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L.L. Lao1, K.H. Burrell1, N.M. Ferraro2, B.A. Grierson2, Y. Huang3, J.E. Kinsey4, K. Li3, O. Meneghini1,G.L. Trevisan5, and B.J. Xiao3

3rd IAEA Technical Meeting on Fusion Data Processing, Validation and Analysis Vienna, Austria May 28-31, 20191General Atomics 2Princeton Plasma Physics Laboratory 3ASIPP 4CompX 4UCSD 5ORAU

Recent Advances in Equilibrium Reconstruction for Fusion Data Analysis and Plasma Control and Prospect for ITER Applications

DIII-D MSE-LS Reconstruction

Lao IAEA TM Meeting 2019

7 MSE-LS

14 MSE-LP

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Equilibrium Reconstruction Is Fundamental to Tokamak Research and Operation

• Provides essential magnetic geometry and current and pressure profiles information necessary for tokamak operation and data analysis

• Contributed to several major tokamak discoveries- Experimental verification of b scaling

- Negative central-shear regime

• This presentation- Review recent important advances

for fusion data analysis and plasma control

- Discuss their prospects for ITER equilibrium reconstruction

Experimental verification of b scaling Stambaugh, IAEA 1984

Strait Phys. Plasmas 1, 1415 (1994)

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Outline

• Equilibrium reconstruction using motional Stark splitting of spectral lines (MSE-LS) show that it could contribute to current profile reconstruction in ITER

• GPU-based reconstructions can accurately reconstruct equilibrium results at a fraction of the serial-based computational cost

• Theory-based pressure and current profiles provide useful physics constraints to guide equilibrium reconstructions

• 3D extended MHD linear simulations can provide good predictive values for plasma 3D displacements and could be used to guide 3D perturbed equilibrium reconstructions

NVDIA TESLA K40 and V100 GPUs

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EFIT Reconstructs Equilibrium by Solving the GS Equation While approximately Conserving Measurements and Constraints

• Inverse problem response Þ source

• Uniqueness of solution ? What information can be reconstructed ?

• Non-linear optimization problem

χ 2 =Mi − Ci

σ i

$

% & '

( )

i∑

2

€

Δ*ψ = −µ0RJϕ (R,ψ)

Jϕ = R & P (ψ) +µ0F & F (ψ)4π 2R

Measured Signals, physics constraints

Computed Signals, Physics Constraints

Measurement/constraint Uncertainty

Lao Nuclear Fusion 25, 1421 (1985)

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EFIT Efficiently Computes Non-Linear Optimization by Transforming into a Sequence of Linear Problems

• Dependence on y linearized with the Picard iteration scheme using the integral form

• The plasma current source is represented in terms of a set of basis functions with linear parameter vector a

• The parameter vector a solved using the singular-value decomposition method to decompose the response matrix into a diagonal form

• Inverting D* while approximately conserving the measured fluxes and magnetic fields.

Ci

m +1(r) = GCij∑ (r,rej )Iej + d " R

V∫ d " Z GCi

(r, " r )Jϕ " R ,ψ m( " r )[ ]

! P (ψ ) = αn xnn∑

€

F " F (ψ) = γ n xnn∑

€

R α = M €

xn

€

R €

Δ*ψm+1 = −µ0RJϕ (R,ψm ,α)

Condition Number = λMAX / λMIN

Type of Basis Functions Boundary Conditions

Number of Fitting ParametersMeasurement Vector

Response Matrix

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Amount of Information That Can Be Reconstructed Increases with Availability of Diagnostic Data

• External magnetics— Plasma boundary, bP, li , and IP

— bP + li / 2 if circular

• External magnetics plus MSE— Plasma boundary, bP, li , and IP

— q profile, some information on internal magnetic geometry

• External magnetics plus MSE and kinetic profiles— Plasma boundary, bP, li , and IP— q profile

— Pressure profile and internal magnetic geometry

• SXR, ECE— Some topological and current profile information

• 3D external magnetics, coils, and edge kinetic profiles— 3D plasma boundary displacement

MSE-LS

GPU and parallel algorithms

Model-based P and J

3D perturbed equilibria

Lao, et al. Nuc. Fusion 25, 1611 (1985)

Lao, et al. Nuc. Fusion 25, 1421 (1985)

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MSE Line Splitting (MSE-LS) Makes Use of Stark Spectrum to Determine |B|

• MSE Line Polarization MSE-LP– Experience on many tokamaks, many possible sources of error such as

background reflection, Faraday rotation, and mirror coating

• MSE Line Splitting MSE-LS– Difficult on current tokamaks, promised to be easier in ITER

– Signals dominate by vacuum toroidal magnetic field

De Bock 2015 ITPA Diagnostics

MSE-LP Makes Use of Line Polarization to Determine Magnetic Field Pitch Angle

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MSE-LS Reconstruction Algorithm Successfully Incorporated into EFIT Response Matrix and Verified

• EFIT MSE-LS model equation

• Numerical scheme- To allow the fitting parameters

to appear linearly, the leading order term is treated implicitly

- All other terms are treated explicitly

θ

Beam Injector Voltage

MSE-LS Magnetic Field

Horizontal NBI

Geometric Coefficients

EFIT Response

Matrix Synthetic Signals

BLS(T)

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MSE-LS Reconstruction Tests Are Based on a Set of DIII-D Discharges with Extensive MSE-LP and Kinetic Profile Data

qmin~ 2 Use Case

q0~ 1 Hybrid Use Case

ITER Baseline

NCS Low BT

Slow IP Ramp

High IP, q95 ~ 4.2

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• Reconstructions using 7 DIII-D MSE-LS channels are tested against those using MSE-LP data

14 MSE

MSE-LS Could Contribute to the Equilibrium Reconstruction of Pressure and q Profiles

7 MSE-LS

3 FF’ Knots, 2 P’ Knots39 Flux Loops and 42 Magnetic Probes

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EFIT Results Indicate MSE-LS Could Contribute to the Equilibrium Reconstruction of Pressure and q Profiles

• 14 MSE, 7 MSE + 7 MSE-LS, 7 MSE reconstruction with spline representations

Experimental MSE-LS Data σ ~ 1% DIII-D160695.04000

14 MSE

7 MSE7 MSE+7MSE-LS

P

q14 MSE

7 MSE + 7 MSE-LS 7 MSE

DIII-D H-mode discharge 160695.4000 qmin ~ 2

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MSE-LS Alone Yields Less Current Profile InformationUse Case 156779.04000 qmin ~ 2

• 14 MSE-LP reconstruction with spline representations

14 MSE-LS σ ~ 3%, 6% 2-knot splines

DIII-D156779.04000Synthetic MSE-LS Data at 14 MSE-LP

Numerically Less Stable if σ < 3%

14 MSE-LS σ ~ 1% + q(0) 3-knot FF’

7 MSE

14 MSE

14 MSE-LS

P

q

7 MSE14 MSE

14 MSE-LS

ρ

14 MSE-LS + q(0)

14 MSE-LS + q(0)

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MSE-LS Alone Yields Less Current Profile InformationUse Case 156779.04000 qmin ~ 2

• 14 MSE-LP reconstruction with FF’ spline P’ splineDIII-D156779.04000

7 MSE

14 MSE

14 MSE-LS

P

q

7 MSE14 MSE

14 MSE-LS

q

7 MSE14 MSE

14 MSE-LS

14 MSE-LS + q(0)

ρ

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MSE-LS Works Best in Conjunction with other ConstraintUse Case 156779.04000 qmin ~ 2

• Full kinetic reconstruction with MSE-LSDIII-D156779.04000

Synthetic MSE-LS Data at 14 MSE-LP

14 MSE-LS σ ~ 1% + q(0) 7-knot P’, FF’

P

14 MSE

14 MSE-LS + q(0)

P

q

ρ14 MSE

14 MSE-LS + q(0)

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MSE-LS Recommendations

• Adequate number of channels and measurement accuracy are criticalØ High accuracy spectrometer dispersion, fiducial, and geometry

necessary

Ø Accurate determination of beam voltage and spatial calibration critical

• Improved numerical treatment of higher order terms could enhanced MSE-LS reconstruction robustness and accuracy

• Explore use of an alternative technique to provide additional central q information to complement MSE-LS reconstruction

• Explore techniques to measure and reconstruct BLS with respect to vacuum toroidal Field BT-VAC: BLS – BT-VAC

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GPU Computation Power Compares Favorably Against CPU

• GPU has evolved into a highly parallel, multi-threaded, many core processor with tremendous computational horsepower and very high memory bandwidth.• GPU provides unique parallel computation

capabibity at modest cost

NVDIA TESLA K40, P100, and

V100 GPUs

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Equilibrium Reconstruction Algorithms Can be Parallelized Efficiently on GPU

P-EFIT is based on EFIT but using massively parallel GPU cores to significantly accelerate the computation:

• Response matrix calculation: Dividing the matrix into small parts, matrix multiplications. (~40 !s, 65�65, GPU)

• Least square fitting: Making use of parallel matrix multiplication, the initial overdetermined equation system is transformed into full rank system. (~30 !s, 65�65, GPU)

• Poloidal flux refreshing (Δ* Inversion), Equilibrium solver: By eigenvalue decomposition, the block tri-diagonal matrix is transformed into independent block diagonal matrix which could be solved in parallel. (~50 !s, 65�65, GPU)

• Boundary search, "P , ℓI , etc parallel algorithm. (~50 !s, 65�65, GPU)

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Full Discharge External Magnetic P-EFIT Equilibrium Reconstruction Results: DIII-D Discharge 133221

• Detailed magnetic, magnetic plus MSE and full kinetic reconstruction results of single time-slice are compared between EFIT and P-EFIT.

• P-EFIT compute the whole discharge of DIII-D shot 133221 from 300ms to 5200ms with 100 time-slices, 257�257 spatial grid.

• Small differences between EFITand P-EFIT results are due to different numerical algorithms used.

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External magnetic plus MSE and kinetic profiles reconstruction results: DIII-D discharge15938 at 3275ms

Case

EFITMagnetic

+ MSE+ Kinetic

P-EFITMagnetic

+ MSE+ Kinetic

Grid Size 257×257 257×257

Ip (MA) -0.978 -0.977

Ω (m3) 19.286 19.320

χ2mag 24.721 26.077

χ2gama 0.397 1.562

q0 1.036 1.034

q95 5.581 5.572

βp 0.780 0.791

li 1.20 1.202

Wp (MJ) 0.474 0.479

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External magnetic plus MSE and kinetic profiles reconstruction results: DIII-D shot 159386@3275ms

CaseP-EFIT

Magnetic

P-EFITMagnetic

+ MSE

P-EFITMagnetic

+ MSE+ Kinetic

Grid Size 257×257 257×257 257×257

Ip (MA) -0.973 -0.975 -0.977

Ω (m3) 18.901 19.021 19.320

χ2mag 20.157 22.437 26.077

χ2gama 0 1.136 1.562

q0 1.004 0.971 1.034

q95 5.571 5.480 5.572

βp 0.840 0.803 0.791

li 1.152 1.210 1.202

Wp (MJ) 0.505 0.488 0.479• Both P’ and FF’ use spline representations

with 5 knots.

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P-EFIT Can Achieve Ten to One Hundred Acceleration Ratio in Whole Discharge Reconstructions

• The main advantage of P-EFIT is the acceleration of computation, the timing test is on a Linux workstation with two Intel(R) Xeon(R) CPU E5-2670 v2 2.50GHz and one NVIDIA Tesla K20c GPU card.

• P-EFIT could complete the calculation in a fraction of the EFIT computational cost (from one-tenth to one-percent cost time).

• With more diagnostics and constraints, the computation cost of P-EFIT does not increase as fast as EFIT on 257�257 spatial grid, which show the potential capacity of real time full kinetic equilibrium reconstructions.

Cost Time of P-EFIT 257�257 with different diagnositcs and constraints

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ISOFLUX/P-EFIT Plasma Shape Control Successfully Tested on EAST

P-EFIT magnetic reconstruction results match RT-EFIT and EFIT results and satisfy the requirement of real-time control.

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Theory-Based Pressure and Current Profiles Provide Useful Physics Constraints to Guide Equilibrium Reconstructions

• Edge bootstrap current- Sauter- NEO kinetic neoclassical, NEO-NN

• NBI fast-ion pressure from Fokker-Planck slowing down- NUBEAM, NFREYA- Difference between theory-

constrained pressure and experimental determined pressure from full kinetic reconstruction provides useful information for interpretation of fast-ion transport

• Pedestal pressure- EPED, EPED-NN

• Core thermal pressure- TGLF, TGLF-NN

Self-consistent TGLF predicted core thermal profiles with EPED pedestal pressure

Meneghini 2016 IAEACritical for burning plasma devices such as ITER where neutron wall loading and radiation is a major concern for diagnostics

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Theory-Based Pressure and Current Profiles Provide Useful Physics Constraints to Guide Equilibrium Reconstructions

Self-consistent TGLF predicted core thermal profiles with EPED pedestal pressure

OMFIT STEP workflow (ONETWO, EFIT, TGYRO, EPED) https://gafusion.github.io/OMFIT-source/usage.html - tutorial-series-videos

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3D Extended MHD Linear Simulations Can Provide Good Predictive Values for Plasma 3D Displacements

• Pedestal measurements clearly show displacements of 1—4 cm in edge when 3D fields are applied- Vacuum modeling predicts separatrix perturbations of a few mm- Linear plasma response modeling with the 3D extended MHD code M3D-C1

shows helical perturbations of comparable magnitude to experiment

Ferraro Nuclear Fusion 53, 073042 (2013)

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3D Extended MHD Modeling Shows Quantitative Agreement with Observed Edge Displacement

• Pedestal measurements clearly show displacements of 1—4 cm in edge when 3D fields are applied- Vacuum modeling predicts separatrix perturbations of a few mm

Ferraro Nuclear Fusion 53, 073042 (2013)

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Summary

• Equilibrium reconstruction using motional Stark splitting of spectral lines (MSE-LS) show that it could contribute to current profile reconstruction in ITER

• GPU-based reconstructions can accurately reconstruct equilibrium results at a fraction of the serial-based computational cost

• Theory-based pressure and current profiles provide useful physics constraints to guide equilibrium reconstructions

• 3D extended MHD linear simulations can provide good predictive values for plasma 3D displacements and could be used to guide 3D perturbed equilibrium reconstructions