A synthetic synchrotron diagnostic for runaways in tokamaks Mathias - PowerPoint PPT Presentation

A synthetic synchrotron diagnostic for runaways in tokamaks Mathias Hoppe 1 Ola Embréus 1 , Alex Tinguely 2 , Robert Granetz 2 , Adam Stahl 1 , Tünde Fülöp 1 1 Chalmers University of Technology, Gothenburg, Sweden 2 PSFC, Massachusetts Institute of Technology, Cambridge, Massachusetts

CHALMERS 2/ 17 Outline 1. Theory of our synthetic diagnostic 2. Geometric effects 3. Image sensitivity to RE parameters 4. Modelling C-Mod discharge

CHALMERS 3/ 17 Synthetic synchrotron diagnostic theory Total power per pixel, per frequency interval d ω : � � � d I ij d ω ( x 0 , ω ) = d x d p d A d n × A N ij � d 2 P ( x , p , x 0 , ω ) × ˆ n · n � r r 2 f ( x , p ) δ r − n d ω d Ω Detector parameters A = Detector surface, d 2 P / d ω d Ω = Angular and spectral n = Line-of-sight distribution of synchrotron radiation, ˆ n = Viewing direction x 0 = Detector position f ( x , p ) = Distribution of runaways, Particle parameters r = | x − x 0 | = Distance between camera and particle.

CHALMERS 4/ 17 Synthetic synchrotron diagnostic theory Three transformations 1. Guiding-center approx., d x d p ≈ d X d p � d p ⊥ d ζ x X Larmor radius ≪ B/ |∇ B |

CHALMERS 4/ 17 Synthetic synchrotron diagnostic theory Three transformations 1. Guiding-center approx., d x d p ≈ d X d p � d p ⊥ d ζ 2. Cylindrical coordinates, z d X = R d R d z d φ R φ

CHALMERS 4/ 17 Synthetic synchrotron diagnostic theory Three transformations 1. Guiding-center approx., d x d p ≈ d X d p � d p ⊥ d ζ τ 2. Cylindrical coordinates, d X = R d R d z d φ 3. Trajectory coordinates ρ ( R , z ) → ( ρ, τ ) , φ ◮ ρ : Major radius of particle in the midplane, at beginning of orbit ◮ τ : Orbit time (a poloidal parameter)

CHALMERS 5/ 17 Synthetic synchrotron diagnostic theory Distribution function independent of: � Toroidal angle φ – Tokamak axisymmetry � Gyrophase ζ – Gyrotropy � Orbit time τ – Liouville’s theorem Guiding-center distribution specified along the line τ = φ = 0 (outer midplane). d I ij � � � d ω = d ρ d τ d φ d p � d p ⊥ × p ⊥ JR × d A d n A N ij � � � d 2 P ( ρ, p � , p ⊥ , x 0 , ω ) × ˆ n · n � r r 2 f gc ( ρ, p � , p ⊥ ) δ r − n d ω d Ω

CHALMERS 6/ 17 Synchrotron radiation Angular and spectral distribution of synchrotron radiation: � ω � 2 � 1 − β cos ψ � 2 d ω d Ω = 3 e 2 β 2 γ 6 ω B d 2 P × 32 π 3 ǫ 0 c ω c β cos ψ 2 / 3 ( ξ ) + ( β/ 2 ) cos ψ sin 2 ψ � � K 2 K 2 × 1 / 3 ( ξ ) 1 − β cos ψ Result of gyro-average:

CHALMERS 7/ 17 SOFT – Sychrotron-detecting Orbit Following Toolkit Computes d I ij / d ω , and outputs � synchrotron images and spectra

CHALMERS 7/ 17 SOFT – Sychrotron-detecting Orbit Following Toolkit Computes d I ij / d ω , and outputs � synchrotron images and spectra � Solves the guiding-center equations of motion using RKF45 in numeric magnetic geometry

CHALMERS 7/ 17 SOFT – Sychrotron-detecting Orbit Following Toolkit Computes d I ij / d ω , and outputs � synchrotron images and spectra � Solves the guiding-center equations of motion using RKF45 in numeric magnetic geometry � Weighted with a given (numeric) runaway distribution function

CHALMERS 7/ 17 SOFT – Sychrotron-detecting Orbit Following Toolkit Computes d I ij / d ω , and outputs � synchrotron images and spectra � Solves the guiding-center equations of motion using RKF45 in numeric magnetic geometry � Weighted with a given (numeric) runaway distribution function � Full distribution runs in 5-10 hours on 4-core Xeon-based desktop, with sufficient resolution

CHALMERS 8/ 17 Comparison with SYRUP [1] � Geometric effects (SOFT) show significant difference in spectrum. � Runaway distribution specified explicitly in outer-midplane (LF-side). � Contributions mostly from HF-side. [1] A. Stahl, et. al. PoP 20, 093302 (2013).

CHALMERS 9/ 17 Parameter scans Magnetic geometry: Alcator C-Mod, 3-8 T � Radiation in the visible range � Camera located 21 cm below midplane Varied parameters: � Energy E Pitch angle θ p � � Initial radius

CHALMERS 10/ 17 Parameter scans – Energy E = 10 MeV E = 25 MeV 100% 80% 60% E = 40 MeV E = 55 MeV 40% Other parameters: Beam radius 16 cm Pitch angle 0 . 15 rad 20% Spectral range 500-1000 nm Magnetic field 3-8 T 0% Camera elevation − 21 cm

CHALMERS 11/ 17 Parameter scans – Pitch angle θ p = 0 . 02 rad θ p = 0 . 10 rad 100% 80% Other parameters: Beam radius 16 cm 60% Energy 30 MeV Spectral range 500-1000 nm θ p = 0 . 18 rad θ p = 0 . 26 rad 40% Magnetic field 3-8 T Camera elevation − 21 cm 20% 0%

CHALMERS 12/ 17 Small pitch angle = small GC Large pitch angle = large GC cone cone = ⇒ small chance of reaching = ⇒ greater chance of reaching detector detector θ p = 0 . 02 rad θ p = 0 . 18 rad

CHALMERS 13/ 17 Parameter scans – Launch radius Other parameters: Beam radius 16 cm Energy 30 MeV Pitch angle 0 . 15 rad 72 cm Spectral range 500-1000 nm 74 cm Magnetic field 3-8 T 76 cm 78 cm Camera elevation − 21 cm 80 cm 82 cm NOTE: Magnetic axis at R = 68 cm . 84 cm Particles at R � 72 cm are invisible in this configuration.

CHALMERS 14/ 17 Distribution function − → � Simulated with CODE [2, 3] � Parameters given on-axis [2] M. Landreman, et. al. CPC 185, 847 (2014). [3] A. Stahl, et. al. NF 56, 112009 (2016).

CHALMERS 15/ 17 Distribution function

CHALMERS 16/ 17 What do we actually see? f ( p � , p ⊥ ) (Distribution function) “ ˆ I × f ( p � , p ⊥ ) ” (Emitted radiation)

CHALMERS 17/ 17 Conclusions � SOFT allows study of synchrotron radiation in arbitrary axisymmetric magnetic configurations

CHALMERS 17/ 17 Conclusions � SOFT allows study of synchrotron radiation in arbitrary axisymmetric magnetic configurations � Pitch angle varies along orbit = ⇒ crucial to be clear about how the runaway distribution is specified.

CHALMERS 17/ 17 Conclusions � SOFT allows study of synchrotron radiation in arbitrary axisymmetric magnetic configurations � Pitch angle varies along orbit = ⇒ crucial to be clear about how the runaway distribution is specified. � Detector placement strongly influences the observed synchrotron radiation.

CHALMERS 17/ 17 Conclusions � SOFT allows study of synchrotron radiation in arbitrary axisymmetric magnetic configurations � Pitch angle varies along orbit = ⇒ crucial to be clear about how the runaway distribution is specified. � Detector placement strongly influences the observed synchrotron radiation. � Sensitivity due to runaway properties helps inferring runaway distribution from image.

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CHALMERS 19/ 17

CHALMERS 20/ 17 Parameter scans – Camera vertical position z = − 21 cm z = − 14 cm 100% 80% Other parameters: Beam radius 16 cm 60% Energy 30 MeV 0 . 15 rad Pitch angle z = − 7 cm z = 0 cm 40% Spectral range 500-1000 nm Magnetic field 3-8 T 20% 0%