Triggerless onset and effect of “natural” rotation on NTMs ... · [9,13-15]. This type of...

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1 EX/P4-32 Triggerless onset and effect of “natural” rotation on NTMs in TCV tokamak E. Lazzaro 1 , S. Nowak 1 , O. Sauter 2 , G. Canal 2 , B. Duval 2 , L. Federspiel 2 , A. N. Karpushov 2 , D. Kim 2 , H. Reimerders 2 , J. Rossel 2 , D. Testa 2 , D. Wagner 2 and the TCV Team 1 Istituto di Fisica del Plasma IFP-CNR, Associazione EURATOM-ENEA-CNR sulla Fusione, 20125, Milano, Italy 2 Ecole Polytechnique Fédérale de Lausanne (EPFL), Centre de Recherches en Physique des Plasmas (CRPP), Association EURATOM-Confederation Suisse, 1015 Lausanne, Switzerland E-mail contact of main author: [email protected] Abstract. Experiments on the TCV tokamak show that central electron cyclotron heating (ECH) and current drive (ECCD) can modify the rotation profile and lead to the onset of neoclassical tearing instabilities, in absence of triggers such as sawteeth, ELMS or relevant ”error” field, but in a regime of unsteady rotation. In turn the growing tearing modes provide a nonlinear magnetic braking that flattens the rotation profile. The experimental results are presented and discussed in the frame of the theoretical models of neoclassical toroidal viscosity and ion inertial effects. 1. Introduction The understanding of the conditions and mechanisms of onset of neoclassical tearing modes (NTM) in tokamaks is a high priority problem for the definition of reliable and controlled operation of future fusion reactors. There are numerous theoretical and experimental works on the possible sources (sawteeth, ELMs, error fields) of “seed” magnetic islands which are deemed necessary to trigger an NTM [1-3]. The onset of NTMs in absence of explicit triggers is a question of principle and of practical interest that must be carefully investigated on the basis of clear experimental evidence [4,5]. This work presents the results of a series of highly reproducible experiments carried out on the TCV tokamak in the same plasma conditions (I p ~150 kA, n e,av ~1.5 10 19 m -3 , T e ~3 keV. T i ~0.25 keV, q 95 ~5.8) with slightly different ramps of Electron Cyclotron Waves (ECW) power injection P EC =1.5 MW. In the TCV tokamak spontaneous plasma toroidal rotation in the absence of external momentum is observed [4,6] and it is found experimentally that central electron cyclotron heating (ECH) and current drive (ECCD) can modify the rotation profile [4,6-8]. In section 2 new experimental evidence of the interplay between the plasma rotation and the presence of m/n=3/2 and 2/1 tearing instabilities in the neoclassical regime is presented and discussed. In section 3 the observed NTM instability conditions in an unsteady rotation regime are discussed. The questions to be eventually addressed for interpretation of observations relate both to the magnetic braking torque of the NTV kind, due to resonant magnetic field perturbations breaking axismmetry [9- 12] and to the consistency with the theory of the ion polarisation current effect [13,17-20].

Transcript of Triggerless onset and effect of “natural” rotation on NTMs ... · [9,13-15]. This type of...

Page 1: Triggerless onset and effect of “natural” rotation on NTMs ... · [9,13-15]. This type of perturbation generates an NTV braking torque € T NTV ∝ν //.ν b ν 2(r)V ϕ −v

1 EX/P4-32

Triggerless onset and effect of “natural” rotation on NTMs in TCV tokamak

E. Lazzaro1, S. Nowak1, O. Sauter2, G. Canal2, B. Duval2, L. Federspiel2, A. N. Karpushov2, D. Kim2, H. Reimerders2, J. Rossel2, D. Testa2, D. Wagner2 and the TCV Team 1Istituto di Fisica del Plasma IFP-CNR, Associazione EURATOM-ENEA-CNR sulla Fusione, 20125, Milano, Italy 2Ecole Polytechnique Fédérale de Lausanne (EPFL), Centre de Recherches en Physique des Plasmas (CRPP), Association EURATOM-Confederation Suisse, 1015 Lausanne, Switzerland

E-mail contact of main author: [email protected] Abstract. Experiments on the TCV tokamak show that central electron cyclotron heating (ECH) and current drive (ECCD) can modify the rotation profile and lead to the onset of neoclassical tearing instabilities, in absence of triggers such as sawteeth, ELMS or relevant ”error” field, but in a regime of unsteady rotation. In turn the growing tearing modes provide a nonlinear magnetic braking that flattens the rotation profile. The experimental results are presented and discussed in the frame of the theoretical models of neoclassical toroidal viscosity and ion inertial effects.

1. Introduction

The understanding of the conditions and mechanisms of onset of neoclassical tearing modes (NTM) in tokamaks is a high priority problem for the definition of reliable and controlled operation of future fusion reactors. There are numerous theoretical and experimental works on the possible sources (sawteeth, ELMs, error fields) of “seed” magnetic islands which are deemed necessary to trigger an NTM [1-3]. The onset of NTMs in absence of explicit triggers is a question of principle and of practical interest that must be carefully investigated on the basis of clear experimental evidence [4,5]. This work presents the results of a series of highly reproducible experiments carried out on the TCV tokamak in the same plasma conditions (Ip~150 kA, ne,av~1.5 1019 m-3, Te~3 keV. Ti~0.25 keV, q95~5.8) with slightly different ramps of Electron Cyclotron Waves (ECW) power injection PEC=1.5 MW. In the TCV tokamak spontaneous plasma toroidal rotation in the absence of external momentum is observed [4,6] and it is found experimentally that central electron cyclotron heating (ECH) and current drive (ECCD) can modify the rotation profile [4,6-8]. In section 2 new experimental evidence of the interplay between the plasma rotation and the presence of m/n=3/2 and 2/1 tearing instabilities in the neoclassical regime is presented and discussed. In section 3 the observed NTM instability conditions in an unsteady rotation regime are discussed. The questions to be eventually addressed for interpretation of observations relate both to the magnetic braking torque of the NTV kind, due to resonant magnetic field perturbations breaking axismmetry [9-12] and to the consistency with the theory of the ion polarisation current effect [13,17-20].

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2 EX/P4-32 2. Interplay of Tearing Modes Onset and Toroidal Rotation

The aim of this paper is to provide factual documentation of the interplay between the plasma toroidal rotation and the neoclassical tearing modes onset. In this section the typical aspects of these TCV experiments are described; Fig.1 (left) shows, for TCV shots, the waveform of the applied ECH power and the associated evolution of the (3,2) modes which appears as a conventional tearing mode driven by the ECH/CD modifications of the plasma equilibrium current profile, while subsequently during the EC power ramp, in a neoclassical collisionality regime (2,1) mode appears, as evident in the spectrogram of Fig.1 (right), presumably as an NTM.

FIG. 1. (left) Time waveforms of PEC power and markings of onset of (3,2) (dotted lines), (2,1) (dashed lines) tearing modes, for shots #45225, #45226, #45246, #45247; (right) Spectrogram of (3,2), (2,1) modes in shots #45225 and #45247.

Evidence of an associated global braking of plasma rotation akin to a neoclassical toroidal viscous effect (NTV) is observed as a pronounced flattening at the onset of these tearing instabilities (3,2) and (2,1). The EC power is applied with slow ramp up along ~ 1 s for #45225, #45226 and with a faster ramp up ~ 0.2 s in #45246, #45247. The onset of the (3,2) mode is observed at 0.8 MW and the (2,1) onset at 1.25 MW in all these shots. The spectrograms of #45226 and #45247 in Fig.1 (right) indicate the (3,2) (light lines around 11-12 kHz and the 2/1 (intense lines) at 4.5-5 kHz. In the latter case during the ramp down (starting at ~1.2 sec) the mode frequency increases slightly while the mode amplitude decreases until the turn off of the power at βp<0.5. No sawteeth or ELMs were present. The time evolution of the toroidal Vφ profiles and of Vφ at different radial locations are shown in Figs.2, 3 (left and right) for #45225 and #45247, respectively. The natural intrinsic positive counter rotation (Ip<0) in absence of external drive (t<0.3 s) changes when central EC heating and co-ECCD are added (t>0.3 s), as already observed in TCV [4]: the plasma is accelerated in the co-Ip direction and the toroidal rotation is reduced. At the onset time of (3,2) mode (t=0.605 s, t=0.555 s for #45225, #45247 respectively) the rotation is significantly modified and changes in sign. When the mode appears, the Vφ profile has a maximum near the corresponding q rational surface (helping also to determine the mode location), which results in a local flattening. This interplay between the plasma toroidal rotation and the tearing onset is observed in the modification of the rotation profiles Vφ evolving in co-current direction, in particular outside the mode location. The gradient inside the mode location reverses sign [4].

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3 EX/P4-32 At the onset of the (2,1) mode (t=1.18 s, t=1s for #45225, #45247) a global rotation braking in the counter direction with the strongest deceleration at q=2 location is present as well.

FIG. 2. #45225: (left) Time slices of measured profiles Vφ(ρ), with markings of rational surfaces (3,2 ) and (2,1); (right) Time evolution of Vφ at different radii.

FIG.3. #45247: (left) Time slices of measured profiles Vφ(ρ), with markings of rational surfaces (3,2 ) and (2,1); (right) Time evolution of Vφ at different radii.

In these experiments the collisionality is typical of the banana regime λ=ν with frequencies in the range νii/ε ≤ qωE

[12,14,15], where ωE is the ExB drift frequency, νii the ion-ion collision frequency, ε the inverse local aspect ratio. Occasionally in few very narrow time intervals the superbanana regime λ=1/ν is entered with νii/ε < qωE < ωti ε

0.5, where ωti = vth,i/(R0q) = (Ti/mi)0.5/(R0q) is the ion transit frequency and vth,i the ion thermal speed.

2.1.Interpretation Model of Braking of Toroidal Rotation

The damping of toroidal rotation observed in these TCV experiments can be mainly related to the action of resonant helical magnetic perturbations, which grow into magnetic islands,

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4 EX/P4-32 with contribution of non-resonant perturbations at noise level. In general, symmetry breaking magnetic perturbations ‘modulate’ the modulus |B| of tokamak magnetic field. The expansion of |B| into helical harmonics, can be expressed in the rectified angular coordinates

ϑ ,ϕ( )by :

B(r + δr) ≈ B0(r) + δr ⋅ ∇B0 +

≈ B0 1−εcosϑ +1B0

[bm,nc cos(mϑ − nϕ) + bm,ns sin(mϑ − nϕ)]m,n∑

The magnetic field toroidal non-uniformity

∂B ∂ϕ causes a non-ambipolar current

Jrna in the

r direction and a toroidal braking force

Fϕ = −JrnaBϑ proportional to the square of the fluid

displacement |dr|. In a fluid description this momentum braking mechanism appears as a neoclassical toroidal viscous force (NTV)

B ⋅ ∇ ⋅Π // ≅ −minν //,νVϕ ∂B ∂ϕ (in the various collisional regimes labeled by λ: ν,1/ν, Plateau), where

Π // is the neoclassical parallel stress tensor. Non resonant helical perturbations (with kink-like displacement

δr R0 ~ br,mn B0 |), without formation of magnetic islands, cause a global toroidal torque with amplitude indicated by

bν2 ∝ bmn B0( )2 and dependent on collisional regime ν while resonant helical

perturbations, with magnetic islands of full width w described by the contours

Ω = 8x 2 w2 − cos mθ − nφ( ) produce a field modulation of the type

B ≈ B0 1−rsR0cosθ ±

w2 2R0

Ω+ cos mθ − nφ( )( )1/ 2

cosθ

where the relevant fluid

displacement is

δr ~ w∝ bmn [9,13-15]. This type of perturbation generates an NTV braking torque

TNTV ∝ν //.νbν2 (r) Vϕ − vϕ 0

ν( )more localized near q–rational resonant surface

and proportional to

bν2 ∝ w R0( )2 . The quantitative evaluation of the braking rate at the

rational surface (Fig. 3) shows an agreement convincing that the process is NTV like.

FIG. 4. Evolution rate of Vφ (km/s) at the q=2 surface in #45225 and #45247: measurement (red solid line) and modelling of NTV braking rate:gray and dashed lines, correspond to estimates of the “offset velocity”

vϕ 0ν = (kc 2πZieBθ )dTi dr , with kc=0 (gray), 0.2 (light blu),

vϕ 0ν ∝ dp dr(green).

The propagation frequency ω = ωMirnov - ωE of the tearing modes appearing in these shots is in the electron drift direction and appears significantly correlated with the local fluid toroidal velocity. For the q=2 modes the relation between the local fluid toroidal rotation frequency ωφ =RVφ and the mode frequency is offset linear ω =ω0 – ωφ, , as shown in Fig.5. For the

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5 EX/P4-32 q=3/2 modes there is a less stringent relation between the local fluid ωφ and ω . On the basis of these observations in the next subsection a phenomenological model is developed, for a better understanding of the mutual role of

FIG. 5. Observed relation of the (2,1) modes rotation frequency with the local plasma fluid rotation (left) # 45225; (right) ) # 45247.

rotation and the tearing mode destabilizing conditions, in such unsteady rotation regime.

2.2.Heuristic 1.5 D Model of TM, NTM and NTV Coupled Evolution The model consists in coupling the 1D toroidal large aspect ratio momentum balance equation with the Rutherford equation for a (3,2) classical TM island width and with the generalized neoclassical version for a (2,1) NTM island, including the ion polarization current term governed by the toroidal flow evolution.

The model embodies an expression of the toroidal viscous force amplitude truncated to contributions of the time evolving resonant modes (3,2) and (2,1) and nonresonant terms:

bν2 (r,t) = n2 bnm

2

B02

n=1,2,3−2≤m≤2∑ Mnm =

b1,12

B02 M1,1 +

w1,22 (t)R02 f1,2

2 (r) +b1,32

B02 M1,3 + 4 w2,3

2 (t))R02 f2,3

2 (r)

The coefficients Mn,m are used as adjustable parameters and wnm(t) are the growing island width of the resonant modes. The space dependent coefficients fnm of these modes are obtained from the vacuum approximation of the linear tearing eigenfunctions. The system of nonlinear coupled equations solved numerically is then:

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

!

RV"

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1

!RV"

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6 EX/P4-32

g1τRrsdw32

dt= rsΔ 03/2

’ 1− w32

wsat

g1τRrsdw21

dt= rs [− Δ 02/1

’ + a1β p εLqLp

w21

w212 +wd

2 −β pε3 / 2 Lq

Lp

2

g ε,ν ii( )′ ω ′ ω −ω*i( )ω*e2 ρθi

2 w21

w214 +wd

4 ]

∂Vϕ∂t

= −SEC r,t( ) +1r∂∂r

µ⊥,eff r∂Vϕ∂r

−ν //.νbν

2 (r,t) Vϕ − vϕ 0ν( )

ω ≈ω0 − RVϕ , ′ ω =ω −ωE , vϕ 0ν = kc

12πZieBθ

dTidr

The initial conditions are taken from the experiment and the usual parameters (Lp~LT=T’/T. Lq=q’/q, VAθ, τR, βp, ρiθ, rs , µ⊥eff , |Δ'0n.m|~ m/rs) are evaluated from the equilibrium. As an example Fig. 6 shows the data for #45225. The NTV braking rate is ν//,ν = νii ω2

ti / ω2E and

vϕ 0ν is the “offset” velocity proportional to the ion diamagnetic velocity with a coefficient kc

~ 0.92 in the ν regime.

FIG. 6. Equilibrium (initial) profiles of #45225;(left)ECE temperature Te[eV];(center) CXSR rotation Vφ,normalized;(right) safety factor q.

The momentum source due to the localized EC power absorption is represented by

SEC (r,t) =4πcJrEC r,t( )

BθVAθ2 c 2

1+VAθ2 c 2( )

, modelled following some ideas of Refs. [8], to be

discussed fully elsewhere. The expression

JrEC r,t( ) VAθ2 c 2

1+VAθ2 c 2( )

is a return current associated

with the asymmetric EC power absorption. Fig.7 shows the results, in dimensionless form, of modelling of # 45225, which show the response of the profile to the EC power deposition and the mode onset causing the NTV braking effect, in qualitative agreement with Fig.2.

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7 EX/P4-32

FIG.7. Qualitative modeling of #45225: (left) Time slices of (dimensionless) magnetic (NTV) braking force ν//bν2 profile evolving with the (3,2) TM and (2,1) NTM Rutherford growth. Active contributions are only from appearance of (3,2) mode at radius x32=0.65, and subsequent onset of mode (2,1) NTM at radius x2=0.76; (right) Time slices of (normalized) Vφ profiles under the effect of the torque due to asymmetric PEC power deposition at xEC=0.28 and the magnetic braking.

3. Instability conditions for NTM in a plasma with non steady rotation

In low collisionality regimes reconnection of magnetic field lines at q rational surfaces can occur as an unstable process triggered by external events, such as sawteeth, ELMs , helical “error” fields, or by rapid modifications of the current density profile, e.g. due rf current drive. The basic theory of neoclassical destabilization by bootstrap current developed in [16] showed the dependence of the saturated width of the magnetic island on βp. Later in addition to the “bootstrap current studies”, a “polarization current branch” theory was developed for rotating magnetic islands. The polarization current is a consequence of the different response of ions and electrons to a rotating island: in fact, if an island rotates at frequency ω in the frame of plasma flow motion ωE a time varying field results since the electrons respond faster to this electric field than do the ions. Thus a polarization current Jpol is generated fulfilling the current closure constraint

∇ //J// = J// (∇B /B) −∇ ⋅ J⊥ and introducing a dependence of the electrostatic potential on the island phase velocity An additional frequency dependent term appears in the Rutherford equation for the island width w (t):

Δ'pol ω( ) =4c16Lqw2Bθ

dx−∞

∫ dξJpol−π

π

∫ x,ξ( )cosξ ∝ βpρiθ2

w3

ω ω*pi −ω2[ ]

ω*pe2

which governs stability and thresholds island width and βp [13,17,18]. According to the latest version of the theory, the predicted stabilizing region is for

0 <ω <ω*pi [17-19]. At issue is both the sign of ω, i.e., whether the island propagates in the ion (>0) or electron (<0) diamagnetic drift direction in the frame where Er=0 and also whether it propagates faster or slower than the ion drift. The stability domain described by the plot (Fig.9 left and right) of

f ω( )∝Δ 'pol ω( ) vs.

ω /ω*pi [19] for the present TCV shots shows that the (2,1) mode starts growing at

ω /ω*pi ~ −1 consistently with the ion polarisation current theory, even considering the standard error in ω (Fig.9 right). Actually the evolution of the modes occurs in the so-called electron–drift waves region [20], where further theoretical understanding is necessary [18]. In absence of other triggers, it may be conjectured that the unsteady rotation regime caused both by the ECH effect and the magnetic braking due to the classical (3,2) mode cause a change of sign of

Δ'pol ω( ) for the (2,1) perturbation consistent with the unstable condition. Since the ECCD is also changing the q profile, it can also be that the (2,1) is destabilized like a classical tearing mode, similar to the (3,2) mode. Further experiments are

0

0.01

0.02

0.03

0.04

0.05

0 0.2 0.4 0.6 0.8 1

!//b2

!

r/a

q=3/2 q=2

time intervals with3/2, 2/1 overlap

time interval with 2/1 only

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

v! /

v! (

0)

r/a

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8 EX/P4-32 needed to clarify this point.

FIG. 9. Domain of stability of the ion polarization current term f (ω) for a (2,1) NTM:(left) for #45225;(right) for #45247, full dots are measured values of f (ω) and upper/lower triangles are values of f (ω±ω error). The (2,1) NTM appears when the local rotation frequency falls in the e-diamagnetic drift waves region.

4. Conclusions

Experiments on TCV show that NTMs can occur without seed island formation, in regimes of non-steady rotation caused by ECH near central power absorption and non-steady pressure and current density profiles. The onset of the NTM is apparently consistent and concomitant with the instability condition associated with the ion polarisation current, although the latter is not necessarily the cause of the onset. The growing tearing modes contribute a nonlinear magnetic braking that consistently modifies the rotation profile. We have shown that the rotation profiles peaks at the mode location surface and that both resonant and non-resonant terms can contribute to NTV braking. The results are significant for the strategies of control of reactor grade tokamaks.

References

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[4] SAUTER O. et al., (Proc. of the 23rd IAEA Fusion Energy Conference, Daejon, Korea, 2010) IAEA, Vienna (2011) CD-ROM file EXS/P2-17

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