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A Proposal for a SWIM Slow-MHD3D Coupled Calculation of the Sawtooth Cycle

in the Presence of Energetic Particles

Josh BreslauGuo-Yong FuS. C. Jardin

Princeton UniversityPlasma Physics Laboratory

presented at the SWIM WorkshopUniversity of Wisconsin

Dec 4th, 2007

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Overview• Sawteeth will occur in ITER with a large energetic particle

population present (due to α-particles)– there is presently no comprehensive 3D model of this

• The M3D (and NIMROD) codes should be able to model this phenomena with minor modifications

• Recent successes in Validation and Verification of these codes will give credibility to this modeling

• We propose to perform this as part of SWIM as it involves the coupling of a heating system and MHD– future step could involve RF as well

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M3D-K Fluid/Particle Hybrid Model

, ,

hinM pt

tp

p p St

PV

V V J B

BE J B E V B J

V V

,

(

(

)

( ) ) ( )

h

i i ii

f R R v v

P P P

P I bb

0**

*0**

* ** *0 0

1(

(

,

i

i

dv B B

dt B q

dv qM B B

dt B q

M vB

q

*0

0

RB b E

B E

B B B b B b

MHD Equations

CGL Pressure Tensor

Gyrokinetic Equations

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Gyrokinetic equation (for particles) can use either f or full-f method

0 0f f f f g w

0

0

1dw f dfw

dt g f dt

0 0( , , )f f P E

0 0 0

1 1

d i

dPdf f dE f

dt dt P dt E

dE dev M B

dt dtdP Pdvd

Pdt dt dt v

E

R

in f, distribution function is sum of equilibrium + perturbed

“weight function” evolves in time

equilibrium distribution is a function of the adiabatic invariants

equilibrium evolves (slowly) as the adiabatic invariants for each particle change due to detailed particle trajectory motions

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M3D Verification and Validation

• Good agreement between M3D and NIMROD for CDX-U sawteeth simulations;

• Good agreement between M3D-K, NOVA2, and NIMROD for energetic particle stabilization of internal kink and excitation of fishbone;

• M3D-K results of beam-driven TAEs are consistent with NSTX observations: mode frequency and its chirping, mode saturation time scale.

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M3D and NIMROD 3D MHD Codes have successfully completed a major nonlinear benchmark: Sawtooth activity in a small tokamak.

• Figure shows kinetic energy vs time the first 10 toroidal modes for NIMROD and M3D with same initial conditions, sources, and boundary conditions

t1 t2 t1 t2

Flux surfaces (Poincaré plots)

t=t1 (NIMROD) t=t2 (NIMROD)t=t1 (M3D) t=t2 (M3D)

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M3D agrees with NOVA2 for n=1/m=1 mode with energetic particle effects

grow

th

rate

freq

uenc

y

ideal kink

fishbone

G. Fu

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Mode Structure: Ideal Kink v.s. Fishbone

G. Fu

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Non-linear single-mode TAE simulations reproduce many features observed in NSTX data

• M3D Nonlinear Hybrid simulations:

– Mode growth and decay times are approximately 50 - 100s

– Bursting/chirping results from:

• Non-linear modification of fast-ion distribution

• Change in mode structure

– Chirp f / f 20% - consistent with experiment

2

0

-2

t = 0.267

108530

200 s

Data

Simulation

n=2

n=2

t=0

t=336

G. Fu

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Recent M3D hybridsimulation results:

(1) Alpha particle stabilizationof n=1 kink in ITER;

(2) Nonlinear frequency chirpingof fishbone;

(3) Beam-driven TAEs in DIII-D;

(4) Beam-driven TAEs in NSTX;

(5) Beam-driven GAM in DIII-D

G.Y. Fu, 2004 IAEA Fusion Energy ConferenceG.Y. Fu et al., Phys. Plasmas, 2006

4m

1.2m

G. Fu

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As flattening region of distribution function increases, the mode frequency chirps down.

distribution

G. Fu

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Domain decomposition is same for particles as for fluid cells

Toroidal decomposition is ok..scales well

Poloidal decomposition is presently only 1D

Need to add radial decomposition as well

typical run will have 25,000 50 zones: 20 particles/zone = 25M particles

should scale well to 1000p

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Work Plan• Goal is to have initial results in the first year using the

current capability of M3D-K and to have more complete results by the third year;

• Preliminary run will just add “test particles” to existing sawtooth simulation as a sanity check.

• Need to upgrade M3D to have more general particle domain decomposition (to run on > 500 p) and to have sources and sinks of particles.

• Initial tests will determine if f (or some improved f method) is sufficient. Do we need to use full-f method ?

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Summary• Sawteeth will occur in ITER with a large energetic particle

population present (due to α-particles)– there is presently no comprehensive 3D model of this

• The M3D (and NIMROD) codes should be able to model this phenomena with minor modifications

• Recent successes in Validation and Verification of these codes will give credibility to this modeling

• We propose to perform this as part of SWIM as it involves the coupling of a heating system and MHD– future step could involve RF as well