MHD Stability Analysis Consistent to Transport Properties

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MHD Stability Analysis Consistent to Transport Properties Powered By Docstoc
					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.