Embed Size (px)
Citation preview
L-‐H遷移の時空間発展研究の新展開 (Recent Progress on the spa3o-‐temporal
evolu3on of the L-‐H transi3on.) 三木 一弘(Kazuhiro Miki)
WCI Center for Fusion Theory, NFRI, Korea Center for Computa3onal Science and e-‐Systems, JAEA, Japan
日本物理学会 26aKC-‐3 招待講演 09/26/2013, Tokushima, Japan
• Collaborators P.H. Diamond, G. Tynan, O.D. Gurcan, T. Estrada, L. Schmitz, G.S. Xu, D. McDonald, Y. Kosuga, H. Jhang, C. Lee, M. Malkov, P. Manz, N. Fedorczak, K.J. Zhao, W.W. Xiao, S.-‐H. Hahn • Thanks N. Miyato, Y. Idomura • Acknowledgements This work is supported by WCI Program, funded by NFR, Korea, and HPCI Strategic Program Field No.4: Next-‐Genera3on Industrial Innova3ons, funded by the MEXT, Japan.
Contents
• Introduc3on to recent progress on L-‐H transi3on.
• Descrip3on of the reduced 1D model for L-‐H transi3on, showing spontaneous transi3on.
• S3mulated transi3on, triggered by transient par3cle perturba3on.
• Concluding remarks
最近のLH遷移に関する論文
• Estrada ‘10 EPL • Conway ’11 PRL • Estrada ‘11 PRL • G.S. Xu ’12 PRL • Schmitz ‘12 PRL • Miki ‘13 PRL • J. Cheng ’13 PRL • Kobayashi ‘13 PRL • Shesterikov ‘13 PRL
近年，LH遷移に関する知見は実験・理論の両面から発展し，注目されている．
L-‐H 遷移とは
• ASDEXで初めて発見 [Wagner ‘82 PRL] – 周辺領域に特有の閉じ込め改善現象 – ExBシアの関連性
• ITERの標準シナリオ
• 理論によるしきい値の評価は未解決
[Mar3n ’08 JoCP]
[Ryter ‘09 NF]
Two kinds of bifurca3on mechanisms – Er bifurca3on due to ion orbit loss etc.
• Ini3ated in late 80s, [Itoh-‐Itoh ‘88 PRL], [Shaing, ’89 PRL].
• Ini3a3ve introduc3on of cusp-‐type bifurca3on. (S-‐curve model)
• incomplete treatment of turbulence fluctua3ons.
6 Itoh-‐Itoh, PRL, ‘88 Shaing, PRL ‘89
Two kinds of bifurca3on mechanisms – turbulence model [Hinton ‘91 PoF B]
7
χ0: neoclassical transport χ1: turbulent transport (1+α<VE>’2)-‐1: suppression by ExB shear
ZF shearing feedback loop is not considered yet.
These model studies are followed by: -‐ Spa3o-‐temporal evolu3on study, dynamics of
slowly propaga3ng barrier [Diamond ‘97], [Lebedev ’97]
-‐ Considera3on of 2-‐field (p, n) model, indica3ng importance of fueling on transi3on [Hinton, Staebler ‘93], [Malkov ’08]
1-‐field barrier dynamics: Turbulence suppression by ExB shear and subsequent posi6ve feedback by mean field.
∂P∂t+
1r∂∂r
(rQ) = S(r), Q = − χ0 +χ1
1+α ∂vE /∂r( )2
#
$%%
&
'((
∂P∂r
, ∂vE∂r
=1eB
∂∂r
1n(r)
∂P(r)∂r
#
$%
&
'(
The framework on the L-‐H transi3on model: 2 predator-‐1 prey(Kim-‐Diamond) model
[E.J. Kim and Diamond ’03, PRL]
8
Turbulence:
Zonal Flow(ZF):
pressure gradient ∇<P>
• Limit-‐cycle oscilla3on prior to the transi3on • Local(0D) model
2 predators
1 prey
ε
VZF Ν=∇<P>
V = dN 2Mean Flow equilibrium:
TJ-‐II revealed physical mechanism behind the L-‐H transi3on from the LCO.
9
T. Estrada et al.
0
5
10
10 -6
10 -5
10 -4
168.5 169.0 169.5 170.0 170.5 171.0
|Er| (
kV/m
) S (a. u.)
time (ms)
Fig. 5: (Color online) Time evolution of Er and densityfluctuation level obtained from the spectrogram shown in fig. 4.
6
8
10
12
10 -7 10 -6 10 -5 10 -4
|Er| (
kV/m
)
S (a.u.)
1
2
3
#23473, t=170-170.4 ms
Fig. 6: (Color online) Relation between Er and density fluctu-ation level. Only two of the cycles shown in fig. 5 are displayed.The time interval between consecutive points is 12.8µs.
The time evolution of both, Er and density fluc-tuations, reveals a characteristic predator-prey rela-tionship: a periodic behaviour with Er (predator)following the density fluctuation level (prey) with90! phase di!erence can be clearly seen. Er is observedto grow and decay at di!erent time rates ((100–150)µsand (50–80)µs, respectively), while the turbulence growsand decays in comparable times ((50–70)µs). These timescales being much faster than the energy confinementtime. The relation between Er and the density fluctuationlevel showing a limit-cycle behavior is represented infig. 6. For the sake of clarity, only two cycles are displayed.The turbulence-induced sheared flow is generated causinga reduction in the turbulent fluctuations (1 in fig. 6), thesubsequent drop in the sheared flow (2 in fig. 6) and theposterior increase in the turbulence level (3 in fig. 6).As has been already discussed, the coupling between
fluctuations and flows, described as a predator-prey evolu-tion, is the basis for some bifurcation theory modelsof the L-H transition [10,11]. The experiments reportedin this letter strongly support these models. However,these results do not necessarily exclude other mechanisms
working together in triggering the transition. In helicaldevices, the negative radial electric field imposed by theNeoclassical ion root condition in the L-mode can act as abias for the transition [16,22], either because a substantialEr-shear already can exist before the transition or becauseof the amplification of zonal flows by Er [27]. On the otherhand, as already mentioned, the magnetic topology caninfluence the H-mode occurrence through mechanisms asflow damping by poloidal viscosity [24] or spin-up of flowsby rational surfaces [28–30]. All these mechanisms mayact together to reach the specific conditions at which theself-regulation between turbulence and flows becomes thedecisive mechanism.To our knowledge, this is the first time that such
an experimental evidence is reported supporting thepredator-prey relationship of turbulence and flows as thebasis for the L-H transition. During the so-called IM modefound in DIII-D power scan experiments [21], the relationbetween electron temperature fluctuations and shearedflows seems to follow a predator-prey behavior, however,as the authors explain, the sheared flow is not directlyobserved but inferred from beam emission spectroscopydata. Two ingredients have been essential in attaining theresults presented in this letter, the possibility of achievinggradual L-H transitions and the capability of measuringsimultaneously both magnitudes, density fluctuations andsheared flows, with good spatiotemporal resolution.
Conclusions. – The dynamics of turbulence andflows has been measured during L-H transitions in thestellarator TJ-II. When operating close to the L-Htransition power threshold, pronounced oscillations inboth, Er and density fluctuation level, are measuredat the transition. The oscillations are measured rightinside the E!B shear layer (located at !" 0.83). Thetime evolution of both, Er and density fluctuation level,shows a characteristic predator-prey behavior, with Er(predator) following the density fluctuation level (prey)with 90! phase delay. These experimental observationsare consistent with L-H transition models based onturbulence-induced sheared/zonal flows.
# # #
The authors acknowledge the entire TJ-II team fortheir support during the experiments. This work has beenpartially funded by the Spanish Ministry of Science andInnovation under contract No. ENE2007-65927.
REFERENCES
[1] Wagner F., Becker G., Behringer K., CampbellD., Eberhagen A., Engelhardt W., Fussmann G.,Gehre O., Gernhardt J., v. Gierke G., Haas G.,Huang M., Karger F., Keilhacker M., KluberO., Kornherr M., Lackner K., Lisitano G., ListerG. G., Mayer H. M., Meisel D., Muller E. R.,Murmann H., Niedermeyer H., Poschenrieder W.,Rapp H. and Rohr H., Phys. Rev. Lett., 49 (1982) 1408.
35001-p4
T. Estrada et al.
0
5
10
10 -6
10 -5
10 -4
168.5 169.0 169.5 170.0 170.5 171.0
|Er| (
kV/m
) S (a. u.)time (ms)
Fig. 5: (Color online) Time evolution of Er and densityfluctuation level obtained from the spectrogram shown in fig. 4.
6
8
10
12
10 -7 10 -6 10 -5 10 -4
|Er| (
kV/m
)
S (a.u.)
1
2
3
#23473, t=170-170.4 ms
Fig. 6: (Color online) Relation between Er and density fluctu-ation level. Only two of the cycles shown in fig. 5 are displayed.The time interval between consecutive points is 12.8µs.
The time evolution of both, Er and density fluc-tuations, reveals a characteristic predator-prey rela-tionship: a periodic behaviour with Er (predator)following the density fluctuation level (prey) with90! phase di!erence can be clearly seen. Er is observedto grow and decay at di!erent time rates ((100–150)µsand (50–80)µs, respectively), while the turbulence growsand decays in comparable times ((50–70)µs). These timescales being much faster than the energy confinementtime. The relation between Er and the density fluctuationlevel showing a limit-cycle behavior is represented infig. 6. For the sake of clarity, only two cycles are displayed.The turbulence-induced sheared flow is generated causinga reduction in the turbulent fluctuations (1 in fig. 6), thesubsequent drop in the sheared flow (2 in fig. 6) and theposterior increase in the turbulence level (3 in fig. 6).As has been already discussed, the coupling between
fluctuations and flows, described as a predator-prey evolu-tion, is the basis for some bifurcation theory modelsof the L-H transition [10,11]. The experiments reportedin this letter strongly support these models. However,these results do not necessarily exclude other mechanisms
working together in triggering the transition. In helicaldevices, the negative radial electric field imposed by theNeoclassical ion root condition in the L-mode can act as abias for the transition [16,22], either because a substantialEr-shear already can exist before the transition or becauseof the amplification of zonal flows by Er [27]. On the otherhand, as already mentioned, the magnetic topology caninfluence the H-mode occurrence through mechanisms asflow damping by poloidal viscosity [24] or spin-up of flowsby rational surfaces [28–30]. All these mechanisms mayact together to reach the specific conditions at which theself-regulation between turbulence and flows becomes thedecisive mechanism.To our knowledge, this is the first time that such
an experimental evidence is reported supporting thepredator-prey relationship of turbulence and flows as thebasis for the L-H transition. During the so-called IM modefound in DIII-D power scan experiments [21], the relationbetween electron temperature fluctuations and shearedflows seems to follow a predator-prey behavior, however,as the authors explain, the sheared flow is not directlyobserved but inferred from beam emission spectroscopydata. Two ingredients have been essential in attaining theresults presented in this letter, the possibility of achievinggradual L-H transitions and the capability of measuringsimultaneously both magnitudes, density fluctuations andsheared flows, with good spatiotemporal resolution.
Conclusions. – The dynamics of turbulence andflows has been measured during L-H transitions in thestellarator TJ-II. When operating close to the L-Htransition power threshold, pronounced oscillations inboth, Er and density fluctuation level, are measuredat the transition. The oscillations are measured rightinside the E!B shear layer (located at !" 0.83). Thetime evolution of both, Er and density fluctuation level,shows a characteristic predator-prey behavior, with Er(predator) following the density fluctuation level (prey)with 90! phase delay. These experimental observationsare consistent with L-H transition models based onturbulence-induced sheared/zonal flows.
# # #
The authors acknowledge the entire TJ-II team fortheir support during the experiments. This work has beenpartially funded by the Spanish Ministry of Science andInnovation under contract No. ENE2007-65927.
REFERENCES
[1] Wagner F., Becker G., Behringer K., CampbellD., Eberhagen A., Engelhardt W., Fussmann G.,Gehre O., Gernhardt J., v. Gierke G., Haas G.,Huang M., Karger F., Keilhacker M., KluberO., Kornherr M., Lackner K., Lisitano G., ListerG. G., Mayer H. M., Meisel D., Muller E. R.,Murmann H., Niedermeyer H., Poschenrieder W.,Rapp H. and Rohr H., Phys. Rev. Lett., 49 (1982) 1408.
35001-p4
TJ-‐II [Estrada ‘10 EPL]
• Time evolu3on of Er and turbulence fluctua3on reveals predator-‐prey behavior.
• Microphysics (fluctua3ons, ZF) should enter the L-‐H transi3on dynamics.
Spa3o-‐Temporal evolu3on of L-‐(I-‐phase)-‐H transi3on.
10
The time-resolved E! B velocity is obtained from theinstantaneous Doppler shift, fD " #!S $!I%=2! "v?k"=#2!%, with v? " vE!B & vph. Neglecting the con-tribution of the fluctuation phase velocity vph, (estimated to
be much smaller than the electron and ion diamagneticvelocities by linear stability calculations for similar plas-mas [18]), one obtains vE!B ' 2! fD=k".
Figure 1(a) shows the time history of theE! B velocityvE!B across the plasma edge (measured by an 8-channelDBS system) in a dithering L-H transition, induced bystepping up the (co-injected) neutral beam power fromPinj " 2:8 MW to 4.5 MW, a value just above the
H-mode transition power threshold (at a toroidal magnetic
field B# " 2 T and plasma current Ip " 1:1 MA), Thecore plasma (line density !n " 2:7! 1019 m$3) is co-rotating, and in the L mode the radial electric field Er 'v#B"=B
2 and the poloidal projection of the E! B veloc-ity shown here are positive except for a narrow radial edgelayer a few cm inside the separatrix, where the contributionof the ion pressure gradient to Er produces weak intermit-tent negative flow. The normalized edge density fluctuationlevel ~n=n peaks near/outside the separatrix [Fig. 1(b)]. ~n=nis measured by DBS at a wave number k" ( 2:7 cm$1,"k"=k" ' 0:3, and k"$s ( 0:4. This wave number rangeoverlaps with the upper wave number range detected bybeam emission spectroscopy (BES) in DIII-D and corre-sponds to the poloidal wavenumber range where the maxi-mum growth rate of the ion temperature gradient mode[ITG], is expected. In addition, resistive balloning modes(RBM) may be active just inside the separatrix. The radia-tive instability, thought to be responsible for limit cycleoscillations in earlier DIII-D experiments with a highertriangularity plasma with lower x-point height [8], is notlikely to be present here due to the higher NBI heatingpower (2.8 vs 0.3 MW) and edge electron temperature, andlow Zeff ' 1:6. Because of backscattering, DBS intrinsi-cally detects modes with kr ' 0. At t( 1271 ms, a strong,periodic limit cycle oscillation (LCO) oscillation in theE!B velocity starts to develop in a 2–3 cm wide layer atand just inside the separatrix. Starting at t0 ( 1271:7 ms,density fluctuations are periodically reduced in the region2:25 m<R< 2:28 m, concomitantly with a sharp reduc-tion in theD% recycling light [Fig. 1(c)]. Edge confinementstarts to improve during the oscillatory phase, as evidencedby the changing rate of increase of the line density[Fig. 1(d)] and edge electron temperature [Fig. 1(e)]. Thefrequency of the E! B flow oscillation decreases gradu-ally from 2.5 to 1.7 kHz. About 15 ms after LCO onset,after a final D% transient, the transition to sustainedH-mode takes place, characterized by a strong, steadyE!B flow layer with Er ($rpi=en, where rpi is theion pressure gradient and n is the plasma density. Anexpanded time history reveals that the E!B velocityoscillations [Fig. 2(a)] at the separatrix lag the densityfluctuation amplitude [Fig. 2(b)] by about 90), as con-firmed by the cross-correlation coefficient [Fig. 2(d)].This phase lag is consistent with the predator-prey modelof the L-H transition advocated previously [11]. ZFs aredriven once density fluctuations reach sufficient amplitude;in turn, ZF shear is thought to quench fluctuations. Thetime delay between the peak fluctuation amplitude (and,presumably, peak radial particle transport flux) and thepeak divertor recycling (D%) light [Fig. 2(c)] is found tobe (120–140 &s, consistent with an estimated plasmaloss time due to parallel scrape-off layer (SOL) flow0:5q95R=0:3cs ' 140–180 &s (the edge safety factor isq95 ( 4:1 and an estimated SOL parallel flow speed of
0:3cs with cs " *#kTe & kTi%=mi+1=2 is used here).
R (m
)
LCFS
R (m
)
1265 1275 1285 1295
140426
D ! (1
015 p
h/s)
Time (ms)
2.53.0
3.5
n e(1
019 m
–3)
0.15
0.33
0.50
kT e
(keV
)
135
P NB
(MW
)
R = 2.2 m2.25 m2.28 m
Limit-CycleOscillations (LCO)L-Mode H-Mode
vExB(km/s)
ñ/n(au)
1.5
2.5
3.5
2.23
2.25
2.27
0.0
0.5
1.0
2.23
2.25
2.27
–6
3
0
6
9
(a)
(b)
(c)
(d)
(e)
(f)
to
FIG. 1 (color). Time evolution of (a) E!B velocity (thedirection of the ion diamagnetic and rB drift is indicated);(b) relative density fluctuation level; (c) divertor D% signal;(d) electron density; (e) edge electron temperature, and(f) neutral beam power across the transition from L modethrough limit cycle oscillations (LCO) to H mode.
PRL 108, 155002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending
13 APRIL 2012
155002-2
DIII-‐D [Schmitz ‘12 PRL]
• Limit-‐cycle oscilla3on with radial propaga3on.
• (At least) one space – one 3me model is necessary.
１次元輸送モデルと自発的遷移
L-‐H時空間構造を調べるために，１次元のメゾスケール輸送モデルを提案しました．
• 5 field reduced mesoscale model(p, n, I, E0, vθ), mo3vated by – 1D transport + 2-‐predator-‐1-‐prey model – Simplified boundary condi3on on p and n at LCFS; no SOL-‐edge interac3on.
• No ion-‐orbit-‐loss (or Er bifurca3on ) • NO MHD ac3vity
• N.B.: No ‘first principle’ simula6ons have reproduced or elucidated the L-‐H transi6on
12
[K. Miki, and P.H. Diamond et al., Phys. Plasmas, 2012]
Descrip3on of the 1D model (1): Predator-‐Prey model (a‘ la Kim-‐Diamond)
13
Driving term Local dissipa3on
ZF shearing MF shearing Turbulence
spreading [Hahm, Lin]
Reynolds stress drive MF inhibi3on in Reynolds cross-‐phase [E. Kim ’03 PRL]
ZF collisional damping
by radial force balance
Turbulence intensity:
Zonal flow(ZF) energy: E0=V’ZF2
Mean flow(MF) shearing:
∂t I = (γL −ΔωI −α0Eo −αVEV )I + χN∂x (I∂xI )
∂tE0 =α0IE0 / (1+ζ0EV )−γdamp
ΕV = (∂xVE×B )2
Assuming ITG turbulence
VnnDDp
xoneon
xoneop
−∂+−=Γ
∂+−=Γ
)(
)( χχ
∂t p(x)+∂xΓ p = H∂tn(x)+∂xΓn = S
Neoclassical transport term
Banana regime χneo ~ χTi ~ εT
−3/2q2ρi2ν ii
Dneo ~ (me /mi )1/2 χTi
D0 ~ χ0 ~τ ccs
2I(1+αt !VE
2 )
Turbulent transport term
pressure
density
)0 ,(
)( ,0,0
<∝⎟⎟⎠
⎞⎜⎜⎝
⎛−≅
+=
TT
TETEP
LILD
RD
vvV
TEP pinch Thermoelectric pinch
Inward
Pinch term
Descrip3on of the 1D model (2):1D transport model
14
ExB shear modifies the cross-‐phase factor in turbulent flux [Z.H. Wang ‘12 NF]
H
S
Descrip3on of the 1D model (3): poloidal flow and radial force balance
−∂uθ∂t
≅α5γLω*
cs2∂xI +ν iiq
2R2µ00 (uθ +1.17csρiLT)Poloidal flow
Evolu3on:
!VE×B =1eB
−1n2
!n !p + 1n
!!p$
%&'
()+
rqR
u||$
%&
'
()!− !uθ
*
+
,,
-
.
//
Pressure curvature
Density gradient
Diamagne3c driq
Toroidal flow (not considered here)
Poloidal flow Radial Force Balance:
Turbulence driven Neoclassical poloidal flow
L-‐H Transi3on in the dynamical systems
• In the local limit, this model is reduced to the local predator-‐prey (Kim-‐Diamond) model.
• Phase-‐portrait on the projec3on of E0=0
p’
I
∂t I = 0∂tE0 = 0
Suppression by mean flow
∂t "p = 0 for low Q
L-‐mode
Unstable H-‐mode
Destabiliza3on of the node due to MF inhibi3on
Ramp up
Cross-‐over in 3D manifold (L-‐H transi3on)
(ref. [Malkov ’08]) for high Q
Limit-‐cycle oscilla3on (stable)
Stable H-‐mode
∂t "p = 0
自発的なL-‐H遷移のケース
17
Model studies recover the spa3o-‐temporal evolu3on of the spontaneous L-‐I-‐H transi3on.
→ At L-‐I transi3on -‐ Limit-‐cycle oscilla3on(LCO) begins.
→ At I-‐H transi3on -‐ MF increases. -‐ Turbulence and ZF
drop in the pedestal
������ ��
���
���
���
����
L�I#phase)/)))Limit#cycle))oscilla2on(LCO)) H#mode�
(∇n/n)-1(r/a=0.95) related Dα�
1D model studies exhibit limit-‐cycle oscilla3on in space of I VS E0 VS Ev
Turbulence intensity VS ZF energy ZF energy VS MF energy
1D model LCO exhibits a gradient oscilla3on, too. [cf. Cheng ’13 PRL]
→ Note: Extended LCO I-‐phase is conceptually and diagnos3cally useful but NOT intrinsic to transi3on
→ Stress driven flow can be excited in burst i.e. cycle → 1 Period
19
turbulence
ZF
MF
3me
r/a
t(a/cs)
r/ar/a
r/a
(a) turbulence
(b) ZF
(c) log(MF)
(d)
(e)
turbulence
MFZF
!0IE
0
"
t(a/cs)
(f)
(A) (B)
FIG. 7. Spatio-temporal evolution of turbulence (a) I, (b) E0
, and (c) ln(EV ) as a function of time
t during a fast power ramp 2 ⇥ 104(a/cs) < t < 4 ⇥ 104(a/cs), and radius (0.5 < r/a < 1.0). No
LCO is seen. The I-phase LCO is compressed into a single burst of ZF at t = 2.72 ⇥ 104(a/cs).
(d) Time evolution of turbulence intensity I (blue chain line), ZF energy E0
(green solid line),
and MF shearing energy EV (red dotted line). This figure shows that at the L!H transition
t = 2.72⇥104(a/cs), the turbulence quenches at a faster rate, the ZF increases before the transition
and damps after the transition, and MF shear rapidly increases at and just after the transition.
(e) The evolution of the product quantity P? = ↵IE0
. The product quantity exhibits a peak just
before the transition and quenches after the transition. (f) An evolution of ⌘, showing a single
burst at t = 2.72⇥ 104(a/cs). This single burst triggers the quench of turbulence and the product
quantity ↵0
IE0
. Thus, the L!H transition occurs. For convenience, we here draw vertical lines
(A) at t = 2.71⇥ 104(a/cs) and (B) at t = 2.72⇥ 104(a/cs).
46
Slow ramp up Fast ramp up
ZF acts as a trigger of the L-‐H transi3on.
• Increasing ZF damping delays L-‐H transi3on.
3me, Q
n
Pthresh(n) Low n High n
Pthresh ∝γZF ∝ν ii ∝n
→ explaining scaling in high density region?
→ Systema3c Iden3fica3on of ZF role on the transi3on
- Useful parameter (from predator-‐prey model):
RT≥ 1 ⇒ turbulence collapse and transi3on
21
RT ≡ vrE vθE ∂ V⊥ / ∂r γeffET
[Manz, PoP ’12]
Useful valida3on on the transi3on
then begins to decay, defining phase (III) in the transition,consistent with recent observations in TJ-II.18 The delaybetween the reduction in the turbulent amplitude and thedrop of the Da (Figs. 2(a) and 2(b)) most likely correspondsto a combination of the time needed for cross-field transportand parallel flow from the midplane region to the divertorregion where the Da emissions are observed. The kineticenergy transfer P? from the turbulence into the shear flowcontinues to increase while the turbulence amplitude isdecreasing (Fig. 2(d)) during phase (III), while the L–H tran-sition is approached as shown in Fig. 2(d)). This confirmedthe earlier conjecture above that the low frequency flow isactually driven by a transfer of energy from the turbulence.About 1 ms before the L–H transition occurs, the kineticenergy transfer P? peaks. To evaluate whether this energytransfer is sufficient to reduce the turbulence level signifi-cantly the transfer rate must be compared to the energy inputrate into the turbulence as discussed above. Using experi-mental data, we find the ratio of the production, P?, normal-ized by ceff h~v2
?i in Figure 2(e). This ratio indicates the powertransfer rate into the shear flow normalized by the power
transfer into the turbulence from the combined effects of thefree energy source and the background Er shearing duringperiods of weak flow. As the Da signal starts to drop (the tra-ditional measure of the start of the L–H transition), the turbu-lence has already reached its minimum. Commensurate withthe drop in Da signal, the flow production rate surpasses theturbulence recovery rate and the turbulence energy collapsesto nearly zero. After the peak of the normalized productionrate, the production P? remains small and the low. But thelow frequency flow begins to recover (Fig. 2(c)), suggestingthat after the transition the flow is sustained by non-turbulentprocesses. The growth of the radial electric field and associ-ated flow shear after the drop in Da light has already beenshown to be associated with the growth of the ion pressuregradient.28 No ion temperature profiles are available here, butthe observation here is consistent with this earlier result. Fur-thermore, in an analysis of similar data obtained just outsideof the separatrix (shown in green in Fig. 2), no such transientbehavior is observed. This indicates that these observationsare isolated to the region inside the LCFS, and that on openfield lines the turbulence amplitude simply collapses after theH-mode transition, as has been reported earlier.3
COMPARISON WITH PRESENT PREDATOR-PREYMODEL
The one-space, one-time multiple shearing predator-prey model22 is used for comparison with the experimentalresults reported here from EAST. A fast ramp of the heatingpower is used to model the regular L–H transition. Underthese conditions, no limit-cycle oscillation is observed.Instead, a single burst of zonal flow energy and turbulencecollapse is observed, followed by a classic L–H transition. Atypical time trace around the L–H transition is shown in Fig.4. Here, Fig. 4(a) depicts the evolution of the amplitudes ofthe turbulence I, the zonal flow E2
0, and the mean flow. Themodest decrease of turbulence I in the early part of the modelevolution is triggered by a rapid growth of the mean flow, ora decorrelation of turbulence drive by mean flow shearing.The coupling between the zonal flow and the turbulence is
FIG. 2. Da (a), turbulent (b), and flow (c) fluctuation amplitude during L–Htransition. Energy transfer between turbulence and shear flow (d) and nor-malized to turbulent fluctuation amplitude and energy recovery time (e).Data from 1.0–1.5 cm inside LCFS denoted by blue and red curves. Datataken 1 cm outside of LCFS in the SOL region denoted by green curve.
FIG. 3. Hodograph of turbulent and flow energies during L-H transition.
072311-4 Manz et al. Phys. Plasmas 19, 072311 (2012)
t(a/cs)
r/a
r/a
r/a
(a) turbulence
(b) ZF
(c) log(MF)
(d)
(e)
turbulence
MFZF
!0IE
0
"
t(a/cs)
(f)
(A) (B)
FIG. 7. Spatio-temporal evolution of turbulence (a) I, (b) E0
, and (c) ln(EV ) as a function of time
t during a fast power ramp 2 ⇥ 104(a/cs) < t < 4 ⇥ 104(a/cs), and radius (0.5 < r/a < 1.0). No
LCO is seen. The I-phase LCO is compressed into a single burst of ZF at t = 2.72 ⇥ 104(a/cs).
(d) Time evolution of turbulence intensity I (blue chain line), ZF energy E0
(green solid line),
and MF shearing energy EV (red dotted line). This figure shows that at the L!H transition
t = 2.72⇥104(a/cs), the turbulence quenches at a faster rate, the ZF increases before the transition
and damps after the transition, and MF shear rapidly increases at and just after the transition.
(e) The evolution of the product quantity P? = ↵IE0
. The product quantity exhibits a peak just
before the transition and quenches after the transition. (f) An evolution of ⌘, showing a single
burst at t = 2.72⇥ 104(a/cs). This single burst triggers the quench of turbulence and the product
quantity ↵0
IE0
. Thus, the L!H transition occurs. For convenience, we here draw vertical lines
(A) at t = 2.71⇥ 104(a/cs) and (B) at t = 2.72⇥ 104(a/cs).
46
RT
transi3on
transi3on
TEXTOR [Shesterikov PRL ’13]: L-‐H transi3on occurs at VbM=150[V]
RT
- Exp. EAST [Manz, PoP ’12] Model [Miki, PoP ’12]
What do we learn? • ZF ‘triggers’ the L-‐H transi3on, (but never exclusive)
– ZF acts as a ‘energy holding pa2ern’ to store energy without increasing transport, allowing mean VE’ to steepen, so the transi3on can develop
• ZF damping can be another factor to determine the L-‐H power threshold. – Explain high density scaling?
• I-‐phase can be dis3nct state, with dis3nct threshold, different from H.
23
誘起遷移の物理
誘起遷移のダイナミクス Mo3va3on Par3cle injec3on to probe and
Control the L→H and H→L transi3ons.
25
[K. Miki, P.H. Diamond et al., PRL ‘13], [K. Miki, P.H. Diamond et al., PoP ‘13]
Previous Work – Par3cle injec3on-‐induced L-‐H transi3on -‐ Askinazi, et al., (1993) – Tuman-‐3
→ Transi3ons triggered by strong, rapid gas puffing
→ Some evidence that H mode triggered by <VE>’ increase near edge.
-‐ Gohil, Baylor, et al., (2001, 2003) – DIII-‐D → Transi3ons triggered by pellet injec3on → Reduc3on of PTh by ~30% → Limited evidence that <VE>’ steepened near
edge. 25
0.08 -f-------- O .08-
0 10 20 30 40 50 60 t, ms 0 10 20 30 40 50 60 t, ms (4 ( W
FIG. 7. (a) Temporal evolution of some plasma parameters in PCH mode and (b) in the shot without transition.
D. The H mode induced by pellet injection
Both active methods for H-mode initiation described above are not feasible for the next generation of tokamaks with hotter plasmas. This is due to the low efficiency of gas puffing in larger tokamaks with divertors, and the impos- sibility of immersing any electrodes in a hot plasma during a longer pulse. So it is highly desirable to find a practical alternative for triggering an H mode using a more subtle method. Pellet injection which is regarded as an effective means for plasma fueling is shown to offer such an option. Pellet injection causes rather steep density and temperature gradients. If the pellet is sufficiently large and the penetra- tion depth is small, the region of steep density gradient occurs close to the LCFS. From Eq. (7) it is obvious that the radial electric field in the L regime increases during the pellet injection due to the growth of the density gradient. klore importantly, it has been shown in Ref. 19 that there is an important causal relationship between the gradient of density profile and the value of the electric-field potential at the separatrix. Steeper profiles cause stronger electric fields and vice versa. Thereafter a strong electric field with a large gradient produces the shear of plasma rotation which can reduce the anomalous transport and cause the transi- tion from L to H regime.
During pellet injection the electron temperature is re- duced while the ion temperature remains approximately the same as before the injection because of the sufficiently long time for heat exchange between ions and electrons compared to the time for pellet evaporation. Therefore, the density dependence of the radial electric field impacts most significantly. Another reason for the improvement of the confinement may be related to the variation of the param- eter vi=d(ln Ti)/d( In ni). The conditions for VI-mode ex- citation are worsened during the pellet injection and the corresponding transport may be reduced.*’ However, this
2425 Phys. Fluids B, Vol. 5, No. 7, July 1993
appears most likely to result from a pellet injection within the inner core.”
Using the regime of injector operation described in Sec. I, an improved confinement regime has been obtained ex- perimentally. This mode has many similarities with the spontaneous H mode and therefore was termed pellet caused H (PCH) mode. The typical traces of some plasma parameters in the PCH mode are shown in Fig. 7(a). For comparison, the shot without transition is presented in Fig. 7(b). In this case, the pellet was larger and therefore pen- etrated deeper into the bulk plasma. This is clearly seen in Fig. 8 where the perturbations of the plasma density in these two cases are shown. The radial profiles of electric field calculated, according to Eq. (7), provided Ti= 70 eV=const, and using experimental n,(r) profiles, are shown in Fig. 9. In the case when the pellet injection trig- gered the L-H transition, the region of the strong shear of the radial electric field dE/dr is shifted outwards closer to
:Ti
;--A 1::
‘0.00 ,I I I 0.10 OY20 r:m ’
FIG. 8. The density profile (solid line) and its perturbation by a pellet which resulted (dashed line) or did not result (dashed-dotted line) in the L-H transition.
Askinazi et a/. 2425
Downloaded 28 Nov 2012 to 203.230.125.100. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions
608 P Gohil et al
1412
10864
864
2
16
14
12
10
Density Interferometer Signal (a.u.)
Pellet Time
Er Measurements
3900 3920 3940 3960Time (ms)
3980 4000 4020 4040
40
30
Er (
kV/m
)
20
10
02.26 2.27 2.28 2.29 2.30
–2 ms0 ms2 ms4 ms6 ms
Separatrix
Shot 102014
R (m)
Upper Divertor D! (a.u.)
8 ms
(a)
(b)
(c)
Figure 4. The time history of a HFS pellet-induced H-mode transition and the radial electric field,Er , profile at the plasma edge. (a) Density interferometer signal indicating the relative change inthe line-average density, (b) upper divertor D! emission (from D! filterscopes), (c) the Er profileat the plasma edge (determined from CER measurements). The green solid vertical line in (a) and(b) indicates the time of pellet injection. The red shaded region in (a) indicates the time periodfor the Er measurements shown in (c) at a sample time of 2 ms. The dashed vertical line in (c)indicates the position of the separatrix.
and after pellet injection from the inside vessel wall i.e. high magnetic field side of the plasma.The interferometer signal increases dramatically, indicating the large increase in density ofpellet injection, and then remains high due to the formation of the edge transport barrier.The D! signal drops simultaneously indicating the H-mode transition and is followed by ashort phase of dithering H-mode, which quickly (<40 ms) turns into an ELM-free H-mode.On pellet injection, the gradient of Er just inside the separatrix increases substantially andthen continues to increase with time and also widens as the edge barrier is firmly established.The values of |Er | were obtained from charge exchange recombination (CER) spectroscopicmeasurements of the CVI impurity ions [51]. The radial electric field is obtained from theradial force balance equation for any plasma species, i, such that
Er = !Pi
niZie+ v"B# " v#B", (1)
where n is the species density, P the pressure, Z the charge number, e the electric charge, v"
the impurity ion toroidal rotation, v# the poloidal rotation, B" the toroidal magnetic field andB# the poloidal magnetic field. The poloidal rotation used is the measured value from CERmeasurements. The greatest changes (i.e. dominant terms in equation (1)) to the total Er at theplasma edge are the result of changes to v" and v# of the impurity ions after pellet injection,whereas the core rotation shows less change. The effect on the individual contributions toEr of !P , v" and v# of the main (bulk) deuterium ions is not determined. However, thereare clear increases in the shear in the edge Er profile for the pellet-induced H-mode whichis identical in behaviour with spontaneous H-mode transitions and is consistent with E # B
velocity stabilization of turbulence leading to the formation of the edge barrier.The large changes in the edge ne, Te, and Ti produced by pellet injection provide a
direct means of testing theories on the formation of the H-mode edge barrier. The theoriesconsidered [24–26] postulate certain critical threshold parameters for the H-mode transition
[Askinazi, ‘93]
[Gohil, ‘03]
E rv 751 1
-25-
I 0.20 r, m
FIG. 9. Radial electric field E calculated for the cases shown in Fig. 8, respectively.
the plasma edge. The shear itself is larger compared to the case without L-H transition.
The PCH mode has some features which distinguish it from the Ohmic H mode. (i) The PCH mode exists in TUMAN-3 only during a short period of about 5 msec, and no way was found to prolong its duration. (ii) Micro- wave reflectometer data suggest that the density fluctuation (and transport) suppression zone is shifted towards the plasma center, as compared to the case of a spontaneous Ohmic H mode. (iii) Since the rate of the SXR intensity increase is higher compared to that for an Ohmic H mode, it is concluded that the improvement of the energy con- finement is enhanced in the PCH mode.
IV. SUMMARY
Three different methods of H-mode initiation in the TUMAN-3 tokamak are described: a short increase of the gas puffing rate, the edge plasma polarization using a bi- ased electrode, and a steep peripheral density gradient brought about by a pellet injected into the boundary plasma.
The improvement of confinement has a common origin in all of these cases, and is clearly caused by a strongly inhomogeneous radial electric field inside the LCFS. The extended version of standard neoclassic theory, invoking anomalous inertia and viscosity driven by a turbulence, has been shown to describe qualitatively all three methods of H-mode initiation.
‘F. Wagner, G. Becker, K. Behringer, D. Campbell, A. Eberhangen, W. Engelhard, G. Fussmann, 0. Gehre, J. Gemhardt, G. v. Gierke, G. Haas, M. Huang, F. Karger, M. Keilhacker, 0. Khmer, M. Komherr, K. Lackner, G. Lisitano, G. G. Lister, H. M. Mayer, D. Meisel, E. R. Muller, H. Murmann, H. Niedermeyer, W. Poschenrieder, H. Rapp, H. Rohr, F. Schneider, G. Siller, E. Speth, A. Stabler, K. H. Steuer, G. Venus, and 0. Vollmer, Phys. Rev. Lett. 49, 1408 (1982).
*K. Toi, K. Kawahata, S. Morita, T. Watari, R. Kumazawa, K. Ida, A. Ando, Y. Oka, M. Sakamoto, Y. Hamada, K. Adati, R. Ando, T. Aoki, S. Hidekuma, S. Hirokura, 0. Kaneko, A. Karita, T. Kawamoto, Y. Kawasumi, T. Kurota, K. Masai, K. Narihara, Y. Ogawa, K. Ohkubo, S. Okajima, T. Ozaki, M. Sasao, K. N. Sato, T. Seki, F. Shimpo, H. Takahashi, S. Tanahashi, Y. Taniguchi, and T. Tsuzuki, Phys. Rev. Lett. 64, 1895 (1990).
‘K. H. Burrell, S. L. Allen, G. Bramson, N. H. Brooks, R. W. Callis, T. N. Carlstrom, M. S. Chance, M. S. Chu, A. P. Colleraine, D. Content,
J. C. DeBoo, R. R. Dominguez, S. Ejima, J. Ferron, R. L. Freeman, H. Fukumoto, P. Gohil, N. Gottardi, C. M. Greenfield, R. J. Groebner, G. Haas, R. W. Harvey, W. W. Heidbrink, F. G. Helton, D. N. Hill, F. L. Hinton, R.-M. Hong, N. Hosogane, W. Howl, C. L. Hsieh, G. L. Jack- son, G. L. Jahns, R. A. James, A. G. Kellman, J. Kim, S. Kinoshita, L. L. Lao, E. A. Lasarus, P. Lee, T. LeHecka, J. Lister, J. M. Lohr, P. J. Lomas, T. C. Lute, J. L. Luxon, M. A. Mahdavi, K. Matsuda, H. Matsumoto, M. Mayberry, C. P. Moeller, Y. Neyatani, T. Ohkawa, N. Ohyabu, T. N. Osborne, D. 0. Overskei, T. Ozeki, W. A. Peebles, S. Perkins, M. Perry, P. I. Petersen, T. W. Petrie, R. Philipono, J. C. Phillips, R. Pinsker, P. A. Politzer, G. D. Porter, R. Prater, D. B. Remsen, M. E. Rensink, K. Sakamoto, M. J. Schaffer, D. P. Schissel, J. T. Scoville, R. P. Seraydarian, M. Shimada, T. C. Simonen, R. T. Snider, G. M. Staebler, B. W. Stallard, R. D. Stambaugh, R. D. Stav, H. St. John, R. E. Stockdale, E. J. Strait, P. L. Tailor, T. T. Taylor, P. K. Trost, A. Turnbull, U. Stroth, R. E. Waltz, and R. Wood, in Plasma Physics and Controlled Nuclear Fusion Research (International Atomic Energy Agency, Vienna, 1989), Vol. I, p. 193.
4R. J. Groebner, K. H. Burrell, and R. P. Seraydarian, Phys. Rev. Lett. 64, 3015 (1990).
5T. Yu. Akatova, L. G. Askinazi, V. I. Afanas’ev, V. E. Golant, V. K. Gusev, E. R. Its, S. V. Krikunov, S. V. Lebedev, B. M. Lipin, K. A. Podushnikova, G. T. Razdobarin, V. V. Rozhdestvenskij, N. V. Sa- harov, A. S. Tukachinskij, A. A. Fedorov, F. V. Chemyshev, S. P. Jaroshevich, L. Kryska, and P. Dvoracek, in Plasma Physics and Con- trolled Nuclear Fusion Research (International Atomic Energy Agency, Vienna, 1991), Vol. I, p. 509.
‘A. N. Arbuzov, L. G. Askinazi, V. I. Afanas’ev, V. E. Golant, V. K. Gusev, E. R. Its, A. A. Korotkov, S. V. Krikunov, B. M. Lipin, S. V. Lebedev, K. A. Podushnikova, G. T. Razdobarin, V. V. Rozhdestven- skij, N. V. Saharov, A. S. Tukachinskij, A. A. Fedorov, and S. P. Jaroshevich, in Proceedings of the 17th European Conference on Con- trolled Fusion and Plasma Heating, edited by G. Briffod, A. Nijsen-Vis, and F. C. Schluller (European Physical Society, Amsterdam, 1990), Vol. 14B, Pt. 1, p. 299.
‘L. G. Askinazi, V. E. Golant, S. V. Lebedev, V. A. Rozhansky, and M. Tendler, Nucl. Fusion 32, 271 (1992).
*R. J. Taylor, M. L. Brown, B. D. Fried, H. Grote, J. R. Liberati, G. J. Morales, P. Pribyl, D. Darrow, and M. Ono, Phys. Rev. Lett. 63, 2365 (1989).
9R. VanNieuwenhove, G. VanOost, R. R. Weynants, J. Boedo, D. Bora, T. Delvigne, F. Durodie, M. Gaigneaux, D. S. Gray, B. Giesen, R. S. Ivanov, R. Leners, Y. T. Lie, A. M. Messiaen, A. Pospieszczyck, U. Samm, R. P. Schom, B. Schweer, C. Stickelmann, G. Telesca, and P. E. Vandenplas, in Proceedings of the 18th European Conference on Con- trolled Fusion and Plasma Physics, edited by P. Backmann and D. C. Robinson (European Physical Society, Berlin, 1991), Vol. 15C, Pt. 1, p.
t”z5Biglari, P. Diamond, and P. Terry, Phys. Fluids B 2, 1 ( 1990). “V. Rozhansky and M. Tendler, Phys. Fluids B 4, 1877 (1992). ‘*B. V. Kuteev, “Voprosy Atomnoj Nauki i Tehniki,” ser. “Termoyad.
Sintez” 3, 242 (1986) (in Russian). 13L. G. Askinazi, V. E. Golant, E. R. Its, D. 0. Komeyev, S. V.
Krikunov, B. M. Lipin, S. V. Lebedev, K. A. Podushnikova, V. V. Rozhdestvenskij, N. V. Sakharov, S. P. Jaroshevich, L. Kryska, and P. Dvoracek, in Proceedings of the 18th European Conference on Controlled Fusion and Plasma Physics, edited by P. Backmann and D. C. Robinson (European Physical Society, Berlin, 1991), Vol. 15C, Pt. 1, p. 401.
14B. Coppi and N. Sharky, Nucl. Fusion 21, 1363 (1981). “A. Yu. Dnestrovsky, L. E. Zaharov, G. V. Pereverzev, K. N. Tarasyan,
P. N. Yushmanov, and A. R. Polevoj (Kurchatov Institute Atomic Energy, Moscow, 1991) (in Russian).
16M. Tendler and V. Rozhansky, Comments Plasma Phys. Control. Fu- sion 13, 191 (1990).
“K. C. Shaing K. G. and E. C. Crume, Phys. Rev. Lett. 63,2369 ( 1989). ‘*H. L. Berk and K. Molvig, Phys. Fluids 26, 1385 (1983). 19M. Tendler, U. Daybelge, and V. Rozhansky, in Proceedings of the 14th
International Conference on Plasma Physics and Controlled Fusion, Wiinburg, Germany, 1992 (International Atomic Energy Agency, Vi- enna), Paper No. IAEA-CN-56/D-4-8.
2426 Phys. Fluids B, Vol. 5, No. 7, July 1993 Askinazi et al. 2426
Downloaded 28 Nov 2012 to 203.230.125.100. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions
ne
Er
Extension: Represen3ng par3cle injec3on Important parameters: δΓSMBI(ISMBI, τSMBI, xdep, Δx, t) ISMBI: par3cle injec3on intensity τSMBI: dura3on of par3cle injec3on
xdep: deposi3on point of injec3on Δx: par3cle deposi3on layer width
∂n∂t− (original term) = δΓSMBI
∂T∂t
− (original term) = −Trefnref
#
$%%
&
'((δΓSMBI
Cooling due to par3cle injec3on
→ Limita3ons of Model Specific to injec3on: -‐ No abla3on, ioniza3on, etc. Injec3on is instantaneous ⇒ 3me delay related to ioniza3on, etc. not accurately represented.
-‐ Source asymmetry ⇒ toroidal and poloidal -‐ Vφ not evolved⇒ model does not include possible benefit from reduc3on in rota3on.
General: -‐ Need separately evolve Te, Ti and ion, electron hea3ng ⇒ low PT(n) behavior
-‐ Generalize turbulence model: ITG+TEM 27
A. ) Model PREDICTS par3cle injec3on-‐induced transi3on Case 1: shallower injec3on triggers turbulence collapse.
Case 2: deeper (xdep=0.95) injec3on does not trigger turbulence collapse.
• Peak of edge <VE>’ is a key. • No apparent evidence for ZF in transi3on!?
Details of Par3cle injec3on-‐induced L→H Transi3on
Change of Profile, (increase of edge grad n and decrease of edge grad T), leads to edge <VE>’ peak.
Correspondence of s3mulated L-‐H transi3on to the dynamical system analyses in Kim-‐
Diamond model.
p’
I
∂t I = 0∂tE0 = 0
∂t "p = 0 for low Q for high Q
Stable H-‐mode
∂t "p = 0 dQ
Stable L-‐mode
B.) Model PREDICTS Effec3ve Hysteresis
31
turbulence
ZF
<VE>’2
Case 3: dQ=.7, ISMBI=100 Strong single injec3on into subcri<cal state can trigger a transient turbulence collapse.
��������
(a)$turbulence�
(b)$ZF�
(c$)$<VE>’2�
Case 4: Sequen<al, repe<<ve injec3on into subcri3cal state can sustain turbulence collapse. ⇒ driven, or ‘ s3mulated H-‐mode’
→ Specula3on: Sequen3al injec3on can enhance effec3ve hysteresis, facilita3ng control of H→L back transi3on.
Correspondence of the driven H-‐mode, to the Kim-‐Diamond model,
in the dynamical systems.
p’
I
∂t I = 0∂tE0 = 0
∂t "p = 0 for low Q for high Q
Unstable H-‐mode
∂t "p = 0
dQ
Stable L-‐mode
Repe33ve par3cle injec3on sustains the unstable H-‐mode.
Op3mizing par3cle injec3on: SMBI VS gas-‐puffing
• Compare the total number of par3cles necessary to trigger L-‐H transi3on by SMBI or gas-‐puffing. 1. SMBI(τSMBI=250cs/a): moderate par3cle injec3on for rdep -‐> 1 ⇒ ΔNSMBI=0.083
2. Gas-‐puffing(τgp=3000cs/a): No injec3on, but par3cle source increment (sta3c and pulsed) i.e. S0-‐> S+δS0 ⇒ ΔNgp=1.08
• Intense, rapid par3cle injec3on is op3mal for transi3on, as opposed to slow gas puffing.
– Because par3cle injec3on causes a large change in edge profiles and <VE>’, which is essen3al for the transi3on.
自発遷移と誘起遷移の定性的な比較
34
RT ≡ vrE vθE ∂ V⊥ / ∂r γeffET =α0E0 / (γL − IΔω)
RH ≡ VE & / γeff =αVEV / (γL − IΔω)
Spontaneous transi3on: RT(edge) leads RH prior to transi3on
S3mulated transi3on: RT(edge) and RH peak simultaneously, at transi3on.
Normalized Reynolds Work
Normalized Shearing Rate
edge Injec3on centroid Dura3on
Order→ Causality: ZFs do not play a cri3cal role in s3mulated transi3ons.
What do we learn? • Subcri3cal transi3ons can indeed occur.
– Zonal flow do not play a key role in such fueling-‐induced transi3ons, in contrast to their contribu3on to spontaneous transi3ons.
• The crucial element for a subcri3cal transi3on appears to be how the injec3on influences the edge <V’E>.
• Below a certain power, subcri3cal injec3on can induce a transient turbulence collapse which later relaxes back to L-‐mode. – However, repe33ve injec3on can sustain subcri3cal improved H-‐mode states.
• Intense, rapid par3cle injec3on is op3mal for transi3on, as opposed to slow gas puffing. – We can op3mize the way to achieve the H-‐mode by facilita3ng par3cle injec3on (depth, intensity, periodicity, etc.)
Concluding remarks: • Mul3ple access to the H-‐mode.
– For spontaneous transi3on, ZF is fundamentally a key to trigger L-‐H transi3on.
• Macro dynamics of PL-‐H(n) can connect with the micro physics of γZF. – For s3mulated case, ZF does not play a role on triggering the L-‐H transi3on.
• Despite of the simple model, obtained implica3ons comprehend the essence of phenomena. – Future works diverge. Selected number of works are effec3ve.
• Toroidal momentum dynamics • SOL-‐edge coupling, i.e. boundary condi3on.
SUPPL. FURTHER MODELS
Further model works • FM3 model (Fundamenski NF ’12)
– SOL Turbulence: LH transi3on happens when
→ Parallel Alfvenic 3me in edge becomes comparable to the perpendicular electron transport 3me. → Enhance inverse cascade and ZF(?)
à Still speculative and underlying physics unclear but..
à Suggests LH transition may be affected by outside (i.e. SOL) à boundary condition?
Phenomenology: Why power threshold is lower for LSN than for USN? Heuris3cs: Interplay of magne3c shear ŝ and ExB shear induced eddy <l<ng Thus, residual Reynolds stress:
39
5 69-TH/P4-02
of ZFs, making it di�cult to recover the H-mode, should a back-transition occur. Thissuggests that proper wall conditioning, or reduction of wall impurity saturation, is neces-sary throughout the long pulse H-mode operation, because the ZF shearing necessary totrigger the transition may be more di�cult to achieve.
3 Physics of the rB-Drift Asymmetry.
Figure 6: Poloidal cross sectionof a LSN shaped plasma. Themagnetic shear e↵ect is noticeableover the whole surface, and theflux expansion close to the X-point.
Figure 7: (a) Radial profile of electro-static velocity for unfavourable (lightgray) and favourable (dark grey)configurations. (b) Associatedelectric shear.
One of the most persistent puzzles in L!H tran-sition phenomenology is why power threshold arelower for LSN configurations (with rB drift intothe X-point) than for USN configurations (withrB-drift away from the X-point). Here, we briefly sum-marise recent progress on a model which links thisasymmetry to the interplay of magnetic shear andE ⇥ B shear induced eddy tilting and its a↵ectionon Reynolds stress generated E ⇥ B flows.
Simply put, either magnetic or electric fieldshear acts to tilt eddys. Thus, the local radial wavenumber is given by,
kr
(✓) = kr
(✓0) + [(✓ � ✓0)s� V 0E
⌧c
]k✓
(1)
where the first term is magnetic shear tilting, whichvaries with angle ✓, and the second term is E ⇥ Bshear tilting which grows in time. Hereafter, we take⌧ = ⌧
c
, the correlation time and thus the eddy lifetime and ✓0 = 0. Given the structure of k
r
(✓), thenon-di↵usive (i.e. ’residual’) Reynolds stress is
hvr
v✓
i = hvr
2(0)iF 2(✓)[�✓s
+ V 0E
⌧c
] (2)
The ✓-dependence of eddy tilting is shown inFig. (6). Now the ’point’ is readily apparent fromobserving that h✓F 2(✓)i
✓
will trend to vanish unlessthere is an imbalance between contributions to theflux surface average from ✓ > 0 and ✓ < 0 – i.e.an up-down asymmetry, as for LSN vs USN! Theremaining question is to determine when the mag-netic shear induced stress adds or subtracts to theE⇥B shear induced stress and the related flow pro-duction. To do that, the electric field shear must becomputed self-consistently, by solving the poloidal momentum balance equation:
Vanishes upon flux surface average, but for X-‐point-‐induced asymmetry between ±θ. → An up-‐down asymmetry effect appears!
[N. Fedorczak, et al., Nucl. Fusion 52, 103013 (2012).]
Physics of the ∇B-‐Driq Asymmetry
Magne3c shear ExB shear
69-TH/P4-02 4
3 Physics of the rB-Drift Asymmetry.
Figure 3: (a) Radial profile ofelectrostatic velocity for unfa-vourable (light gray) and favou-rable (dark grey) configurations.(b) Associated electric shear.
One of the most persistent puzzles in L!H transi-tion phenomenology is why power threshold are lowerfor LSN configurations (with rB drift into the X-point)than for USN configurations (with rB-drift away fromthe X-point). Here, we briefly summarise recent progresson a model which links this asymmetry to the interplayof magnetic shear and E ⇥ B shear induced eddy tilt-ing and its a↵ection on Reynolds stress generated E⇥Bflows[8]. Simply put, either magnetic or electric fieldshear acts to tilt eddys. Thus, the local radial wavenumber is given by, k
r
(✓) = kr
(✓0)+ [(✓� ✓0)s�V 0E
⌧c
]k✓
,where the first term is due to magnetic shear tilting,which varies with angle ✓, and the second term is E⇥Bshear tilting, which grows in time. Hereafter, we take⌧ = ⌧
c
, the correlation time and thus the eddy life time,and ✓0 = 0. Given the structure of k
r
(✓), the total non-di↵usive (i.e. ’residual’) Reynolds stress is
hvr
v✓
i = hvr
2(0)iF 2(✓)[�✓s+ V 0E
⌧c
] (1)
Now the point here is readily apparent from observing that h✓F 2(✓)i✓
will trend tovanish unless there is an imbalance between contributions to the flux surface average from✓ > 0 and ✓ < 0 – i.e. an up-down asymmetry, as for LSN vs USN! The remaining questionis to determine when the magnetic shear induced stress adds or subtracts to the E ⇥ Bshear induced stress and the related flow production. To do that, the electric field shearmust be computed self-consistently, by solving the poloidal momentum balance equation:@t
hv✓
i+ @r
hvr
v✓
i = ��CX
hv✓
i where hvr
v✓
i = ��✓
@r
hv✓
i+ ⇧s
+ ⇧V
0Eso
@t
v✓
+ @r
(⇧s
+ ⇧V
0E) = �(�
CX
� @r
�✓
@r
)[VE
+ V⇤i] (2)
Here we have used radial force balance while neglecting toroidal flow, have accountedfor turbulent viscosity (�
✓
) and frictional damping (�CX
), and retained magnetic shear(⇧
s
) and electric field shear (⇧V
0E) driven residual stresses. Of course, hV
E
i0 in the lat-ter also must satisfy radial force balance. Equation (??) is solved while imposing theboundary conditions E
r
= �3@r
Te
(i.e. determined by SOL physics) at the LCFS andVE
= �V⇤i in the core. Assuming GB turbulence and using standard parameters, wecalculate V
E
/cs
and V 0E
, as shown in Fig. (??). It is readily apparent that favourable(i.e. LSN) configurations give a larger and stronger edge electric field shear layer than dounfavourable (i.e. USN) configurations. The e↵ect is significant – maximum shears aretwice as strong for LSN than for USN. The corresponding Reynolds force is obtained, too.We also note that the e↵ect is not poloidally symmetric, when variation in ✓ is considered.Thus, the flux surface averaged Reynolds stress cannot be inferred from a single pointmeasurement.
To determine self-‐consistent ExB shear induced by the magne3c shear ŝ stress, poloidal momentum balance is used.
40
69-TH/P4-02 6
@t
hv✓
i+ @r
hvr
v✓
i = ��CX
hv✓
i (3)
where
hvr
v✓
i = ��✓
@r
hv✓
i+ ⇧s
+ ⇧V
0E
(4)
so
@t
v✓
+ @r
(⇧s
+ ⇧V
0E) = �(�
CX
� @r
�✓
@r
)[VE
+ V⇤i] (5)
Figure 8: Radial derivative of theReynolds stress across the profile for afavourable configuration with aballooning envelope, averaged overthe flux surface (full curve) or takenlocally at the outboard midplane(dashed line).
Here we have used radial force balance whileneglecting toroidal flow, have accounted forturbulent viscosity (�
✓
) and frictional damp-ing (�
CX
) and retained by magnetic shear (⇧s
)and electric field shear (⇧
V
0E) driven residual
stresses. Of course, hVE
i0 in the latter alsosatisfies radial force balance. Equation (5)is solved while imposing the boundary condi-tions E
r
= �3@r
Te
(i.e. determined by SOLphysics) at the LCFS and V
E
= �V⇤i in thecore. Assuming GB turbulence and using stan-dard parameters, we calculate V
E
/cs
and V 0E
, asshown in Fig. (7). It is readily apparent thatfavourable (i.e. LSN) configurations give a largeand stronger edge electric field shear layer thando unfavourable (i.e. USN) configurations. Thee↵ect is significant – maximum shears are twice as strong for LSN than for USN. The cor-responding Reynolds force is plotted in Fig. (8). We also note that the e↵ect is notpoloidal symmetric, when variation in ✓ is considered. Thus, the Reynolds stress cannotbe inferred from a single point measurement.
Future work will aim to integrate this analysis into the model discussed in Section 2.Along the way, the treatment of the SOL boundary conditions will be generalised.
4 Intrinsic Variability, Heat Avalanches, and Noisey
Transitions
It is well known that the L!H transition exhibits significant variability. The scatter inPT
trends is large. Also, many anecdotes exist describing sudden transitions in stationary,slightly subcritical states. In particular, I-phase!H-phase transition appear noisey andunpredictable. Many observations have been reported, which describe prolonged (i.e.
ŝ induced residual stress
Er driven residual stress
Fric3onal damping
Turbulent viscosity
Boundary condi3on: determined by SOL physics
69-TH/P4-02 6
@t
hv✓
i+ @r
hvr
v✓
i = ��CX
hv✓
i (3)
where
hvr
v✓
i = ��✓
@r
hv✓
i+ ⇧s
+ ⇧V
0E
(4)
so
@t
v✓
+ @r
(⇧s
+ ⇧V
0E) = �(�
CX
� @r
�✓
@r
)[VE
+ V⇤i] (5)
Figure 8: Radial derivative of theReynolds stress across the profile for afavourable configuration with aballooning envelope, averaged overthe flux surface (full curve) or takenlocally at the outboard midplane(dashed line).
Here we have used radial force balance whileneglecting toroidal flow, have accounted forturbulent viscosity (�
✓
) and frictional damp-ing (�
CX
) and retained by magnetic shear (⇧s
)and electric field shear (⇧
V
0E) driven residual
stresses. Of course, hVE
i0 in the latter alsosatisfies radial force balance. Equation (5)is solved while imposing the boundary condi-tions E
r
= �3@r
Te
(i.e. determined by SOLphysics) at the LCFS and V
E
= �V⇤i in thecore. Assuming GB turbulence and using stan-dard parameters, we calculate V
E
/cs
and V 0E
, asshown in Fig. (7). It is readily apparent thatfavourable (i.e. LSN) configurations give a largeand stronger edge electric field shear layer thando unfavourable (i.e. USN) configurations. Thee↵ect is significant – maximum shears are twice as strong for LSN than for USN. The cor-responding Reynolds force is plotted in Fig. (8). We also note that the e↵ect is notpoloidal symmetric, when variation in ✓ is considered. Thus, the Reynolds stress cannotbe inferred from a single point measurement.
Future work will aim to integrate this analysis into the model discussed in Section 2.Along the way, the treatment of the SOL boundary conditions will be generalised.
4 Intrinsic Variability, Heat Avalanches, and Noisey
Transitions
It is well known that the L!H transition exhibits significant variability. The scatter inPT
trends is large. Also, many anecdotes exist describing sudden transitions in stationary,slightly subcritical states. In particular, I-phase!H-phase transition appear noisey andunpredictable. Many observations have been reported, which describe prolonged (i.e.
at LCFS → Favorable (LSN) configura3on yields a
stronger edge V’E than does unfavorable (USN) configura3on, other parameters comparable
→ LSN vs USN asymmetry explained.
R R
Another candidate: Pfirsch-‐Schlüter currents [Aydemir ‘12 NF]
