilize NTM. Important issues for the NTM stabilization by ECCD (1) Sensitivity of the stabilization to the EC current location (2) Necessary ECCD power for stabilization For the design of ECCD in ITER, EC power required for the stabilization should be examined. To analyzed the NTM, the modiﬁed Rutherford equation should be analyzed with the transport property, the bootstrap current and the current drive of ECCD. The following three topics have been studied. (1) Estimation of parameters in modiﬁed Rutherford eq. by comparing with JT-60U experiments (2) Sensitivity of the stabilization to the EC current location (3) EC current necessary for stabilization ( Power evaluation on ITER ) 2. Numerical model Modiﬁed Rutherford equation ECCD code EC current Physical values Current proﬁle Transport Geometry & Additional at rational surface Bootstrap degradation Plasma proﬁles current source current, etc 1.5D tokamak simulation code ( TOPICS ) Normalized minor radius deﬁned by toroidal ﬂux : ρ = (Φ(ρ)/Φ(1))0.5 (0 < ρ < 1) 1D transport equations for density and temperatures ∂ ∂Ψ ∂ ∂ ∂Ψ 1D current diffusion equation ρ = Dc E c ∂t ∂Φ ∂ρ ∂ρ ∂Φ − Sc ( j BS, j EC ) Bootstrap current : Matrix Inversion method for Hirshman & Sigmar formula Neoclassical resistivity : Hirshman & Hawryluk model 2D MHD equilibrium : Grad-Shafranov equation ( Fixed boundary ) NTM model: the modiﬁed Rutherford equation µ0 a 2 dW Helical angle = ΓΔ ′ + ΓBS + ΓGGJ + Γ pol + ΓECCD magnetic η dt island ΓΔ ′ : Classical tearing stability index term O-point ECCD ΓBS : Disappearance of bootstrap current due to plasma proﬁles ﬂattened in the magnetic island destabilizes the mode. 0 ΓGGJ : Stabilization by magnetic well W (Glasser-Green-Johnson effect) jEC Γ pol: Stabilization or destabilization by ion polarization current X-point ΓECCD : EC current compensating bootstrap current lost in the ρ magnetic island stabilizes the mode. Degradation model in TOPICS χ : Diffusivities Di, χe, χi Flattening effect model: Ampliﬁed diffusivities is enlarged inside island χ in island (ρ) = C(ρ) • χ(ρ) ρ − ρ 2 2 C( ρ) = 1 + C0 1− s W 2 [ C0 = Min 20, 10 (W −W init ) 4 3 ] ECCD model: EC current proﬁle based on results of EC code EC code of ray tracing method and Fokker-Planck eq. JT-60U (K.Hamamatsu, et al., Plasma Phys.Control. Fusion 42(2000)1309.) 2 Injection angles Beam divergence 1 0.5 0.7 jEC (MA/m2) ρEC 1° Z (m) 0 WEC 2° 3° -1 0 0 0.45 0.5 ρ 0.55 0.6 0.45 0.5 ρ 0.55 0.6 -2 2 3 4 5 R (m) Based on the result of the EC code, Total EC current is evaluated by the EC power. EC current proﬁle can be modeled by a Gaussian distribution. ρ − ρ 2 ρEC : Peak location of EC current EC j EC ∝ exp − WEC WEC : Full width at half maximum 3. Results of Analysis NTM growth and its stabilization by ECCD in JT-60U experiments NTM island growth ( E36705 ) Stabilization by ECCD ( E41666 ) Fundamental O-mode EC wave of 110GHz m/n=3/2 mode NTM grew from t=6.4 s was injected to 3/2 mode NTM. and was saturated at t=7.2 s. Real-time control of the EC current location Normalized beta was decreased by about ( island is detected from Te perturbation ) 10%. could completely stabilize the NTM. A.Isayama, et al. 19th IAEA conf. E036705 Ip=1.5MA, Bt=3.7T Ip=1.5MA, Bt=3.7T, q95=3.9 |B|[arb.] PNB [MW] E041666 |B|[arb.] PNB [MW] 20 30 ECH(~1-1.5MW) ECCD(~3MW) 0 0 n=2 1 (m=3) n=2 (m=3) 0 1.8 2 βN βN 1 1.5 1.67 1.3 0 6 7 8 9 10 11 5 6 7 8 9 time[s] time[s] Determination of coefﬁcients of Modiﬁed Rutherford eq. 2 µ0 dW 2 ∇ρ W Lq 1 2 1 = kc Δ′ (W ) ∇ρ + kBSµ 0Lq jBS − kGGJε s2β p 1− 2 ∇ρ η dt B p W 2 + Wd 2 ρsL p q W Classical Bootstrap 2 GGJ 1.5 ρpi Lq 2 1 Lq ∇ρ I 1 0.1 −k polε s β p ∇ρ − kEC µ 0 ηEC EC BS without ECCD Lp W3 ρs Bp 2 a W2 Polarization ECCD dW/dt (s-1) W : Magnetic island width in ρ coordinate 0 Classical kc=1.2, Δ'(W) : Cylindrical model Polarization Lq = q / ( dq/dρ) , ρs : Rational surface position GGJ -0.1 Wd : Finite χ⊥/ χ// effect (R.Fitzpatrick, 1995) 0.1 with ECCD ηEC : Localized efficiency of EC current (s-1) IEC : Total EC current 0 Parameters of kBS, kGGJ, kpol, kEC are constant of dW/dt order unity. ECCD Value of Wd depends on theoretical models -0.1 0 0.05 ( to limit the parallel heat transport ). W These coefﬁcenets should be estimated by kBS can be estimated from large W. ﬁtting to experiments. kGGJ, kpol, Wd from small W Unknown Parameters is estimated by ﬁtting of JT-60 experiments EC current ( IEC=52 kA, WEC=0.12 ) kBS=4.5,kGGJ=kpol=1,Wd=0.02 kBS ~ 4-5 from the saturated island width at 7.5s 0.15 Δρ=0 kEC 0.12 Real-time control of the EC current location was 0.09 W applied in the experiment. 2.5 0.06 Misalignment between the EC current location and the rational surface due to an interval of Te 0.03 4 3 2.9 measured points (2 cm). 0 7.5 8 8.5 9 9.5 In numerical calculations, time (s) EC current location traces the rational surface with 0.15 and without a constant misalignment of 0.025 (~2 cm). |Δρ|=0.025 kEC 0.12 Δρ = ρEC − ρs 0.09 Δρ, kEC=0.025, 3.4 W ρEC : peak location of EC current proﬁle 0.06 −0.025, 4 0, 2.9 kEC ~ 3-4 from ﬁtting to experimental results 0.03 0 7.5 8 8.5 9 9.5 Other parameters (kpol, Wd) also can be estimated. time (s) Evaluation of EC current necessary for stabilization in ITER kBS=10, k GGJ=kpol=1, kEC=6, Wd=0.008 Based on the JT-60 expeiments, ITER Plasma was 0.05 simulated for the inductive scenario #2 : 3/2 mode I p = 15 MA, BT = 5.3 T, R 0 = 6.2 m, a = 2 m, IEC/ Ip(x10-2 )=0.73 β N ≈ 1.8, ne = 1.0 × 1020 m-3, Te = 9.1 keV, Ti = 8.3 keV W ne, Te, Ti proﬁles : Fixed proﬁles 1.2 0.99 EC current ( WEC=0.04 ) EC 1.0~Ifs0/ Ip Fundamental O-mode EC wave of 170 GHz 0 EC current location is assumed to be just island center. 0 2 4 6 time (s) EC wave was injected during the island growing: Early stabilization (Winj < WES) The EC current necessary for the full stabilization, Ifs , can be reduced. Early stabilization 1.2 0.1 Ifs / Ip (x10-2) w/o EC Ifs0 dW/dt (s-1) jEC/jBS~2 Ifs<Ifs0 Ifs=Ifs0 0.6 0 with EC WES WES -0.1 0 0 0.01 Winj 0.02 0 0.01 0.02 0.03 0.03 W Results : Necessary power for stabilization was obtained Dependence on the parameters of the modiﬁed Rutherford eq. 2 2 0.03 kpol=1, Wd=0.01 WES Ifs0 / Ifs00 Ifs0 / Ifs00 0.02 WES 1 1 Ifs0 original 0.01 WES~0.01 Wd with flux-limit 0 0 0 2 3 4 5 6 7 0 0.01 0.02 kBS Wd For kBS~4, kpol~1, kEC~4, Wd~0.01 estimated from JT-60U experiment, Ifs0 ~ 74 kA for 3/2 mode and ~ 54 kA for 2/1 mode on ITER ( error~20% ) . ECCD power necessary for both 3/2 and 2/1 modes NTM stabilization on ITER is 30 MW. Necessary ECCD power can be reduced to 12 MW when the EC current width decreased by optimizing both toroidal and poloidal injection angles. 4. Summary of NTM analysis This method is useful to evaluate the suppression by ECCD. It is need to develop the model including the basic model, for example, to clarify the mechanism of the mergence of the island. Beta limits Analysis 1. Purpose of CH Analysis Current hole (CH) with nearly zero toroidal current in the central region has been observed form MSE measurements. In the CH plasma, high conﬁnement performance is achieved due to the formation of strong internal transport barrier (ITB) in the reversed- shear (RS) region. E36639, 5.4s 10 8 ITB Ti The CH plasma has the autonomous [keV] 6 property because the pressure proﬁle and 4 Te the current proﬁle are strongly coupled 2 each other. And discharges are mostly 0 0 0.2 0.4 0.6 0.8 1 terminated by the disruption frequently. 1.2 20 1 q j 15 [MA/m2] 0.8 The physical mechanism of proﬁle 0.6 current 10 formation and sustainment of the 0.4 hole 5 0.2 CH plasma which is consistent 0 0 0 0.2 0.4 0.6 0.8 1 MHD stability should be clariﬁed. ρ T.Fujita, et al., Phys. Rev. Lett. 87(2001)245001. 2. Simulation model Proﬁle formation and sustainment of CH plasma by 1.5D time-dependent transport simulations for JT-60U parameters are investigated. 1.5D transport code 1D transport equations : Normalized minor radius deﬁned by toroidal ﬂux : ∂n i ∂ 2 ∂n ρ = (Φ(ρ)/Φ(1))0.5 (0 < ρ < 1) = V ′ ∇ρ Di i + S ∂t V ′∂ρ ∂ρ ∂ 3 ∂ 2 ∂T j dV nT = V ′ ∇ρ n j χ j + Pj ( j = e,i) V′ = ∂t 2 j j V ′∂ρ ∂ρ dρ 1D current diffusion equation : Parallel current density ∂ ∂Ψ ∂ ∂ ∂Ψ 2Φ1 −2 1/ 2 ∂ ∂Ψ ρ = D E − S( j BS ) j= D R ρ E ∂t ∂Φ ∂ρ ∂ρ ∂Φ µ0 ∂ρ ∂Φ Impurity : C6+, Timp=Ti, assumed proﬁle Zeff NBI source : Fixed proﬁle, Given deposition ratio of ion to electron, RNB Neutral : Monte-Carlo method, Given recycling coefﬁcient, R 2D MHD equilibrium : Grad-Shafranov equation ( Fixed boundary ) Model of current limit inside CH is applied. ( qlimit= constant ) 2D Equilibrium data low n MHD stability: ERATO/MARG2D, High n ballooning and Interchange Model of CH was proposed: Axisymmetric Tri-Magnetic- Islands (ATMI) equilibrium T.Takizuka, et al., J. Plasma Fusion Res. 78(2002)1282. ATMI has three islands along the R direction ( a central-negative- current island and two side-positive-current islands ) and two x-points along the Z direction. ATMI equilibrium is stable with the elongation coils when the current in the ATMI region is limited to be small. 2I + 1+ κ 1 2 Stable condition : I− ≈ κ 12 qATMI Zc 2 z I- > κ12 κ1 = 1 qa a(Zc − Zx ) r1 qATMI : Effective safety factor at the surface of ATMI qa : Engineering safety factor at the I+ surface of a whole plasma Zx, Zc : Position of x-points and IATMI= 2I+-I- ~ I->0 elongation coils (κ1~1) JT-60U parameters : qATMI>30 for Zc=2, Zc-Zx=0.6, a=0.8, qa=4, κ 1 = 1 Upper limit of q inside the current hole is modeled based on the ATMl model Safety factor at the surface of stable ATMI equilibrium : qATMI Safety factor inside CH is limited by qlimit = qATMI in the calculation of the MHD equilibrium and the neoclassical transport. CH radius is at q=qlimit and moves in the simulation. 1D Transport qeq (ρ), n( ρ),T( ρ) 2D MHD equilibrium Negative current inside Neoclassical transport CH ( Bootstrap current, diffusivity, resistivity ) Geometry qlimit j qlimit jeq Bootstrap current Neoclassical q diffusivity qeq 0 Neoclassical 0 0 ρqlimit ρ 1 resistivity 0 ρqlimit ρ 1 Model of transport: Neoclassical inside of s~0 and anomalous outside of s~0 Diffusivities in the transport eqs. : χ i = χ e = χ neo,i + χ ano Di = CD Dneo,i + Dano Neoclassical transport : Diffusivity and bootstrap current : Matrix inversion method for Hirshman & Sigmar formula ( M.Kikuchi, et al., Nucl. Fusion 30(1990)343. ) Neoclassical resistivity : Hirshman & Hawryluk model ( Nucl. Fusion 17(1977)611. ) Inside the CH region, model of current limit is applied. ( qlimit= constant ) Anomalous transport : Negative magnetic shear is effective to stabilize the ballooning mode and micro-instabilities. CDBM-type model 1 α >0 χ ano = Dano = χ 0 F (s − k α ) F(s-kα) χ 0 : constant anomalous diffusivity k<0 k=0 k : arbitrary constant k>0 Function F depends on s : magnetic shear 0 -2 -1 0 1 2 α : normalized pressure gradient s CDBM model : F with k=1 was originally developed for the ballooning mode turbulence. (A.Fukuyama, et al., Plasma Phys. Control. Fusion 37(1995)611.) 3. Results Simulation was done for the shot of E36639. Neutral-beam (NB) is injected during the current ramp-up. Initial current proﬁle is assumed to be off-centered like that measured in a similar shot. Bt0 = 3.7 T, R0 = 3.3 m, a = 0.8 m Initial plasma proﬁles are assumed to be parabolic. (ne0=1019m-3, Ti0=Te0=1keV) 1.5 20 E36639 1.5 20 Ip PNB (MW) Ip (MA) Ip 1 Ip PNB P NB 10 1 [MA] 0.5 [MW] 0 0 10 2 PEC 1 PNB βN PEC 0.5 βN, βp 1 0.5 βp [MW] 0 0 0.06 0 0 Bp 0.04 ρ~0.37 ρ~0.27 0 1 2 3 4 5 6 7 ρ~0.47 Bt 0.02 ρ~0.17 time (s) 0 ρ~0.08 0.3 40 -0.02 1 q t=0s contour plot of (MA/m2) 0.2 30 ρ 0.5 current q 20 density current hole 0.1 j 0 10 10 3 4 5 6 7 8 9 j time [s] Simulation in this paper 00 0 0.2 0.4 0.6 0.8 1 ρ Transport simulation reproduces features of a JT-60U experiment. To validate the model, the simulation for a shot E36639 has been done. Parameters set to simulate the experiment : k = 0, χ 0 = 2.6 m2 s, CD = 1, RNB = 2, R = 0.978 for ITB for Ti / Te for ne Normalized beta, poloidal beta, contour plot of current density and proﬁles almost agree with those in E36639. 2 10 βN Ti 2s p N 8 qaxis,qlimit β , β 1 T (keV) βp 6 exp. 0 80 qlimit 2 Zc 2 4 Te qATMI = κ1 q a(Zc − Zx ) a 2 qaxis 0 0 4 ne (x1019m-3) 1 j (MA/m2) 3 0.8 0.6 2 ρ 0.4 1 0.2 qlimit 0 0 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 time (s) ρ Physical mechanism of current hole formation (a) Off-central bootstrap current 4 1 increases due to the ITB growth 0s 2s (a) 3 (b) in the reversed-shear region. 0.8 jBS(MA/m2) (1) E (mV/m) 2 (b) (1) Large bootstrap current 0.6 decreases the parallel electric 1 0.4 1.6 ﬁeld to negative. (2) The negative 0.8 0 electric ﬁeld diffuses into the 0.2 0.4 -1 1.2 central region. 0.2 (2) ∂E η 2 ∂j 0 -2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 = ∇ E − η BS ρ ρ ∂t µ0 ∂t (2) (1) 1.4 60 (c) The current in the central region drops due to the negative electric 1.2 2s (c) (d) ﬁeld and becomes negative at 1 j(MA/m2) 40 t=1.2 s. Outer current increases 0.8 qlimit due to the bootstrap current. q 0.6 0.4 20 (d) The safety factor increases in 0.2 the central region. The minimum q 0 decreases and the radius at the 0 -0.1 0 minimum q surface moves inward. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ρ ρ Stability Analysis for results of transport simulation Disruption Low n modes stability was checked by E27302 ERATO-J: Equilibrium of each time steps in E36639 (previous discharge) are stable for low n mode. These are consistent to the observation. However, the discharge of E27302 terminated by the disruption. Disruption: βN~1.6 Steep pressure grad. t~4sec Stability analysis of low n modes 30.0 20.0 TOTPR QQT Result of the transport simulation *10 4 of the case of disruption P-proﬁle 15.0 t=5s 20.0 Disruption observed t=4.5s in the experiment t=4s 10.0 TOTPR QQT 20.0 TCUR BETN QSURF q-proﬁle 10.0 5.00 Ip=1.5MA βN~1.5 1.50 15.0 qmin~1.5 0.00 0.00 2.00 βN 0.00 0.20 0.40 0.60 RO 0.80 1.00 t=4.5s 1.00 10.0 QSURF TCUR BETN 10-3 t=4s t=5s t=5.0s n=5 UNSTABLE ^ 1.00 Stabilities γ 0.50 5.00 are checked n=4 UNSTABLE qmin~1.5 10-3 t=4.5s ^ γ 0.00 0.00 0.00 0.00 1.00 2.00 3.00 4.00 0 TIME 10-3 t=4.0s STABLE The unstable mode was obtained by the ^ γ stability analysis. But results are very 0 sensitive to proﬁle. 1 2 3 4 5 6 Further investigations are required. Toroidal mode No. n Inﬂuence of transport property on the MHD Stability: Case of k=0.3 30.0 20.0 TOTPR QQT χ ano = Dano = χ 0 F (s − k α ) *10 4 15.0 P-proﬁle 1 Pressure proﬁles 20.0 α >0 become broader, βN F(s-kα) 10.0 TOTPR k<0 k=0 QQT increases and qmin q-proﬁle k>0 10.0 increases increase. 5.00 qmin 0 -2 -1 0 1 2 ~1.7-1.8 s 0.00 0.00 t=5s 2/1 mode unstable 0.00 0.20 0.40 0.60 0.80 1.00 20.0 RO BETN QSURF βN~1.7 Internal and external modes βN modes become unstable 1.50 15.0 t=5s m=2 t=5.5s 1.00 10.0 m=3 QSURF BETN m=3 qsurf m=2 0.50 5.00 qmin~1.8 m=4 m=4 0.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 TIME Inﬂuence of transport property on the MHD Stability: Case of k=-0.5 30.0 20.0 TOTPR QQT *10 4 Pressure proﬁles become more peaked and the qmin P-proﬁle 15.0 20.0 decreased down to ~1.3 10.0 TOTPR q-proﬁle QQT Internal modes of m/n=1/1,2/1,3/1 10.0 becomes unstable 5.00 qmin due to the low qmin (~1.2). ~1.2-1.3 20.0 BETN QSURF 0.00 0.00 2/1 mode unstable 0.00 0.20 0.40 0.60 0.80 1.00 RO 1.50 15.0 βN~1.3 m=1 βN 1.00 10.0 QSURF BETN t=4.5s qsurf m=2 m=2 0.50 5.00 qmin~1.2 m=3 0.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 TIME 4. Summary of beta limits analysis There is a possibility to sustain the strongly reversed proﬁle autonomously, but the beta limit due to the ﬁnite n mode may be low. It is need to explore the suitable current proﬁle control to sustain the high beta. More careful analysis is required because the accuracy of the equilibrium and the stability analysis. Edge Stability Analysis (planing) MARG2D code was developed for low-n and high-n mode stability [S.Tokuda, Phys. Plasmas 6 (8) 1999] MARG2D solves the 2D Newcomb equation d dξ d t dξ Nξ := − L − ( M ξ ) + M + Kξ = 0 dr dr dr dr associated with eigenvalue problem Nξ = −λRξ R : diagonal matrix with Rm,m ∝ (n / m −1 / q ) 2 Properties of the code - This method can avoid problems due to the continuum spectrum. - Applicable for high n modes stabilities (more than n=50) - Very fast calculation time (~85sec for n=40, NR=2800, NV=280,m=90 by Origin 3800,128cpu) - Results of the stability of n=1 are agree to those of ERATO-J Possibility of utilization Consistent stability analyses from low-n to high-n modes Apply to the burning plasma simulation as the ELM model Plan: Iterative calculation during 1.5D time-dependent transport simulations 1.5D transport code 1D transport equations : 1D current diffusion equation : 2D MHD equilibrium : Grad-Shafranov equation ( Fixed boundary ) Model of current limit inside CH is applied. ( qlimit= constant ) ELM/Pedestal : Modelling due to stability results ?? 2D Equilibrium data Growth rate and eigenfunction Finite n mode: MARG2D, High n ballooning Summary Issues of the integration of the transport and MHD analyses: Issues of modeling: - Modeling of the inﬂuence of the MHD instability on the transport? Mixing length approximation? - Difference of the dimension. How models 2D structure of the island to 1D transport model? 3D structure is more complicated. Issues of numeical calulation: - MHD code requires the high accuracy of the equilibrium, however the transport code does not. Small modulation of the proﬁles due to the transport effects are large inﬂuence on the stability calculation. Issues of integration: - Physically meaningful modeling is difﬁcult. - Large integration is time consuming. Further investigation is required.

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mhd stability, mhd modes, pressure gradient, plasma edge, plasma phys, stability analysis, beta value, mhd equilibrium, magnetic shear, magnetic field, aspect ratio, plasma boundary, tokamak plasmas, current density, mode structure

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posted: | 6/15/2010 |

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