Flux Surfaces According to Killing Magnetic Vectors in Riemannian Space $\mathbb{S}ol3$

Year 2023, , 89 - 100, 30.06.2023

Nourelhouda Benmensour , Fouzi Hathout

https://doi.org/10.33401/fujma.1163741

Abstract

In this paper, we define flux surface as surfaces in which its normal vector is orthogonal to the vector corresponding to a flux with its associate scalar flux functions in ambient manifold M. Next, we determine, in 3-dimensional homogenous Riemannian manifold $\mathbb{S}ol3$, the parametric flux surfaces according to the flux corresponding to the Killing magnetic vectors and we calculate its associate scalar flux functions. Finally, examples of such surfaces are presented with their graphical representation in Euclidean space.

Keywords

Flux, Flux surface, Killing magnetic, Scalar flux function, $\mathbb{S}ol3$ manifold

References

• [1] A. H. Boozer, Physics of magnetically confined plasmas, Rev. Mod. Phys., 76 (2004), 1071–1141.
• [2] H. M. Dida, F. Hathout, Killing magnetic flux surfaces in the Heisenberg three group, Facta Universitatis (NIS) Ser. Math. Inform., 37(5) (2022), 975–991.
• [3] Z. Erjavec, J. Inoguchi, Killing magnetic curves in Sol space, Math. Phys. Anal. Geom., 21(2018), 15.
• [4] Z. Erjavec, J. Inoguchi, Magnetic curves in Sol3, J. Nonlinear Math. Phys., 25(2)(2018), 198-210.
• [5] R. D. Hazeltine, J. D. Meiss, Plasma Confinement, Dover Publications, inc. Mineola, New York, 2003.
• [6] T. Körpinar, R. C. Demirkol, Z. Körpinar, Approximate solutions for the inextensible Heisenberg antiferromagnetic flow and solitonic magnetic flux surfaces in the normal direction in Minkowski space, Optik, 238 (2021), 166403.
• [7] T. Körpinar, R. C. Demirkol, Z. Körpinar, New analytical solutions for the inextensible Heisenberg ferromagnetic flow and solitonic magnetic flux surfaces in the binormal direction, Phys. Scr., 96 (2021), 085219.
• [8] Talat Körpinaret, R. C. Demirkol, V. Asil, Z. Körpinar, Magnetic flux surfaces by the fractional Heisenberg antiferromagnetic flow of magnetic b-lines in binormal direction in Minkowski space, J. Magn. Magn. Mater., 549 (2022), 168952.
• [9] Z. Ozdemir, İ. Gök, Y. Yaylı, F. N. Ekmekçi, Killing magnetic flux surfaces in Euclidean 3-space, Honam Math. J. 41(2) (2019), 329–342.
• [10] T.S. Pedersen, A. H. Boozer, Confinement of nonneutral plasmas on magnetic surfaces, Phys. Rev. Lett. 88 (2002), 205002.
• [11] M. Barros, A. Romero, Magnetic vortices, EPL 77 (2007), 34002.
• [12] R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, Wiley, ISBN 0-471-07392-X, 1960.
• [13] S. R. Hudson, E. Startsev, E. Feibush, A new class of magnetic confinement device in the shape of a knot, Phys. Plasmas, 21(1) (2014), 010705.
• [14] W. Thurston, Three-dimensional geometry and topology, Princenton Math. Ser. 35, Princenton Univ. Press, Princenton, NJ, (1997).
• [15] M. Troyanov, L’horizon de SOL. Expo. Math., 16 (1998), 441–479.
• [16] Walter. A. Strauss, Partial differential equations: An introduction, Math. Gaz., 77(479) (1993), 286–287 ISBN 0-471-57364-7, Wiley.