A Collection of Slides To Stimulate Discussion on the

Simulation of Feedback Stabilization of Neoclassical

Tearing Modes

S.E. Kruger, D.D. Schnack, C.C. Hegna, E.D. Held

Where Do We Want To Be?

Driving question:

How much RF power does ITER need to stabilize the expected performance-limiting

NTMs?

What do we need to answer that question?• Codes that can simulate these nonlinear instabilities in realistic geometry

• Fluid models to handle the disparate time scales of NTMs

• Accurate and reliable closures for tokamak problems• Codes to accurate model RF driving fields and how the energy and momentum is deposited into the plasma

• Integration of RF sources into fluid model

What is Needed To Model NTM’s?Current Understanding from Analytic

Theory• Experimental/Theory Comparisons Framed In Terms of Modified Rutherford Equations:

Hegna PoP 6 (1999) 3980Lutjens et.al. PoP 8 (2001) 4267Waelbroeck

€

dW

dt=k 0η

* Δ* + k1

DR /(α s − H)

W 2 + 0.65Wd2

+DncW

W 2 + Wd2

+Dpol

W 3

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

MHD e Only valid with Aniso Heat Conduction

Anisotropic Heat Conduction

Two-fluid

I (both neo.,gy

ro.

Hard part (theoretically): e

Minimal requirement: Get same physics as analytic theories

Desired goals: Include kinetic effects in fundamental way at realistic parameters

To date: Simulations have only included e and anisotropic heat conduction

Features of NTMs Make It Challenging Computationally

Mode is inherently nonlinear Need to simulate as initial-value problem

ITER-relevant experiments are in “long-pulse” regime:

Plasma is near marginality at all timesModes are slow growing: ~100 msec to

saturation

“Near marginality” is difficult for initial-value codes

Sensitive to equilibriumComputationally expensive

Simple cases have been done since 1996. Most challenging case attempted to date:

Try and obtain seeded island from sawtooth with approximate closure scheme

(don’t need to go all the way to saturation).

Simulations of DIII-D #86144 Show Resistive MHD Computations Cannot Explain

Observations

•Secondary islands are much smaller than experimentWexp ~ 6-10 cm

• 3/2 island width decreases with increasing SSexp ~ 108

1/1

2/1

3/2

Plasma Parameters Severely Constrained By Ordering of Length Scales

•Need Wd >> v (lin. layer) High S

•High S means smaller secondary island Need smaller threshold

•To get smaller threshold, need higher anisotropy

•Anisotropy also slows 1/1 growth rate

•Quickly leads into realistic plasma parameters

Nonlinear NTM calculations are extremely challenging!

⇒

⇒

Lessons Learned From All Previous Simulations

•Heuristic closure works•Parallel heat conduction works

•Simulations of DIII-D Experiments• Parameters needed are aggressive• Cannot easily decouple 1/1 physics from seed island physics

•Classical tearing mode studies• Experience using experimental equilibria is robust, and can be used to help guide how to choose model equilibria

• Many experiments displaying NTM’s are near ideal limit

’~0 is important

•Improved Fluid Models

• Two-fluid to get the effect of Dpol term

• Focus of current CEMM SciDAC• Almost there!

•Improved closures• Local closures

•Landau fluid heat flux•Heuristic closure

• Non-local closure•Improve heat flux calculation•Develop non-local stress tensor that contains trapped particle effects

•DEKIS

Near-Term Development for NTM Physics

Briefly discussed next

Not discussed

• Has “neo-classical” effects relevant for long mean-free-path regime:

• Poloidal flow damping• Enhancement of polarization current• Bootstrap current • Neoclassical resistivity

• Has other nice features:• Dissipative and energy conserving• No toroidal damping

•To Do List:• Repeat previous MHD/NTM calculations with updated algorithm• Investigate “2x2” moment version of heuristic closure

“Heuristic Closure” Meets Minimal Requirements

• Simplified model captures most neo-classical effects (T. A. Gianakon, S. E. Kruger, C. C. Hegna, Phys. Plasmas 9 (2002) 536)

20 2

0

m n Bθαμ θα α α αθ⎛ ⎞

⎜ ⎟⎝ ⎠

⋅∇∇⋅ = ∇⋅∇

v

B

Benchmarked against modified Rutherford Equation:

Non-local Closure for Heat Flux More Rigorously Includes Electron Physics

•Parallel dynamics of electrons is fast - on MHD time scale they have farther than an MHD length scale

•Capturing this physics via a Chapman-Enskog-like method:

If K(L,L’)=1,

•Gives arbitrary collisionality, geometry effects

To Do:Improve numerical implementation

Implement Landau-like closure

€

q|| =neqv th

π 3 / 2K(L',L) T(−L') − T(L')[ ]∫ dL'

€

) q || ≈

neqv th

π 3 / 2

k||

k||

) T

Landaufluid closures

Held, PP 8, 1171 (2001)Held, PP 11, 2419 (2004)

Non-local Closure Can Be Extended To Calculate Parallel Stress Tensor

•In a manner similar to parallel heat flux:

To Do:Complete inclusion of trapped particle effectsImplement numerically

€

|| = nmμ ||

∂u||

∂L

Held, PP 10, 4708 (2003)

RF Feedback Stabilization of NTM’s a Critical

Part of ITER’s Planned Operation

Center island current out of

plane

•Localized electron momentum deposition differential on the particles

•Currents can be induced• Local: produces helical current that affect island region physics

• Global: counteract island drive (′, q)

•Not much current required (IRF/Iplasma ~ 3%)

€

dw

dt=

η

μ0

′ Δ + ...+ ΔRF (w)[ ]

See: Rutherford, Varenna 86; Kurita, NF 94; Hegna, PP 97; Lazzaro, PP 96; Perkins, EPS 97; Giruzzi NF, 99

Rutherford equation shows that effects can be treated independently:

Can study RF stabilization of islands without NTM physics

Probably easiest MHD mode to study effects of RF sources

Work Already Exists on Computational Modeling of RF Stabilization of Tearing

Modes

•Simplest model:

•Slightly more complicated (developed by Giruzzi):

•Models fit well into fluid approximation• Numerical analysis poorly understood, but in general time scale of QL diffusion is slow (explicit OK?)

•Where do we go from here?• How do we integrate with CQL3D or other RF code?• Do the hot particles modify the neoclassical closures

€

E + V × B = η (J − JRF (x))

€

E + V × B = ηJ + η RF JRF (x, t))

€

∂JRF

ζ

∂t=

1

ωe

B • ∇P −ν f J RF

ζ − SRF + D ∇ 2JRF

ζ + E ||ne2

me

Yu et.al., PP 2000

Gianakon, PP 2001

Where Do We Go From Here?Short Term (Post-doc)

•Produce a more rigorous Giruzzi-like current source equation

•Implement simplified analytic model into NIMROD• Source from ECCD modeling code? (TORAY, …)• Analytic source term?

Needs to be function of time and 3D space

•Perform simulations of feedback stabilization of classical tearing mode (“JCP case”)

time (s)

En(J)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

10-6

10-4

10-2

100

102

104

106

n=2

n=1

n=0 + ss

Where Do We Go From Here?Longer term

•Understand relationship of RF hot particle distribution and NTM closures in more detail.• What is best way of adding RF sources to fluid equations?

SRF(x,t), SRF(x), JRF(x,t)•Integrate RF/Classical TM studies with CEMM NTM studies•RF and ions? (Much harder. Not necessary for NTM, but …)

•Computer science aspects• What is the best way of interfacing RF codes to MHD codes?

•Applied Math• What is numerical stability properties of coupled system?

• Fluid codes could always use better solvers

Dalton is leading effort to write white paper on an NTM roadmap. If you are interested in participating,

please email.

Backup Slides

Why is the 3/2 seen instead of the 2/1 mode?

Why does the 3/2 mode appear without an obvious “seed”?

Why does the 3/2 mode appear on the nth sawtooth?

Computational Studies Aim to Answer Experimental Questions

Modeling of Instabilities Use Extended MHD Models

[ ]( 1) ( :)T

n V T n V QT V qtα

α α α α α α α α

∂γ

∂⎛ ⎞

+ ⋅∇ = − − ∇⎜ ⎟⎝ ⎠

+ Π ∇ + +∇g g

•Momentum Equation

• Generalized Ohm’s law:

pt

∂ρ∂

⎛ ⎞+ ⋅∇ = × −∇⎜ ⎟⎝

∇⋅⎠

−v

v v J B

{ { ( )

( ) ( )

Re

20

1 1 1

1

1

(1 )

(1 )

1

(1 )

e i

Hall Effect DiamagneticEffe

Ide

pe

Electron Inert

al MHD sisti

e i

Closures ia

t

ve MHD

c

p pne

n

n

t

e

e

ν

∂

ε ω νν

ν

ην

∂

νν

− ∇⋅ ⎡ ⎤+ +∇⋅Π

−+

− +⎢

× − ∇ −

⎥+

+=

⎣+

+

⎦

− × +

Π

E v

JvJ J

B BJ J

v

1 44 2 4 43 1 4 4 4 2 4 4 4 3

1 4 4 44 2 4 4 4 43 1 4 24 4 4 44 4 4 4 4 4 43

• Temperature Equations:

Experimentally RF Successfully Stabilized

NTM’s and Other MHD Instabilities

ICRF modifies sawteeth in C-Mod

S.J. Wukitch et al., PP, 2005

ECCD suppresses NTM’s in DIII-D

R. Prater et al., NF, 2003

ICRF modifies ELM behavior in JETG.P. Maddison et al PPCF 2002

Anisotropic Heat Conduction Needed For Experiments Possible With High-Order

Elements

•Perpendicular thermal conduction can compete with parallel thermal conduction at rational surfaces where parallel gradient = 0.

•Analytic predictions from balancing two terms give scale length that scales as (||perp))-1/4

[Fitzpatrick, PoP 2, 825 (1995)]

•Fit of computational results is ||perp=3.0x103 (wd /a)-4.2.

•Result is for toroidal geometry.

•Run with biquartic elements. wd (cm)

χ||/χperp

2 3 4 5 6 7

108

109

1010