Overview of Physics Research in the SciDAC Center for

Simulation of Wave-Plasma Interactions

Paul Bonoli on behalf of the SciDAC Center for Simulation of Wave-Plasma Interactions

20th Meeting of the American Physical SocietyDivision of Plasma Physics

Dallas, TexasNovember 17-21, 2008

Poster CP6.00009

Participants in the Center for Simulation of Wave – Plasma Interactions (CSWPI)

L.A. Berry, D.B. Batchelor, D. Green,E.F. Jaeger, E. D`Azevedo, M. Carter

C.K. Phillips, E. ValeoN. Gorelenkov, H. Qin

P.T. Bonoli, J.C. WrightA. S. Richardson

R.W. Harvey, A.P. SmirnovN.M. Ershov

M. Choi

M. BrambillaR. Bilato

D. SmitheJ. Carlsson

D. D’Ippolito, J. Myra

Politecnico di Torino R. MaggioraV. Lancellotti

• Mathematical statement of the wave-particle interaction problem.

• 3-D field reconstructions for ICRF heating and current drive in ITER, NSTX, and Alcator C-Mod.

• Inclusion of finite ion-orbit width effects in ICRF simulations.

• Coupled full-wave / Fokker Planck simulations of lower hybrid current drive (LHCD).

• Analysis and simulation of nonlinear ICRF sheath effects.

Outline

antp iic

JJEE 00

2

2

ωμωε

ω−=⎟⎟

⎠

⎞⎜⎜⎝

⎛++×∇×∇−

I. Mathematical formulation of the Wave Propagation and Absorption Problem

( )( ) ( )tttEftddt s

t

sp ′′⋅′′′′= ∫∑∫

∞−

,,,,,)( ,0 rErrrr,J σ

The plasma current (Jp) is a non-local, integral operator (and non-linear) on the rf electric field and conductivity kernel:

The long time scale response of the plasma distribution functionis obtained from the bounce averaged Fokker-Planck equation:

For time harmonic (rapidly oscillating) wave fields E with frequency ω,Maxwell’s equations reduce to the Helmholtz wave equation:

Need to solve this nonlinear, integral set of equations for wave fields and velocity distribution function self-consistently. This requires an iterative process to attain self-consistency.

( ) RSft

++Γ⋅∇=∂∂

00 uu0λ where ),()( 00 fQfC Euu +=Γ⋅∇

Wave Solvers(AORSA)(TORIC)

SIGMAD

} Plasma Response(CQL3D)0

• Second harmonic tritium absorption in ITER using AORSA.

• Fast wave driven current density in NSTX using AORSA and TORIC.

• Ion cyclotron heating in Alcator C-Mod using coupled AORSA-CQL3D.

II. Reconstruction of 3-D ICRF wave fields from full toroidal mode spectrum

of antenna loops

Extension of ICRF Full-Wave / Fokker Planck Calculation to 3-D

• Sum the quasilinear diffusion coefficients (D_ql) over all toroidalharmonics of the antenna:

• Since D_ql is flux surface averaged, by Parseval’s theorem it can summed directly over nφ’s.

• AORSA utilizes 2048 processors for 8 hours on Franklin for one iteration with the Fokker Planck solver - writes a 4 GB file with Dql to be summed.

• Using the summed QL operator, CQL3D calculates the evolving ion distribution function(s) consistent with the 3-D fields.

• Process is repeated until convergence is achieved.

3-D visualization of the ICRF wave fields in ITER shows “hot spots” near the antenna surface where the wave amplitude is high

• AORSA simulation used 100 toroidal modes of the ICRF antenna.• Calculation done on 2048 processor cores in 2 hours on Jaguar facility.

Parameters for 3-D ICRF Simulation of ITER Scenario 2 Elmy H-Mode

• Plasma:– R0 = 6.2m, a = 1.85m– B0 = 5.3 T, Ip = 15 MA– Te(0) = 24.8 keV, Ti(0) =

21.2 keV31.7% D: 50% T: 2% Be4, 4.4% He4, 0.7% α’s

– ne(0) = 1.02 × 1020 m-3

• ICRF:– f0=53 MHz, PIC=20 MW

Absorption Results for 3-D ITER Simulation are Dominated by Electron Landau Damping

and Second Harmonic Tritium Absorption

• P(ELD) = 48.7%, P(2ΩT) = 42.7%, P(α)=5.9%, P(Be) =1.2%

Iteration Procedure for two Non-Maxwellian Ions

• Alternate the iterative steps between the two species:

• AORSA and CQL3D are first iterated for ion species #1, assuming the distribution function for species #2 remains fixed.

• Repeat the iteration for species #2, assuming the ion distribution function for sepcies #1 remains fixed.

• This process neglects collisions between the two developing energetic tails by assuming the density of the tails is low.

• Both ion tails are included self-consistently in the wave solution, and both contribute to the total power absorption.

Two Ion Species Calculation (AORSA/CQL3D) for ITER Scenario 2 with a 3% He3 Minority Component:

[f0=53 MHz, nφ=-27: P(He3) = 53%, P(T) = 7%, P(E) = 40% ]

r/a=0.27

• AORSA simulation of HHFW wave fields in NSTX using 101 toroidal antenna modes (k// = 8m-1)

• AORSA - 101 toroidal modes (26 kA)• TORIC – single mode (37 kA)• Experiment – (34 kA)

• Comparison above provides opportunity to validate full-wave solvers and current drive calculations.

3D Reconstructions of HHFW Fields for Current Drive Calculations in NSTX

AORSA- CQL3D Simulations of ICRF Minority Heating in Alcator C-Mod:

Single vs. Summed Toroidal Mode Analysiskφ Spectrum

Heating

Fields

Alcator C-Mod: Full-wave (AORSA) heating profiles show differences between single and summed modes

P(H) =0.526 MW

AORSA – 2D(nφ = 10)

P(H) =0.6614 MW

Iteration #4

H

2ΩcD

AORSA – 3D(nφ = -50 : 50)

H

2ΩcD

Self consistent quasilinear operator ( r/a = 0.306 ) shows significant “smoothing” of phase space structure with

multiple toroidal modes -> 3D field reconstruction is to essential to see this feature!

2D: Single mode (nφ = 10) 3D: All modes (nφ = -50 : 50) weighted by 2 strap antenna

v//N (mid-plane) v//N(mid-plane)

v⊥N (mid-plane) v⊥N (mid-plane)

Time Dependent Simulations of minority ICRF Heating in Alcator C-Mod have Begun

• TSC used to simulate a C-Mod discharge with minority (H) ICRH:

– χ(Coppi-Tang)– Porcelli Sawtooth trigger– Model electron and ion

heating sources for ICRH

• Time dependent AORSA-CQL3D simulation of the ICRF tail turn-on has been performed for this plasma:

– Used fixed background density and temperature profiles at t=1.1 sec.

0 0.2 0.4 0.6 0.8 1.0Time (s)

0

1.0

2.0

T e,i(0

) (ke

V)

Ti(0) (keV) - TSC(0) (keV) - TSC

Te(0) (keV) - TSC(0) (keV) - TSC Te(0) - C-Mod Experiment

0.86 0.88 0.9 0.92 0.94 0.96 Time (s)

T e(0) (

keV

)

1.0

1.5

2.0

2.5

3.0

1

2

0

P icrf

(MW

)

Time Dependent AORSA – CQL3D Simulation Performed for Minority Tail Formation in

Alcator C-Mod Case Using the SWIM Framework AORSA - CQL3D Minority Tail Energy at AORSA - CQL3D Minority Tail Energy at ρ=0.1=0.1CNPA Count Rate for Channel at R = 69 cm

o

Time (s)

Ener

gy (k

eV)

Ener

gy (k

eV)

Counts per Second / 10 4

0

100

200

0 0.02 0.040

0.5

1.0

Picrf = 2.65 MW @ 80 MHz

Square wave pulse for ICRF power [30 ms-on and 30 ms-off.

CNPA count rate binned over successive square wave pulses to increase signal-to-noise.

Good agreement on tail turn-on.Agreement not good on tail decay time (no spatial losses included in CQL3D simulation).

III. We are investigating finite ion drift orbit effects using two approaches:

• The diffusion coefficient (D) has been evaluated by a direct orbit integration using electric fields from AORSA:

– The “DC” code computes averages of the changes in velocity, pitch angle, and radial position over a complete bounce orbit, to obtain a set of RF induced diffusion coefficients.

– Diffusion Coefficient calculations done on CRAY XT3 (ORNL) using256 processors @ 10 min.

• The Monte Carlo code ORBIT RF has been combined with the AORSA ICRF solver:

– Self-consistent iteration not yet carried out.– ORBIT RF code ported to JAGUAR where good scaling to > 1000

processor cores has been demonstrated. – We are now examining best way to pass statistical distribution from

ORBIT RF to TORIC & AORSA to do self-consistent iteration.

Orbit topology modifies wave-particle resonance

• Shown at right are trajectories for 12 particles in the C-Mod case:– 4 equi-spaced || velocities– 3 equi-spaced ⊥ velocities– 409,600 complete poloidal

orbits• Particle cyclotron resonances and

strong quasilinear diffusion occur in roughly vertical planes in zero-orbit width description.

• But orbit topology can move particles away from (or towards) resonances that would be sampled (not sampled) in full-wave solver.

Investigation of Finite Ion Orbit Effects Using the DC Orbit Code

• Used AORSA electric fields for a C-Mod minority heating case from an AORSA-CQL3D calculation (zero orbit width limit).

• Integrate Lorentz force equation for ions and compute change in velocity (u) after one complete poloidal plane transit.

Orbit effect turned off Orbit effect turned on

Finite ion drift orbit effects are also being studied using the coupled AORSA and ORBIT-RF codes

• The Monte Carlo code ORBIT RF has been combined with the AORSA ICRF solver for a simplified set of conditions for a minority heating case in C-Mod:– Single k// = (nφ / R) and k⊥ from fast wave Disp. Rel.– AORSA computes E+(√ψp, θ), where θ=tan-1 (Z/(R-R0), assuming

a minority Maxwellian ion distribution function with temperature Tmin.

– ORBIT-RF determines resonance locations, evaluates Dql(E+) using the AORSA fields, and then computes fmin

– Test particles are not updated in energy and pitch after “kicks.– Single passing of resonances for each of 1 million particles

• Next steps:– Pass Dql(v⊥, v//, R, Z) to ORBIT-RF to account for k// exactly. – Examine best way to pass the statistical distribution from

ORBIT-RF to AORSA to do self-consistent iteration.

• C-Mod 11051206002

Quantitative Agreements Are Found in ICRF Hydrogen Minority Maxwellian Heating at Fundamental Harmonic

P abs

(MW

/m3 )

AORSA with EORBIT-RF with E

AORSA with E+ORBIT-RF with E+

ψ pψ p

25 keV Maxwellian 60 keV Maxwellian

1.0

2.0

3.0

4.0

0.00 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

1.0

• E- effect on Pabs is minor at fundamental harmonic regime due to small J2 (Bessel function).

• Successful implementation and testing of a parallel LH field solver TORLH with waveguide BC for the parallel electric field:– Simulation with resolution of 2047 (Nm) × 980 (Nr)

satisfies 2008 DoE JOULE Theory Milestone.– Effort underway to understand full-wave results using

advanced ray tracing techniques.

• Closed loop coupling between TORIC and CQL3D has been completed:– Implementation of nonthermal ions and electrons in

TORIC.– Calculation of Dql by TORLH

IV. Coupled Full-Wave / Fokker Planck Simulations of Lower Hybrid Current Drive

Convergence of LH wave fields demonstrated- Satisfies 2008 JOULE Theory Milestone -

• Must retain enough poloidal modes to resolve the electron Landau resonance limit of ω / (k// vte) ≈ 2.5:

• n// = 1.55, Nm = 2047, Nr = 980, 104 cpu-hours @ 256 processor cores

V. Nonlinear ICRF Antenna Sheath Work• Goal: determine sheath potential and power dissipated in sheaths

(parasitic losses) by wave energy near the wall from a sheath boundary condition (BC) in TORIC that accounts for misalignment between metal wall and B:

– Compute the linear ICRF fields from combined TOPICA-TORIC codes.– Proof of principle: metal wall BC with estimate of wall-magnetic field

mismatch allows estimate of sheath potential and resulting powerdissipation in sheaths: (BC implemented but needs testing).

– Full implementation: iterate TORIC fields until nonlinear sheath BC is satisfied; compute sheath power dissipation from converged fields.

• Investigate sheath dissipation for minority heating schemes in C-Mod:

– Strong single pass damping – D (H)– Weak single pass damping – D (3He)

• Time domain simulations using 3D EM field solver – VORPAL– Fully implicit time domain dielectric response module has been

implemented for electrons and ions.– Use PIC-treatment for ion response (fully nonlinear).

ICRF Launchers in Contact with Plasma are Subject to Nonlinear Effects, Leading to Parasitic Power

Losses

Alcator C-Mod Dipole Antenna RF sheaths can form due to mismatch between equilibriumB and the antenna structure (E//), resulting in power dissipation:

sh eANT rfP n V∝

Electrons are preferentiallyaccelerated out of the sheath region.A DC voltage Vrf is set up to maintain ambipolarity.

VORPAL Time Domain Simulation of Antenna Loop

VORPAL Time Domain Simulation of Antenna Loop

RF B-field Surface E-fieldSurface E-field onLoop antenna

Wavefronts

Radiation Pattern Radiation from Behind

Time Domain LHRF Plasma Simulations

• Has found use outside SciDAC RF, in support of LHRF MST experiment– Providing opportunity for more feedback, debugging, verification, and

validation.– LH launcher analysis, with fully realistic detail of all components.– Up to 24 million cells on local Madison cluster (34 CPU’s, 136 Gig memory

total).– Realistic plasma (based upon parametric fit to EQDisk equilibirum data).– Engineering diagnostics added & validated (Bdot-loops, Poynting flux).– Benchmarking against Microwave studio (vacuum cases, only, of course)– May serve as test-bed for Tech-X’s new ADI algorithm.

Plots show electric field and plasma current at midplane of MST antenna. 3-D simulation geometry shown at right.

• AORSA utilization of 154 TF @ 30, 000 processors on CRAY XT-4 Jaguar NLCF.

• Algorithm for iterative solution of LH full-wave problem.

VI. Algorithmic and performance enhancements

AORSA scaling on the quad-core Cray XT-4 (Jaguar)ITER Benchmark (2ΩCT and 3He, nφ = 27)

SSE optimization essential to retain linear scaling

350x350

512x512

350x350

512x512

154 TF = 62 %

113 TF = 45 %

Sparsity of the wave operator in LHRF may permit ITER scale problems

• The TORLH stiffness matrix is block tridiagonal.

• The equilibrium poloidal variation is smooth, coupling m tom'=m±j , with j << m, i.e., the block matrices are banded with bandwidth ∆m<<m .

• A test solver has been implemented to take advantage by decomposing blocks to banded blocks and solving iteratively.

• Operation count is reduced from O(m3) to O(∆m)2m. Convergence takes a few iterations.

• To do: test scaling to larger relevant sized problems, introduce sparse representation.

ipsi

width

Block width is very small except at boundary. Typical block size wouldbe in the 1000s (in this case it's 124).

VII. Summary• Capability has been developed for relatively routine reconstruction of

3-D ICRF wave fields, from full toroidal mode spectrum of antenna loops:

– Fast wave driven current density in NSTX and ITER using AORSA.– Ion cyclotron heating in Alcator C-Mod and ITER using coupled

AORSA-CQL3D.• Major upgrades to physics models were completed:

– Preliminary coupling of AORSA and ORBIT-RF coupling, including extensive verification tests through an inter-code comparison.

– AORSA and time dependent CQL3D simulations were performed for C-Mod minority heating: – (collaboration with SWIM FSP Project):

• Simulations will be repeated for DIII-D and NSTX HHFW – fast ion interaction experiments.

– Successful implementation and testing of a parallel LH field solver TORLH with waveguide BC for the parallel electric field:

• Simulation with resolution of 2047 (Nm) × 980 (Nr) satisfies 2008 DoE JOULE Theory Milestone.

– Implementation of nonthermal ions and electrons in TORIC in and closed loop coupling between TORLH and CQL3D.

– Implementation of a far field sheath BC in the TORIC solver.

Summary

• Achieved performance enhancements and initiated algorithmic improvements:

– AORSA utilization of 154 TF @ 30, 000 processors on CRAY XT-4 Jaguar NLCF.

– Studies of iterative solution techniques for the LH full-wave problem were started.

• New collaborations with SAP Projects will start very soon:– E. D’Azevedo (ORNL):

• Work on ORBIT-RF coupling to full-wave codes– Transfer of “noisy” statistical distribution functions to full-wave

solvers.– Transfer of large data sets (Dql in 4D) from full-wave to orbit code.

• Iterative solver techniques for full-wave LH field solver.– A. Sanderson (University of Utah):

• Reconstruction of 3-D electric fields from ICRF antennas.• 3-D mapping of field lines from ICRF antennas to vessel structure (for RF

sheath studies).

Related Posters and Talks

• CP6.00001: D. L. Green, “Towards including finite orbit effects in self-consistent calculations of ion cyclotron heating in non-Maxwellian plasmas”.

• CP6.00003: E. J. Valeo, “Full-wave simulations of lower hybrid wave propagation in toroidal plasma with nonthermal electron distributions”.

• CP6.00011: D. B. Batchelor, “Time Dependent Simulation of Energetic Ion Tail Formation Coupled to Thermal Plasma Transport”.

• CP6.00012: E. F. Jaeger, “Self-Consistent Simulation of High Power Electromagnetic Wave Heating in Three Dimensional Tokamak Geometry”.

• CP6.00013: D. A. D’Ippolito, “Analytic Model of Antenna Sheaths”.

• CP6.00014: D. N. Smithe, “Time-Domain Simulation of Edge Plasma, Sheaths, and RF Couplers, on NERSC Supercomputers”.

Related Posters and Talks• CP6.00015: J. R. Myra, “Resonance Cone Interaction

with a Self-Consistent Radio Frequency Sheath”.• GP6.00044: A. S. Richardson, “Reconstructing RF Fields

from Ray Tracing Data”.• JP6.00097: M. Choi, “Ion Finite Drift Orbit Effect in

ICRH Modeling Using Quasi-Linear Full Wave and Particle”.

• NP6.00106 : C. K. Phillips, “Numerical Modeling of HHFW Heating and Current Drive on NSTX”.

• NP6.00107: R. W. Harvey, “Comparison of a New Self-Adjoint Calculation of NSTX High Harmonic Current Drive with CQL3D”.

• VI2.00003: J. C. Wright, “An assessment of full-wave effects on the propagation and absorption of lower hybrid waves”.