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.

[2]

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

[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.

[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).

[5]

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

[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.

[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.

[8]

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

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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).

[30]

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

[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).

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[42]

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

[43]

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

[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.

[45]

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

[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.

[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.

[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.

[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.

[50]

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

[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.

[52]

B. I. Cohen, B. B. Afeyan, A. E. Chou and N. C. Luhmann, Computational study of ultra-short-pulse reflectometry,, Plasma Phys. Control. Fusion, 37 (1995), 329. doi: 10.1088/0741-3335/37/3/011.

[53]

A. Taflove and S. C. Hagness, "Computational Electrodynamics: The Finite-Difference Time-Domain Method,", Second edition, (2000).

[54]

E. Blanco, S. Heuraux, T. Estrada and J. Sanchez, 2D full-wave code for reflectometry simulations in TJ-II,, Rev. Sci. Instrum., 75 (2004), 3822. doi: 10.1063/1.1783605.

[55]

S. Liu, M. Liu and W. Hong, Modified piecewise linear current density recursive convolution finite-difference time-domain method for anisotropic magnetised plasmas,, IET Microw. Antennas Propag., 2 (2008), 677. doi: 10.1049/iet-map:20070291.

[56]

L. Xu and N. Yuan, FDTD Formulations for Scattering From 3-D Anisotropic Magnetized Plasma Objects,, IEEE Ant. Wir. Prop. Lett., 5 (2006), 335. doi: 10.1109/LAWP.2006.878901.

[57]

S. Liu and M. Liu, Runge-Kutta exponential time differencing FDTD method for anisotropic magnetized plasma,, IEEE Ant. Wir. Prop. Lett., 7 (2008), 306. doi: 10.1109/LAWP.2008.921370.

[58]

J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves,, J. Comput. Phys., 114 (1994), 185. doi: 10.1006/jcph.1994.1159.

[59]

S. Zhao and G. W. Wei, High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces,, J. Comput. Phys., 200 (2004), 60. doi: 10.1016/j.jcp.2004.03.008.

[60]

E. Kashdan and E. Turkel, High-order accurate modeling of electromagnetic wave propagation across media-Grid conforming bodies,, J. Comput. Phys., 218 (2006), 816. doi: 10.1016/j.jcp.2006.03.009.

[61]

D. A. D'Ippolito and J. R. Myra, Analytic model of near-field radio-frequency sheaths part I. Tenuous plasma limit,, Phys. of Plasmas, 16 (2009).

[62]

M. Schubert, S. Heuraux, T. Gerbaud and F. da Silva, On the absolute value of the density fluctuation from reflectometry,, EPD Sciences, (2007), 141.

[63]

M. Schubert, A. Popov, E. Gusakov and S. Heuraux, On the deconvolution of the phase fluctuation profile measured by reflectometry to determine the local density fluctuation,, $35^{nd}$ EPS Conf. Control. Fusion and Plasma Physics, (2008), 9.

[64]

M. Colin, S. Heuraux, G. Leclert, F. Clairet, R. Sabot and S. Hacquin, Full-size simulations of a fluctuation reflectometer in tokamak plasmas,, $28^{th}$ EPS Conf. Controlled Fusion and Plasma Physics, (2001), 18.

[65]

E. Gusakov, N. V. Kosolapova and S. Heuraux, Modelling of the turbulence wave number spectra reconstruction from the radial correlation reflectometry data,, $36^{th}$ EPS Conference on Plasma Physics, (2009).

[66]

E. Blanco and T. Estrada, Study of Doppler reflectometry capability to determine the perpendicular velocity and the k-spectrum of the density fluctuations using 2D full-wave code,, Plasma Phys. Control. Fusion, 50 (2008).

show all references

References:
[1]

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

[2]

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

[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.

[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).

[5]

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

[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.

[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.

[8]

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

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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).

[30]

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

[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).

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

[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.

[42]

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

[43]

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

[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.

[45]

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

[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.

[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.

[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.

[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.

[50]

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

[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.

[52]

B. I. Cohen, B. B. Afeyan, A. E. Chou and N. C. Luhmann, Computational study of ultra-short-pulse reflectometry,, Plasma Phys. Control. Fusion, 37 (1995), 329. doi: 10.1088/0741-3335/37/3/011.

[53]

A. Taflove and S. C. Hagness, "Computational Electrodynamics: The Finite-Difference Time-Domain Method,", Second edition, (2000).

[54]

E. Blanco, S. Heuraux, T. Estrada and J. Sanchez, 2D full-wave code for reflectometry simulations in TJ-II,, Rev. Sci. Instrum., 75 (2004), 3822. doi: 10.1063/1.1783605.

[55]

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