April  2012, 5(2): 307-328. doi: 10.3934/dcdss.2012.5.307

Simulations on wave propagation in fluctuating fusion plasmas for Reflectometry applications and new developments

1. 

IJL, UMR-CNRS 7198, BP 70239, 54506 VANDOEUVRE Cedex, France

2. 

Associação EURATOM/IST-IPFN Instituto Superior Técnico, 1046-001 Lisboa, Portugal

Received  September 2009 Revised  December 2009 Published  September 2011

The problems associated to the optimization of the systems of a future fusion reactor require new developments due to the limits of existing models, which are unable to describe the experimental behaviour of diagnostics with the required accuracy. This is also true for the study of the coupling antenna-plasma or the computations of the deposits of power for the plasma heating. Simulations on wave propagation on full ITER size is a key issue properly to take into account all the possible effects arising during the wave propagation. These effects should be the scattering processes, back- and forward- scattering, absorption, multi-reflections, diffraction, interference, depolarisation, and mode conversion. Each phenomenon requires an adapted description having its own numerical conditions, which are functions of mesh size, density of modes associated to given plasma fluctuations to name a few. The numerical requirements to fulfil the theoretical modelling cannot always be reached especially when all the space dimensions are needed to have a realistic description of the wave propagation in fluctuating plasmas. A rapid review of each model with its limitations and the specific tools associated to the different kinds of reflectometry diagnostics is detailed. A discussion on the problems and the works underway in the plasma reflectometry community concludes this review.
Citation: Stéphane Heuraux, Filipe da Silva. Simulations on wave propagation in fluctuating fusion plasmas for Reflectometry applications and new developments. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 307-328. doi: 10.3934/dcdss.2012.5.307
References:
[1]

ITER Physics Basis editors, et al., Progress in the ITER Physics Basis,, Nuclear Fusion, 39 (1999), 2137. Google Scholar

[2]

W. M. Tang, Scientific and computational challenges of the Fusion Simulation Project (FSP),, Journal of Physics: Conference Series, 125 (2008). Google Scholar

[3]

A. E. Costley, D. J. Campbell, S. Kasai, K. E. Young and V. Zaveriaev, R&D: Auxiliary Systems: Plasma Diagnostics,, Fusion Engineering and Design, 55 (2001), 331. doi: 10.1016/S0920-3796(01)00200-9. Google Scholar

[4]

J. Garcia, G. Giruzzi, J. F. Artaud and V. Basiuk, et al., Analysis of DEMO scenarios with the CRONOS suite of codes,, Nuclear Fusion, 48 (2009). Google Scholar

[5]

K. Tobita, S. Nishio, M. Enoeda and H. Kawashima, et al., Compact DEMO, SlimCS: design progress and issues,, Nuclear Fusion, 49 (2009). Google Scholar

[6]

N. Katsuragawa, H. Hojo and A. Mase, Computational study on cross polarization scattering of ultrashort-pulse electromagnetic waves,, J. Phys. Soc. Jpn., 67 (1998), 2574. doi: 10.1143/JPSJ.67.2574. Google Scholar

[7]

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[8]

B. Eliasson and P. K. Shukla, Numerical and theoretical study of Bernstein modes in a magnetized quantum plasma,, Phys. Plasmas, 15 (2008). Google Scholar

[9]

A. Hakim, J. Loverich and U. Shumlak, A high resolution wave propagation scheme for ideal two-fluid plasma equations,, Journal of Computational Physics, 219 (2006), 418. doi: 10.1016/j.jcp.2006.03.036. Google Scholar

[10]

David N. Smithe, Finite-difference time-domain simulation of fusion plasmas at radiofrequency time scales,, Phys. Plasmas, 14 (2007). Google Scholar

[11]

M. Masek and K. Rohlena, Novel features of non-linear Raman instability in a laser plasma,, Eur. Phys. J. D, (2009), 00271. Google Scholar

[12]

G. J. Kramer, R. Nazikian, E. J. Valeo, R. V. Budny, C. Kessel and D. Johnson, 2D reflectometer modelling for optimizing the ITER low-field side X-mode reflectometer system,, Nucl. Fusion, 46 (2006). doi: 10.1088/0029-5515/46/9/S21. Google Scholar

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M. A. Irzak and A. Yu Popov, 2D Modeling of the O-X conversion in toroidal plasmas,, Plasma Phys. Control. Fusion, 50 (2008). Google Scholar

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L. Colas, X. L. Zou, M. Paume and J. M. Chareau, et al., Internal magnetic fluctuations and electron heat transport in Tore Supra tokamak: Observation by cross-polarization scattering,, Nucl. Fusion, 38 (1998), 903. doi: 10.1088/0029-5515/38/6/308. Google Scholar

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E. Westerhof, M. D. Tokman and M. A. Gavrilova, Ray-tracing through EC resonance and the wave energy flux,, Fusion Engineering and Design, 53 (2001), 47. doi: 10.1016/S0920-3796(00)00475-0. Google Scholar

[20]

A. N. Saveliev, The virtual beam tracing method for microwave beams in an inhomogeneous plasma,, Plasma Phys. Control. Fusion, 51 (2009). Google Scholar

[21]

C. Honoré, P. Hennequin, A. Truc and A. Quéméneur, Quasi-optical Gaussian beam tracing to evaluate Doppler backscattering conditions,, Nucl. Fusion, 46 (2006), 809. doi: 10.1088/0029-5515/46/9/S16. Google Scholar

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C. Fanack, I. Boucher, S. Heuraux, G. Leclert, F. Clairet and X. L. Zou, Ordinary mode reflectometry: Modifications of the backscattering and cut-off responses due to shape of localized density fluctuations,, Plasma Phys. Control. Fusion, 38 (1996), 1915. doi: 10.1088/0741-3335/38/11/004. Google Scholar

[23]

E. J. Valeo, G. J. Kramer and R. Nazikian, Two-dimensional simulations of correlation reflectometry in fusion plasmas,, Plasma Phys. Control. Fusion, 44 (2002). doi: 10.1088/0741-3335/44/2/101. Google Scholar

[24]

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[25]

J. T. Mendoça, Time refraction in expanding plasma bubbles,, New Journal of Physics, 11 (2009). Google Scholar

[26]

B. B. Afeyan, A. E. Chou and B. I. Cohen, The scattering phase shift due to Bragg resonance in one-dimensional fluctuation reflectometry,, Plasma Phys. Control. Fusion, 37 (1995), 315. doi: 10.1088/0741-3335/37/3/010. Google Scholar

[27]

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[28]

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[29]

E. Z. Gusakov, S. Heuraux and A. Yu. Popov, Nonlinear regime of Bragg backscattering leading to probing wave trapping and time delay jumps in fast frequency sweep reflectometry,, Plasma Phys. Control. Fusion, 51 (2009). Google Scholar

[30]

F. da Silva, S. Heuraux and M. Manso, Studies on O-Mode reflectometry spectra simulations with velocity shear layer,, Nucl. Fusion, 46 (2006). Google Scholar

[31]

A. Casati, V. Grangirard and C. Bourdelle, et al., Turbulence in the TORE SUPRA tokamak: Measurements and validation of nonlinear simulations,, Phys. Rev. Lett., 102 (2009). Google Scholar

[32]

L. Vermare, S. Heuraux, F. Clairet, G. Leclert and F. da Silva, Density fluctuations measurements using X-mode fast sweep reflectometry on Tore Supra,, Nucl. Fusion, 46 (2006), 743. doi: 10.1088/0029-5515/46/9/S10. Google Scholar

[33]

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[34]

G. Vayakis, C. I. Walker and F. Clairet, et al., Status and prospects for mm-wave reflectometry in ITER,, Nucl. Fusion, 46 (2006), 836. doi: 10.1088/0029-5515/46/9/S20. Google Scholar

[35]

A. J. H. Donné, S. H. Heijnen and C. A. J. Hugenholtz, Pulsed radar reflectometry and prospects for fluctuation measurements,, Fusion Eng. Design, 34-37 (1997), 34. Google Scholar

[36]

Y. Yokota, A. Mase, Y. Kogi, L. G. Bruskin, T. Tokuzawa and K. Kawahata, Measurement of edge density profile of LHD plasmas using an ultrashort-pulse reflectometer,, Rev. Sci. Instrum., 79 (2008), 056106. doi: 10.1063/1.2917579. Google Scholar

[37]

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[38]

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[39]

F. da Silva, S. Heuraux, S. Hacquin and M. Manso, Unidirectional transparent signal injection in finite-difference time-domain electromagnetic codes,, J. of Computational Physics, 203 (2005), 467. doi: 10.1016/j.jcp.2004.09.002. Google Scholar

[40]

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[41]

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[42]

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[43]

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[44]

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[45]

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[47]

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[48]

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[49]

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show all references

References:
[1]

ITER Physics Basis editors, et al., Progress in the ITER Physics Basis,, Nuclear Fusion, 39 (1999), 2137. Google Scholar

[2]

W. M. Tang, Scientific and computational challenges of the Fusion Simulation Project (FSP),, Journal of Physics: Conference Series, 125 (2008). Google Scholar

[3]

A. E. Costley, D. J. Campbell, S. Kasai, K. E. Young and V. Zaveriaev, R&D: Auxiliary Systems: Plasma Diagnostics,, Fusion Engineering and Design, 55 (2001), 331. doi: 10.1016/S0920-3796(01)00200-9. Google Scholar

[4]

J. Garcia, G. Giruzzi, J. F. Artaud and V. Basiuk, et al., Analysis of DEMO scenarios with the CRONOS suite of codes,, Nuclear Fusion, 48 (2009). Google Scholar

[5]

K. Tobita, S. Nishio, M. Enoeda and H. Kawashima, et al., Compact DEMO, SlimCS: design progress and issues,, Nuclear Fusion, 49 (2009). Google Scholar

[6]

N. Katsuragawa, H. Hojo and A. Mase, Computational study on cross polarization scattering of ultrashort-pulse electromagnetic waves,, J. Phys. Soc. Jpn., 67 (1998), 2574. doi: 10.1143/JPSJ.67.2574. Google Scholar

[7]

N. J. Sircombe and T. D. Arber, VALIS: A split-conservative scheme for the relativistic 2D Vlasov-Maxwell system,, Journal of Computational Physics, 228 (2009), 4773. doi: 10.1016/j.jcp.2009.03.029. Google Scholar

[8]

B. Eliasson and P. K. Shukla, Numerical and theoretical study of Bernstein modes in a magnetized quantum plasma,, Phys. Plasmas, 15 (2008). Google Scholar

[9]

A. Hakim, J. Loverich and U. Shumlak, A high resolution wave propagation scheme for ideal two-fluid plasma equations,, Journal of Computational Physics, 219 (2006), 418. doi: 10.1016/j.jcp.2006.03.036. Google Scholar

[10]

David N. Smithe, Finite-difference time-domain simulation of fusion plasmas at radiofrequency time scales,, Phys. Plasmas, 14 (2007). Google Scholar

[11]

M. Masek and K. Rohlena, Novel features of non-linear Raman instability in a laser plasma,, Eur. Phys. J. D, (2009), 00271. Google Scholar

[12]

G. J. Kramer, R. Nazikian, E. J. Valeo, R. V. Budny, C. Kessel and D. Johnson, 2D reflectometer modelling for optimizing the ITER low-field side X-mode reflectometer system,, Nucl. Fusion, 46 (2006). doi: 10.1088/0029-5515/46/9/S21. Google Scholar

[13]

M. A. Irzak and A. Yu Popov, 2D Modeling of the O-X conversion in toroidal plasmas,, Plasma Phys. Control. Fusion, 50 (2008). Google Scholar

[14]

L. Colas, X. L. Zou, M. Paume and J. M. Chareau, et al., Internal magnetic fluctuations and electron heat transport in Tore Supra tokamak: Observation by cross-polarization scattering,, Nucl. Fusion, 38 (1998), 903. doi: 10.1088/0029-5515/38/6/308. Google Scholar

[15]

S. Hacquin, S. Heuraux, M. Colin and G. Leclert, Fast computations of wave propagation in an inhomogeneous plasma by a pulse compression method,, Journal of Computational Physics, 174 (2001), 1. Google Scholar

[16]

D. G. Swanson, "Plasma Waves,", 2nd edition, (2003). Google Scholar

[17]

T. H. Stix, "Waves in Plasmas,", Springer-Verlag, (1992). Google Scholar

[18]

S. Weinberg, Eikonal method in Magnetohydrodynamics,, Phys. Rev., 126 (1962), 1899. doi: 10.1103/PhysRev.126.1899. Google Scholar

[19]

E. Westerhof, M. D. Tokman and M. A. Gavrilova, Ray-tracing through EC resonance and the wave energy flux,, Fusion Engineering and Design, 53 (2001), 47. doi: 10.1016/S0920-3796(00)00475-0. Google Scholar

[20]

A. N. Saveliev, The virtual beam tracing method for microwave beams in an inhomogeneous plasma,, Plasma Phys. Control. Fusion, 51 (2009). Google Scholar

[21]

C. Honoré, P. Hennequin, A. Truc and A. Quéméneur, Quasi-optical Gaussian beam tracing to evaluate Doppler backscattering conditions,, Nucl. Fusion, 46 (2006), 809. doi: 10.1088/0029-5515/46/9/S16. Google Scholar

[22]

C. Fanack, I. Boucher, S. Heuraux, G. Leclert, F. Clairet and X. L. Zou, Ordinary mode reflectometry: Modifications of the backscattering and cut-off responses due to shape of localized density fluctuations,, Plasma Phys. Control. Fusion, 38 (1996), 1915. doi: 10.1088/0741-3335/38/11/004. Google Scholar

[23]

E. J. Valeo, G. J. Kramer and R. Nazikian, Two-dimensional simulations of correlation reflectometry in fusion plasmas,, Plasma Phys. Control. Fusion, 44 (2002). doi: 10.1088/0741-3335/44/2/101. Google Scholar

[24]

B. I. Cohen, T. B. Kaiser and J. C. Garrison, One and two-dimensional simulations of ultra-short pulse reflectometry,, Rev. Sci. Instrum., 68 (1997), 1238. doi: 10.1063/1.1147896. Google Scholar

[25]

J. T. Mendoça, Time refraction in expanding plasma bubbles,, New Journal of Physics, 11 (2009). Google Scholar

[26]

B. B. Afeyan, A. E. Chou and B. I. Cohen, The scattering phase shift due to Bragg resonance in one-dimensional fluctuation reflectometry,, Plasma Phys. Control. Fusion, 37 (1995), 315. doi: 10.1088/0741-3335/37/3/010. Google Scholar

[27]

E. V. Gusakov and A. Yu Popov, Non-linear theory of fluctuation reflectometry,, Plasma Phys. Control. Fusion, 44 (2002), 2327. doi: 10.1088/0741-3335/44/11/303. Google Scholar

[28]

G. Leclert, S. Heuraux, E. Z. Gusakov, A. Yu. Popov, I. Boucher and L. Vermare, Full-wave test of the radial correlation reflectometry analytical theory in linear and nonlinear regime,, Plasma Phys. Control. Fusion, 48 (2006), 1389. doi: 10.1088/0741-3335/48/9/008. Google Scholar

[29]

E. Z. Gusakov, S. Heuraux and A. Yu. Popov, Nonlinear regime of Bragg backscattering leading to probing wave trapping and time delay jumps in fast frequency sweep reflectometry,, Plasma Phys. Control. Fusion, 51 (2009). Google Scholar

[30]

F. da Silva, S. Heuraux and M. Manso, Studies on O-Mode reflectometry spectra simulations with velocity shear layer,, Nucl. Fusion, 46 (2006). Google Scholar

[31]

A. Casati, V. Grangirard and C. Bourdelle, et al., Turbulence in the TORE SUPRA tokamak: Measurements and validation of nonlinear simulations,, Phys. Rev. Lett., 102 (2009). Google Scholar

[32]

L. Vermare, S. Heuraux, F. Clairet, G. Leclert and F. da Silva, Density fluctuations measurements using X-mode fast sweep reflectometry on Tore Supra,, Nucl. Fusion, 46 (2006), 743. doi: 10.1088/0029-5515/46/9/S10. Google Scholar

[33]

Thomas Gerbaud, "Étude de la Microturbulence par Réflectométrie dans un Plasma de Fusion sur le Tokamak Tore-Supra," (French),, Ph.D thesis, (2008). Google Scholar

[34]

G. Vayakis, C. I. Walker and F. Clairet, et al., Status and prospects for mm-wave reflectometry in ITER,, Nucl. Fusion, 46 (2006), 836. doi: 10.1088/0029-5515/46/9/S20. Google Scholar

[35]

A. J. H. Donné, S. H. Heijnen and C. A. J. Hugenholtz, Pulsed radar reflectometry and prospects for fluctuation measurements,, Fusion Eng. Design, 34-37 (1997), 34. Google Scholar

[36]

Y. Yokota, A. Mase, Y. Kogi, L. G. Bruskin, T. Tokuzawa and K. Kawahata, Measurement of edge density profile of LHD plasmas using an ultrashort-pulse reflectometer,, Rev. Sci. Instrum., 79 (2008), 056106. doi: 10.1063/1.2917579. Google Scholar

[37]

J. Sanchez, B. Branas, T. Estrada, E. de La Luna and V. Zhuravlev, Amplitude modulation reflectometry for large fusion device,, Rev. Sci. Instrum., 63 (1992), 4654. doi: 10.1063/1.1143651. Google Scholar

[38]

G. R. Hanson, J. B. Wilgen and T. S. Bigelow, et al., Differential-phase reflectometry for edge profile measurements on Tokamak Fusion Test Reactor,, Rev. Sci. Instrum., 66 (1995), 863. doi: 10.1063/1.1146187. Google Scholar

[39]

F. da Silva, S. Heuraux, S. Hacquin and M. Manso, Unidirectional transparent signal injection in finite-difference time-domain electromagnetic codes,, J. of Computational Physics, 203 (2005), 467. doi: 10.1016/j.jcp.2004.09.002. Google Scholar

[40]

F. Simonet, Measurement of electron density profile by microwave reflectometry on tokamaks,, Rev. Sci. Instrum., 56 (1985), 664. doi: 10.1063/1.1138200. Google Scholar

[41]

F. Clairet, C. Bottereau, J. M. Chareau, M. Paume and R. Sabot, Edge denstity profile measurements by X-mode reflectometry on Tore Supra,, Plasma Phys. Cont. Fusion, 43 (2001), 429. doi: 10.1088/0741-3335/43/4/305. Google Scholar

[42]

E. Mazzucato, Density fluctuations in adiabatic toroidal compressor,, Bull. Am. Phys. Soc., 20 (1975), 1241. Google Scholar

[43]

L. Cupido, J. Sanchez and T. Estrada, Frequency hopping millimeter wave reflectometer,, Rev. Sci. Instrum., 75 (2004), 3865. doi: 10.1063/1.1788834. Google Scholar

[44]

E. J. Doyle, T. Lehecka and N. C. Luhmann, et al., Reflectometry density fluctuation measurements on DIIID,, Rev. Sci. Instrum., 61 (1990), 3016. doi: 10.1063/1.1141973. Google Scholar

[45]

A. Krämer-Fleken, V. Dreval and S. Soldatov, et al., Turbulence studies with means of reflectometry at TEXTOR,, Nucl. Fusion, 44 (2004), 1143. Google Scholar

[46]

V. A. Vershkov, V. V. Dreval and S. Soldatov, A three-wave heterodyne correlation reflectoemeter developed in T-10 tokamak,, Rev. Sci. Instrum., 70 (1999), 1700. doi: 10.1063/1.1149654. Google Scholar

[47]

G. Conway, J. Schirmer and S. Klenge, et al., Plasma rotation profile measurements using Doppler reflectometry,, Plasma Phys. Control. Fusion, 46 (2004), 951. doi: 10.1088/0741-3335/46/6/003. Google Scholar

[48]

K. Shinohara, R. Nazikian, T. Fujita and R. Yoshino, Core correlation reflectometer at the JT-60U tokamak,, Rev. Sci. Instrum., 70 (1999), 246. doi: 10.1063/1.1150061. Google Scholar

[49]

M. Gilmore, W. A. Peebles and X. V. Nguyen, Investigation of dual mode (O-X) correlation reflectometry for the determination of the magnetic field strength,, Plasma Phys. Control. Fusion, 42 (2000), 655. doi: 10.1088/0741-3335/42/6/304. Google Scholar

[50]

G. C. Cohen, "Higher-Order Numerical Methods for Transient Wave Equations,", With a foreword by R. Glowinski, (2002). Google Scholar

[51]

T. A. Davis, Algorithm 832: UMFPACK V4.3-an unsymmetric-pattern multifrontal method,, ACM Trans. Math. Softw., 30 (2004), 196. doi: 10.1145/992200.992206. Google Scholar

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