interpretation and control of helical perturbations …ifa-mg.ro/euratom/documentatie/days 2006/oral...

Association Day Cluj, 9-11 October 2006

Association EURATOM – MEdC MHD

INTERPRETATION AND CONTROL OF HELICAL PERTURBATIONS IN TOKAMAKS

C.V. Atanasiu

National Institute for Lasers, Plasma and Radiation Physics (NILPRP) Plasma Physics and Nuclear Fusion Department

Mathematical Modelling for Fusion Plasmas GroupMagurele - Bucharest

Collaboration with other associations: IPP Garching, Germany (since 1996)JET, Culham, UK (since 2003)

Collaboration outside the associations:I.V. Kurchatov Institute, Russian Federation (since 1984)PPPL, Princeton, USA (since 1999)

Contact person: C.V. Atanasiu, [email protected]; [email protected]



OUTLINE:

1. Plasma models for feedback control of helical perturbations (Resistive Wall Modes) (with IPP)

2. Profiles reconstruction at JET in eddy current environment (with PPPL & JET)

3. Work programme 2007

4. ReferencesMAIN PURPOSE:

• to understand the resistive wall mode instability as the present limitation for fusion applications, in order to stabilise it;

• to reconstruct current and pressure profiles at JET from magnetic measurements & MSE, by taking into account the parasitic influence of the eddy currents and iron core in the tokamak



1. Plasma models for feedback control of Resistive Wall Modes

why RWMs investigation ?

• the maximum achievable β in "advanced tokamaks" is limited by the pressure gradient driven ideal external-kink modes (EKM) (10-6s) → no possible feedback !;

• if the plasma is surrounded by a close fitting resistive wall, the relatively fast growing ideal EKM is converted into the far more slowly growing "resistive wall mode" (RWM) which grows on the characteristic time of the wall τw= L/R Pfirsch 1972, Bondeson 1997;

Present status of the problem

• experimentally: one can exceed the no-wall β limit for time periods much longer than τw, provided that the plasma is rotating sufficiently rapidly Taylor 1995, Okabayashi 1996;



• a plausible stabilization mechanism is a combined effect of plasma rotational inertia & dissipation due to interaction with the sound wavecontinuum at a toroidally coupled resonant surface lying within the plasma A. Bondeson, 1994, Betti, 1995.

• in the latest RWM theories Betti, 1998, Gregoratto, 1994 plasma dissipation is associated with internal Alfvén resonances;

• Fitzpatrick & Aydemir Fitzpatrick, 1996 have developed a simplified mechanism where the plasma dissipation is provided by the edge

plasma viscosity;

• an extensively investigated RWM was on DIII-D Garofalo, 1999

• the only theory on the effect of the RWM on plasma rotation was given by Gimblet and Hastie Gimblet, 2000, but Ωc predicted by this model, based on dissipation via an internally resonant linear tearing layer, is too low withrespect to the real one;



there is, at present, no clear consensus of opinion as to what are the necessary ingredients for the stabilization of thismode;

• open problems:

- what physics determines the critical rotation rate needed to stabilize the mode,

- whether some form of plasma dissipation (e.g., absorption of sound waves, viscosity) is always required for stabilization,

- what the optimum properties of the conducting shell are for achieving stabilization at low plasma rotation rates,

- the stabilization of RWM in ITER, where it is probably not possible to maintain a very fast plasma rotation is still an open problem.



• our objectives:

- to understand the physics of macroscopic performance limiting of RWMs and their passive and active control,

- to have a simple and fast "instrument" to consider as much as possible factors and to estimate their influence on the mode growth rate.



we have considered two models of the RWM :

- a 2D semi-analytic model Atanasiu, 2005, 2006 :- cylindrical tokamak approximation - arbitrary poloidal and “toroidal” disposals, i.e. without

symmetries, of wall, feedback and detector systems. - the dissipation of plasma rotation via anomalous

viscosity has been taken into account;

- a 3D numerical model:- real axisymmetrical tokamak model with arbitrary:

- cross-section and plasma parameters, - poloidal and toroidal disposals of wall,

feedback and detector systems.



our model: generalization of Fitzpatrick’s model Nucl. Fusion, 36, 11 (1996).



Reduced MHD equations

- standard right-handed cylindrical polar co-ordinates (r , θ, z)

- plasma is periodic in the z direction with periodicity length 2пR0,

- “classical” large aspect ratio & low β ordering scheme,

- linearized force balance,

- linearized Ohm’s law

- assumed for the perturbed quantities an exp(mθ-nφ)= exp(mθ-nz/R0) dependence

- in the main current carrying plasma, the perturbation is governed by the marginally stable ideal-MHD equations (plasma inertia, resistivity and viscosity are negligible)



• the m/n mode rational surface assumed to lie outside the current carrying plasma (rs > a)

• in the edge region (inertial layer & skin current layer) the reduced MHD equations can be written as a pair of layer equations

• The dispersion relation obtained by writing:

- the jump conditions of Ψ at the wall;

- the jump of Ψ across the feedback coils;

- the inductive voltage generated in the klth feedback coil;

- the inductive voltage generated in the klth detector loop.



• the applied feedback voltage Vfkl, related to the Vd

kl voltage measured by the klth detector loop

• Finally, we have obtained the following homogeneous equation with Ψw

jk as unknowns

• ( 2 ) = spurious roots which do not characterize the RWM, • γ0= γ + iω0 represents the variation rate of the RWM and the rotation

velocity of the plasma edge: ω0=m Ω,θ-nΩφ, ,• complex coefficients Ajk0

mn Ajk4mn, Kjk

wf, Qjkwf, qjk0 qjk2=f(j, k, a, R0, qa,

q0, Bza, τA, τR, τV, rf, rw, rd, dw1, dw2, Δθf,d, Δφf,d, M, N, ηw, ηf, Gp, Gd, Ωφ,θ). m,j=m1 m2, n,k=n1 n2

( )( ) ( )22 4 3 20 2 0 1 0 0 4 0 3 0 2 0 1 0

,

1 0mn mn mn mn mn mn mn mn jk jk jkjk jk jk jk jk jk jk jk wf wf w

j k

q q q A A A A A K Qk

γ γ γ γ γ γ+ + + + + + Ψ =∑ % % % % %



0

z

R aB

τ μ ρΑ 0= ( )

2

( )R

aa

μτη

0=

2 ( )( )V

a aa

ρτμ⊥

=

w w w wrτ μ σ δ0=

8 3/ 22.7 10 ex Tη −=

edge plasma Alvén time scale

edge plasma resisitive time scale

edge plasma viscous time scale

characteristic vessel time constant

resistivity of the plasma edge (Spitzer)



Normalizations:

- the plasma edge rotation Ωφ has been normalized to the Alfvén time τA=9.6086-10 s

-the growth rate γ has been reported to the resistive wall time τwAl=3.1078x10-02 s, τwSs=1.6057x10-03 s


Results


- a wall that is close to the plasma gives rise to a RWM with a real frequency growth rate (Re[γ0]) closer to zero,- initially, the plasma rotation has a destabilizing effect on the RWM and that only when γ0i > γ0r, does the stabilization of the mode start,-somewhat counter intuitively, the optimum configuration is to place the passive shell as far away from the plasma. The descending rate of γ0 is lower for a close-fitting passive shell in the presence of a strong edge plasma rotation. The reason for this fact consists in an easier mode decoupling from the shell in the case of a far-fitting resistive shell. -it can be seen an initially destabilizing effect of the edge plasma rotation on the RWM; as long as the shell is able to lock the mode, the angular rotation “feeds” energetically the RWM which grows radially. When a sufficient angular rotation is provided, it unlocks the mode from the shell, the RWM starting to stabilize itself.


-the two types of passive shell are alternatively disposed and the feedback coils are disposed over the shell of index 2.

- it can be seen that the best stabilizing choice corresponds to case (1) and the worst to case (3) for the same values of amplification factors: - the aluminum passive shell, having a lower resistivity, stabilize better the RWM in the absence of feedback coils, whereas the stainless steel shell stabilize better in the presence of feedback coils, allowing the feedback signal to penetrate easier the shell against the plasma instability.



rc = the radius up to which the shell converts the ideal EK mode into a non-rotating RWM rc=1.381m)-placing the feedback coils and detectors between plasma and the passive shell seems to be the best choice for RWM stabilization: curves (3), (4)

|______|_____|_____| (2)pl w d f

|______|_____|_____| (3)pl d w f

|______||_____| (4)pl d,f w

we define a figure of merit



- the area of the stable region for the resistive wall mode is an increasing function of dissipation by viscosity in the plasma; -at the same time, the minimum edge rotation required for the stabilization decreases as the dissipation takes higher values; -κ is plotted between the no-wall plasma beta (corresponding to κ = 0) and ideal-wall plasma beta (corresponding to κ = 1).



Fig.5 γ(Gd)|Gp=0 for different disposals of the Al & Ss. -best choice is (2) ≡ in the absence of the most important amplification factor (Gp ) the role of the active feedback system is weaker and the Al-Al stabilize better the RWM. As in the previous case, the SS-SS case is the worst choice.

Fig.6 γ(Gp)|Gd=0: the most stabilizing choice is the Al-SS wall disposal



a destabilizing effect by approaching the wall radius to the critical Newcomb radius with no rotation has been foundrc=1.381 m



Plasma model for RWM with a neoclassical dissipation mechanism (preliminary results only !)

- used the same analytical model (with the anomalous viscositydissipation mechanism). Here the normalized mode growth rate at edge g was given by the equation:

- in addition, plasma flows are damped by the magnetic surface averaged symmetry breaking plasma viscosity induced by the distorted magnetic surfaces due to the presence of the RWMs,

- applying the perturbed neoclassical plasma viscosity to our model, the normalized mode growth rate at edge g has the expression

0 A

ag

cnsγ τ

=



0 0 ( )z

RA B aτ μ ρ=

/ a s

a a

m n q r acs q a− −

= ≅

ar dqsq dr

=

3/ 21 00.778 / 1.4iiμ εν γ ε≈ +

2 202

20

111

A

a

Bgcns Bθ

γ τ μμ γ

⎡ ⎤⎛ ⎞= +⎜ ⎟ ⎢ ⎥+⎝ ⎠ ⎣ ⎦

edge plasma Alfvén time scale

rs = rational surface

magnetic shear

νii is the ion-ion collision frequency (νii =71s-1 for ASDEX-Upgrade param.)



Fig. 8 γr(Ωφ) dependence for two dissipation mechanisms: (1) due to anomalous plasma viscosity at the edge μ=9 x 10-14 kg/m/s and (2) due to the perturbed neoclassical viscosity driven current. Plasma parameters: a=1m, R0=6m, qa=2.9, q0=1.3, Bza=2.1 T, ρa=9 x 10-9 kg/m3, rw=1.2 m, dw=.001m, ηAl=0.465 x 10-07 Ωm, N=10, m0/n0=3/1.

Ωφ normalized to the Alfvén time τA=3.0385 x 10-07 s, γr reported to the resistive wall timeτw=3.1078 x 10-02 s.

RESULT: the critical toroidal rotation necessary for the stabilization of the RWM is reduced by a factor of the order B/Bθ.

this reduction is a consequence of the parallel momentum equation when neoclassical viscosity becomes important.


2. Participation in the exploitation of the JET facilities(with PPPL & TF-D JET)

Main purpose:

Current and pressure profiles reconstruction on JET, by taking into account the parasitic influence of the induced eddy currents

Equilibrium reconstruction: to find the r.h.s of the GSh eq. + magnetic measurements & MSE


given J(r,Ψ) ↔ measurements of Ψ(l) are sufficient

measurements of Bi are used to limit the freedom in J(r,Ψ)



in an axisymmetric configuration (n=0), the n≠0 components are generated by the eddy currents and the iron core



- an arbitrary function- averaged & oscillatory parts

-flux loops measure the averaged component of Aφ- only B0 is appropriate for equilibrium reconstruction

the time history is necessary for eliminating the eddy current effects



Accomplished objectives:• determination of the transfer function s(t);

• elimination of contribution of eddy currents generated by the PF coils into signals;

• elimination of contribution of eddy currents generated by the plasma;

• the effect of eddies on equilibrium, by including the time history should be included into equilibrium reconstruction



there where necessary “dry runs” ≡ Ipl=0



3. Work programme 2007

3.1 Plasma models for feedback control of helical perturbations (RWM) (continuation with IPP)

up to now, we succeeded to investigate the anomalous viscosityand the neoclassical dissipation mechanisms (neo – preliminary res. !)

• other dissipation mechanisms

why necessary ?

for RWMs in ITER it is probably not possible to maintain a very fast plasma rotation → which forms of plasma dissipation are always required for stabilization is still an open problem


• milestones:

• investigation of possible dissipation mechanisms in a rotating tokamak plasma due to:

- anomalous viscosity dumping, - neoclassical flow-damping, - sound-wave damping, - charge-exchange with neutrals,

• 3D RWM model

why necessary?

- to find the necessary conditions to be fulfilled by a 3D wall with arbitrary holes;

- investigate the modes coupling effects due to real toroidal effects and plasma shape



milestones:

- to consider a real 2D equilibrium model (with axisymmetrical geometry) for a 3D perturbation of the RWM,

- to describe the vacuum field by given the normal component of the perturbed field on the plasma boundary for a 2D axisymmetrical tokamak geometry, and by using our concept of a surface current Boozer, 1998, Atanasiu, 1999,

- to describe of the fields across thin layers and to derive the feedback circuit equations



3.2 Numerical Data Analysis for current and pressure profiles reconstruction on JET and calibrating magnetic measurements in eddy current and iron core environments (continuation with TF–D & PPPL)

• the profiles reconstructions on JET with the help of the ESC code will continue, by using corrected magnetic signals input data

- Determination of the response functions by eliminating the n ≠ 0 eddy currents and iron core contributions, with n the toroidal wave number

a correlation matrix between sensors located outside and inside the vacuum vesselhas to be introduced in order to determine the parasitic n 0 perturbation inmagnetic fields generated by the iron core.

- Determination of the correlation time dependent matrix of response functions

an other time dependent matrix of response functions has to be introduced in orderto eliminate the n 0 perturbation generated by the eddy currents – necessarycalibration shots (without plasma) only - we have already performed a number ofspecific “dry runs” at JET.



4. References:[1] D. Pfirsch, H. Tasso, Nucl. Fusion, 11, 259 (1972).[2] A. Bondeson, M. Benda, M. Persson, M.S. Chu, Nucl. Fusion, 37,1419 (1997).[3] T.S. Taylor, E.J. Strait, L.L. Lao, et al, Phys. Plasmas, 2, 2390 (1995).[4] M. Okabayashi, N. Pomphrey, J. Manickam et al, Nucl. Fusion, 36, 1167 (1996).[5] A. Bondeson, D. Ward, Phys. Rev. Lett., 72, 718 (1994).[6] R. Betti, J.P. Freidberg, Phys. Rev. Lett., 74, 718 (1995).[7] R. Betti, Phys. Plasmas, 5, 3615 (1998).[8] D. Gregoratto, A. Bondeson, M.S. Chu et all., Plasma Phys. Control. Fusion,

43, 1425 (1999).[9] R. Fitzpatrick, A.Y. Aydemir, Nucl. Fusion, 36, 11 (1996). [10] Bondeson, M.S. Chu, Phys. Plasmas, 3, 3013 (1996).[11] E. Strumberger, S.Günter, P. Merkel et al, Nucl. Fusion, 45, 1156 (2005).

[12] Y.Q. Liu, A. Bondeson, M.S. Chu, et al, Nucl. Fusion, 45, 1131 (2005).[13] C.V. Atanasiu and I.G. Miron, "An analytical model for resistive wall modes

stabilization“,11th European Fusion Theory Conference, 26-28 September 2005, Aix-en-Provence, France.


[14] C.V. Atanasiu and I.G. Miron, “A model for resistive wall mode control”, EPS30 (P5.157),Rome, Italy, 19-23 June, 2006.[15] C.V. Atanasiu, S. Günter, K. Lackner et all., Phys. Plasmas, 11, 3510 (2004).[16] J.A. Wesson, Nucl. Fusion 18, 87 (1978). [17] S. Günter, C. Angioni, C.V. Atanasiu, et al., Nucl. Fusion, 10 (2005).[18] C.V. Atanasiu and I.G. Miron, "An analytical model for resistive wall modes stabilization“,11th European Fusion Theory Conference, 26-28 September 2005, Aix-en-Provence,France.[15] C.V. Atanasiu and I.G. Miron, “A model for resistive wall mode control”, EPS30 (P5.157),Rome, Italy, 19-23 June, 2006.[16] C.V. Atanasiu, I.G. Miron, S.Günter, K. Lackner, A.Moraru, L.E. Zakharov and A.A. Subbotin, "MHD Modelling in Diverted Tokamak Configurations", 13th Conference on Plasma Physics and Applications, Iassy, Romania, 27-29 October 2005 (invited paper).[17] C.V. Atanasiu and I.G. Miron, "An Analytical Model to Control Resistive Wall Modes", 13th Conference on Plasma Physics and Applications, Iassy, Romania, 27-29 October 2005.[18] I.B. Bernstein, E.A. Frieman, M. Kruskal, and R.M. Kulsrud, Proc. R. Soc. London A 244,17 (1958).




[19] C.V. Atanasiu, A.H. Boozer, L.E. Zakharov, et al, Phys. Plasmas 6, 2781 (1999).[20] C.V. Atanasiu, S. Günter, K. Lackner, A. Moraru, L.E. Zakharov, and A.A. Subbotin,Phys. Plasmas, 11, 5580 (2004).[21] A.H. Boozer, Phys. Plasmas 5, 3350 (1998).

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