7 th Workshop on Fusion Data Processing, Validation and Analysis Recent Developments of Integrated Data Analysis at ASDEX Upgrade R. Fischer, L. Barrera, A. Burckhart, M.G. Dunne, C.J. Fuchs, L. Giannone, J. Hobirk, P.J. McCarthy, M. Rampp, S.K. Rathgeber, R. Preuss, W. Suttrop, P. Varela, M. Willensdorfer, E. Wolfrum, and ASDEX Upgrade Team Max-Planck-Institut für Plasmaphysik, Garching EURATOM Association Frascati, Mar 26-28, 2012
Multi-diagnostic profile reconstruction ➢ Lithium beam impact excitation spectroscopy (LiB) ➢ Interferometry measurements (DCN) ➢ Electron cyclotron emission (ECE) Forward modelling of the electron cyclotron radiation transport → T e at optically thin plasma edge ➢ Thomson scattering (TS) ➢ Reflectometry (REF) → n e at plasma edge ➢ Equilibrium reconstructions for diagnostics mapping
1. ECE forward modelling • Magnetic fusion: Understanding and control of plasma edge • Large variations of plasma parameters within a very thin layer • Reliable electron density, temperature and pressure profiles with high spatial and temporal resolution • Workhorse ECE: (+) plasma core , (-) plasma edge • ECE assumptions: local emission and black-body radiation (optically thick plasma) • Optically thin plasma edge → EC emission depends on density → combination with data from density diagnostics → calculate broadened EC emission and absorption profiles depending on T e and n e → solve radiation transport equation → forward modelling in the framework of Integrated Data Analysis
ECE forward modelling S. K. Rathgeber, et al., to be published S. K. Rathgeber, PhD thesis
Electron cyclotron intensity Radiation transport equation: s LOS coordinate dI s spectral intensity I = j s − s I s emissivity j ds reabsorption Emissivity: 2 c m th harmonic m 2 m 2 j m = e 2 m − 1 m 2 m − 1 cos 2 1 2 sin angle to magn.field 2 0 m = meB 8 m − 1 ! freq. cold resonance m e × ∫ [ 1 − ∥ cos ] − m 2 2m ⊥ − 2 = 1 − 2 = 1 − ∥ 2 − ⊥ 2 f ∥ , ⊥ 2 ⊥ d ⊥ d ∥ ∥ , ⊥ = v ∥ , ⊥ c f ∥ , ⊥ Maxwell distribution = j Absorption I BB (Kirchhoff's law for thermal equilibrium) : 2 Black-body intensity I BB = 2 k B T e 3 c 8 (in Rayleigh-Jeans approximation) :
Electron cyclotron emissivity Emissivity for 2 nd harmonic in X-mode j 2X = 2X j n e 1 4 cos 2 2 nd harmonic in X-mode fraction only 8 sin 2X = 1 2 2 1 cos 2 1 cos 4 16 sin 2 n e j n e = 4 e 2X = m e c² 2 cos 2 1 0 c sin total emissivity 2 k B T e ∫ [ 1 − ∥ cos ] − 2X exp −[ ⊥ 7 / 2 = 2 ∥ 5 d ⊥ d ∥ 2 ] ⊥ shape function Shape function can be integrated analytically ...
Electron cyclotron emissivity Shape function: 5 [ exp − 1 − F ] 1 2 3 / 2 2 2 2 − sin − 3 − 2 sin = cos = m e c² 2 = 1 − ∥ cos 2 k B T e = 2X 2 2 1 / 2 = 1 − ∥ ,1 / 2 cos ∥ ,1 / 2 = cos ∓ 1 − sin 2 2 = 1 cos 2 1 cos F Dawson integral (efficient subroutines and approximations) → fast forward model for the EC radiation transport in the plasma S. K. Rathgeber, et al., to be published
IDA: LIB + DCN + ECE(radiation transport) ✔ Efficient ECE radiation transport forward modeling ✔ High temporal (32 μs) and spatial (5 mm) resolution of edge temperature profiles ✔ Combination with density diagnostics (LIB, DCN) ✔ Quantitative reproduction of EC emission for all frequencies resonant in between the vessel walls ✔ s hine-through peak explained without needing supra-thermal electrons ✔ s hine-through peak provides important information about the pedestal T e gradient
IDA: LIB + DCN + ECE(radiation transport) ✔ Reveals steeper pedestal T e gradients compared to conventional analysis ✔ Pressure profiles and gradients at plasma edge
IDA: LIB + DCN + ECE(radiation transport) ✔ Provides information about diagnostics alignment ➔ Relative shift between n e and T e → minimum in data residues ✔ Sensitivity study: ➔ Profile parameterization with cubic spline (number and position of knots) ➔ Amplitude and alignment of electron density ➔ Reflection on the vessel wall (tungsten) ➔ Antenna pattern ➔ Additional 1 st O-mode contribution S. K. Rathgeber et al., to be published; PhD thesis
2. Reflectometry forward modelling • Goal: n e profiles, plasma position control (ITER) ● Classical analysis: Abel inversion (O-mode) → location of cutoff layer ● Problems: ● Multiple analysis steps (phase of reflected wave → group delay → density) ● error treatment/propagation; profile uncertainties ● density initialization outside first cutoff layer ● unphysical profiles • IDA ● Forward modelling of measured data for given density profile ● Benefit: ● Additional data (density initialization, complementary at pedestal top) ● Alignment
Reflectometry forward modelling Time delay of the reflected f = 1 ∂ beam (group delay): 2 ∂ f Phase of reflected beam: r 1 = 4 f r dr − ∫ c 2 r c f r = 2 0 m e f 2 n c f ; n c f = 4 1 − n r Refractive index: 2 e r c − r 1 Forward model for group delay f , n r = 2 2 x c ∫ dx 1 − n r c − x for a given density profile: 2 0 n c
IDA: LIB + DCN + Reflectometry IDA: LIB+DCN RPF (Abel inv.) IDA: • Only physically reasonable LIB+DCN+RPF profiles possible (spline) • Alignment is ok (< 5 mm) • Modification of n e at pedestal top
IDA: LIB + DCN + Reflectometry • Systematic deviance in REF residue • Minor changes in LIB residue due to modification of n e at pedestal top • SNR(REF) < SNR(LIB)
3. IDA and the Magnetic Equilibrium ➢ Combine profile diagnostics LIB, DCN, ECE, TS, REF (new) → n e and T e profile fits to all data at once (IDA shotfile) ➢ Mapping on a common coordinate grid using an existing equilibrium (EQH/EQI/FPP) ➢ Inconsistency: Equilibrium is not evaluated with kinetic profiles from IDA ✗ Position of magnetic axis, separatrix, inner flux surfaces? ✗ DCN: H2-H3 vertical plasma position often seems to be wrong up to ~1cm. ✗ ECE: (r,z) depends on equilibrium ✗ TS: vertical system relies very much on equilibrium ✗ Alignment of TS, ECE, LIB (with separatrix T e ) → uncertainties in the equilibrium ??? ➢ Goal: combine data from profile diagnostics with magnetic data for a joint estimation of profiles and the magnetic equilibrium ➢ Needs equilibrium code: ✗ CLISTE very successful, but code too sophisticated to be adapted to the IDA code ✗ New code based on the ideas (success) of CLISTE (P. McCarthy, L. Giannone, P. Martin, K. Lackner, S. Gori) ✗ Extra: Parallel Grad-Shafranov solver (R. Preuss, M. Rampp, K. Hallatschek, L. Giannone)
Grad-Shafranov solver Grad-Shafranov equation: Ideal magnetohydrodynamic equilibrium for poloidal flux function Ψ for axisymmetric geometry R ∂ ∂ z² =− 2 ² 0 R 1 ∂ R ∂ ² ∂ 2 P ' 0 FF ' ∂ R R 1) Grid MxN (typically: radially 65 x vertically 128) 2) (a) Garchinger Equilibrium Code (GEC: Lackner et al, 1976) based on cyclic reduction (b) Garchinger Parallel Equilibrium Code (GPEC: Preuss et al, 2012) 3) CLISTE “Fast Mode ” : solve Ψ individually for N p +N F basis functions π and Ф (cubic spline N p p ' = ∑ (CLISTE), Bernstein polynomials (Giannone), Fourier-Bessel polynom.) for p' and FF' c h h h = 1 [P.J. Mc Carthy, W. Schneider, P. Martin, "The CLISTE interpretive equilibrium code", N F FF ' = ∑ IPP laboratory report 5/85 (1999).] d k k k = 1 4) SOL: P' and FF' ≠ 0 c , 5) Linear regression to data (B pol , D psi , I ext , pressure profile, ...) d
IDE: data and residues #25764 B pol measured B pol fitted B pol residue Residues in the order of ~mT D psi measured D psi fitted D psi residue
IDE: external currents and residues I ext measured I ext fitted I ext residue I ext measured and fitted: V1o, V1u, V2o, V2u, V3o, V3u, OH1 = OH3o = OH3u = OH, OH2o = OH + dOH2s, OH2u = OH + dOH2s + dOH2u, Coiu, Coiu, Ipslon, Ipslun
Comparison EQH/IDE: pressure and poloidal current Grad-Shafranov: p' FF' Pressure profile and Net poloidal plasma current IDA pressure constraints (wo external currents) (T i = T e , Z eff = 1.5, uncertainty 50%) ➢ Center: Pressure constraints reduce I pol,net ➢ Center: Pressure gradient → p' ➢ Edge: p and p' ➢ Edge: Opposite direction
Comparison EQH/IDE: flux contour and separatrix EQH IDE
Comparison EQH/IDE: Temperature and density #25764, 2.0s IDA(EQH) IDA(IDE) ~5cm core edge ~5mm
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