Time Dependent evolution of RF-generated non-thermal particle distributions in fusion plasmas
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1 Time Dependent evolution of RF-generated non-thermal particle distributions in fusion plasmas John C. Wright 1 J.P. Lee 1 P. T. Bonoli 1 A.E. Schmidt 1 and E.J. Valeo 2 1 Plasma Science and Fusion Center Massachusetts Institute of Technology 2 June EPS - Strasbourg
2 Introduction Contents 1 Introduction 2 Numerics 3 Automatic iteration 4 Synthetic HXR 5 Iteration improvements 6 Next steps and Summary 2 EPS2011 Strasbourg
3 Introduction Lower Hybrid wave introduction Frequency range: ω/2 > ω LH Ω ce Ω ci Unmagnetized ions Strongly magnetized electrons Two propagating modes: fast and slow wave 10 4 Cold Plasma DR slow( ) and fast Wavelengths are very short λ ω ω pe λ 1mm Predicts an accessibility criterion: n >= n a ω pe Ω ce + S n 2 _ n a Accessibility x (cm) 3 EPS2011 Strasbourg
4 Introduction Iteration for self-consistent coupling!"#$%&'"()*#(+,%*"-*&*.($/%,+*0$%%* 1&2,*&)+*3(44,567%&).4*-8-',# T($).,*&2,5&B,+*3(44,567%&).4*.(+,* 'C&'*-(%2,-*0(5*),1*+"-'5"U$'"()*0$).'"() 05(#*;3*R$&-"%"),&5*0%$SG* 0 2 VV!!"#$%&'"()-*$-,*O3<9*5,.()-'5$.',+*#&B),'".*,R$"%"U5"&Q L",%,.'5".*0(5*>?*$-,- P,5(*3>;*,%,.'5()-*&)+ $)#&B),'"P,+*"()-Q =&%.$%&',-*'C,*50*0"",%+-*&)+ 'C,*R$&-"%"),&5*0%$SQ*! O%,.'5()*+"-'5"U$'"()*0$).'"()-*05(#*"',5&'"()*1"'C*&*3(44,567%&).4*.(+,*!&5,*$-,+*0(5*+",%,.'5".Q! "" A fixed point iteration scheme is used: f n+1 = FP(f n, Dql n ) Dql n+1 = WE(f n ) This technique is known to converge slowly in many instances. It can also be unstable in cases of weak (n 2 < 40/T e) absorption. 4 EPS2011 Strasbourg
5 Numerics Python driving scripts 3D reconstruction python driven with batch jobs dispatched for each toroidal mode. Separate script for reconstruction and export in VTK format for visualization by VisIT. Python script for handling automation of CQL3D-TORLH iteration. 1 Runs CQL3D [Harvey and McCoy, 1993] and TORLH [Wright et al., 2010] in batch jobs with locking to enforce run order 2 Copies files, makes directories and links for next iteration 3 Ensures input consistency across multiple input files for physics and numerical parameters. 5 EPS2011 Strasbourg
6 Numerics Iteration with Fokker-Planck done manually pol <ExB> Peld Peld <ExB> Power_e Poynting Power_e <ExB> pol Poynting Power_e 80 <ExB> Peld Poynting 90 Poynting Peld Power_e pol pol Iteration # Converges after some oscillations in damping strength. Power of 350 kw generates 150 ka. (nk = 2.5, Te = 2.5keV ) Resolution, TORLH 400Nr 255Nm, CQL3D 60Nr 88Nµ 160Nu 6 EPS2011 Strasbourg
7 Numerics Converged at Step 5. Poloidal spectrum on labeled flux surfaces 2 3 log10 scale m ant P eld 50 <ExB> Power_e Poynting pol 7 EPS2011 Strasbourg
8 Numerics Contexts used to increase parallelism. A schematic of the 3-dimensional parallelization for a block-tridiagonal system. The size of each block, L, D and R is n2 n2,and there are n1 rows of the blocks. The rows are divided by P1 groups, and the element of each block is assigned to P2P3 processors. So, every element has a 3-dimensional index of the assigned processor among total number of processors, Ptot =P1*P2*P3. Block rows are distributed among several parallel contexts to form a 3D processor grid for solving the block tri-diagonal system. A combination of block-cyclic and divide and conquer is used [Garaud and Garaud, 2008], [Lee and Wright, 2011] Floating point work is increased but communication intensity is decreased improving overall strong scaling. 8 EPS2011 Strasbourg
9 Automatic iteration Iteration example with python driver Strong pass case after iteration. Alcator C-Mod like geometry. 3D reconstruction used 20 toroidal modes with damping from iteration using a single peak toroidal mode Electric field of LH Wave in TORLH 2D E field with n =-1.9 3D E field Reconstruction with 20 toroidal modes Radial power absorption profile by 3D reconstruction 9 EPS2011 Strasbourg
10 Automatic iteration C-Mod nk = 1.55 Upper left initial Lower left converged Upper right converged spectrum Lower right converged power 10 EPS2011 Strasbourg
11 C-Mod n = 3.1 Automatic iteration Upper left initial Lower left converged Upper right converged spectrum Lower right converged power 11 EPS2011 Strasbourg
12 Synthetic HXR HXR diagnostic on Alcator C-Mod C-Mod Cross Section HXR Camera D=123 cm d=40 cm Figure Courtesy John Liptac Figure 2-8: HXR camera schematic and viewing chords with respect to the poloidal cross-section of the tokamak. Figure courtesy John Liptac. 32 chords, measures X-rays from LH accelerated electrons that have been pitch angle scattered A synthetic hard X-ray diagnostic in CQL3D uses the same geometry to produce a signal for comparison EPS2011 Strasbourg
13 Synthetic HXR HXR comparison using synthetic HXR Count Rate (s 1 kev 1 ) 3.00e e e e e e+03 Modeled HXR Spectra 44 kev(cql3d) 53 kev(cql3d) 63 kev(cql3d) [Schmidt Diss. 2011] 0.00e Chord Number Comparison of measured and modeled HXR profiles for shot with n = Modeled profiles from ray tracing (left panel) have been scaled by a factor of 1/5 and those from full wave (right panel) have not been scaled. May be linked to interference effects in Dql from full wave. Both and are narrower in spatial extent than experiment. 13 EPS2011 Strasbourg
14 Iteration improvements Effect of n on iterations output power (W) iteration # C-Mod n = 1.55 C-Mod n = 3.1 Target power of 500kW for C-Mod Large oscillations in weak absorption case which eventually settle. 14 EPS2011 Strasbourg
15 Iteration improvements Vector Acceleration methods for iteration Improves rate of convergence of fixed point iteration methods. Similar to Newton-Krylov and GMRES [Sidi CMA 2008] Ψ(x = s) = 0 x n+1 = F (x n ) F (x) = x + ωa(x)ψ(x) x n+1 = F (s) + F (s)(x n 1) x n+1 = Tx n + b This is a linear system of size N. Need the inverse. Sample the system with k << N iterations. Construct inverse through least squares constraints to optimally project to solution. Solution is a linear superposition of iterations, s = k i=1 c ix i. 15 EPS2011 Strasbourg
16 Next steps and Summary Acceleration of iteration reduces error n\ = 1.55 C-Mod extrapolated iteration normal iteration normal error iteration number log 10 of L2 norm of error shown. Refinement of solution takes less than 10 minutes with gain in accuracy of 100 times. Power is now accurate to within 1% 16 EPS2011 Strasbourg
17 Next steps and Summary Summary Automated iteration enables and improved matrix solver greatly improve time to self consistent solution. New vector acceleration method increases accuracy and reduces iterations needed. Converged full wave solutions are starting to elucidate physics differences from ray tracing. 17 EPS2011 Strasbourg
18 Next steps and Summary References and further reading Garaud, P. and Garaud, J. (2008). Dynamics of the solar tachocline II. The stratified case. Mon.Not.R.Astron.Soc. 391, Harvey, R. W. and McCoy, M. G. (1993). The CQL3D Fokker-Planck Code. In Proc. of the IAEA Tech. Committee Meeting (Montreal,1992), pages , Vienna. IAEA, Institute of Physics Publishing; USDOC/NTIS Doc. DE Schmidt, A. E., Bonoli, P. T., Meneghini, O., Parker, R. R., Porkolab, M., Shiraiwa, S., Wallace, G., Wright, J. C., Harvey, R. W., and Wilson, J. R. (2011). Investigation of lower hybrid physics through power modulation experiments on Alcator C-Mod. Phys. Plasmas 17, to be published. Wallace, G., Parker, R., Bonoli, P., Hubbard, A., Hughes, J., LaBombard, B., Meneghini, O., Schmidt, A., Shiraiwa, S., Whyte, D., Wright, J., Wukitch, S., Harvey, R., Wilson, J., Smirnov, A., and the Alcator C-Mod Team (2011). Lower Hybrid Current Drive at High Density in Alcator C-Mod. Nucl. Fusion 51, to be published. Wright, J., Lee, J., Valeo, E., Bonoli, P., Phillips, C., Jaeger, E., and Harvey, R. (2010). Challenges in Self-Consistent Full-Wave Simulations of Lower Hybrid Waves. Plasma Science, IEEE Transactions on 38, For copies, or further information contact or see jwright 18 EPS2011 Strasbourg
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