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Calculations of Gyrokinetic Microturbulence and Transport for NSTX
and C-MOD H-modesMartha Redi
Princeton Plasma Physics Laboratory
Transport Task Force Meeting
April 2-5, 2003
Madison, Wisconsin
MOTIVATION: Investigate turbulent microinstabilities in NSTX and CMOD H-mode plasmas exhibiting unusual plasma transport - Remarkably good ion confinement and Resilient Te profiles on NSTX - ITB formation on CMOD- Identify underlying key plasma parameters
for control of plasma performanceAcknowledgement:
R. Bell, D. Gates, K. Hill, S. Kaye, B. LeBlanc, J. Menard, D. R. Mikkelsen, G. Rewoldt (PPPL)
C. Fiore, P. Bonoli, D. Ernst, J. Rice, S. Wukitch (MIT), W. Dorland (U. Maryland),
J. Candy, R. Waltz (General Atomics), C. Bourdelle (Association Euratom-CEA, France)
METHOD- GS2 and GYRO flux tube simulations- Complete electron dynamics. 3 radii, 4 species.- Linear electromagnetic; nonlinear, electrostatic calculations (CMOD)
Gyrokinetic Model Equations
Kotschenreuther, et al Comp. Phys. Comm. 88 128 (1995)
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˜ Φ (r,θ,ζ , t) = exp[inζ − inq(r)θ] ˜ φ (θ − 2πp,r, t)exp[inq(r)2πp]p=−∞
p=∞
∑Perturbed electrostatic potential:
Linearized gyrokinetic equation, ballooning representation, “s-” MHD equilibrium:
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∂∂t
˜ g s + v//
qR
∂
∂θ˜ g s + iωds ˜ g s + C( ˜ g s) =
es
Ts
FmsJ0(∂
∂t+ iω*s
T )[ ˜ φ (θ) −v //
c˜ A //(θ)] + [δB //terms]
Where
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˜ g s ≡ ˜ f s + (es
Ts
)Fms˜ φ (θ),ω*s = (k⊥Ts /ZsB)(d lnNs /dr)
ωds = ω*s(Lns R)(E Ts)(1+ v //2 v 2){cosθ + [˜ s θ −α cosθ]sinθ}
kθ = −nq /r,k⊥ = kθ {1+ [˜ s θ −α sinθ]2}1 2
˜ s ≡ (r /q)(dq dr),α ≡ −q2R(dβ dr)
ω*sT ≡ ω*s{[1+ η s[E Ts) − 3 2]},J0 ≡ J0(k⊥v⊥ Ωs),
NSTX H-mode: Electron Temperature Profile Resiliency
During H-modeTe(r) remains resilientelectron density increasesion temperature decreases
Examine microinstability Growth rates at 3 zones
What clampsElectron temperature profile?
0 0.5 1.0
5
10
15
0 0.5 1.0
2
4
6
0 0.5 1.0
1.0
0.5
0 0.5 1.0
1.5
1.0
0.5
q profile: partial kinetic EFIT
NSTX: Examine Microinstability Growth Rates at 3 Zones
0
5 104
1 105
1.5 105
2 105
2.5 105
3 105
0 0.2 0.4 0.6 0.8 1
ITG Range of WavelengthsStronger growth near plasma edge
Weak instabilities insideMicrotearing modes observedLater plasma has stronger ITG
r/a
NSTX 108730ITG-TEM range of frequenciesH-mode
t=0.6 sec
t=0.4 sec
classicitg g(aky)andeven parityeigenfunction
microtearing g(aky)tearing parityeigenfunction
stable
numerical?
Circles denote ExB shearing rate ITG may be stabilized by shearing at all radii
0
5 105
1 106
1.5 106
2 106
0 0.2 0.4 0.6 0.8 1
ETG Range of WavelengthsStronger growth near plasma edge
Weak or stable modes insideLater plasma has weaker ETG
r/a
NSTX 108730ETG range of frequenciesH-mode
stable
t=0.4 sec
t=0.6 sec
classic etg g(aky)even parity eigenfunction
Circles denote ExB shearing rateETG stabilized by shearing rateexcept near edge at r/a=0.8
NSTX: Critical Gradient Below or At Marginal Stability for ITG
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15
Experimental Temperature Gradient
and above Marginal Stability for TEM Drift Modes.Drift mode with maximum growth rate
changes from ITG to TEM as grad Ti/a/Ti decreased.Find two critical gradients, for distinct ITG and TEM rootsExB shearing rate ~ maximum growth rate: ITG likely stable
a(grad Ti)/Ti
NSTX 108730t=0.6, r/a=0.8
Maximum growth rate
kperp rho-ifor fastest growingITG-TEM drift mode
TEM ModeCritical value
Measuredvalue
ITG ModeCritical Gradientvalue
ExB Shearing Rate ~ Maximum Growth Rate
0.2
0.4
0.6
0.8
1
1.2
1.4
Fastest Growing ITG Drift Mode Wavelengths
Drift Modes far below Marginal Stability when ExB Shearing Rate Subtracted
Hybrid root changes from ITG to TEM characterbelow experimental a(grad Ti)/Ti.
Change little as grad Ti/Ti is reduced
a(grad Ti)/Ti
NSTX 108730t=0.4 sec, r/a=0.8
Maximum ITG growth rate
ITG-TEM Critical
kperp rho-ifor fastest growingITG-TEM drift mode
Measured value
00 10 20
value
ExB Shearing Rate
NSTX: Far Above Critical Gradient for ETG ModesExB Shearing Rate<<Maximum Growth Rate
Fastest Growing ETG Drift Mode Wavelengthsand Growth Rates Decrease as gradTe/Te is Reduced
Higher Critical Gradient for ETG than ITG
0
20
40
60
80
100
120
0 5 10 15 20
a(grad Te)/Te
MaximumETG growth rate
Kperp-rho-i for fastest growing ETG drift mode
NSTX 108730t=0.4 sec, r/a=0.8
Measured ValueETG
Critical Gradientvalue
ExB Shearing Rate ~1/10 Maximum Growth Rate
0
20
40
60
80
100
120
0 5 10 15 20
a(grad Te)/Te
ETG CriticalGradientValue
MeasuredValue
MaximumETG Growth Rate
kperp rho-ifor fastest growingETG drift mode
NSTX 108730t=0.6, r/a=0.8
ExB Shearing Rate~1/4 Maximum Growth Rate
What is the Instability at 0.65r/a on NSTX?
What Effect Does It Have on Transport?
-2
-1
0
1
2
3
0 1 2 3 4 5
Character of fastest growing mode changes to ITG/TEM. This is an ETG-type microtearing mode, driven by (gradTe)/Te. If a(gradNs)/Ns and a(gradTi)/Ti=0, mode ~unchanged.
Scaling factor on a(gradTe)/Te
Growth Rate
Real Frequency
Summary: NSTX H-mode Gyrokinetic ResultsGood ion transport appears due to stabilized ITG
Poor electron transport and resilient Te profiles as yet unexplained
0.25
r/a
0.80
0.65
i eITG ETG
< neo stable stable
Likely ExB stabilized
stable
< neoExB stabilized Likely ExB
stabilized
ExB stabilized stable
< neo ExB stabilized unstable
Likely ExB stabilized
unstable
>> it=0.4s
t=0.6s
t=0.4s
t=0.6s
t=0.4s
t=0.6s
>> i
>> i
CMOD Internal Transport Barrier TRIGGER time: Examine Microinstability Growth
Rates at 3 Zones
Ne Te
Ni(deut)Ti
ITB Trigger Time:Linear, Electromagnetic Gyrokinetic Calculations with GS2:Drift wave Microturbulence at ki = 0.1 to 80.Low kI: ITG => I
anomalous outside ITB TEM and ITG: already stabilized at and within ITBHigh ki: ETG driven by strong Te => e
anomalous at and outside ITB
-25
-20
-15
-10
-5
0
5
0.1 1 10
Real frequencies (~10**6/sec)zones 5,9,13
kperp rho-i from 0.1 through 80
kperp rho-i
Plasma core
At ITB
Outside ITB
-1
0
1
2
3
4
0.1 1 10 100
Growth rates at zones 5,9,13for kperp rho-i from 0.1 to 80
ITG stabilized in plasma core and near ITBeta-i small, TE drive weak; ITG and TEM stable
~10**6/sec
kperp rho-i
Outside ITB
At ITB
Plasma core
Electron drift direction
Ion drift direction
NONLINEAR GS2 Simulations reproduce linear result ITB TRIGGER: Before ne peaks, region of reduced transport and stable ITG microturbulence is established without ExB shear
Quiescent, microturbulence in ITB regionModerate microturbulence in plasma coreHigh microturbulence level outside half-radius
Just inside ITB
Outside ITB
In plasma core
dV2
-Strongest driving force: grad Te/Te
NSTX r/a=0.8: ITG Range of FrequenciesOutside Core, ITG Range of FrequenciesGrowth Rates and Eigenfunction at Most Unstable Wavelength
NSTX r/a=0.65: ITG Range of Frequencies
Growth Rates and Eigenfunction of Most Unstable Mode - Tearing Parity
Growth rate (10^4/sec)
Real frequency (10^4/sec)electron diagmagnetic drift direction
kperp-rho-i
kperp-rho-i
Θ
SUMMARY:
Linear calculations of drift wave instabilities in the ion temperature gradient and electron temperature gradient range of frequencies Roughly consistent with improved ion confinement in NSTX andimproved confinement within and at ITB in CMOD H-mode plasmas
Remarkably good ion transport in NSTX H-mode (Gates, PoP 2002) would be expected from stable ITG throughout plasmaProfile effects (GYRO) may fully stabilize ITG everywhere. Electron transport => q monotonic so unstable ETG at all r…MSE?
Resilient temperature profiles on NSTX may be maintained through ETG instabilities, Nonlinear calculations needed. Tearing parity microturbulence found - in contrast to tokamaks - effects on transport to be determined.
Internal transport Barrier on CMOD appears after off-axis RF heating, where microstabilities quiescent. Nonlinear calculations in ~agreement with linear.Sawtooth propagation measurements confirm low transport in the region at the trigger time (Wukitch, PoP, 2002).