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)

€

˜ 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:

€

∂∂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

€

˜ 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).