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Mathematical Models of Equilibrium Configurations of Plasma Surrounding Current-Carrying Conductors

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Abstract

This work belongs in the field of mathematical modeling and numerical studies of equilibrium configurations of plasma, magnetic field, and electric current in Galathea traps with current-carrying conductors immersed in a plasma volume. Models of permissible configurations in the annular neighborhood of a straight conductor that are not in contact with its surface are constructed and investigated. Such configurations can be considered a common basic element of any Galathea. The dependence of the configurations on the position of the outer boundary of the neighborhood is considered, and the MHD-stability of equilibrium configurations with respect to three-dimensional perturbations in the linear approximation is studied. Quantitative laws governing the restrictions on the plasma pressure are found that ensure the existence and stability of the configurations under consideration.

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Notes

  1. The ideal MHD model without allowance for dissipative processes is isentropic; i.e., \(p=p(\rho ) \).

REFERENCES

  1. Dnestrovskii, Yu.N. and Kostomarov, D.P., Matematicheskoe modelirovanie plazmy (Mathematical Modeling of Plasma), Moscow: Nauka, 1982.

    Google Scholar 

  2. Artsimovich, L.A., Upravlyaemye termoyadernye reaktsii (Controlled Thermonuclear Reactions), Moscow: Fizmatgiz, 1961.

    Google Scholar 

  3. Morozov, A.I., Galathea-plasma confinement systems in which the conductors are immersed in the plasma, Sov. J. Plasma Phys., 1992, vol. 18, no. 3, pp. 159–165.

    Google Scholar 

  4. Morozov, A.I. and Savel’ev, V.V., On Galateas—magnetic traps with plasma-embedded conductors, Phys.-Usp., 1998, vol. 41, no. 11, pp. 1049–1089.

    Article  Google Scholar 

  5. Morozov, A.I. and Pustovitov, V.D., Stellarator with levitating windings,Sov. J. Plasma Phys., 1991, vol. 17, p. 740.

    Google Scholar 

  6. Brushlinskii, K.V., Zueva, N.M., Mikhailova, M.S., and Petrovskaya, N.B., On the uniqueness and stability of solutions of two-dimensional plasmostatic problems, Mat. Model., 1995, vol. 7, no. 4, pp. 73–86.

    MathSciNet  Google Scholar 

  7. Morozov, A.I. and Frank, A.G., Galateya toroidal multipole trap with azimuthal current, Plasma Phys. Rep., 1994, vol. 20, pp. 879–886.

    Google Scholar 

  8. Brushlinskii, K.V. and Goldich, A.S., Mathematical model of the Galathea-belt toroidal magnetic trap, Differ. Equations, 2016, vol. 52, no. 7, pp. 845–854.

    Article  MathSciNet  MATH  Google Scholar 

  9. Frank, A.G., Kyrie, N.P., and Markov, V.S., Experiments on the formation of Galatea–belt magnetoplasma configurations, Plasma Phys. Rep., 2019, vol. 45, no. 1, pp. 28–32.

    Article  Google Scholar 

  10. Morozov, A.I., Bugrova, A.I., Bishaev, A.M., Lipatov, A.S., and Kozintseva, M.V., Plasma parameters in the upgraded Trimyx-M Galathea, Tech. Phys., 2007, vol. 52, no. 12, pp. 1546–1551.

    Article  Google Scholar 

  11. Medvedev, S.Yu., Martynov, A.A., Savel’ev, V.V., and Kozlov, A.N., Calculations of ideal MHD-stability in Trimyx Galathea-trap, Preprint of Keldysh Inst. Appl. Math, Russ. Acad. Sci., Moscow, 2018, no. 253.

  12. Tao, B., Jin, X., Li, Z., and Tong, W., Equilibrium configuration reconstruction of multipole Galatea magnetic trap based on magnetic measurement, IEEE Trans. Plasma Sci., 2019, vol. 47, no. 7, pp. 3114–3123.

    Article  Google Scholar 

  13. Shafranov, V.D., On magnetohydrodynamic equilibrium configurations,Sov. Phys. JETP, 1958, vol. 6, pp. 545–554.

    MathSciNet  MATH  Google Scholar 

  14. Grad, H. and Rubin H., Hydrodynamic equilibria and force-free fields, inProc. 2nd United Nations Int. Conf. Peaceful Uses At. Energy, Geneva, 1958; New York, 1959; vol. 31. pp. 190–197.

  15. Brushlinskii, K.V., Matematicheskie i vychislitel’nye zadachi magnitnoi gazodinamiki (Mathematical and Computational Problems of Magnetic Gas Dynamics), Moscow: Binom, 2009.

    Google Scholar 

  16. Brushlinskii, K.V., Matematicheskie osnovy vychislitel’noi mekhaniki zhidkosti, gaza i plazmy (Mathematical Basics of Computational Mechanics of Fluid, Gas, and Plasma), Moscow: Intellekt, 2017.

    Google Scholar 

  17. Dudnikova, G.I., Morozov, A.I., and Fedoruk, M.P., Numerical simulation of straight belt-type Galatheas, Plasma Phys. Rep., 1997, vol. 23, no. 5, pp. 357–366.

    Google Scholar 

  18. Brushlinskii, K.V. and Chmykhova, N.A., Plasma equilibrium in the magnetic field of Galatea traps, Math. Models Comput. Simul., 2010, vol. 3, pp. 9–17.

    Article  Google Scholar 

  19. Brushlinskii, K.V. and Chmykhova, N.A., Numerical model of the formation of plasma quasi-equilibrium in the magnetic field of Galathea traps, Vestn. Nats. Issled. Yad. Univ. MIFI , 2014, vol. 3, no. 1, pp. 40–52.

    Google Scholar 

  20. Brushlinskii, K.V., Two approaches to the stability problem for plasma equilibrium in a cylinder, J. Appl. Math. Mech., 2001, vol. 65, no. 2, pp. 229–236.

    Article  MathSciNet  Google Scholar 

  21. Shafranov, V.D., Plasma equilibrium in a magnetic field, Rev. Plasma Phys., Leontovich, M.A., Ed., New York, 1966, no. 2, pp. 103–152.

  22. Kadomtsev, B.B. Hydromagnetic stability of a plasma, Rev. Plasma Phys., Leontovich, M.A., Ed., New York, 1966, no. 2, pp. 153–206.

  23. Solov’ev, L.S., Symmetric magnetohydrodynamic flows and helical waves in a circular plasma cylinder, Rev. Plasma Phys., Leontovich, M.A., Ed., New York, 1967, no. 3, pp. 277–325.

  24. Solov’ev, L.S., Hydromagnetic stability of closed plasma configurations,Vopr. Teor. Plazmy, Leontovich, M.A., Ed., Moscow, 1972, no. 6, pp. 210–290.

  25. Brushlinskii, K.V., Krivtsov, S.A., and Stepin, E.V., On the stability of plasma equilibrium in the neighborhood of a straight current conductor, Comput. Math. Math. Phys., 2020, vol. 60, no. 4, pp. 686–696.

    Article  MathSciNet  Google Scholar 

  26. Bateman, G., MHD Instabilities, Cambridge, MA: MIT Press, 1980. Translated under the title: MGD-neustoichivosti, Moscow: Energoizdat, 1982.

    Google Scholar 

  27. Gerlakh, N.I., Zueva, N.M., and Solov’ev, L.S., Linear theory of helical instability, Magn. Gidrodin., 1978, no. 4, pp. 49–54.

  28. Bernstein, L.B., Frieman, E.A., Kruskal, M.D., and Kulsrud, R.M., An energy principle for hydromagnetic stability theory, Proc. R. Soc. London, Ser. A, 1958, vol. 224, pp. 17–40.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Moscow, 125047, Russia

    K. V. Brushlinskii & E. V. Stepin

  2. National Research Nuclear University MEPhI, Moscow, 115409, Russia

    K. V. Brushlinskii & E. V. Stepin

  3. Lomonosov Moscow State University, Moscow, 119991, Russia

    K. V. Brushlinskii

Authors
  1. K. V. Brushlinskii
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  2. E. V. Stepin
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Correspondence to K. V. Brushlinskii or E. V. Stepin.

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Translated by V. Potapchouck

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Brushlinskii, K.V., Stepin, E.V. Mathematical Models of Equilibrium Configurations of Plasma Surrounding Current-Carrying Conductors. Diff Equat 56, 872–881 (2020). https://doi.org/10.1134/S001226612007006X

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  • DOI: https://doi.org/10.1134/S001226612007006X

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