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Simulation as a tool to improve wave heating in fusion plasmas

  • S. Heuraux (a1), F. da Silva (a2), T. Ribeiro (a3), B. Despres (a4), M. Campos Pinto (a4), J. Jacquot (a5), E. Faudot (a1), S. Wengerowsky (a1), L. Colas (a5) and L. Lu (a1) (a5)...


Firstly, a brief overview will be given on different models that are able to describe the behaviour of wave propagation as a function of specific frequency ranges. Each range corresponds to different heating systems, namely, 20–100 MHz for the ion cyclotron resonant heating, 2–20 GHz for lower-hybrid heating or current drive, and 100–250 GHz for electron cyclotron resonant heating or current drive systems. The specification of every system will be explained in detail, including the typical set of equations and the assumptions needed to describe the properties of these heating or current drive systems, as well as their specific domains of validity. In these descriptions, special attention will be paid to the boundary conditions. A review of specific physical problems associated with the wave heating systems will also be provided. The review will detail the role of simulation in answering questions that arise from experiments on magnetized plasma devices devoted to fusion. A few examples that will be covered are the impact of edge turbulence on wave propagation and its consequences on heating system performance, the effects of fast particles and ponderomotive effects, among others. A study that is more focused on radio-frequency sheath effects will also be discussed. It shows that such simulations require very sophisticated tools to gain a partial understanding of the observations undertaken in dedicated experiments. To conclude this review, an overview will be given about the requirements and progress necessary to obtain relevant predictive simulation tools able to describe the wave heating systems used in fusion devices.


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Artaud, J. F., Basiuk, V., Imbeaux, F., Schneider, M., Garcia, J., Giruzzi, G., Huynh, P., Aniel, T., Albajar, F., Ané, J. M., Bécoulet, A., Bourdelle, C., Casati, A., Colas, L., Decker, J., Dumont, R., Eriksson, L. G., Garbet, X., Guirlet, R., Hertout, P., Hoang, G. T., Houlberg, W., Huysmans, G., Joffrin, E., Kim, S. H., Kochl, F., Lister, J., Litaudon, X., Maget, P., Masset, R., Pégourié, B., Peysson, Y., Thomas, P., Tsitrone, E. & Turco, F. 2010 The CRONOS suite of codes for integrated tokamak modelling. Nucl. Fusion 50, 043001.
Bai, B., Li, X., Liu, Y., Xu, J., Shi, L. & Xie, K. 2014 Effects of reentry plasma sheath on the polarization properties of obliquely incident EM waves. IEEE Trans. Plasma Sci. 42, 33653372.
Batanov, G. M., Borzosekov, V. D., Kovrizhnykh, L. M., Kolik, L. V., Konchekov, E. M., Malakhov, D. V., Petrov, A. E., Sarksyan, K. A., Skvortsova, N. N., Stepakhin, V. D. & Kharchev, N. K. 2013 Backscattering of gyrotron radiation and short-wavelength turbulence during electron cyclotron resonance plasma heating in the L-2M stellarator. Plasma Phys. Rep. 39, 444455.
Bérenger, J.-P. 1994 Numerical modeling of the coupling of an ICRH antenna with a plasma with self-consistent antenna current in the ion cyclotron range of frequencies. J. Comput. Phys. 114, 185200.
Bertelli, N., Maj, O., Poli, E., Harvey, R., Wright, J. C., Bonoli, P. T., Phillips, C. K., Smirnov, A. P., Valeo, E. & Wilson, J. R. 2012 Paraxial Wentzel–Kramers–Brillouin method applied to the lower hybrid wave propagation. Phys. Plasmas 19, 082510.
Bindslev, H. 1991 Dielectric effects on Thomson scattering in a relativistic magnetized plasma. Plasma Phys. Control Fusion 33, 17751804.
Bobkov, V., Braun, F., Dux, R., Herrmann, A., Giannone, L., Kallenbach, A., Rohde, V., Schweinzer, J., Sips, A., Zammuto, I. & ASDEX Upgrade Team 2010 Assessment of compatibility of ICRF antenna operation with full W wall in ASDEX upgrade. Nucl. Fusion 50, 035004.
Bonoli, P. 1985 Linear theory of lower-hybrid wave in tokamak plasmas. In Wave Heating and Current Drive in Plasmas, pp. 175218. Gordon and Breach.
Bonoli, P. T. 2014 Review of recent experimental and modeling progress in the lower hybrid range of frequencies at ITER relevant parameters. AIP Conf. Proc. 1580, 1524.
Bornatici, M., Cano, R., de Barbeiri, O. & Englemann, F. 1983 Electron cyclotron emission and absoprtion in fusion plasmas. Nucl. Fusion 23, 11531257.
Brambilla, M. 1999 Numerical simulation of ion cyclotron waves in tokamak plasmas. Plasma Phys. Control. Fusion 41, 134.
Brambilla, M. & Bilato, R. 2009 Advances in numerical simulations of ion cyclotron heating of non-Maxwellian plasmas. Nucl. Fusion 49, 085004.
Budny, R. V., Berry, L., Bilato, R., Bonoli, P., Brambilla, M., Dumont, R. J., Fukuyama, A., Harvey, R., Jaeger, E. F., Indireshkumar, K., Lerche, E., McCune, D., Phillips, C. K., Vdovin, V. & Wright, J. 2012 Benchmarking ICRF full-wave solvers for ITER. Nucl. Fusion 52, 023023.
Cairns, R. A. 1991 Radiofrequency Heating of Plasmas. chap. 3–5, Adam Hilger.
Campos Pinto, M., Sonnendrücker, E., Friedman, A., Grote, D. P. & Lund, S. P. 2014 Noiseless Vlasov–Poisson simulations with linearly transformed particles. J. Comput. Phys. 275, 236256.
Casati, A., Grangirard, V., Bourdelle, C., Hennequin, P., Gerbaud, T., Heuraux, S., Vermare, L., Aniel, T., Clairet, F., Garbet, X., Imbeaux, F., Candy, J. & Waltz, R. 2009 Turbulence in the Tore Supra tokamak measurements and validation of nonlinear simulations. Phys. Rev. Lett. 102, 165005.
Clairet, F., Colas, L., Heuraux, S. & Lombard, G. 2004 ICRF coupling and edge density profile on Tore Supra. Plasma Phys. Control. Fusion 46, 15671581.
Coelho, R., Äkäslompolo, S., Dinklage, A., Kus, A., Sundén, E., Blanco, E., Conway, G., Hacquin, S., Heuraux, S., Lechte, C., Silva, F. & Sirinelli, A. 2013 Synthetic diagnostics in the EU-ITM simulation platform. Fusion Sci. Technol. 63, 19.
Colas, L., Jacquot, J., Heuraux, S., Faudot, E., Crombé, K., Kyrytsya, V., Hillairet, J. & Goniche, M. 2012 Self consistent radio-frequency wave propagation and peripheral direct current plasma biasing: simplified three dimensional non-linear treatment in the wide sheath asymptotic regime. Phys. Plasmas 19, 092505.
Colas, L., Zou, X. L., Paume, M., Chareau, J. M., Guiziou, L., Hoang, G. T., Michelot, Y. & Gresillon, D. 1998 Internal magnetic fluctuations and electron heat transport in Tore Supra. Nucl. Fusion 38, 903918.
Comisso, L. & Lazzaro, E. 2010 Two-dimensional effects in the problem of tearing modes control by electron cyclotron current drive. Nucl. Fusion 50, 125002.
Courant, R., Friedrichs, K. & Lewy, H. 1928 Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 3274.
Crouseilles, N., Lemou, M. & Méhats, F. 2013 Asymptotic preserving schemes for highly oscillatory Vlasov–Poisson equations. J. Comput. Phys. 248, 287308.
Decker, J., Peysson, Y. & Coda, S. 2012 Effect of density fluctuations on ECCD in ITER and TCV. Eur. Phys. J. 32, 01016.
Després, B., Imbert-Gérard, L.-M. & Weder, R. 2014 Hybrid resonance of Maxwell’s equations in slab geometry. J. Math. Pure Appl. 101, 623659.
D’Ippolito, D. A. & Myra, J. R. 2006 A radio-frequency sheath boundary condition and its effect on slow wave propagation. Phys. Plasmas 13, 102508.
Dumont, R. J. & Zarzoso, D. 2013 Heating and current drive by ion cyclotron waves in the activated phase of ITER. Nucl. Fusion 53, 013002.
van Eester, D. 2012 Modeling particle and current drive in fusion machines: brief review of adopted techniques. Trans. Fusion Sci. Technol. 61, 347354.
van Eester, D. & Koch, R. 2001 Integrating the finite-temperature wave equation across the plasma/vacuum interface. Plasma Phys. Control. Fusion 43, 779794.
Falchetto, G. L., Coster, D., Coelho, R., Scott, B. D., Figini, L., Kalupin, D., Nardon, E., Nowak, S., Alves, L. L., Artaud, J. F., Basiuk, V., Bizarro, J. P. S., Boulbe, C., Dinklage, A., Farina, D., Faugeras, B., Ferreira, J., Figueiredo, A., Huynh, Ph., Imbeaux, F., Ivanova-Stanik, I., Jonsson, T., Klingshirn, H.-J., Konz, C., Kus, A., Marushchenko, N. B., Pereverzev, G., Owsiak, M., Poli, E., Peysson, Y., Reimer, R., Signoret, J., Sauter, O., Stankiewicz, R., Strand, P., Voitsekhovitch, I., Westerhof, E., Zok, T. & Zwingmann, W. 2014 The European Integrated Tokamak Modelling (ITM) effort: achievements and first physics results. Nucl. Fusion 54, 043018.
Fanack, C., Boucher, I., Heuraux, S., Leclert, G., Clairet, F. & Zou, X. L. 1996 Ordinary mode reflectometry: modifications of the backscattering and cut-off responses due to shape of localized density fluctuations. Plasma Phys. Control. Fusion 40, 19151930.
Faudot, E., Heuraux, S. & Colas, L. 2006 Peaking criterion for rectified potential in front of ICRF antennas in fusion plasma. Phys. Plasmas 13, 042512.
Faudot, E., Heuraux, S., Colas, L. & Gunn, J. P. 2010 Broadening of rectified potential structures induced by RF currents in a magnetized plasma application to ITER scrape-off layer. Phys. Plasmas 17, 042503.
Faudot, E., Heuraux, S., Kubic, M., Gunn, J. & Colas, L. 2013 Fluid modeling of RF and DC currents in a biased magnetized plasma. Phys. Plasmas 20, 043514.
Figini, L., Decker, J., Farina, D., Marushchenko, N. B., Peysson, Y., Poli, E., Westerhof, E. & ITM-TF contributors 2012 Benchmarking of electron cyclotron heating and current drive codes on ITER scenarios within the European Integrated Tokamak Modelling framework. Eur. Phys. J. 32, 01011.
Fuchs, V., Ram, A. K., Schultz, S. D., Bers, A. & Lashmore-Davies, C. N. 1995 Mode conversion and electron damping of the fast Alfvén wave in a tokamak at ion–ion hybrid frequency. Phys. Plasmas 2, 16371647.
Gerbaud, T., Clairet, F., Sirinelli, A., Sabot, R., Heuraux, S., Vermare, L. & Leclert, G. 2006 Comparison of density fluctuation measurements between O-mode and X-mode reflectometry on Tore Supra. Rev. Sci. Instrum. 77, 10E928.
Gnesin, S.2011 Electron cyclotron heating and suprathermal electron dynamics in the TCV tokamak. PhD thesis, EPFL Laussane.
Gusakov, E. Z., Heuraux, S., Irzak, M. & Popov, A. Yu. 2011 Possibility of giant scattering enhancement due to wave trapping in a reflectometry experiment. Phys. Scr. 84, 04504.
Gusakov, E. Z., Heuraux, S. & Popov, A. Yu. 2009 Nonlinear regime of Bragg backscattering leading to probing wave trapping and time delay jumps in fast frequency sweep reflectometry. Plasma Phys. Control. Fusion 51, 065018.
Gusakov, E. Z., Popov, A. Yu. & Saveliev, A. N. 2014 Trapping of electron Bernstein waves in drift-wave eddies and parametric decay instability at second harmonic ECRH in toroidal devices. Plasma Phys. Control. Fusion 56, 015010.
Heuraux, S. & da Silva, F. 2012 Simulations on wave propagation in fluctuating fusion plasmas for reflectometry applications and new developments. Discrete Continuous Dyn. Syst. S 5, 307328.
Heuraux, S., Faudot, E., da Silva, F., Jacquot, J., Colas, L., Hacquin, S., Teplova, N., Sysoeva, E. V. & Gusakov, E. Z. 2014 Study of wave propagation in various kinds of plasmas using adapted simulation methods, with illustrations on possible future applications. C. R. Phys. 15, 421429.
Heuraux, S., Gusakov, E. Z., Popov, A. Yu., da Silva, F. & Irzak, M. 2010 Simulations on the role of the resonance of the probing wave on reflectometry measurements in fluctuating plasmas. IEEE Trans. Plasma Sci. 38, 21502158.
Heuraux, S., Leclert, G. & Hadjoudj, Y. 1994 Low-frequency responses of a plasma to a large amplitude high-frequency wave polarized on the extraordinary mode. J. Phys. III 4, 839848.
Hillairet, J., Voyer, D., Ekedahl, A., Goniche, M., Kazda, M., Meneghini, O., Milanesio, D. & Preynas, M. 2010 ALOHA: an Advanced LOwer Hybrid Antenna coupling code. Nucl. Fusion 50, 125010.
Irzak, M. A. & Popov, A. Yu. 2008 2D modeling of the O-X conversion in toroidal plasmas. Plasma Phys. Control. Fusion 50, 025003.
Jackson, J. D. 1999 Classical Electrodynamics, pp. 1922. Wiley.
Jacquet, P., Colas, L., Mayoral, M.-L. & Arnoux, G. et al. 2011 Heat-loads on jet plasma facing components from ICRF and LH wave absorption in the sol. Nucl. Fusion 51, 103018.
Jacquot, J., Colas, L., Clairet, F., Goniche, M., Heuraux, S., Hillairet, J., Lombard, G. & Milanesio, D. 2013 2D and 3D modeling of wave propagation in cold magnetized plasma near the Tore Supra ICRH antenna relying on the perfectly matched layer technique. Plasma Phys. Control. Fusion 55, 115004.
Jacquet, P., Colas, L., Mayoral, M.-L., Arnoux, G., Bobkov, V., Brix, M., Coad, P., Czarnecka, A., Dodt, D., Durodie, F., Ekedahl, A., Frigione, D., Fursdon, M., Gauthier, E., Goniche, M., Graham, M., Joffrin, E., Korotkov, A., Lerche, E., Mailloux, J., Monakhov, I., Noble, C., Ongena, J., Petrzilka, V., Portafaix, C., Rimini, F., Sirinelli, A., Riccardo, V., Vizvary, Z., Widdowson, A., Zastrow, K.-D. & JET EFDA Contributors 2014 Radio-frequency sheath physics: experimental characterization on Tore Supra and related self-consistent modeling. Phys. Plasmas 21, 061509.
Jaeger, E. F., Batchelor, D. B. & Stallings, D. C. 1993 Influence of various physics phenomena on fast wave current drive in tokamaks. Nucl. Fusion 33, 179195.
Jandieri, G. V., Ishimaru, A., Jandieri, V. G. & Zhukova, N. N. 2011 Depolarization of metric radio signals and the spatial spectrum of scattered radiation by magnetized turbulent plasma slab. Prog. Electromag. Res. 112, 6375.
Javan, N. S. & Adli, F. 2013 Polarization effect on the relativistic nonlinear dynamics of an intense laser beam propagating in a hot magnetoactive plasma. Phys. Rev. E 88, 043102.
Jucker, M., Graves, J. P., Cooper, W. A., Mellet, N., Johnson, T. & Brunner, S. 2011 Integrated modeling for ion cyclotron resonant heating in toroidal systems. Comput. Phys. Commun. 182, 912925.
Kohno, H., Myra, J. R. & D’ipolitto, D. A. 2012 Numerical modeling of the coupling of an ICRH antenna with a plasma with self-consistent antenna current in the ion cyclotron range of frequencies. Comput. Phys. Commun. 183, 21162127.
Kuhn, S., Tskhakaya, D. D. & Tskhakaya, D. Jr. 2007 The magnetized plasma–wall transition (PWT) and its relation to fluid boundary conditions. Comput. Phys. Commun. 177, 8083.
Kuley, A., Wang, Z. X., Lin, Z. & Wessel, F. 2013 Verification of particle simulation of radio frequency waves in fusion plasmas. Phys. Plasmas 20, 102515.
Lancellotti, V., Milanesio, D., Maggiora, R., Vecchi, G. & Kyrytsya, V. 2006 TOPICA: an accurate and efficient numerical tool for analysis and design of IRRF antennas. Nucl. Fusion 46, S476S499.
Laqua, H. P., Erckmann, V., Hartfuss, H. J. & Laqua, H. 1997 Resonant and nonresonant electron cyclotron heating at densities above the plasma cutoff by O-X-B mode conversion at the W7-AS stellarator. Phys. Rev. Lett. 78, 34673470.
Ledenev, V. G., Tirsky, V. V., Tomozov, V. M. & Zlobec, P. 2002 Polarization changes in solar radio emission caused by scattering from high-frequency plasma turbulence. Astron. Astrophys. 392, 10891094.
Marushchenko, N. B., Turkin, Y. & Maassberg, H. 2014 Ray-tracing code TRAVIS for ECR heating, EC current drive and ECE diagnostic. Comput. Phys. Commun. 185, 165176.
Mazzucato, E. 1998 Microwave reflectometry for magnetically confined plasmas. Rev. Sci. Instrum. 69, 22012217.
Milanesio, D., Meneghini, O., Maggiora, R., Guadamuz, S., Hillairet, J., Lancellotti, V. & Vecchi, G. 2012 TOPLHA: an accurate and efficient numerical tool for analysis and design of LH antennas. Nucl. Fusion 52, 013008.
Mirnov, V. V., Brower, D. L., Hartog, D. J. & Ding, W. X. et al. 2014 Electron kinetic effects on interferometry, polarimetry and Thomson scattering measurements in burning plasmas. Rev. Sci. Instrum. 85, 11D302.
Noterdaeme, J.-M. & van Oost, G. 1993 The interaction between waves in the ion cyclotron range of frequencies and the plasma boundary. Plasma Phys. Control. Fusion 35, 14811511.
Pécoul, S., Heuraux, S., Koch, R. & Leclert, G. 2002 Numerical modeling of the coupling of an ICRH antenna with a plasma with self-consistent antenna current. Comput. Phys. Commun. 146, 166187.
Perkins, W. 1989 Radiofrequency sheaths and impurity generation by ICRF antennas. Nucl. Fusion 29, 583592.
Peysson, Y. & Decker, J. 2014 Numerical simulations of the radio-frequency-driven toroidal current in tokamaks. Fusion Sci. Technol. 65, 2242.
Peysson, Y., Decker, J. & Morini, L. 2012 A versatile ray-tracing code for studying RF wave propagation in toroidal magnetized plasmas. Plasma Phys. Control. Fusion 54, 045003.
Poli, F. M., Kessel, C. E., Bonoli, P. T., Batchelor, D. B., Harvey, R. W. & Snyder, P. B. 2014 External heating and current drive source requirements towards steady-state operation in ITER. Nucl. Fusion 54, 073007.
Popov, A. Yu. 2015 Anomalous reflection of electromagnetic waves at O–X mode conversion in 2D inhomogeneous turbulent plasma. Plasma Phys. Control. Fusion 57, 025010.
Prater, R., Farina, D., Gribov, Yu., Harvey, R. W., Ram, A. K., Lin-Liu, Y.-R., Poli, E., Smirnov, A. P., Volpe, F., Westerhof, E. & Zvonkov, A. 2008 Benchmarking of codes for electron cyclotron heating and electron cyclotron current drive under ITER conditions. Nucl. Fusion 48, 035006.
Shiraiwa, S., Meneghini, O., Parker, R. & Bonoli, P. et al. 2010 Plasma wave simulation based on a versatile finite element method solver. Phys. Plasmas 17, 056119.
Smithe, D. N. 2007 Finite-difference time-domain simulation of fusion plasmas at radiofrequency time scales. Phys. Plasmas 14, 056104.
Smithe, D., Myra, J. R. & D’Ippolito, D. A. 2014 Quantitative modeling of ICRF antennas with integrated time domain RF sheath and plasma physics. AIP Conf. Proc. 1580, 8996.
Stangeby, P. C. 2012 The chodura sheath for angles of a few degrees between the magnetic field and the surface of divertor targets and limiters. Nucl. Fusion 52, 083012.
Stix, T. H. 1992 Waves in Plasmas. American Institute of Physics.
Swanson, A. 2003 Plasma Waves, 2nd edn. IOP Publishing.
Sysoeva, E. V., da Silva, F., Gusakov, E. Z., Heuraux, S. & Popov, A. Yu. 2014 ECRH microwave beam broadening in the edge turbulent plasma. AIP Conf. Proc. 1522, 522525.
Sysoeva, E. V., da Silva, F., Gusakov, E. Z., Heuraux, S. & Popov, A. Yu. 2015 ECRH beam broadening in the edge turbulent plasma of fusion machines. Nucl. Fusion 55, 033016.
da Silva, F., Campos Pinto, M., Després, B. & Heuraux, S. 2015 Stable coupling of the Yee scheme with a linear current model. J. Comput. Phys. 295, 2445.
da Silva, F., Heuraux, S., Gusakov, E. Z. & Popov, A. Yu. 2010 A numerical study of forward- and back-scattering signatures on Doppler reflectometry signals. IEEE Trans. Plasma Sci. 38, 21442149.
da Silva, F., Heuraux, S., Hacquin, S. & Manso, M. 2005 Unidirectional transparent signal injection in finite-difference time-domain electromagnetic codes. J. Comput. Phys. 203, 467492.
da Silva, F., Heuraux, S. & Manso, M. 2006a Developments on reflectometry simulations for fusion plasmas: applications to ITER position reflectometry. J. Plasma Phys. 72, 12051211.
da Silva, F., Heuraux, S. & Manso, M. 2006b Studies on O-mode reflectometry spectra simulations with velocity shear layer. Nucl. Fusion 46, S816S823.
da Silva, F., Heuraux, S., Manso, M. E. & Varela, P. 2003 A 2D FDTD full-wave code for simulating the diagnostic of fusion plasmas with microwave reflectometry. In Computational Methods in Engineering and Science, pp. 233238. A. A. Balkema.
Taflove, A. & Hagness, S. C. 2000 Computational Electrodynamics: the Finite-Difference Time-Domain Method. pp. 175235. Artech House.
Tenouxa, T. & Lostanlen, Y. 2012 ECRH beam broadening in the edge turbulent plasma of fusion machines. Phys. Commun. 5, 338351.
Ticos, C. M., Stoica, D. S. & Delzanno, G. L. 2012 Generation of dust projectiles passing over an obstacle in the plasma sheath. Phys. Plasmas 19, 083701.
Tierens, W. & de Zutter, D. 2012 Finite-temperature corrections to the time-domain equations of motion for perpendicular propagation in nonuniform magnetized plasmas. Phys. Plasmas 19, 112110.
Tsironis, C., Peteers, A. G., Isliker, H., Strinzi, D. & Chatziantonaki, I. 2009 Electron cyclotron wave scattering by edge density fluctuations in ITER. Phys. Plasmas 16, 112510.
Wilson, J. R. & Bonoli, P. 2015 Progress on ion cyclotron range of frequencies heating physics and technology in support of the International Tokamak Experimental Reactor. Phys. Plasmas 22, 021801.
Wright, J. C., Valeo, E. J., Phillips, C. K., Bonoli, P. T. & Brambilla, M. 2008 Full wave simulations of lower hybrid waves in toroidal geometry with non-Maxwellian electrons. Commun. Comput. Phys. 4, 545555.
Wukitch, S. J., LaBombard, B., Lin, Y., Lipschultz, B., Marmar, E. & Reinke, M. L. 2009 ICRF specific impurity sources and plasma sheaths in Alcator C-Mod. J. Nucl. Mater. 390–391, 951954.
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