Well-posedness and generalized plane waves simulations of a 2D mode conversion model. (English) Zbl 1349.78037
Summary: Certain types of electro-magnetic waves propagating in a plasma can undergo a mode conversion process. In magnetic confinement fusion, this phenomenon is very useful to heat the plasma, since it permits to transfer the heat at or near the plasma center. This work focuses on a mathematical model of wave propagation around the mode conversion region, from both theoretical and numerical points of view. It aims at developing, for a well-posed equation, specific basis functions to study a wave mode conversion process. These basis functions, called generalized plane waves, are intrinsically based on variable coefficients. As such, they are particularly adapted to the mode conversion problem. The design of generalized plane waves for the proposed model is described in detail. Their implementation within a discontinuous Galerkin method then provides numerical simulations of the process. These first 2D simulations for this model agree with qualitative aspects studied in previous works.
MSC:
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 82D10 | Statistical mechanics of plasmas |
References:
| [1] | Preinhaelter, J.; Kopecky, V., Penetration of high-frequency waves into a weakly inhomogeneous magnetized plasma at oblique incidence and their transformation to Bernstein modes, J. Plasma Phys., 10, 1-12 (1973) |
| [2] | Bonoli, P. T., Review of recent experimental and modeling progress in the lower hybrid range of frequencies at ITER relevant parameters, (AIP Conf. Proc., vol. 1580 (2014)), 15-24 |
| [3] | Laqua, H. P.; Erckmann, V.; Hartfuß, H. J.; Laqua, H., 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, 3467-3470 (1997) |
| [4] | Laqua, H. P.; Hartfuß, H. J.; Team, W.-A., Electron Bernstein wave emission from an overdense plasma at the w7-as stellarator, Phys. Rev. Lett., 81, 2060-2063 (1998) |
| [5] | Laqua, H. P.; Maassberg, H.; Marushchenko, N. B.; Volpe, F.; Weller, A.; Kasparek, W., Electron-Bernstein-wave current drive in an overdense plasma at the Wendelstein 7-AS stellarator, Phys. Rev. Lett., 90, 7, Article 075003 pp. (2003) |
| [6] | Batchelor, D. B.; Berry, L. A.; Bonoli, P. T.; Carter, M. D.; Choi, M.; D’Azevedo, E.; D’Ippolito, D. A.; Gorelenkov, N.; Harvey, R. W.; Jaeger, E. F.; Myra, J. R.; Okuda, H.; Phillips, C. K.; Smithe, D. N.; Wright, J. C., Electromagnetic mode conversion: understanding waves that suddenly change their nature, J. Phys. Conf. Ser., 16, 1, 35 (2005) |
| [7] | Volpe, F., Analytical solution of the o-x mode conversion problem, Phys. Lett. A, 374, 15-16, 1737-1741 (2010) · Zbl 1236.76115 |
| [8] | Weitzner, H., O-x mode conversion in an axisymmetric plasma at electron cyclotron frequencies, Phys. Plasmas, 11, 3, 866-877 (2004) |
| [9] | Imbert-Gérard, L.-M.; Després, B., A generalized plane-wave numerical method for smooth nonconstant coefficients, IMA J. Numer. Anal., 34, 3, 1072-1103 (2014) · Zbl 1301.65121 |
| [10] | Imbert-Gérard, L.-M., Interpolation properties of generalized plane waves, Numer. Math., 1-29 (2015) |
| [11] | Herman, R., A Treatise on Geometrical Optics (1900), University Press · JFM 31.0797.02 |
| [12] | Weitzner, H.; Batchelor, D. B., Conversion between cold plasma modes in an inhomogeneous plasma, Phys. Fluids, 22, 7, 1958-1988 (1979) · Zbl 0402.76097 |
| [13] | Babuska, I. M.; Sauter, S. A., Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., 34, 6, 2392-2423 (1997), Reprint of · Zbl 0894.65050 |
| [14] | Gittelson, C. J.; Hiptmair, R.; Perugia, I., Plane wave discontinuous Galerkin methods: analysis of the \(h\)-version, ESAIM: Math. Model. Numer. Anal., 43, 297-331 (2009) · Zbl 1165.65076 |
| [15] | Trefftz, E., Ein Gegenstuck zum Ritzschen Verfahren, (Proceedings of the 2nd International Congress of Applied Mechanics. Proceedings of the 2nd International Congress of Applied Mechanics, Zurich (1926)), 131-137 · JFM 52.0483.02 |
| [16] | Pluymers, B.; Desmet, W.; Vandepitte, D.; Sas, P., Trefftz-based methods for time-harmonic acoustics, Arch. Comput. Methods Eng., 14, 343-381 (2007) · Zbl 1170.76332 |
| [17] | Després, B., Sur une formulation variationnelle de type ultra-faible, C. R. Math. Acad. Sci. Paris, Sér. I, 318, 10, 939-944 (1994) · Zbl 0806.35026 |
| [18] | Ciarlet, P., The Finite Element Method for Elliptic Problems (2002), Society for Industrial and Applied Mathematics · Zbl 0999.65129 |
| [19] | Cessenat, O.; Després, B., Application of an ultra weak variational formulation of elliptic PDEs to the two dimensional Helmholtz problem, SIAM J. Numer. Anal., 55, 1, 255-299 (1998) · Zbl 0955.65081 |
| [20] | Buffa, A.; Monk, P., Error estimates for the ultra weak variational formulation of the Helmholtz equation, ESAIM: Math. Model. Numer. Anal., 42, 925-940 (2008) · Zbl 1155.65094 |
| [21] | Hiptmair, R.; Moiola, A.; Perugia, I., Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the \(p\)-version, SIAM J. Numer. Anal., 49, 1, 264-284 (2011) · Zbl 1229.65215 |
| [22] | Huttunen, T.; Malinen, M.; Monk, P., Solving Maxwell’s equations using the ultra weak variational formulation, J. Comput. Phys., 223, 2, 731-758 (2007) · Zbl 1117.78011 |
| [23] | Huttunen, T.; Monk, P.; Kaipio, J. P., Computational aspects of the ultra-weak variational formulation, J. Comput. Phys., 182, 1, 27-46 (2002) · Zbl 1015.65064 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
