Geodesic
Investigation of Beltrami fields by methods of integral geometry
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 3-14.

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In this paper, we propose a tomographic method for studying linear Beltrami fields in a bounded domain of space based on the expansion of vector fields and their ray transforms by basic vector functions. In addition, we develop a numerical algorithm and present the results of numerical simulation.
Keywords: inverse problem, Beltrami field, ray transform, vector spherical harmonic. 
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A. L. Balandin. Investigation of Beltrami fields by methods of integral geometry. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 3-14. http://geodesic.mathdoc.fr/item/INTO_2021_196_a0/

[1] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1970 | MR

[2] Balandin A. L., Murata Y., Ono Y., “Radial velocity profile reconstruction by Doppler spectroscopy measurements”, Eur. Phys. J. D., 27 (2003), 125–130 | DOI

[3] Balandin A. L., Ono Y., “Tomographic determination of plasma velocity with the use of ion Doppler spectroscopy”, Eur. Phys. J. D., 17 (2001), 337–344 | DOI

[4] Bellan P. M., Fundamentals of Plasma Physics, Cambridge Univ. Press, Cambridge, 2006

[5] Biedenharn L. C., Louck J. D., Angular Momentum in Quantum Physics, Addison-Wesley, Reading, MA, 1981 | MR | Zbl

[6] Chandrasekhar S., “On force-free magnetic fields”, Proc. Natl. Acad. Sci. U.S.A., 42 (1956), 1–5 | DOI | MR | Zbl

[7] Chandrasekhar S., Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, New York, 1961 | MR | Zbl

[8] Chandrasekhar S., Kendall P. C., “On force-free magnetic fields”, Astrophys. J., 126 (1957), 457–460 | DOI | MR

[9] Colton D, Kress R., Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1992 | MR | Zbl

[10] Constantin P., Majda A., “The Beltrami spectrum for incompressible fluid flows”, Commun. Math. Phys., 115 (1988), 435–456 | DOI | MR | Zbl

[11] Edmonds A. R., Angular Momentum in Quantum Mechanics, Princeton Univ. Press, Princeton, New Jersey, 1957 | MR | Zbl

[12] Freeden W., Gervens T., Schreiner M., Constructive Approximation on the Sphere with Applications to Geomathematics, Clarendon Press, Oxford, 1998 | MR | Zbl

[13] Furth H. P., Jardin S. C., Montgomery D. B., “Force-free coil principles applied to high-temperature superconducting materials”, IEEE Trans. Magn., 24 (1998), 1467–1468 | DOI | MR

[14] Hansen W. W., “A new type of expansion in radiation problem”, Phys. Rev., 47 (1935), 139–143 | DOI

[15] Howard J., “Vector tomography applications in plasma diagnostics”, Plasma Phys. Control Fusion., 38 (1996), 489–503 | DOI

[16] Kawamori E., Sumikawa T., Ono Y., Balandin A. L., “Measusremnt of global instability of compact torus by three-dimensional tomography”, Rev. Sci. Instr., 77 (2006), 093503 | DOI

[17] Lakhtakia A., Beltrami fields in Chiral Media, World Scientific, 1994

[18] Marsh G. E., Force-Free Magnetic Fields. Solution, Topology and Applications, World Scientific, 1996

[19] McLaughlin D., “Some notes on periodic Beltrami fields in Cartesian geometry”, J. Math. Phys., 32 (1991), 797–804 | DOI | MR | Zbl

[20] Moffatt H. K., Magnetic Field Generation in Electrically Conducting Fluids, Cambridge Univ. Press, London–New York, 1978

[21] Morse P. M., Feshbach H., Methods of Theoretical Physics, McGraw-Hill, New York, 1953 | MR | Zbl

[22] Natterer F., Wübbeling F., Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, 2001 | MR | Zbl

[23] Norton S. J., “Tomographic reconstruction of 2-D vector fields: application to flow imaging”, J. Geophys., 97 (1987), 161–168 | DOI

[24] Schuster Th., “An efficient mollifier method for three-dimensional vector tomography: convergence analysis and implementation”, Inverse Problems., 17 (2001), 739–766 | DOI | MR | Zbl

[25] Sharafutdinov V., Integral Geometry of Tensor Fields, Utrecht VSP, 1994 | MR

[26] Sparr G., Strahlen K., Lindstrom K., Persson H. W., “Doppler tomography for vector field”, Inverse Problems., 11 (1995), 1051–1061 | DOI | MR | Zbl

[27] Stratton J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941 | Zbl

[28] Osman N. F., Prince J. L., “3D vector tomography on bounded domains”, Inverse Problems., 14 (1998), 185–196 | DOI | MR | Zbl

[29] Varshalovich D. A., Moskalev A. N., Khersonskii V. K., Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988 | MR

[30] Watson G. N., A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, New York, 1945 | MR

[31] Wiegelmann T., Sakurai T., “Solar force-free magnetic fields”, Living Rev. Solar Phys., 9 (2012), 5–40 | DOI

[32] Winters K. B., Rouseff D., “Tomographic reconstruction of stratified fluid flow”, IEEE Trans. Ultrason. Ferroelectr. Freq. Control., 40 (1993), 26–33 | DOI

[33] Woltjer L., “A theorem on force-free magnetic fields”, Proc. Natl. Acad. Sci. U.S.A., 44 (1958), 489–491 | DOI | MR | Zbl