Association DayIasi, 2 July 20101 Association EURATOM – MEdC MHD MHD STABILITY RESEARCH IN TOKAMAKS C.V. Atanasiu National Institute for Laser, Plasma

Association Day Iasi, 2 July 2010 1

Association EURATOM – MEdC MHD

MHD STABILITY RESEARCH IN TOKAMAKS

C.V. AtanasiuNational 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



OUTLINE:

PROJECT: Plasma models for feedback control of helical perturbations (since 2000)

1. Tearing modes calculations in tokamaks1.1. 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 tokamak2.1. Status of the EKM - RWM problem 2.2. Progress in RWM calculation2.3 Application of RWMs calculation to ASDEX Upgrade and JET

3. Conclusions and next steps



• 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.


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].

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.

1'' ( ') Y f G Y+V Y


• 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


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


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:

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










2. Resistive Wall Modes (RWM) calculations in tokamak

2.1 Status of the problem2.1.1 Stabilization of the external kink instability

• Stabilizing the external kink instability allows higher β andhigher bootstrap current fraction, leading to more economicaltokamak power plants;

• If the instability can not be stabilized higher toroidal fieldsare necessary to compensate for lower β, however, the lowerbootstrap fraction would remain;

• At present we don’t know if we can stabilize the external kink mode !


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 2-10% of Alfvén speed)

– and there is a dissipation mechanism in the plasma (sound waves, viscosity,…)

• Or– using feedback coils located behind the shell andsensors in front of the shell (no rotation is required)


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!


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), quasi-continuous phase variation for n =1- 3.

2.2 Calculation of the wall response to the EKMFP7 - 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.]


polo

idal

dire

ctio

n toroidal direction

Fig.4 A typical toroidal wall structure




2 +i( - )0; ; epl eddy pl t m nB

U d B B B B Bt

+i( - ) i( ) +i i

+i( - ) i( ) +i i i

( , , ) e e

( , , ) e e

pl pl plt m n m n t t t

eddy eddy eddyt m n m n t t t

t

t

i( ) i i( )0 0 0e i e epl eddyU d B B

2.1. Fixed plasma: time domain formulation

2.2 Rotating plasma: frequency domain formulation

• The diffusion eq. in the frequency-domain formulation (all values are complex)



( ) ( ) 1 1 1wvv uv uu uvg g g gn B r B U U U U

dt t D u w D u D v v w D v D u

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

- therefore, any wall geometry can be described


2.5. Test problem

0

( , ) 2sin sin ;

[0 : ]; [0 : ]; 0.

U x y x y

x y U

without holes, there exists an analytical solution: U(x,y)=sin x sin y

Fig.5 Wall with 3 holes

For the wall case with holes we developed a new very fast solving method to determine the U stream function



2.6. Comparison between different metods

Solving method No. of grid points Running time[s]

Classical 101 x 101 (u,v) 103

New meth. 101 x 101 (u,v) 3

Classical 151 x 151 (u,v) 690

New meth. 151 x 151 (u,v) 14


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

analytic 64.93939402267 1753.363638612 88.57368818753

numeric

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

Table. 2 Accuracy test. 101 x 101 /3 : 101 grid intervals along u, and v; 3 : O(h3)

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.



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].



Fig. 1 Time dependencies of different plasma parameters for the considered shot # 40523: a) plasma boundary, b) safety factor q, Other considered parameters: plasma pressure p, plasma current Ipl, toroidal magnetic field Bt and internal plasma inductivity li

Fig.2 The surface charge distribution along the plasma boundary for a flux perturbation Y3/1. Shot # 40523.


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).

Documents

Association DayIasi, 2 July 20101 Association EURATOM – MEdC MHD MHD STABILITY RESEARCH IN TOKAMAKS C.V. Atanasiu National Institute for Laser, Plasma