Transcript MHD stability research in tokamaks

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MHD STABILITY RESEARCH IN TOKAMAKS
C.V. Atanasiu
National Institute for Laser, Plasma and Radiation Physics
External contributors:
• S. Günter, K. Lackner (1994-present) : MP-IPP Garching,
Tokamakphysics Department (Theory 3)
• A.H.Boozer (1998), Applied Physics Department of Columbia U,
USA
• L.E. Zakharov(1999-present): PPPL, Theory Department, USA
• L.E. Zakharov (1983-1991), A.A. Subbotin (1983-2005) V.D.
Pustovitov (2007): I.V. Kurchatov, Theory Department, Moscow,
Russia
• S. Gerasimov (2003-2008), M.Gryaznevich (2008): JET, Culham,
UK
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OUTLINE:
PROJECT: Plasma models for feedback control of helical
perturbations (since 2000)
1. Tearing modes calculations in tokamaks
1.1. Determination of the influence of the plasma triangularity on the
tearing mode stability parameter ' for ASDEX Upgrade;
1.2. Tearing modes calculations, for specific shots of ASDEX Upgrade
2. Resistive Wall Modes (RWM) calculations in tokamak
2.1. Status of the EKM - RWM problem
2.2. Progress in RWM calculation
2.3 Application of RWMs calculation to ASDEX Upgrade and JET
3. Conclusions and next steps
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• The present contribution deals with Tearing modes and
RWMs investigations for toroidal configurations with
direct application to ASDEX Upgrade and JET
• It represents a continuation of our research activity
performed during the last ten years under the frame of a
wide range of co-operations with different fusion centers
around the world
• Our research activity was mainly oriented to the
“Provision of support to the advancement of the ITER
Physics Basis” EFDA coordinated activities, shared under
Task Force ITM and Topical Group MHD.
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1. Calculation of Tearing modes in a diverted
tokamak configuration
1.1. Potential energy calculation and boundary conditions for
EKM
•
writing the expression for the potential energy in terms of the
perturbation of the flux function, and performing an Euler
minimization, we have obtained a system of ordinary coupled
differential equations in that perturbation [1].
Y ''  f 1  (G  Y + V  Y ')
f , V and G are matrices, and Y is the flux function perturbation
vector, with the non-diagonal terms representing both
toroidicity and shape coupling effects
•
this system of equations describes a tearing mode or an
external kink mode, the latter if the resonance surface is
situated at the plasma boundary.
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• we have considered a "natural" boundary condition just
at the plasma boundary [1]. From potential theory we
know that a continuous surface distribution of simple
sources extending over a not necessarily closed Liapunov
surface ∂D and of density σ(q), generates a simple-layer
potential at p, in ∂D.
• for a unit perturbation Y2/1 (m = 2, n = 1) and Y3/2 (m = 3,
n = 2) the corresponding surface charge distributions are
Fig. 1 σ(q) for Y2/1
and Y3/2 for shot
no. 13476 at 5.2 s
ASDEX Upgrade
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1.1 Determination of the influence of the plasma triangularity on the
tearing mode stability parameter ' for ASDEX Upgrade
Δ’ dependences on triangularity δ
and ellipticity κ for ASDEX Upgrade
(shot no. 13476 at 5.2s)
3
different modes have been
considered:m/N=2/1, 3/2 and 4/3;
 the stabilising influence of the triangularity
is at least in qualitative agreement with
measurements on ASDEX Upgrade
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1.2 Tearing modes calculations, for specific
shots of ASDEX Upgrade:
observed islands of type m/n=2/1 for shots ?
1)
# 20049 at t=4.79 … 4.82 s
2)
# 20823 at t=2.234 … 2.419 s
3)
# 20833 at t=3.698 … 4.782 s
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2. Resistive Wall Modes (RWM) calculations in
tokamak
2.1 Status of the problem
2.1.1 Stabilization of the external kink instability
• Stabilizing the external kink instability allows higher β and
higher bootstrap current fraction, leading to more economical
tokamak power plants;
• If the instability can not be stabilized higher toroidal fields
are necessary to compensate for lower β, however, the lower
bootstrap fraction would remain;
• At present we don’t know if we can stabilize the external
kink mode !
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2.1.2 Theoretical Predictions
• It has been shown theoretically that the external kink mode
can be stabilized if a conducting shell is present and ...
– the plasma (or shell) is rotating sufficiently fast (roughly 210% of Alfvén speed)
– and there is a dissipation mechanism in the plasma (sound
waves, viscosity,…)
• Or
– using feedback coils located behind the shell and
sensors in front of the shell (no rotation is required)
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2.1.3 Experimental Observations
• DIII-D seem now to be the best experimental tokamak
examining kink stabilization. ASDEX Upgrade with its
new wall will be the closest to ITER for RWMs studies;
• Experiments with plasma rotation (from neutral beams)
showed that although the β was increased above the
predicted limit with no rotation, the plasma consistently
found a way to slow itself down, allowing the kink mode
to become unstable;
• Experiments with feedback coils show that the instability
can be affected, holding off the instability temporarily, but
the plasma does not appear to be stabilized indefinitely,
• Until now!
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2.2 Calculation of the wall response to the EKM
FP7 - ASDEX Upgrade plays the best “step-ladder” role for
RWMs control in ITER:
[O. Gruber, “Status and Experimental Opportunities of ASDEX Upgrade in 2009”,
Ringberg Meeting, 2008.]
2010/11
201
22222
AUG: 3 x 8 coils
ITER: 3 x 9 coils
ASDEX Upgrade wall
Fig. 3 Internal Control coils of AUG (3 x 8) and ITER (3 x 9) coils: to generate
up to n=4 error fields with different poloidal structure. Three poloidal coil sets:
flexible m spectrum. Eight toroidal coils: n=4 (small core islands), quasicontinuous phase variation for n =1- 3.
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poloidal direction
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toroidal direction
Fig.4 A typical toroidal wall structure
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2.1. Fixed plasma: time domain formulation
 2U   d
B
; B  Bpl  Beddy ; Bpl  B0 e t +i( m -n )
t
2.2 Rotating plasma: frequency domain formulation
   e t +i it
  (t , ,  )    e
eddy
eddy  t +i( m - n )  i( m  n ) t
eddy  t +i  i  i t
  e
  (t , ,  )    e
pl
pl
 t +i( m - n )  i( m  n ) t

pl

• The diffusion eq. in the frequency-domain
formulation (all values are complex)
 U 0ei(   )   d   i   B0pl ei  B0eddy ei(   ) 
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2. 3. Moving wall and fixed plasma
2.4. Coordinate system for a real toroidal wall
- we developed a new curvilinear coordinate system (u,v,w) for wall:
- 2 covariant basis vectors (ru, rv) are tangential to the wall
surface and rw is normal to the wall surface
 (n  B)
(r w  B) 1   1  gvv U guv U
d
 


t
t
D  u  w  D u
D v
  1  guu U guv U  
  v  w  D v  D u  



- therefore, any wall geometry can be described
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2.5. Test problem
U ( x, y )  2sin x sin y;
x  [0 :  ]; y  [0 :  ]; U 0  0.
For the wall case with holes we developed a new very
fast solving method to determine the U stream function
without holes, there exists an analytical solution:
U(x,y)=sin x sin y
Fig.5 Wall with 3 holes
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2.6. Comparison between different metods
Solving method
Classical
New meth.
No. of grid points
101 x 101 (u,v)
101 x 101 (u,v)
Running time[s]
103
3
Classical
New meth.
151 x 151 (u,v)
151 x 151 (u,v)
690
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2.7. Accuracy check
1) Stokes theorem: by performing some line integrals and
verifying the corresponding surface integrals (overlapping
up to the 6th significant digit);
2) simple cases permitting an analytical solution.
METHOD
No. of grid
points/order
of approx.
O(hn )
Contour Integral
around the wall
Contour integral
around a hole
Scalar potential
of the surface
current U
64.93939402267
1753.363638612
88.57368818753
51x51 / 2
64.93939402246
1753.163995570
88.58057153492
51x51 / 3
64.93939402296
1753.297066133
88.56891335580
101x101 / 2
64.93938664805
1753.312709730
88.57543866452
101x101 / 3
64.93939488308
1753.346663857
88.57247069322
151x151 / 3
64.93912067249
1753.340452717
88.57441239957
analytic
numeric
Table. 2 Accuracy test. 101 x 101 /3 : 101 grid intervals along u,
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2.3 Application of RWMs calculation to JET
• a) On the advice of our JET colleagues we considered the
shots no. 40523, 40418 and 40183 where the RWMs were
present, while at shots performed during the S/T S2-2.3.1
the presence of RWMs was not evident. Considering
discharges with low wave numbers m/n at the boundary
(46.3676s – 46.993 s for shot # 40523) only, we have
found that in the no-wall limit, discharges are not stable (Δ’
≥ 0) but in the ideal wall limit (superconducting wall) are
all stable.
• b) The influence of the wall on the boundary conditions of
the external kink mode equations is now under
consideration for a 3D wall with holes [3].
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Fig.2 The surface
charge distribution
Fig. 1 Time dependencies of different plasma parameters along the plasma
boundary for a flux
for the considered shot # 40523: a) plasma boundary,
perturbation Y3/1.
b) safety factor q, Other considered parameters: plasma
Shot # 40523.
pressure p, plasma current Ipl, toroidal magnetic field Bt
and internal plasma inductivity li
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3. Conclusions and next steps:
•
the tearing modes can be stabilized both theoretically and
experimentally;
• the EKMs have not been stabilized experimentally with plasma
rotation or feedback control, although theoretical results indicate it is
possible !
• little can be done more analytically for RWM - main part
has to be numerical !
• to put some feedback, error field and sensor coils;
• to run the code on the GATEWAY;
• to take dissipation into account in our numerical approach.
References
[1] C.V. Atanasiu, S. Günter, K. Lackner, et al., Phys. Plasmas 11, 5580 (2004).
[2] C.V. Atanasiu, A.H. Boozer, L.E. Zakharov, et al, Phys. Plasmas 6, 2781(1999).
[3] J. Adamek, C. Angioni, G. Antar, C.V. Atanasiu, M. Balden, W. Becker, Review
of Scientific Instruments, 81, 3, 033507 (2010).
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