1. Development and Application of Real-Time Magnetic Coordinate Mapping System in LHD C. Suzuki, K. Ida, Y. Suzuki, M. Yoshida, M. Emoto and M. Yokoyama National Institute for Fusion Science 322-6 Oroshi-cho, Toki 509-5292, Japan S12.4

2. Contents  Introduction  Background / objectives  Equilibrium database  VMEC calculations / inverse mapping  User tools to access  "Line-of-sight" database  Equilibrium mapping  Real-time mapping to "Thomson" data  Automation / viewer program  Current status  Possible applications

3. Introduction

4. Background  Experimental profile data (e.g., T e /n e profile) are originally expressed in real coordinates.  It is important to map real coordinates into magnetic coordinates by specifying a proper equilibrium.  For this inverse problem, it is necessary to construct a database composed of numerous 3D equilibria in wide ranges of plasma parameters (pressure, plasma current). Real coordinates Magnetic coordinates equilibrium mapping

5. Objectives  Construction of equilibrium database  Automatic generation of a new large-scale database in 7 dimensional parameter space (Rax, Bq, gamma, p0, pf, ip, ipf)  Coverage of special cases (e.g., strongly peaked pressure profiles)  Development of user-friendly tools  Routines to retrieve mapping results (r eff, B, etc.) by specifying the 7 parameters and real coordinates  Automatic real-time mapping  Real-time mapping for all the time slices of "Thomson" data  Automatic generation and registration of mapping data table for any position-dependent diagnostics (e.g., CXS, MSE)

6. Equilibrium database

7. Database in 7 (=3+4) dimensional parameter space 7865 (=11x5x13x11) equilibria have been calculated for each vacuum configuration (combination of Rax, Bq, gamma). 3 parameters to specify vacuum magnetic configuration 4 parameters to specify plasma condition Calculation sometimes does not converge specifically for extremely high beta or peculiar current profile. descriptionrangestandard value Rax magnetic axis position3.50 – 4.00 [m]3.60 Bq quadruple field-50 – 200 [%]100 gamma pitch parameter1.129 – 1.2621.254 descriptionrangemesh points p0 peak beta0 – 10 [%]11 pf pressure peaking factor1.41 – 5.005 ip toroidal current-150 – 150 [kA/T]13 ipf current peaking factor-12.60 – 4.0011

8. Equilibrium calculation and inverse mapping  Free-boundary LHD equilibria by the latest VMEC code* in the 7-dimensional parameter space  Magnetic flux through LCFS (phiedge) same as that in vacuum configuration (No specific definition of LCFS for finite beta)  Fourier coefficients automatically extrapolated up to s = 2.0  Relational database composed of equilibrium parameters including R 00 (0), R 00 (1), etc. *S. P. Hirshman and J. C. Whitson: Phys. Fluids 26 (1983) 3553. Original flux surface Extrapolated (imaginary) flux surface (R, Z, Φ) Inverse mapping (r eff, θ, φ)

9. Library for database access  Library routine for the server access via PV-WAVE, C and Fortran  The server program retrieve equilibrium parameters from relational database, and run inverse mapping to retrieve output parameters for given (R, Z, Phi).  Any intermediate or excess input values are allowed, and output parameters are automatically interpolated or extrapolated. Library (written by C) PV-WAVE C VMEC output files Inverse mapping solver （ written by Fortran ） Fortran Client domain Server domain Relational database Server program Rax, Bq, gamma p0, pf, ip, ipf R, Z, Phi r eff, theta, b r, b z, b phi, iota, di/ds, p, du/dv, phi Input parameters Output parameters R 00 (0), R 00 (1), etc. path of VMEC output files

10. Line-of-sight database for real-time processing  Inverse mapping solver is not fast enough for real-time processing.  "Line-of-sight" database : pre-calculated mapping results along several lines of sights (Thomson, CXS, FIR, etc.)  Retrieve output parameters by designating name of line-of-sight  Output parameters are automatically interpolated. start pointend point Diag.R (m)Z (m) Phi (deg.) R (m)Z (m) Phi (deg.) # of Points YAG2.3000.000234.05.3000.000234.0301 CXS (ary1) 2.3000.000199.35.3000.000199.3301 MSE (NB3A) 4.3260.000134.764.5410.000201.07512 FIR (3669) 3.6691.20072.03.669-1.20072.0121 FIR interferometer lines of sights

11. Equilibrium mapping

12. Initial Thomson scattering mapping program (TSMAP)  Choice of an equilibrium minimizing the discrepancy between inboard and outboard sides of the electron temperature profile*  In search of best-fitted equilibrium...  Peaking factors (pf, ipf): proper initial values  Plasma current (ip) : Rogowski coil data  Retrieving line-of-sight database,  Search for p0 minimizing the discrepancy for each time slice p0=0.00, pf=2.14, ip=8.0, ipf=2.00, a99= 0.629, avmec=0.637, chi=280. Best Fit p0=4.00, pf=2.45, ip= 8.0, ipf=2.00, a99=0.612, avmec=0.625, chi=4.188 *K. Narihara, I. Yamada, H. Hayashi, and K. Yamauchi: Rev. Sci. Instrum. 72 (2001) 1122. inboard outboard

13. Real-time mapping and automation  Real-time processing to follow the sequence (3 min. shot interval) using 2 servers and 6 virtual machines  Automatic registration of "TSMAP" results for our Kaiseki Data Server System*  Mapping Results (R ↔ r eff )  Additional equilibrium parameters : e.g., W pe, R 00 (0)  Mapped data for CXS and MSE  Development of a viewer program to display T e / n e profiles as functions of effective minor radius *M. Emoto, S. Ohdachi, K. Watanabe, S. Sudo, and Y. Nagayama: Fusion Eng. Des. 81 (2006) 2019.

14. Viewer program inboard/outboard side effective minor radius stored energy (diamagnetic) and electron kinetic energy minor radius and magnetic flux in vacuum calibrated electron density

15. Current status & application

16. Current status of database generation  The equilibrium calculation takes about 5 days for each vacuum configuration under multi-CPU environment.  Database generation (incl. line-of-sight database) has been completed for 24 vacuum configurations.  Coverage of more than 95% of the shots in the LHD experiment Bq=100% gamma 1.1291.1511.1741.1971.2081.2541.2591.262 Rax (m) 3.5 3.55 3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.95 4.0 Databases for (Rax, Bq, gamma) = (3.7, 50, 1.254), (3.75, 50, 1.254), (3.7, 0, 1.254) have also been prepared.

17. How many percentage of shots can be mapped ? RaxBqgamma# of shots 3.61001.25383557 3.751001.25381075 3.91001.2538482 3.531001.2538457 3.61001.1967187 3.81001.2538167 3.75531.253884 3.551001.253881 3.5751001.253878 3.611001.253867 3.61001.12963 3.631001.253856 3.581001.253851 3.851001.253846 3.661001.253845 3.71001.253844 3.651001.253842 3.61001.255341 3.81001.196736 3.751001.196736 3.71001.196735 41001.253817 3.591001.253816 3.91001.196714 3.751001.25532 3.581001.25532 3.551001.25532 3.651001.25532 3.5751001.25532 3.611001.25531 3.91001.25531 total6789 14th cycle (FY2010) 13th cycle (FY2009)15th cycle (FY2011)

18. Summary & application to further analysis  We have newly developed a large-scale LHD equilibrium database and user tools to retrieve the results of inverse mapping program.  We have also developed a real-time magnetic coordinate mapping system available for almost all of the LHD shots by using "Thomson" data.  Any points in real coordinates can be mapped into magnetic coordinates by specifying shot number and time. This can be applied to:  Inversion of line-integrated data (e.g., spectrometer, bolometer, etc.)  Simulation of millimeter wave ray tracing in a plasma  The "TSMAP" data is utilized for generation of input T e / n e profiles for transport analysis codes  Evaluations of particle/heat transport, confinement, neutral beam deposition, power balance... (mentioned by Dr. Yokoyama on Wed. and Dr. Ida this morning)

19. Convergence of VMEC calculation 012345678910 1.41 2.14 3.00 4.00 5.00 R ax = 3.75 m, B q = 100%, gamma = 1.2538, ip = 0 kA, ipf = 2.0 ■ : Converged ■ : Converged if residual tolerance is mitigated (10 -11 -> 10 -8 ) ■ : No convergence / Error p0 pf

20. #108200 f 204/ 292: 4.050s Rax= 3.600m Bq= 100% gamma=1.254 avac= 0.637m denl= 480.581 [a.u.] nel_fir= 4.937 nel_mmw= 9.005 [10^19 m^-2] kk= 0 k= 0 p0= 0.00 pf= 2.14 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.629 avmec= 0.637 chi= 280. kk= 0 k= 1 p0= 1.00 pf= 2.14 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.627 avmec= 0.635 chi= 184. kk= 0 k= 2 p0= 2.00 pf= 2.14 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.623 avmec= 0.631 chi= 96.0 kk= 0 k= 3 p0= 3.00 pf= 2.14 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.619 avmec= 0.628 chi= 28.3 kk= 0 k= 4 p0= 4.00 pf= 2.14 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.612 avmec= 0.626 chi= 6.41 kk= 0 k= 5 p0= 5.00 pf= 2.14 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.608 avmec= 0.625 chi= 50.6 kk= 1 k= 0 p0= 3.40 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.616 avmec= 0.626 chi= 13.9 kk= 1 k= 1 p0= 3.50 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.615 avmec= 0.626 chi= 11.6 kk= 1 k= 2 p0= 3.60 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.615 avmec= 0.625 chi= 9.07 kk= 1 k= 3 p0= 3.70 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.614 avmec= 0.625 chi= 7.10 kk= 1 k= 4 p0= 3.80 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.613 avmec= 0.625 chi= 5.38 kk= 1 k= 5 p0= 3.90 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.613 avmec= 0.625 chi= 4.20 kk= 1 k= 6 p0= 4.00 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.612 avmec= 0.625 chi= 4.06 kk= 1 k= 7 p0= 4.10 pf= 2.48 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.611 avmec= 0.624 chi= 4.80 kbest= 6 p0= 4.00 pf= 2.45 ip= 8.0 ipf= 2.00 sts= 0 err= 0 a99= 0.612 avmec= 0.625 chi= 4.188 Wpe= 9835.5 geom_center= 3.735 Rax_vmec= 3.896 nl_thomson_mid= 53.37 nl_thomson_R3579= 89.24 nl_thomson_R3669= 89.86 nl_thomson_R3759= 94.89 nl_thomson_R3849= 89.73 nl_thomson_R3939= 86.30 Example of mapping for high beta discharge p0 coarse scan p0 fine scan Best fit Fixed experimental value renew initial value scan by 0.1% step χ-square error

21. Plasma sometimes spreads beyond VMEC boundary p0 = 1.20, pf = 1.94, ip = 2.7, ipf = 2.00 a99 = 0.544, avmec= 0.555 R ax = 3.6 m, B q = 100%, gamma = 1.197 p0 = 1.00, pf = 2.14, ip = 0.0, ipf = 2.00 p0 = 5.00, pf = 2.14, ip = 0.0, ipf = 2.00 p0 = 5.10, pf = 2.35, ip = 28.8, ipf = 2.00 a99 = 0.544, avmec= 0.552

22. 'reff': Effective minor radius (m) 'a99':Plasma minor radius defined by the region covering 99% kinetic energy (m) 'p0':Central beta for the best-fitted equilibrium (%) 'pf':Pressure peaking factor for the best-fitted equilibrium 'ip':Plasma current per 1 Tesla measured by Rogowski coil (kA/T) 'ipf':Current peaking factor (fixed at 2.0) 'nl_thomson_mid':Line-integrated electron density along midplane from Thomson (arb) 'nl_mmw_mid':Line-integrated electron density along midplane from MMW interferometer (e19m-2) 'nl_thomson_3669':Line-integrated electron density along FIR (R3669) from Thomson (arb) 'nl_fir_3669':Line-integrated electron density from FIR (R3669) (e19m-2) 'nl_thomson_3579':Line-integrated electron density along FIR (R3579) from Thomson (arb) 'nl_fir_3579':Line-integrated electron density from FIR (R3579) (e19m-2) 'nl_thomson_3759':Line-integrated electron density along FIR (R3759) from Thomson (arb) 'nl_fir_3759':Line-integrated electron density from FIR (R3759) (e19m-2) 'nl_thomson_3849':Line-integrated electron density along FIR (R3849) from Thomson (arb) 'nl_fir_3849':Line-integrated electron density from FIR (R3849) (e19m-2) 'nl_thomson_3939':Line-integrated electron density along FIR (R3939) from Thomson (arb) 'nl_fir_3939':Line-integrated electron density from FIR (R3939) (e19m-2) 'chi':Best minimum mean square error of mapping 'Wpe':Electron kinetic energy calculated from Thomson and best-fitted equilibrium (arb) 'dVdreff':Derivative of plasma volume versus reff [dV/d(reff)] (m^2) 'geom_center':Geometric center of the best-fitted equilibrium (m) 'Rax_vmec':Magnetic axis position of the best-fitted equilibrium (m) 'Te_median_smooth':Median smoothed electron temperature profile used for mapping (keV) 'ne_median_smooth':Median smoothed electron density profile used for mapping (arb) 'mask':Reserved for masking 'Wp':Stored energy from diamagnetic loop (kJ) Registered parameters For n e calib. Pressure / current parameters for the best-fitted equilibrium kinetic Energy Geometric center / Axis position

23. Preset pressure/current profiles Pressure profile 1 2 3 4 5 r eff /a

24. Input/output of the library (PV-WAVE) Library name Structure definition (vacuum magnetic field, real coordinates, output parameters) Definition of structure array Input parameters Input real coordinates output parameters Library call effective minor radius, poloidal angle, 3 components of magnetic field, rotational transform, pressure, specific volume, toroidal flux Required time: 5 seconds per 100pointsNot enough for real-time process of Thomson data

25. Required time to process one shot #90982 (132 frames)#90983 (132 frames) PC1 only188 s PC2 only155 s156 s Both (PC1, PC2)198 s167 s Both (PC2, PC1)158 s189 s Run multiple jobs from separate PCs at the same time PC1: Celeron E1400 @2.00GHz PC2: Core2 Duo E7200 @2.53GHz If we use high-spec multiple clients, real-time mapping between shot intervals may be possible.

26. # [Parameters] # Name = TSMAP # ShotNo = 90000 # Date = '12/17/2010 11:59' # # DimNo = 2 # DimName = 'Time', 'R' # DimSize = 267, 136 # DimUnit = 's', 'm' # # ValNo = 33 # ValName = 'Te', 'dTe', 'n_e', 'dn_e', 'laser', 'laser number', 'reff', 'a99', 'p0', 'pf', 'ip', 'ipf', 'nl_thomson_mid',... # ValUnit = 'keV', 'keV', 'arb', 'arb', 'arb', 'arb', 'm', 'm', '%', 'arb', 'kA/T', 'arb', 'arb',... # # [Comments] # Bt = -1.4300000 # Rax = 3.6000000 # Bq = 100.000 # Gamma = 1.2538000 # phiedge = -3.28050 # avac = 0.63900000 # ExpDate = 20081127 # Cycle = 12 # density (ne) is only for very rough information # Do not use data for ne_bar < 5*1-^12cm-3 # High Voltage is set at 0.90 times # Te limit = 22keV # Laser#1, #3, #5, # # [data] 0.000, 2.532, 0.016, 21.780, 0, 1, 2365, 1, 10.000, 0.000, 0.000, 0.000, -0.000,... 0.000, 2.572, 0.090, 21.780, 2, 1, 2365, 1, -0.688, 0.000, 0.000, 0.000, -0.000,... 0.000, 2.611, 0.000, 21.780, 0, 1, 2365, 1, -0.677, 0.000, 0.000, 0.000, -0.000,... 0.000, 2.648, 0.008, 21.780, 0, 1, 2365, 1, -0.666, 0.000, 0.000, 0.000, -0.000,... 0.000, 2.683, 2.646, 19.357, 0, 0, 2365, 1, -0.655, 0.000, 0.000, 0.000, -0.000,... Registered data format same as Thomson dataadded parts Diagnostic name: tsmap (mapped to T e profile) or tsmap_pe (mapped to p e profile) Information on magnetic field configuration